TRENDS IN MATHEMATICAL PSYCHOLOGY
ADVANCES IN PSYCHOLOGY 20 Editors
G . E. STELMACH
P. A . VROON
NORTH-HOLLAND AMSTERDAM ' N E W YORK'OXFOKD
TRENDS IN MATHEMATICAL PSYCHOLOGY
Edited by
E. DEGREEF and
J. VAN BUGGENHAUT Centrefor Statistics and Operational Research Brussels, Belgium
1984
NORTH-HOLLAND AMSTERDAM. NEW YORK . OXFORD
0 Elsevier Science
Publishers B.V.. 1984
All rights reserved. No part of this publication may be reproduced. stored in a retrieval system. or transmitted. in any form o r by any means. electronic. mechanical. photocopying. recording o r otherwise. without the prior permissionof thecopyright owner.
ISBN: 0 4 4 87512 3
Publishers: ELSEVIER SCIENCE PUBLISHERS B . V . P.O. Box 1991 1ocN) B Z Amsterdam The Netherlands
Sole disiribur0r.rfor the U.S.A. und (briadu. ELSEVIER SCIENCE PUBLISHING COMPANY. INC. 52 Vanderbilt Avenue New York. N.Y. 1OUI7 U.S.A.
Library of Comgmm Catmlogl~g11 PmbUeathm Datm
MBin.entry under title: Trends in mathematical psychology. (Advances in psychology ; 2 0 ) Papers presented at the 14th European Mathematical Ssychology Group Meeting, held in Brussels, Sept. 12-14,
1983. Includes indexes. Psychometrics--Congresses. I. Degreef, E., 1953TI. Buggenhaut, J. van, 1937111. European Mathematical Psychology Group. IV. European Mathematicd' Psychology Group. Meeting (14th : 1983 : Brussels, Belgium) V. Series: Advances in psychology ( h t e r d a m , Netherlands) 4 20.
.
BF39.T74 1984 150' .28'7 ISBN 0-444-87512-3 ( U . S . )
84-6032
PRINTED IN THE NETHERLANDS
V
PREFACE
T h i s volume g a t h e r s most o f t h e papers p r e s e n t e d a t t h e 1 4 t h Eurooean Mathem a t i c a l Psychology Grouo Meetinq. Sentember 12 t o t e i t Brussel.
The m e e t i n g took p l a c e i n B r u s s e l s f r o m
September 14, 1983 and was h o s t e d by t h e V r i j e U n i v e r s i -
F i n a n c i a l s u p n o r t was a l s o p r o v i d e d by; t h e p e l q i a n " N a t i o -
n a a l Fonds voor wetenschanpeli j k Dnderzoek", t h e B e l g i a n " t c i n i s t e r i e van N a t i o n a l e Opvoeding", t h e E e l F i a n " C u l t u r e l e B e t r e k k i n g e n " and t h e " N a t i o n a l e Bank van B e l g i e " . The i d e a o f t h e Groun i s t o b r i n g t o g e t h e r once a y e a r , neoDle i n Europe w o r k i n g i n t h e f i e l d o f V a t h e m a t i c a l Psychology,
Taken i n t o a c c o u n t t h e
growing importance a t t a c h e d t o mathematical models i n human s c i e n c e s , t h e exaeriences a c q u i r e d i n Drevious meetings and t h e presence, besides o t h e r we1 1-known p a r t i c i p a n t s , o f f o u r i n v i t e d l e c t u r e r s , namely S. Crossberg (Boston U n i v e r s i t y ) , F.S. and V.F.
Venda
-
V.Y.
Roberts (Rutoers, S t a t e U n i v e r s i t y o f New J e r s e y )
K r y l o v (Floscow, USSR Academy o f Sciences), gave us
t h e i d e a t o make a s e l e c t i o n o f t h e c o n t r i b u t i o n s and t o g a t h e r them i n a book. I n o r d e r t o s t r u c t u r e t h e whole, we t o o k t h e l i b e r t y t o groun t h e papers i n t o t h r e e p a r t s , knowing t h a t t h e c l a s s i f i c a t i o n can be discussed; o f t h e papers, indeed, can f i n d a p l a c e i n more than one p a r t .
some
So y o u w i l l
find: Part I
: P e r c e p t i o n , l e a r n i n g and memory
P a r t I 1 : Order and measurement P a r t 111: Data a n a l y s i s . ble hope t h a t t h e s t u d i e s c o l l e c t e d here, f a i r l y r e o r e s e n t t h e d i f f e r e n t p e r s p e c t i v e s and t h a t t h e volume as a whole w i l l be a dynamic r e s o u r c e f o r those who want t o keeD a b r e a s t o f f l a t h e m a t i c a l Psychology i n g e n e r a l and t h e Eurooean one i n p a r t i c u l a r . We f i n a l l y w i s h t o thank F r a n c i s Gheys f o r t h e e x c e l l e n t t y p i n g o f t h e manuscript.
I t was a l o t o f work, b u t we e n j o y e d i t .
The e d i t o r s B r u s s e l s , February 1984
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Vii
CONTENTS
v
Preface P a r t i cioants
xi
PART I: PERCEPTION, LFARNINI; AND MEMORY
1
Tree r e n r e s e n t a t i o n s o f a s s o c i a t i v e s t r u c t u r e s i n semantic and e n i s o d i c memory r e s e a r c h 11. b b d i , J.-P.
FivLRhUtrny,
Y.
Luong
3
Task-denendent r e p r e s e n t a t i o n o f c a t e g o r i e s and memory-ouided inference duri nq c lassi f ication
H. R U ~ ~ W L . ~ti.-(;. ,
&?hAeet
33
O u t l i n e o f a t h e o r y o f b r i g h t n e s s , c o l o r , and f o r m n e r c e n t i o n
s.
h044bUg
59
A t t e n t i on i n d e t e c t i o n t h e o r v E.C. b!icbo
a7
I m n o s s i b l e o b j e c t s and i n c o n s i s t e n t i n t e r p r e t a t i o n s
E. TLtoianne
105
I n s e a r c h i n g o f general r e g u l a r i t i e s o f a d a p t a t i o n dynamics: on t h e t r a n s f o r m a t i o n l e a r n i n g t h e o r y V.F.
Venda
121
A D r o h a b i l i s t i c choice model adanted f o r t h e a n g l e D e r c e n t i o n experiment
N.P. VenheL5.t
159
PART I 1 : ORDER AND MEASUREPIEKT
175
About t h e a s y m t r i e s o f s i m i l a r i t y judgments: an o r d i n a l n o i n t o f view
J.-P. Rah.thaePrnu
177
The a x i o m a t i z a t i o n o f a d d i t i v e d i f f e r e n c e models for p r e f e r e n c e judgments
M.A. Cnoon
193
Contents
Viii
Generalizations o f i n t e r v a l orders
1.- 7 .
209
poiqnon
Reseaux s o c i a u x g e n e r a l i s e s : en combinant graphes e t hyperoranher C.
2 19
FLnment
On conceots o f t h e dimension o f a r e l a t i c n and a g e n e r a l i z e d r e copnition e x p c r i m n t
227
K . IJe?rh4t I s o t o n i c r e g r e s s i o n a n a l y s i s and a d d i t i v i t y
239
‘Inc Patiold
7.p.
T e s t i n g Fechnerian s c a l a b i l i t y b v Faximum l i k e l i h o o d e s t i m a t i o n
of o r d e r e d b i n o m i a l narameters
25 5
?. “ n t ~ 6 4 e f d
Prnbahi l i s t i c c o n s i s t e n c y , homoaeneous fami l i e s o f r e 1 a t i o n s and
1i n e a r 1-re1 a t i ons 271
B. V o n j m d c t A o p l i c a t i o n s o f t h e t h e o r v o f meaninafulness t o o r d e r and matching e x o e r i men t s
283
F.S. Robem2 A new d e r i v a t i o n o f t h e Rasch model
E . E . Ra.ikn~i, P.C.W.
Jnmen
293
A d e f i n i t i o n o f p a r t i a l i n t e r v a l orders
? I . ?oubeno, Ph. VLncke
309
Causal l i n e a r s t o c h a s t i c deDendencies: t h e f o r m a l t h e o r y
317
?. Sfelreh
F a c t o r i z a t i o n and a d d i t i v e decomposition o f a weak o r d e r
”.
347
Suck
PADT
363
111: DATA A N A L Y S I S
The oroblem o f r e p r e s e n t a t i o n based u m n two c r i t e r i a
365
G. De Ueut, hl. G ~ 4 n e ~X. , HubaLLt Tree r e p r e s e n t a t i o n s o f r e c t a n g u t a r p r o x i m i t y m a t r i c e s
G, Oe S o e t e , W.S. DCa.tbo, C.Pl.
FUMLU,
J.D. CmoU
377
ifleak and s t r o n g models i n o r d e r t o d e t e c t and measure o o v e r t y ?.
Picken
393
Contents
ix
Am1 ic a t i o n s o f a Bayes an Poisson model f o r misreadings M.G.f/.
Jan6en
405
The polychotomous Rasch model and d i c h o t o m i z a t i o n o f graded resnonses P.G.P!.
.TanAeen, E.E. Qobkam
413
Comnonent a n a l y s i s o f t r a n s i t i o n n r o b a b i l i t i e s and i t s a n o l i c a t i o n to prisoner's d i l e m a
K. R a i n i o
433
An a n p l i c a t i o n o f mu1 t i d i m e n s i o n a l s c a l i n g on D r e d i c t i o n : t h e Radex structure
F. Van O v W e
449
Author i n d e x
473
Subject index
477
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xi
PARTICIPANTS
H. A B D I , L a b o r a t o i r e de p s y c h o l o g i e , Ancienne Facul t e , Rue Chabot Charny, 21000 Di j o n , France
H. ALGAYER, D o n n d o r f o r s t r . 93, D-8580 Bayreuth, Federal R e p u b l i c Germany J. ANDRES, R a t h a u s s t r . 53, 53 Bonn 3, Federal R e p u b l i c Germany J.-P. BARTHELEMY, ENST, Deoartement d ' I n f o r m a t i q u e , 46 Rw B a r r a u l t 75634 P a r i s Cedex 13, France A. BOHRER. CRS, S e c t i e voor P s y c h o l o g i s c h Onderzoek, Kazerne K l e i n K a s t e e l t j e , 1000 B r u s s e l , Belgium H. F. J .M. BUFFART, P s y c h o l o g i sch L a b o r a t o r i urn , Kathol i e k e U n i v e r s i t e i t Nijmegen, Postbus 9104, 6500HE Nijmegen, The N e t h e r l a n d s M.A. CROON, K a t h o l i e k e Hopeschool T i l b u r g , Hogeschoollaan 225, T i l b u r g , The Nether1ands E. DEGREEF, CSOO, V r i j e U n i v e r s i t e i t B r u s s e l , P l e i n l a a n 2, 1050 B r u s s e l , Belgium L. DELBEKE, P s y c h o l o g i s c h I n s t i t u u t , K a t h o l i e k e U n i v e r s i t e i t Leuven, T i e n s e s t r a a t 102, 3000 Leuven, Belgium G. DE MEUR, U n i v e r s i t e L i b r e de B r u x e l l e s , CP 135, 1050 B r u x e l l e s , Belgium G. DE SOETE, D i e n s t v o o r Psychologie, R i j k s u n i v e r s i t e i t Gent, H e n r i Dunantlaan 2, 9000 Gent, Belgium M. DESPONTIN, CSOO, V r i j e U n i v e r s i t e i t B r u s s e l , P l e i n l a a n 2, 1050 B r u s s e l , Be1 gium P. DICKES, U n i v e r s i t e de Nancy 11, Bd A l b e r t I, BP 3397, 54015 Nancy-Cedex, France J.-P. DOIGNON, U n i v e r s i t e L i b r e de B r u x e l l e s , CP 216, 1050 B r u x e l l e s , Belgium C. FLAMENT, Departement de P s y c h o l o g i e , U n i v e r s i t e de Provence, Les Blaques, Cereste, F-04110 R e i l l a n n e , France H.-G. GEISSLER, S e k t i o n Psychologie, K a r l - M a r x - U n i v e r s i t a t L e i p z i g , T i e c k s t r . 2, 7030 L e i p z i g , German Democratic R e p u b l i c M. GASSNER, U n i v e r s i t e L i b r e de B r u x e l l e s , CP 135, 1050 B r u x e l l e s , Belgium C.A.W. GLAS, C.I.T.O., Postbus 1034, 6801 MG Arnhem, The N e t h e r l a n d s S. GROSSBERG, Center f o r A d a p t i v e Systems, Boston U n i v e r s i t y , 111 C u m i n g t o n S t r e e t , Boston, Massachusetts 02215, USA P.K.G. GUNTHER, S i e b e n g e b i r g s t r . 11, D-5330 K o n i g s w i n t e r 41, F e d e r a l R e p u b l i c Germany M. HAHN, M o l l w i t z s t r . 5, D-1000 B e r l i n 19, Federal R e p u b l i c Germany K. HERBST, I n s t i t u t filr Psychologie, U n i v e r s i t a t Regensburg, U n i v e r s i t a t s t r . 31, D-8400 Regensburg, Federal R e p u b l i c Germany D. HEYER, I n s t i t u t f i r P s y c h o l o g i e , U n i v e r s i t a t K i e l , Ohlshausenstr. 40/60, D-2300 K i e l , Federal R e p u b l i c Germany M.G.H. JANSEN, I n s t i t u u t voor Onderwi jskunde, R i j k s u n i v e r s i t e i t Groningen, Westerhaven 16, 9718 AW Groningen , The N e t h e r l a n d s V.Y. KRYLOV, Department o f Mathematical Psychology, I n s t i t u t e o f Psychology, USSR, Academy o f Sciences, Moscow 129366, 13 Yaroslavskaya, USSR N. LAKENBRINK, A u f dem Kampf 11, D-2000 Hamburg 63, Federal R e p u b l i c Germany X. LUONG, L a b o r a t o i r e de Mathematiques, U n i v e r s i t e de Besancon, Besancon, France R.R. MAC DONALD, Department o f Psychology, U n i v e r s i t y o f S t i r l i n g , S t i r l i n g FK94LA. S c o t l a n d
xii
A.W.
Participants
MAC RAE, U n i v e r s i t y o f Birmingham, Box 363, Birmingham 615 ZTT, Great B r i t a i n R. MAUSFELD, U n i v e r s i t a t Bonn, Auf den S t e i n e n 13A, 53 Bonn 1, Federal R e p u b l i c Germany H.C. MICKO, I n s t i t u t f u r P s y c h o l o g i e , Technische U n i v e r s i t a t Braunschweig, Spielmannstrasse 12A, 0-3300 Braunschweig, F e d e r a l R e p u b l i c Germany B. MONJARDET, C e n t r e de M a t h h a t i q u e S o c i a l e , 54 Bd R a s p a i l , 75270 P a r i s Cedex 06, France C. MULLER, L e h r s t u h l fur P s y c h o l o g i e 111, U n i v e r s i t a t Regensburg, Postfach 397, D-8400 Regensburg, F e d e r a l R e p u b l i c Germany p . PLENNIGER. Seminar fiir P h i l o s o n h i e und E r r i c h u n g s w i s s e n s c 5 a f t d e r U n i v e r s i t a t F r e i b u r g , IJerthmannnlatz, D-7800 F r e i b u r o , F e d e r a l R e p u b l i c Germany B. ORTH. I n s t i t u t f u r P s y c h o l o g i e , U n i v e r s i t a t Hamburg, Von-Melle-Park 6, D-2000 Hamburg 13, F e d e r a l R e p u b l i c Germany M. PIRLOT, U n i v e r s i t e de 1 ' E t a t a Mons, 17 P l a c e Warocque, 7000 Mons, Belgium J .G.W. RAAIJMAKERS, P s y c h o l o g i s c h L a b o r a t o r i u m , K a t h o l i e k e U n i v e r s i t e i t Nijmegen, Postbus 9104, 6500 HE Nijmegen. The N e t h e r l a n d s K. R A I N I O , U n i v e r s i t y o f H e l s i n k i , P e t a k s e n t i e 44, 00630 H e l s i n k i , F i n l a n d F.S. ROBERTS. Qutqers, S t a t e U n i v e r s i t y c f f'ew Jersey. Deep. o f 'lath. a t New Brunswick, H i l l Center f o r t h e '4athematical Sciences, Rush Camous, New Rrunswick, New J e r s e y 08903, 1J.S.A. E.E. ROSKAM, Vakgroep Mathematische P s y c h o l o g i e , K a t h o l i e k e U n i v e r s i t e i t Nijmegen. Postbus 9104, 6500 HE Nijmegen, The N e t h e r l a n d s M. ROUBENS, F a c u l t e P o l y t e c h n i q u e de Mons, 9 Rue de Houdain, 8-7000 Mons, Belgium U. SCHULZ, F a k u l t a t f u r P s y c h o l o g i e und S p o r t w i s s e n s c h a f t , A b t . f u r e x p e r i m e n t e l l e und angew. P s y c h o l o g i e , U n i v e r s i t a t B i e l e f e l d , P o s t f a c h 8640, 4800 B i e l e f e l d , Federal R e p u b l i c Germany A . J . SMOLENAARS, P s y c h o l o g i s c h L a b o r a t o r i u m , U n i v e r s i t e i t Amsterdam, WeesDerolein 8 . 1018 XA Amsterdam, The N e t h e r l a n d s R. STEYER,' F a c h b e r e i c h I - P s y c h o l o g i e , U n i v e r s i t a t T r i e r , Schoeidershof, 0-5500 T r i e r , Federal R e p u b l i c Germany R. SUCK, U n i v e r s i t a t Osnabruck, P o s t f a c h 4469, 45 Osnabruck, F e d e r a l Republ i c Germany E . TEROUANNE, UER Mathematiques, U n i v e r s i t e Paul V a l e r y , BP 5043, 34032 M o n t p e l l i e r - Cedex, France P. VAN ACKER, 1 Chaussee de Wavre, 1050 B r u x e l l e s , B e l g i u m J . VAN BUGGENHAUT, CSOO. V r i j e U n i v e r s i t e i t B r u s s e l , P l e i n l a a n 2. 1050 B r u s s e l , B e l g i u m A . VAN DER WILDT, CSOO, V r i j e U n i v e r s i t e i t B r u s s e l , P l e i n l a a n 2, 1050 B r u s s e l , Be1g i um A. VAN DINGENEN, CSOO, V r i j e U n i v e r s i t e i t B r u s s e l , P1 e i n l a a n 2 , 1050 B r u s s e l , Belgium L . VAN LANGENHOVE, F a c u l t e i t P s y c h o l o g i e en Opvoedkunde, V r i j e U n i v e r s i t e i t B r u s s e l , P l e i n l a a n 2, 1050 B r u s s e l , Belgium F. VAN OVERWALLE, EDUCO, V r i j e U n i v e r s i t e i t B r u s s e l , P l e i n l a a n 2 , 1050 B r u s s e l , B e l g i u m J.C. VAN SNICK. F a c u l t 6 des Sciences Economiques e t S o c i a l e s . U n i v e r s i t e de 1 ' E t a t a Mons, 17 P l a c e Warocque, 7000 Mons, Belgium V.F. VENDA. Department o f E n g i n e e r i n g Psychology, I n s t i t u t e of Psychology, USSR Academy o f Sciences, Moscow 129366, 13 Yaroslavskaya. USSR R . VERHAERT, CSOO, V r i j e U n i v e r s i t e i t B r u s s e l , P l e i n l a a n 2 , 1050 B r u s s e l , Belgium
Participants
N . D . VERHELST, Subfacul t e i t der psychologie, vakgroep PSM, R i jksuniversit e i t Utrecht, Sint-Jacobsstraat 14, 3511 BS Utrecht, The Netherlands Ph. VINCKE, I n s t i t u t de Statistique, Universite Libre de Bruxelles, CP 210, 8-1050 Bruxelles, Belgium
xiii
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PARTI PERCEPTION, LEARNING AND MEMOR Y
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TRENDS IN MATHEMATICAL PSYCHOLOGY E. Degreef and J . Van Bu genhaut (editors) 0 Elsevier Science Dublisfers B. V. (North-Holland), 1984
3
TREE REPRESENTPTIONS OF ASSOCIATIVE STPUCTUpE5 I N SFFWNTIC AND E P J SOD1 C VFF1ORY RE SF ARCH Herve Abdi Laboratoi r e de Ps.vcholoaie, D i j o n Jean-Pierre Barth12l&ny F.PJ.q.T., Paris Xuan Luona L a b o r a t o i r e de Mathematioues, Besanqon lnle exnose some research i n the area o f psycholorry o f memory i n v o l v i n q o r o x i m i t v o r d i s t a n c e m a t r i c e s . nronose some ways o f b u i l d i n g LIP such r a t r i c e s .
Then
we d e t a i l an a l a o r i t h m allowinc! the r e n r e s e n t a t i o n o f p r o x i m i t y matrices b y an a d d i t i v e t r e e , and c o n t r a s t t h i s new a l o o r i thm w i t h orevious ones,
F i n a l l v , we
examine some r e s u l t s obtained w i t h t h s method.
1. INTRODUCTION The general ournose o f t h i s naner i s t o emphasize the u t i l i t y and describe the use o f a d d i t i v e t r e e s i n o r d e r t o describe data c o l l e c t e d i n the f i e l d of the nsychology o f Femory.
This naner i s t h r e e f o l d : we f i r s t describe
some research l e a d i n o t o the c o n s t r u c t i o n o f distance o r n r o x i m i t y matrices; secondly we expose and d e t a i l the c o n s t r u c t i o n o f an a d d i t i v e t r e e as a r e n r e s e n t a t i o n o f the o r i o i n a l r i a t r i x : f i n a l l y , we examine the r e s u l t s o b t a i n ed. The u t i l i z a t i o n o f c l u s t e r i n q methods f o r a t t e s t i n a t h e o r a a n i r a t i o n o f memor v o r r e v e a l i n g i t s s t r u c t u r e has been s t r o n y l y advocated r e c e n t l y by some authors i n d i f f e r e n t areas o f c o g n i t i v e Fsycholooy (see, among others:
M i l l e r (1969) (1982)).
, Henley
(1969), F r i e n d l y (19781, Rosenbera e t a1 (1968)
,
(1972),
Most o f the used methods amount t o r e n r e s e n t the o r i g i n a l m a t r i x
by an U l t r a w t r i c Tree.
Recently, t h e r e has been an attempt t o b u i l d sow
methods l e a d i n g t o r e n r e s e n t a t i o n s l e s s s t r i n g e n t than the c l a s s i c a l U1 t r a m e t r i c Tree, i.e. the A d d i t i v e Tree (see Carroll & Chang (1973), Cunningham (1974), (1978); S a t t a t h R Tversky (1977)).
Ye nronose h e r e a f t e r an (econo-
m i c a l ) h e u r i s t i c g i v i n a an A d d i t i v e Tree from a o r o x i m i t y m a t r i x and i l l u s t r a te i t with c a l oaners.
some examples borrowing from our c u r r e n t research o r from c l a s s i -
H. ,4bdi, J.-P. Burthhlhmy und X. Long
4
2. A RUNCH OF EXAWLES
2.1. RAPTLET': : " ' i l ' f
T C THE GHOSTS"
I n 1932, R a r t l e t t asked
i:
few suhdects t o read an l r w r i c a n I n d i a n f o l k t a l e
(named "The '.tar o f the chosts"! and t o r e c a l l the s t o r i t on r e v e r a l occasions ( a w t h o d c a l l e d "rerleatec' r e n r o d u c t i o n " ) ; i n a v a r i a n t o f the method ( i . e . "yerial
r e n r o d u c t i o n " ) a chain o f d i f f e r e n t s u h i e c t s i s used, the f i r s t
heinq shown the o r i c l i n a l t e x t and then r e c a l l i n n f o r the second s u b j e c t who
would nass i t t o a t h i r d and s o on.
These c l a s s i c a l e x n e r i w n t s o f P a r t l e t t
w i l l serve here t o i l l u s t r a t e a s e t o f i n f o r m a t i c nrocedures, the aim of which i s t o b u i l d s o w distances hetween t e x t s . " ) I n f o m a t i c orocedurw
For reasons o f c o m p a t i h i l i t v , the nroorams are w r i t t e n i n Standard V i c r o s o f t and a t l e a s t 64 K-Bytes o f
Pasic (Vnder C P p ) ,
RAP1
and a d i s k U n i t are need-
Although these n r o n r a m i n c l u d e some various n o s s i h i l i t i e s , we w i l l r e -
ed.
s t r i c t o u r s e l f t o the n a r t d e a l i n a s n e c i f i c a l l y w i t h the c o n s t r u c t i o n o f
metri cs between t e x t s . I t must be c l e a r t h a t when we sneak of t h e t e x t oiven by a s u b i e c t , we c o u l d sqeak as w e l l o f a s e t o f themes o r ideas aiven by a suhdect D r o v i dino an adequate codinc; o f the raw data. The t e x t s are f i r s t t r a n s f o m d i n a d i s k f i l e , then f o r each t e x t we b u i l d the Lexicon associated w i t h i t . This Lexicon c o u l d be e i t h e r a Boolean Lexicon (i.e. i t rnerelv i n d i c a t e s the Presence o r the Absence o f the i t e m
o f Vocabulary) o r an i n t e o e r Lexicon ( i . e . rences o f each i t e m ) .
-
build
by union
-
i t i n d i c a t e s t h e numher o f Occur-
From the d i C f e r e n t Lexicons (Boolean or I n t e g e r ) we
a General Lexicon t h a t d e f i n e s the Vocabularv shared hy
the d i f f e r e n t t e x t s .
R ) Construction o f distances hetween t e x t s k n e n d i n q on the p o i n t o f view adonted, we c o u l d d e f i n e d i f f e r e n t distances; as an i l l u s t r a t i o s we examine t h r e e ways: (i\
the t e x t s as suhsets of the Vocabulary
( i i ) the t e x t s as R i - o a r t i t i n n s o f the Vocahularv ( i i i)a "orobabi 1i s ti c " q e n e r a 1 i t a t i o n . Dpncte by L i the Lexicon associated w i t h a t e x t T i , the aeneral Lexicon hy
V =
0
i
L i . and by
the comlement o f L i i n V.
5
Tree representations of associative structures
( i ) Each (Boolean) Lexicon* i s a subset o f t h e Vocahularv and we c o u l d use, f o r examnle , t h e we1 1-known d i s t a n c e between s e t s , t h e so c a l l e d c a r d i n a l o f t h e symmetric d i f f e r e n c e : d(Ti,Tj) ( i i ) {Li,E}
= ILi A L j l = ILi
nnl
+ l E n Ljl.
d e f i n e s a R i - P a r t i t i o n o f V ( i . e . a P a r t i t i o n w i t h two classes), So, we c o u l d use sow d i s t a n c e s between P a r t i t i o n s
and so does { L j , a l .
( c f . A r a b i e & Roorman ( 1 9 7 3 ) ) o r S i - P a r t i t i o n s , e . q . symmetric d i f f e r e n c e hetween ' i - p a r t i t i o n s d(Ti,Tj)
= =
the distance o f the
t h a t can be expressed as:
2( l L i n L i l + l E q n l ) ( l L i ~ ~ l + l E n L j l ) 2( lT"Kil)(I L i A L j I )
( i i i ) I n o r d e r t o take e x n l i c i t l y account o f t h e I n t e g e r Lexicons we c o u l d l o o k f o r an e x t e n s i o n of ( i ) . Ielith each T i i s a s s o c i a t e d a p r o b a b i l i t y measure on V ( i .e. t h e frequency o f t h e d i f f e r e n t words); denote t h e p r o b a h i l i t y o f i t e m x o f t e x t T i by P i ( x ) ; then we f i n d a f a m i l y o f d i s t a n c e s by d,(Ti
,Tj) =
c
[Pi(x)-Pj(x)I?
*V 2.2.
FFnTLJRES OF PFRSONALITY
T h i s r e s e a r c h l a y s on t h e b o r d e r between t h e work on t h e o r o a n i z a t i o n o f t h e semantic memory and t h e work unon the " i m p l i c i t Dsycholoqy".
The ourpose i s
t o describe the subjective orsanization o f the q u a l i f i e r s o f the character. As a m a t t e r o f f a c t , i t has o f t e n been n o t e d t h a t we t e n d t o qroup s u b j e c t i v e l y some f e a t u r e s o f c h a r a c t e r as i f we has an " I m o l i c i t Theory o f Pers o n a l i t y " ( c f . e.a.,
Rosenherrr e t a1 (1972), 'Veoner and V a l l a c h e r ( 1 9 7 7 ) ) .
I n t h i s e x p e r i m e n t we s e l e c t f i f t y e i q h t q u a l i f i e r s o f t h e c h a r a c t e r ( u s i n a some Thesauruses and a b i t o f l i t e r a t u r e , . ..). These q u a l i f i e r s a r e then o r i n t e d on s e o a r a t e cards and a i v e n i n d i v i d u a l l y t o t w e n t y - e i o h t s u b j e c t s w i t h t h e r e q u e s t t h a t he o r she s o r t t h e cards i n t o D i l e s w i t h t h e c o n s t r a i n t t h a t " t h e cards i n a same w i l e g i v e t h e f e e l i n g t o no t o o e t h e r " ; s u b j e c t s were f r e e t o choose t h e numher o f D i l e s f o r s o r t i n n ( f o r a r e v i e w o f t h e p r o and c o n t r a o f t h e s o r t i n a method, see Rosenbern ( 1 9 8 2 ) ) .
*N o t i c e
fuzzy " t h e "boolean" d i s t a n c e i n (i) and ( i i ) by t a k i n g t h e f u z z y e q u i v a l e n t o f t h e u n i o n and i n t e r s e c t i o n , i .e. Min and Max. t h a t one c o u l d
"
6
H. Abdi, j . 2 . Borthdl6my and X. Luong
Hence, each subject exoresses his oninion by a nartition on the s e t of the n u a l i f i e r s , and f o r usino as a 0-rlethodoloay ( c f . Kerlinoer (1973)) the afore evoked distances between nartitions could e a s i l y be used. The p a r t i tion given by a subject i s associated with a matrix whose rolls and columns ren-esert the Q u a l i f i e r s , a n d where we p u t a 1 a t the intersection of a row and a column i f the nualifiers are not sorted i n the same n i l e by the subi e c t . Obviously t h i s i s a distance matrix ( c f . W l l e r ( 1 9 6 9 ) \ , and so will he the matrix definetiby the sum of the matrices of tbe d i f f e r e n t subjects. I n this matrix we simnly count the number of suhjects who do n o t n u t tooether the q u a l i f i e r s . Pually vie could have & f i r e d a matrix o f co-occurences by the sum of the so-called incidence matrix (where a 1 means t + a t the q u a l i f i e r s are i n the same n i l e ) , f n r commodity reasons t h i s i s the m a t r i x we qive l a t e r ( c f . Table 3 ) . I t must be noted, in nassinq, t h a t the Data obtained a n d consenuently the distance matrix denend uoon the w t h o d s desinned f o r o b t a i n i n n such Data. I n n a r t i c u l a r , other wthods ( e . 0 . word associations, o r distances between words in free-recall, e t c . ) lead to other results (see Pbdi (1383)). 2.3. OLRIEP RUT
Tt could be useful t o comnare different methods on well-known d a t a . examine here three sets of data (included two " c l a s s i c s " ) .
Isle will
The f i r s t comes from a study by f i i l l e r (1969) nioneerinn the structure of semantic memory. Fiftv college students, each sorted fortv-eiaht words according t o t h e i r similarity o f waninq ( t h e suhjects were qiven a brief definition of each noun taken from a dictionary). The co-occurence matrix (similar t o the m a t r i x Dreviously discussed, c f . 2.2) was subjected t o hierarchical clusterinn usinq Johnson's (1967) connectedness and diarceter mthod. These results are recalled i n 4 ( c f . Table 4 and Fioure 5 ) . The second i s extracted from a study conducted bv Henlev (1969). She ohtained from twenty one subjects an estimation of the distance between twelve animal terms ( f o r each subject she simoly counts the number of terms separ a t i n g the terms of i n t e r e s t i n a l i s t given by the subiect nrovided w i t h the instruction t o " l i s t a l l the animals thev could"); the matrices are then standardized and the mean f o r each cell (e.4. across subjects) give the entrv o f a dissimilarity matrix. Finallv, Friendly (1979), i n a naoer akind t o the two orevious ones,
7
Tree representations of associative structures
s t r o n c r l y advocate t h e use o f h i e r a r c h i c a l c l u s t e r i n ? t o diaaram t h e memory He nronosed a n r o x i r i t y m a t r i x o h t a i n e d s o n w h a t l i k e Henley
orpanization.
e x c e p t t h a t t h e o r i o i n a l d a t a came from f r e e r e c a l l exDeriments.
The data
were "a n r i o r i " o r q a n i z e d i n t h r e e subsets: animal terms, n a r t s o f t h e huran body, vegetables.
The a i r o f t h e s t u d y was t h e r e c o v e r y o f t h i s a D r i o r i
s t r u c t u r e ( c f . R o u s f i e l d (1353)
, Tulving
(1972)).
3. PlETHnDS P r o a d l y sneaking, t h e methods d e a l i n n w i t h D r o x i m i t v d a t a can he d i v i d e d under two aeneral c a t e g o r i e s : The rlans and t h e Cranhs. The llaps encompass
-
i n narticular
-
the d i f f e r e n t variants o f h l t i d i w n -
s i o n a l S c a l i n a ( c f . K r u s k a l l (1978), Ychiffman e t a1 (19811, Young e t a l , t o annear), Ootimal o r Dual S c a l i n g o r Corresnondence A n a l y s i s dependina on t h e a u t h o r s ( c f . Benzecri ( 1 9 7 3 ) , H i l l (1974), N i s h i s a t o ( 1 9 8 0 ) ) o r c l a s s i c a l s c a l i n a o r " t r i n l e m a l v s i s " i f t h e p r o x i m i t y c o u l d he judqed as d i s tance (Toraerson ( 1958) , Renzecri (1973) )
.
The a e n e r a l c h a r a c t e r i s t i c o f t h e Graohs i s t o r e p r e s e n t t h e o h i e c t s as v e r t i c e s and t h e D r o x i r n i t y ( o r t h e d i s t a n c e ) between them by arcs ( w i t h o r without valuation).
One c o u l d n o s s i b l y e x p l o r e t h e s t r u c t u r e o f these
qraohs b y l o o k i n g f o r t h e connected comDonents, o r i t s c e n t r o i d ( c f . Abdi ( 1 9 8 0 ) ) o r b y imposina some t h r e s h o l d f o r drawing t h e arcs between v e r t i c e s ( c f . The A n a l y s i s o f S i m i l i t u d e developed by Flament e t a1 (1962), and Degenne and Verges ( 1 9 7 3 ) ) .
One can e v e n t u a l l y imnose some r e s t r i c t i o n s on
t h e Sranh, i . e . b e i n q a t r e e .
bfe d e t a i l t h i s p o i n t i n a moment.
3.1. TRFFS
A q e n e r a l d e f i n i t i o n o f a Tree can be a c y c l e f r e e , connected, u n d i r e c t e d aranh; f o r convenience assume a v a l u a t i o n on t h e edqes, and d e f i n e t h e d i s tance between two v e r t i c e s as t h e l e n a t h o f t h e n a t h f r o m one v e r t e x t o t h e other. The U l t r a m e t r i c Tree c o u l d he c h a r a c t e r i z e d the classical
"
ultrametric inequality".
-
besides o t h e r c o n d i t i o n s
-
-
on
Three v e r t i c e s
say x,y,z
-
by
an U l t r a m e t r i c Tree v e r i f y : d(x,y)
s
Flax [d(x,z)
, d(y,z)I
.
T h i s n r o D e r t y i s q u i t e d r a s t i c and b a r e l y v e r i f i e d by s i m i l a r i t y measures o b t a i n e d f r o m an exDeriment (see Y i l l e r (1969) f o r a s t i m u l a t i n q discussion).
8
H. Abdi, ].-P. Bartliildmy
and X. Luong
ci-
So t h e n r o b l e m i s t o h u i l d un t h e h e t i e r u i t r a m e t r i c a n n r o x i m a t i o n of a ven s i m i l a r i t y
m a t r i x , and a l o t o f a l a o r i t h m s a r e a v a i l a h l e f o r t h i s j o h
( c f .
Instead o f imnosin? the
u l t r a w t r i c i n e q u a l i t y on t h e r e n r e s e n t a t i o n o f a d i s s i m i 1 a r i t . v ( o r d i s t a n c e ) matrix,
some a u t h o r s have nronosed t o weaken t h e u l t r a m t r i c i n e q u a l i t y i n
o r d e r t o o b t a i n a m r e o e n e r a l and more n a t u r a l r e D r e s e n t a t i o n ( c f . C a r r o l and Chano ( 1 9 7 3 ) , Cunnincham (1?174), (1978) ; S a t t a t h and Tverskv (1977) cal1c.C' f o l l o w i n g t h e a u t h n r s we? rlhtcd t r e e , f r e e t r e e , n a t h l e n g t h t r e e o r u n r o o t e d t r e e ( c f . amonn o t h e r s : Cicrr'an nuneman (1971), Dobson (1974)).
(1958)
, Fakami
t
d(u,v)
(1964)
Val!
,
T h i s k i n d of a t r e e i s c h a r a c t e r i z e d by t h e
following i n e o u a l i t y , h o l d i n q for every four-unlet d(x,y)
and
s 'lax [d(x.u) + d ( y , v ) ;
-
say x,y,u,v
d(x,v)
-:
.
+ d(y,u)l
I t can be shown t h a t t h e u l t r a m e t r i c i n e q u a l i t y i s s t r o n ? e r than the a d d i t i v e i n e q u a l i t y which, i n t u r n , i m o l i e s t h e t r i a n g l e i n e o u a l i t y . The a l g o r i t h m s f o r f i t t i n r l an a d d i t i v e t r e e t o a n r o x i m i t v m a t r i x a r e l e s s
numerous than those desioned f o r t h e s a e c i e s u l t r a m e t r i c t r e e .
The c o n s t r u c -
t i o n o f t h e a d d i t i v e t r e e can be s e a a r a t e d ( f o r t h e c l a r i t y of t h e exolanat i o n ) i n t o two p a r t s :
1) t h e f i n d i n g o f t h e t r e e - s t r u c t u r e (we darc! n o t use t h e t e r m s k e l e t o n . . . ) 2) t h e e s t i m a t i o n o f t h e v a l u a t i o n o f t h e branches o f t h e t r e e . The c l a s s i c a l apnroach ( a s i l l u s t r a t e d b y Cunninqham (1974),
(1978);
S a t t a t h and Tversky ( 1 9 7 7 ) ) makes a d i r e c t use o f t h e a d d i t i v e i n e q u a l i t v f o r the f i n d i n g o f t h e t r e e - s t r u c t u r e , and then e s t i v a t e s t h e v a l u a t i o n
with a l e a s t - s q u a r e method.
F o r o u r n a r t , we pronose a n o t h e r anproach t h a t
we d e t a i l i n a moment and c o n t r a s t w i t h t h e c l a s s i c a l one.
3.2.
CONSTR!KTION Or TIIE :REF
STRUCTURE
( A ) S a t t a t h and T v e r s k y (1977) i n t r o d u c e d o f "loose c l u s t e r " . L e t E he
3
s e t and
denoted as ( E , d ) .
-
i n a seminar naner
-
the notion
r' a d i s s i m i T a r i t y f u n c t i o n d e f i n e d on E x E , h e r e a f t e r A s u h s e t A f r o m E i s a l o o s e c l u s t e r i f , f o r a l l {x,y}
f r o m A and f o r a l l (u,v}
f r o v E ( w i t h x+u, x+v, ypu, y p v ) , t h e f o l l o w i n g
inenuality i s verified. d(x,y)
If
t
d(u,v)
<
f l i n Id(x,u)
+ d(y,v);
d(x,v)
+ d(.v,u)l
.
(1)
E corresponds t o t h e t e r m i n a l v e r t i c e s o f a t r e e , t h e n t h e f o l l o w i n g
9
Tree representations of associative structures
i n e q u a l i t v h o l d s f o r a l l f o u r - u n l e t (x,y,u,v): + d(u,v)
d(x,y)
i d(x,u)
+ d(y,v)
( o r any o f t h e f i v e i n e q u a l i t i e s t h a t can he
s d(x.v) + d(v,u)
(2)
deduc?d f r o m t h i s one by p e r -
mutation). Then i f { x , y l
i s a l o o s e c l u s t e r on a t r e e , from ( 2 ) and (1) f o 1ows : + d(u,v)
d(x,y)
<
d(x,u)
+ d(y,v)
= d(x,v)
+ d(y,u)
So, i n t h e f o l l o w i n p t r e e w i t h t e r m i n a l v e r t i c e s x,y,u,v
Y
{x,y} on t h e one hand, and { u , v l on t h e o t h e r a r e l o o s e c l u s t e r s , and ( 2 ) i s then d e f i n e d .
I n o a r t i c u l a r , x and y, u and v a r e t o adont t h e f o l l o -
w i n o c o n f i y r a t i o n and a r e s a i d t o merge.
coulr' be r e o r e s e n t e d hv a t r e e , we g e t
Converssly, knowinp t h a t x,y,u,v o n l y f o u r D o s s i h l e cases:
Y
v
u
v
(b)
(a)
v
u
(c)
u
V
(4
Here, t h e knowledce o f a l o o s e c l u s t e r i s s u f f i c i e n t t o c h a r a c t e r i z e f u l l y the renresentation.
(i)I f a t l e a s t one D a i r i s l o o s e then: I u , v l loose * t r e e a
(x,y}
loose
{x,u}
loose * {y,vj
loose *
tree b
{x,v)
loose
{y,u}
loose *
tree c
0
4
( i i ) No o a i r i s l o o s e , and t h e n we g e t t h e t r e e d. These p r e v i o u s remarks c o u l d he used i n o r d e r t o h u i l d t h e t r e e s t r u c t u r e fitting
a (E,d).
F o r each { x , Y l f r o m E t h e S a t t a t h and T v e r s k y ' s a l g o r i t h m counts t h e number
L
10
H. Abdi, J.-R BarfkiEmy a d X. Luong
o f p a i r s { u , v l v e r i f y i n a (1); c a l l t h i s number t h e s c o r e o f t!:e p a i r {x,!!). The a l g o r i t h m t h e n merges t h e h i s h e s t s c o r e n a i r ment
-
z
say
-.
The d i s s i m i l a r i t y between
-
say x,y
-
i n t o a new e l e -
z and t h e r e m a i n i n ? elements i s
s i m p l y o b t a i n e d by t h e average o f t h e d i s s i m i 1 a r i t . y o f x,y. then r e i t e r a t e d u n t i l 1 a l l elements have been merced.
The process i s
I t can e a s i l y be
v e r i f i e d t h a t t h i s aloorithm gives the exact t r e e o f (F,d) i f i t e x i s t s , n r o v i d i n g t h a t t h e i n t e r i o r v e r t i c e s o f t h e t r e e a r e o f decrree 3 .
(B)
f i r s t g i v e some d e f i n i t i o n s .
(i)
S c o r i n g s u b s e t and s c o r e .
G!e use t h e same n o t a t i o n s as before. dinality of
L e t k be an i n t e p e r l e s s than t h e c a r -
E. {a,bl i s element o f E x E, and
F* = F.
-
{a,bl.
Then we sa.v
that
-
a k-subset
say S
-
o f E* i s a s c o r i n g subset f o r ( a , b l i f f o r a l l
x,y f r o m S t h e f o l l o w i n p i n e q u a l i t y h o l d s : d(a,b)
t
Max d(x,y)
<
rlax[d(a,x)
t
The number o f k - s c o r i n g subsets f o r t h e p a i r {a,b)
d(,hy)l
.
i s c a l l e d the k-score o f
{ a , b l and i s denoted k-Sc(a,b). Let
(
E,d) be an a d d i t i v e t r e e and k equal t o 2.
The s o - c a l l e d c a n o n i c a l
d i s t a n c e g i v e s t o e v e r y branch o f t h e t r e e t h e l e n g t h 1.
tlw we can g i v e
the: Prooosi t i o n : I f t h e c a n o n i c a l d i s t a n c e between trio t e r m i n a l v e r t i c e s
- say
a,b
-
i s two, then k-Sc(a,h)
i s maximal and k-Sc(a,u)=k-Sc(b,u)
f o r e v e r y u of E . The p r o o f i s l e f t t o t h e r e a d e r , b u t can e v e n t u a l l y be found i n Luong (1983) w i t h some o t h e r p r o o e r t i e s o f t h e s c o r e s .
( i i ) Construction o f the t r e e s t r u c t u r e . S e t k=2. F o r e v e r y p a i r x,v we c o u n t t h e number o f s c o r i n a subsets. compute then a s c o r e m a t r i x .
-
say a,b
-,t h e n
Me
Me l o o k f o r t h e p a i r w i t h t h e maximal s c o r e
we merne a,b t o F i v e c and we Dose: k-Sc(c,u)
= [k-Sc(a,u)tk-Sc(b,u)l
/ 2
,
The nrocess i s then r e i t e r a t e d u n t i l a l l t h e r e m a i n i n g elements have been merged.
T h i s procedure i s j u s t i f i e d b y t h e o r e v i o u s n r o p o s i t i o n .
Notice
i n n a s s i n g t h a t we c o u l d s t r a i g h t f o r w a r d l y e x t e n d t h i s Drocedure t o any k ( c f . Luong (1983)).
11
Tree representations of associative structures
3.3. ESTIMATIOK OF THE VALUPTION OF THE EDGES
( P ) S a t t a t h and Tversky (1977) , Cunningham ( l 9 7 4 ) , (1978), nroDose a l e a s t square method f o r e s t i m a t i n a the l e n g t h o f t h e branches o f t h e t r e e .
Pre-
c i s e l y , i f d a r e t h e d i s t a n c e s between t e r m i n a l v e r t i c e s on t h e t r e e and 6 t h e d i s s i m i l a r i t y observed, d i s t o s a t i s f y : WIN [
-
Id(x,y)
I:
X ,*€
6(x,v)121
This nroblem i s equivalent t o the m a t r i c i a l equation: Ct
c
Y = Ct 6 ,
rlhere 6 i s t h e v e c t o r o f t h e ohserved d i s s i m i l a r i t v , C t h e i n c i d e n c e m a t r i x o f t h e branches and t denotes t h e transDosed m a t r i x . Voreover S a t t a t h and Tverskv i n d i c a t e t h e e x i s t e n c e it
- o f a method a v o i d i n n t h e
-
but without giving
comnutation o f t h e i n v e r s e o f a m a t r i x .
( 8 ) The q e n e r a l i d e a o f o u r nrocedure i s t o make a o e m t r i c a l e s t i m a t i o n o f t h e d i s t a n c e ; i . e . we embed t h e d i s s i m i l a r i t i e s i n an E u c l i d e a n space, and use t h e g e o m e t r i c a l D r o n e r t i e s o f t h i s sDace i n o r d e r t o o b t a i n an e s t i m a t i o n o f t h e d i s t a n c e s on t h e t r e e .
I f d i s a distance
The t r e e o b t a i n e d as d e s c r i b e d above, i s a b i n a r y t r e e . d e f i n e d on a b i n a r v t r e e , r e n r e s e n t i n q t + a o a i r !a,bl
by z, t h e b e s t pos-
s i b l e r e n r e s e n t a t i o n n u t s z on t h e n e a r e s t v e r t e x f r o m a,b. t h e d i s t a n c e s d(z,u)
- u
d(z,u)
belonoino t o E
+
= [d(a,u)
-
I n t h i s case,
are e a s i l y estimated:
d(b,u)
-
d(a,b)l
/ 2
.
I n t h e more g e n e r a l case where d i s a d i s s i m i l a r i t y , we i n t r o d u c e a " c e n t r a l p o i n t " c a l l e d 9, w i t h : d(g,u)
= [
r
/ n,
d(v,u)l
E E f o r a l l u i n E, w i t h n = c a r d i n a l i t v o f E . We t h e n embed t h e d i s s i m i l a r i t i e s i n an E u c l i d e a n snace. maximal score, we b u i l d t h e s u h t r e e a z b.
I f { a , b l has a
The d i s t a n c e s on t h e t r e e w i l l
then be e v a l u a t e d b y t h e r e m a i n i n a d i s s i m i l a r i t i e s d(a,u) from E.
Now z r e o r e s e n t s { a , b l ;
brouqht nearer t o
0.
d(b,u)
for all u
we make t h e h y n o t h e s i s t h a t z must be
According t o t h i s , we can determine by t h e geometry
t h e new d i s s i m i l a r i t y d(z,u)
f o r aT1 u f r o m
E
-
Ia,bl.
The nrocessus i s
t h e n r e i t e r a t e d p a r a l l e l l y w i t h t h e s c o r e procedure d e s c r i b e d p r e v i o u s l y .
H. Abdi. J:P. Barthifkmy and X . h o n g
12
3.4. EVALUATInN OF THE ALGnRITHt’ The nrocedure we oronosed i s c l e e r l v ‘ i c l r i s t i c , and so, i t c o u l d he i n t ? r e s t i n o t o have an e v a l u a t i o n o f t h e q u a l i t y o f i t .
One n o s s i b l e way can be
some random d i s t a n c e m a t r i c e s , t o h u i l d t h e a d d i t i v e t r e e , t h e n
t n oenerate
t o r e h u i l d the d i s t a n c e m a t r i x and t o n e a s u m C . k c‘ecirec c f f i t between t h e o r i g i n a l m a t r i x and t h e r e h u i l t m a t r i x .
And so we d i d .
1le decided t o
choose as a measure o f f i t t h e c l a s s i c a l o r o d u c t mment c o r r e l a t i o n ! h u t i t
i s we11 known t h a t r i s s t r o n a l y r e l a t e d t o some o t h e r measures o f f i t , e . 0 . t h e s t r e s s o f K r u s k a l l . The r e s u l t s o f o u r s i w l a t i o n s a r e Siven i n t a 5 l e 1. Number o f set averacle standard rrin wax trials size r d e v i a t i on 25
12
.
.C?f
.R4
.95
20
18
.884
.034
.G33
.930
16
24
.861
.029
.811
.914
ia
30
.863
.016
.843
.a8
10
36
.860
-022
.8?
.89
8
42
.855
.018
.El
.905
4
43
.840
.021
.82
.875
Tahle 1 Goodness o f f i t measure between random E u c l i d e a n d i s t a n c e m a t r i c e s and r e b u i l t d i s t a n c e on an a d d i t i v e t r e e .
I t c o u l d he seen t h a t t h e o v e r a l l degree o f f i t i s q u i t e m o d .
I n fact,
t h e r e s u l t s o b t a i n e d f a l l c l o s e l y n e a r t h e r e s u l t s o b t a i n e d b y Pruzansky e t a1 (1982) w i t h t h e o r i o i n a l ADDTREE o f T v e r s k y .
3 . 5 . !“/!
rJtIL1TY
n u r a l n o r i t h m i s t r a n s l a t e d i n t o s t a n d a r d R a s i c (CP/hI) and a l i s t i n q o f t h e nroaram i s a v a i l a b l e on r e q u e s t . 4 . S O T QESULTS The d i f f e r e n t f i g u r e s and t a b l e s a r e q i v e n a t t h e end o f t h e oaDer.
J . 1 . THE WAR OF THF GHOSTS !,!e examine h e r e t h e r e s u l t s o b t a i n e d w i t h t h e d i s t a n c e between t e x t s seen as B i - p a r t i t i o n s ( a s d e f i n e d i n 2 . 1 . R . i i ) .
Figure 1 gives the a d d i t i v e
t r e e o b t a i n e d w i t h t h e “ r e p e a t e d r e n r o d u c t i o n ” method.
have i n c l u d e d
,.lree representations of associative structures t h e o r i g i n a l t e x t f o r ease o f comparison.
13
F o r convenience, we g i v e t h e
d i s t a n c e m a t r i x (Table 1) an6 t h e "ao r e s u l t i n g o f a F a c t o r i a l A n a l y s i s o f D i s t a n c e (F.A.D.)
aoolied t o t h i s m a t r i x (Figure 2).
One s t r i k i n g r e s u l t i s t h e f i d e l i t y o f t h e s u b j e c t s t o themselves as c o u l d he seen on the a d d i t i v e t r e e .
Moreover, one c o u l d e a s i l y d e t e c t t h e accu-
r a t e s u b . i e c t ( s ) here, f o r examole, t h e s u b j e c t LP i s f a i r 1 . v a c c u r a t e b o t h on t h e same day o r 120 days a f t e r .
These r e s u l t s a r e w o r t h n o t i n g , because
t h e emnhasis i n t h e l i t e r a t u r e i s o e n e r a l l y n u t on t h e G i s t of t h e r e c a l l f o r these k i n d o f d a t a i n s t e a d of 'Verhatim r e c a l l .
Nevertheless, the f i -
d e l i t y o f t h e s u b j e c t s t o themselves c o u l d be due t o t h e s t a b i l i t y o f t h e i r h a b i t s of language as w e l l as t o t h e " q u a l i t v " o r constancy o f t h e i r nemory. TIE
FLD
Flap g i v e s e s s e n t i a l l y t h e same c o n c l u s i o n , athouoh t h e dimensions
r e v e a l e d by t h e Map a r e n o t e a s i l y i n t e r o r e t e d .
Nevertheless a d i s c r e p a n c y
must he noted: t h e Man i n d i c a t e s L 120 f a r f r o m P R I G and L 1 c o n s t r a t i n g w i t h t h e a d d i t i v e t r e e which F i v e s L 120 n e a r O R I C and L1.
I n f a c t t h e ex-
a m i n a t i o n o f t h e d a t a m a t r i x r e v e a l s t h a t L 120 and L 1 a r e c l o s e t o each o t h e r , so do L 1 and UPIC b u t O R I G and L 120 a r e q u i t e f a r f r o m each o t h e r .
So t h e two methods d i f f e r when a g e n e r a l view i s n o t o b v i a u s . The a d d i t i v e t r e e o b t a i n e d f r o m t h e d i s t a n c e m a t r i x f o r t h e " s e r i a l r e p r o d u c t i o n " method a l l o w s t h e r e c o v e r v o f t h e s e r i a l s t r u c t u r e ( i . e . a s e n i o r d e r ) o f t h e d a t a ( a l t h o u g h t h e a d d i t i v e t r e e model i s p r o b a b l y n o t t h e b e t t e r one f o r t h i s k i n d o f p r o b l e m ) .
As p r e v i o u s l y , a s i m n l e l o o k a t t h e
t r e e s u f f i c e s t o d i s t i n g u i s h t h e i n a c c u r a t e "Eaos" (e.g.
between R3 and R4,
and R6 and R7).
4.2. THE OUALIFIERS OF THE CHARACTER The d a t a m a t r i x 58 x 58 r e p r e s e n t s a c t u a l l y o u r b i p g e s t examole t r e a t e d by o u r methods.
The e f f i c i e n c y o f o u r a l g o r i t h m apoears here: we needed seven
h o u r s (on a m i c r o comouter) t o o b t a i n t h e t r e e and t h e v a l u a t i o n s on t h e arcs; i n c o n t r a s t , t h e c l a s s i c a l a l g o r i t h m would have needed - a t t h e v e r y least
-
t h r e e days i n t h e same c o n d i t i o n s .
Ye have i n c l u d e d h e r e i n a d d i t i o n t o t h e a d d i t i v e t r e e , t h e three-dimension a l r e s u l t o f a F.A.D. o f t h e same data.
A l t h o u p h t h e r e i s no m a j o r d i s -
crepancy between t h e two approaches , t h e addi t i ve t r e e i s c l e a r l y more readable.
P r o b a b l y because t h e i m p l i c i t model h e r e i s one o f c l u s t e r r a t h e r
t h a n one o f d i f f e r e n t dimensions.
To be more o r e c i s e , t h e s o r t i n g i n s t r u c -
t i o n induces t h e c o n s t r u c t i o n o f c l u s t e r s and t h e h i g h dearee o f agreement
H.Abdi, J.-P. Barthilimy and X. Luong
14
between s u b j e c t s ( c f . t h e d a t a m a t r i x ) l e a d s o h v i o u s l y t o a " c l u s t e r " s o l u tion.
N e v e r t h e l e s s , s o r e d i s t i n c t c l u s t e r s o f q u a l i f i e r s can he i d e n t i f i e d
and he i n t e r o r e t e d i n t e r m n f " i m a l i c i t p s y c h o l o n y " , h u t t h i s i n t e r p r e t a t i o n remains t o h e v e r i f i e d e x n e r i m e n t a l l y , l a t e r on.
4.3. OLDIES RUT GOLDIES F o r t h e d i f f e r e n t examDles evoked, vie g i v e t h e o r i n i n a l d a t a m a t r i x , t h e h i e r a r c h i c a l s c l u t i o n p r o v i d e d by the? a u t h o r s and t h e a d d i t i v e t r e e ( e x c e n t f o r Henley f o r w h i c h we q i v e a trap r e s u l t i n o f r o m a F.A.D.).
The comparison
i s l e f t t o t h e r e a d e r as a n a s s t i m e . . . .
TABLES AND FIGURES
ORIG N 1 h15 HP1 HP8
L1 L12G
P1 P45 P105 P1145
R1 R15 R45 XlFn
ORIG
0 21 0 1115 el 32 HP1 119 102 121 102 HP8 L1 78 85 ~ 1 2 0 110 91 118 83 P1 P45 109 88 P105 113 92 P1145 115 80 R1 80 59 R15 91 C.2 R45 101 6 8 XI80 113 7E 111
0 106 108 89 99 101 94 104 84 65 68 68 5?
0 0 42 c! 107 113 88 97 99 91 97 102 100 102 107 94 100 105 76 80 107 83 92 75 84 93 76 78 92 103 84 90 111
c 92 95 97 87 82 79 77 87
0
59 49 71 80 75 79 83
0 52 80 77 78 86 92
0 76 87 86 90 90
0 53 46 42 58
0 23 37 65
0
?C
0
58 I r,r,
0
Table 2 D i s t a n c e m a t r i x between 15 t e x t s o h t a i n e d by S a r t l e t t (1932) w i t h t h e metho:! o f "reeeated reproduction". The d i s t a n c e i s t h e d i s t a n c e o f t h e s y m m e t r i c a l d i f f e r e n c e between R i - D a r t i t i o n s (see t e x t ) .
The s u b j e c t s a r e denoted by t h e l e t t e r ( s ) b e g i n n i n g t h e
L a b e l , t h e f i g u r e s f o l l o w i n o t h e l e t t e r ( s ) i n d i c a t e t h e number o f days between t h e f i r s t p r e s e n t a t i o n o f " t h e war o f t h e Ghosts" and t h e r e c a l l .
Tree representations of associative structures
C R I ~ N A L LI
Fioure 1 A d d i t i v e t r e e o b t a i n e d f r o m t h e d i s t a n c e m a t r i x of Table 2. The s u b j e c t s a r e denoted by t h e l e t t e r ( s ) b e q i n n i n o t h e l a b e l , t h e f i g u r e f o l l o w i n ? t h e l e t t e r ( s ! - i n d i c a t e t h e number o f days between t h e f i r s t P r e s e n t a t i o n n f t h e '%r o f t h e Ghosts" and t h e r e c a l l .
16
P.ll=1/Cx\’=2 P 105
12
--
-1
I
a45
P1
I
P1145
N15 1
X180
R15
I
R
R45
’8 ’1
I I
__I_ ___ _ Fiqure 2
Euclidean representation ( f a c t o r i a l analysis o f d i s t a n c e ) o f the distance m a t r i x of Table 2. The s u b j e c t s a r e denoted bv the l e t t e r ( s ) beginnincl the l a h e l , the figures follorrina the l e t t e r ( s ) i n d i c a t e the numher o f days between the f i r s t nresentation of “ t h e ‘Jar of the Ghosts” and the r e c a l l .
17
Fiaure 3 Additive t r e e obtained from the distance matrix b u i l t f r o p the d a t a o f R a r t l e t t (1932): "Serial reproduction". The l a b e l s g i v e the s e r i a l o r d e r .
18 H. Abdi, J.2. Borthifimy and X. Luong
V
m
w "re o
Tree representotions uf assuciatiue structures
19
a
Figure 4 Additive free obtained from the distance matrix o f Table 3 . the character).
(Qualifiersof
20
H . Abdi, J:P. Bartliilemy and X . L u o q
an;u
FIN
FAlW
TWE
SIM TOLE
Fiwl YRJ
PESS
m
Fiaure 5 E u c l i d e a n r e p r e s e n t a t i o n ( f a c t o r i a l a n a l y s i s of d i s t a n c e ) o f t h e distance m a t r i x o f Table 3. ( n u a l i f i e r s o f the c h a r a c t e r ) . Pxe I end P.xe 2.
Tree representations of acsociative structures
FRlU
UISP
)#IF
!
PESS
7011
I
rwls
YC
FMI
1
SNE
Figure 6 Euclidean representation ( f a c t o r analysis o f distance) o f the distance matrix o f t a b l e 3. ( q u a l i f i e r s o f the c h a r a c t e r ) . Axe 1 and Axe 3 .
21
s
5
Co-occurence m a t r i x for 48 Enalish nouns. Data from W l l e r (1969). The c e l l s indicate the number o f subjects who o u t the words i n the same p i l e (number of subjects: 5 0 ) .
23
Tree representations of associative structures
Diometer method
Connectedness method
f " " '
1
'
I
I
I
I
I
:
'
j
*-----,J
VAP:::Sii
I
I
7
I 3ATTLf
KILL
om. LABOR PLAY
AID COUNSEL
vow HONOR
EASE REGRET T11RliL
C
10
20
30
40 50
50
40
30
20
10
0
Number of subjects
Fiwre 7 Hierarchical solutions for the data of Table 4 . (1969).
From Miller
H.Abdi, ] . - P BartlihMmy atid X. Luong
1.
1
Figure 8
Additive Tree obtained from the data o f Table 4 .
25
Tree representations of associative structures
Bear Cat cow Deer
Cat
Cow
Deer
noo
coat
Horse L i o n
47.2
27.7 30.9
40.1 56.1 43.6
49.6 2.0 30.2 50.9
19.1 29.0 11.0 44.5 17.0
29.0 25.3. 7.7 43.0 24.0 7.2
DO0
Coat Horse Lion Mouse Pi9 Rabbit
22.6 24.1 24.5 44.7 26.9 23.1 28.6
blouse P i ?
29.5 24.8 34.1 39.9 27.5 39.6 32.6 33.2
E a b b i t Sheen
21.4 43.0 17.0 41.1 45.0 19.5 25.7 29.3 34.9
20.3 41.5 27.9 19.9 33.4 21.8 30.1 33.3 22.6 25.9
16.1 47.1 8.2 53.1 46.8 1.8 15.2 35.0 51.9 19.6 32.5
Table 5 P r o x i m i t y m a t r i x between animal terms f r o m l!enley (1969).
St'EEP
r.r CC PIG
r !!ORSE LION
BEAR
1 CAT DrY
RAPBIT MOUSE
DEER
Fioure 9 Euclidean representation ( f a c t o r i a l analysis of distance) o f the d i s t a n c e m a t r i x o f Table 5 . ( H e n l e y ' s animal
terms).
F 3
i
I L--,PlOUSC
I-
rD
n
W
Q5
L
4.
dog
X
leg
100 91 88 86 88 5.5 62 63 62 58
iooc
5s
celery spinach lettuce
69 69
potdl0
68 66 66
('dl
lion tiger > heep godl _. ldCe
mouth arm iinger
rice Vam
70
a
93 91 88 89 57 65 66 64 61 57 67 68 69 67 65 65
x
100 92 94 57 62 65 63 62 59 56 60 60
60 54 57
x
91 94 57 61 65 63 62 59 56 58 58 59 53 56
x
100
x
i2 60 61 56 56
51 56
51 61 65 64 64 53 60
60 58 57 53 63 64 64 65 58 62
X
95 90 94 89 93 60 61 60 65 66 63
x
92 94 92
90 60 62 61 65 66 64
x
93 96 92 57 60 59 64 65 62
x
90 91 62 62 62 67 70 67
x 98 48 56 54 60 58 58
x
49 57 55 60 60 57
I
Table 5 P r o x i m i t y M a t r i x t:e+weer:
18 e n g l i s h Nouns o b t a i n e d by F r i e n d l y
(1979) i n f r e e r e c a l l e x p e r i m e n t s .
2
X
96 94 87 88 91
e, R
x
97 95 86 93
x
93 85 92
0
0
p -.
x
92 97
x
99
(D
x
28
H.Abdi, /.-I? BartliilCrny and X . Luong
C O t e Q O r i r e d List I I '
--
Prori mi t y
F i y r e 11 H i e r a r c h i c a l s o l u t i o n for the d a t a of T a b l e 6 (from F r i e n d l y ( 1%9) )
.
1
Fiqure 12 A d d i t i v e tree o b t a i n e d from t h e d a t a o f T a b l e G ( F r i e n d l y ' s d a t a ) .
29
Tree representations of associative structures
RE FEREFICES
Abdi , t ; , h a r o c h e s nreordinales en analyse des donnees nsychologiques , these, Aix-en-"rovence, ( 1980) . A h d i , H., Structuration des n u a l i f i c a t i f s d u c a r a c t e r e , Multigraph, Di ,jon ( 1983) . Arabie, P . a n d Roorman, S . , Vultidimensional s c a l i n q of measures of distances hetween Darti t i o n s , -!curnal of P a t h . Psychol , 10 (1973), 148-203. B a r t l e t t , F.C., Petwmberino, Cambridge University Press, Cambridge (1932). Benzecri, J.P., L'analyse des donnees, Dunod, Paris (1973). Boorman, S . A . , P l i v i e r , D . C . , Vetrics on soaces o f f i n i t e t r e e s , Journal of !lath. Psycho1 , 10 (1973\, 26-59. Bousfield, [-I./'., The occurence of c l u s t e r i n o in the r e c a l l of rilndclrly arranged a s s o c i a t e s , Journal of Ceneral Psvcllol , 49 (1353), 229-240. Runeman, P . , A note on the metric pronerties of t r e e s , Journal of Comh i n a t o r i a l theory, 1 7 ( b ) , 48-50. C a r r o l l , J . D . , Chan?, J . J . , P. method f o r f i t t i n ? a c l a s s of h i e r a r c h i cal t r e e s t r u c t u r e models t o d i s s i m i l a r i t y data and i t s anolication t o some body Darts data of Y i l l e r ' s , i n Proceedings of the 81th Annual Convention Awrican Psycholonical Asscciation, 8 (1973). 1097-1098. Colonius, H., and Schulze, P.H., Tree s t r u c t u r e s f o r Droximitv d a t a , a r i t i s c h Journal of bcath. and S t a t . Psychol 34 (1981). 167-180. Cunningham, J . P . , Finding an optimal t r e e r e a l i z a t i o n of a Droximity matrix, paper presented a t the mathematical asycholoay meeting, Ann Arbor (1974). Cunningham, J .P., Free t r e e s and b i d i r e c t i o n a l t r e e s as renresentations of psychological distances, Journal of Math. Psychol , , 17 (1978), 165-188. Cegenne, A . , and Vernes, P . , Introduction a l ' a n a l y s e de s i m i l i t u d e , Revue FranGaise de Socioloqie (1973). Dobson, J . , Unrooted t r e e f o r nunerical taxonom.y, Journal of Ppnlied P r o b a b i l i t y , 11 (1974), 32-42. F l a m n t , C . , L'analyse de s i m i l i t u d e , Cahiers du C . R . O . , Brussels(l962). Friendly, M., Yethods f o r findinq graphic reoresentations of associat i v e memory s t r u c t u r e s , i n Puff,C.R. (Ed.), Nemorv organization and s t r u c t u r e , Academic Press, New York (1969).
.
.
.
.,
H. Abdi, J.-P. B m t k h h i y
30
I171 I1p.1
Hakami, S.L., and Yau, S.S., Distance matrix o f a rranh a n d i t s r e a l i z a b i l i t , y , ' ! u a r b r l y o f Annlied Matkcmatics, f: (!cG4), 305-317. Harti?an, J . F . , Renresentations o f s i m i l a r i t v matrices by t r e e s , tl.A.S.A.,
1191
1201 [211
and X. Luong
6 2 ( 1 9 6 7 ) , 1140-1152.
Iiartioan, J . A . , Clusterino a l y r i t h m s , J . ' J i l e v . New vork ( 1 9 7 5 ) . Iknley, N . Y . , P nsvcholoaical study of the semantics o f animal t e r m , ,Icurnal of Verb. Learn. and Verb. ?e'?avior, E ( 1 9 6 9 ) , 176-184. H i l l , ?'.O., Corresoondence a n a l y s i s , nnol?t+ed method, Amlied S t a t i s t i cs , 23 ( 1974) , 340-354. Johnson, Z . C . ,
I!iF:rarchical c l u s t e r i n q s c h e m e s , P ~ ' r c h o ~ t r i k a32 ,
( 19G7), 241-254. 1231
Kerlinger, F . N . , Foundations of behavioral research, t b l t . New York
1241
Kruskall, J.P., and 'dish, 'I., r!ultidimensional s c a l i n n , Sane. Beverly Hi 11s (1978) . Luonq, X., Voisinaoe l l c h e , score e t famille s c o r a n t e , rlultigranh, BesanGon ( 1 9 8 2 ) . Luona, X., t o a m e a r ( 1 9 8 3 ) . ? f i l l e r , G . A . , P asvcholonical method t o i n v e s t i a a t e verbal conceots, ,lournal of Math. Psychol . , 6 ( 1 9 6 9 ) , 169-191. Nishisato, S . , Analysis of categorical c!ata: dual s c a l i n o a n d i t s a n p l i c a t i o n s , llniversi ty o f Toronto Press, Toronto (19SO). Pruzansky, S . , Tversky, P . , and C a r r o l l , J.R., S n a t i a l versus tree rearesentation of nroximitv d a t a , Psychometrika, 4 7 ( 1 ) ( 1 9 8 2 ) , 3-24. Riordan, J . , An introduction t o combinatorial a n a l y s i s , ,I. !Jiley, Pew York ( 1 9 5 6 ) . Rosenbera, S . , The method of s o r t i n a in m u l t i v a r i a t e research, in Hirschberg, N . , and Humnhreys, L . G . , ( E d s . ) , U u l t i v a r i a t e aDDlications in the s o c i a l s c i e n c e s , L . Erlbaum, H i l l s d a l e ( 1 9 8 2 ) . Rnsenbera, S . , Nelson, C . , and Vivekananthan, P . S . , A multidimensional amroach t o the s t r u c t u r e o f personal i t y imoressions, Journal of Pers . and Soci a1 Psychol ., 9 (19681, 233-294. S a t t a t h . S., and Tversky, A . , Additive s i m i l a r i t y t r e e s , Psychometrika,
(1973).
1251
1261 [271
[281 I291 1301 [3 11
1321
1331
42( 3 ) ( 1 9 7 7 ) , 319-345. 1341 I351
qchiffman, S . , Reynolds, l d . L . , and Young, F.I.I., Introduction t o multidimensional s c a l i n a , Academic Press, New York (1902). ?neath, P.H., Sokal , R.R., Numerical taxonomv, Freeman, San Francisco (1973).
Tree representations of associative structures
[361
Toroerson, !J.S.,Theory
[37]
T u l v i n g , E.,
[30]
l+gner,
31
and methods o f s c a l i n g , ,I. Ialiley, New York
(1958).
and Donaldson, N. (Eds.),
m i c Press, New v o r k (1972)
D.P.,
Younq, F.V.,
.
and V a l l a c h e r , R.R.,
s i t v Press , O x f o r d (1977)
[39]
De Leeuw,
n r a a n i z a t i o n o f w m o r y , Acade-
I m p l i c i t osycholopv, o x f o r d U n i v e r -
. Takane, Y.,
t o amear: h t i d i m e n s i o n a l
S c a l i n g : t h e o r y and methods, L. Erlbaum, H i l l s d a l e .
This Page Intentionally Left Blank
TRENDS IN MA T H E M A TICAI. PSYCHO LOG Y E. Degree/ and J. Von nu enhout (editors) @Ekevier Science Publisgrs 8. V . (North-€lolland),I984
33
TASK-DEPENDENT REPYESENTATION OF CATE'VRIES AND lFfIOPV-c:I!I DEn INFERENCE DURING CLASYI F I CCTInN Hans B u f f a r t U n i v e r s i t v o f N i imeoen, The N e t h e r l a n d s
-
Hans Geo r q Cei s s 1e r Karl-rlarx-Uni v e r s i tat, L e i q z i g, Deutsche D e m k r a t i s c h e Renubli k
Tt:c f o r m a l i s m o u t l i n e d p r e s e r v e s t h e most e s s e n t i a l constituents o f guided inference. P r o c e s s i n q i s s p e c i f i e d i n terms o f o p e r a t i o n s a c t i n 9 on c a t e g o r y r e p r e s e n t a t i o n s t h a t i n c l u d e task-cons t r a i n t s and " r e d e f i n i t i o n o f f e a t u r e s " . Our r u l e s can be c o n s i d e r e d as examples o f dimensiona l o r d e r i n g as d i s c u s s e d b y q a r n e r (1978).
1. II~TR~DIJCTION The nroblem o f how i n f o r m a t i o n i s arocessed i n c l a s s i f y i n g o b j e c t s a c c o r d i n q t o conceatual c a t e g o r i e s has r e c e i v e d much a t t e n t i o n t h r o u g h o u t t h e h i s t o r y o f e x n e r i m e n t a l nsycholony.
Thouoh t h i s t o n i c ranges o v e r a d i v e r s i t y o f
e x n e r i m e n t a l n a r a d i qms t h e r e seems t o be sow t h e o r e t i c a l agreement amon? r e s e a r c h e r s t h a t we summarize i n t h e f o l l o w i n g f i v e a s s e r t i o n s .
1.1. REPRESENTATIONS a ) The r e n r e s e n t a t i o n o f conceots i s based on d i s c r e t e f e a t u r e s .
b ) Concents can be r e n r e s e n t e d i n terms o f l o a i c a l f u n c t i o n s r e f e r r i n g t o these f e a t u r e s , most commonly by c o n j u n c t i o n s and d i s j u n c t i o n s o f t r u t h values
.
1.2. THE CLATSIFICATION-PROCESS c ) There i s a d i s t i n c t i o n between r e l e v a n t and i r r e l e v a n t f e a t u r e s .
d) The process o f c l a s s i f i c a t i o n i m p l i e s a s e r i a l t e s t i n g o f t h e r e l e v a n t f e a t u r e s o r grouos o f such f e a t u r e s . e ) The process o f c l a s s i f i c a t i o n comprises an e n c o d i n g s t a o e t r a n s l a t i n g objects i n t o the feature format f o r s e r i a l arocessing.
34
H. Buffart and H . G . Geissler
One ma!' d e v i s e s e v e r a l naradioms t o measure t h e c l a s s i f i c a t i o n - o r o c e s s n a r t l v deoendent on t h e view on t h i s n r o c e s s .
F o r examnle, a qaradiqm
c o m o n l y used i n a c l a s s i f i c a t i o n t a s k i s t o measure r e a c t i o n - t i m e s .
It i s
based on the i d e a t h a t r e a c t i o n - t i m e i n c r e a s e s w i t h t h e number o f d e c i s i o n s involved.
IJe do n o t d i s a n r e e w i t h such a h y n o t h e s i s i n n e w r a l , h u t we
l i k e t o emnhasize t h a t t h e r e a c t i o n - t i m e s do n n t i n f o r m about t b e c h a r a c t e r o f t h e d e c i s i o n s and t h a t thus c o n c l u s i o n s from e x n e r i m e n t a l r e s u l t s h e a v i l v l e a n unon b o t h t h e t h e o r e t i c a l model and tCle e x o e r i m e n t a l t a s k . flanv o f such models assume t h a t t h e d e c i s i o n - s t r a t e n y used h y s u b j e c t s i s most rlarsimonious w i t h r e s r l e c t t o t h e c l a s s i f i c a t i o n t o be s o l v e d . such a n o i n t
From
of view t h e s t r a t e q y can be d e r i v e d from t h e l o o i c o f t h e
c l a s s i f i c a t i o n nrohlems and r e a c t i o n - t i m e s mav be n r e d i c t e d . an ' o o t i m a l d e c i s i o n ' view.
It i s called
P s A l i e n t example i s t h e s e r i a l e l i m i n a t i o n
mode1 nroqosed b y Wick (1952) who c o n s i d e r s c l a s s i f i c a t i o n i n a m u l t i n l e c b o i c e s i t u a t i o n t o be based on a sequence o f b i n a r y d e c i s i o n s .
I t can be
e a s i l y oroven t k a t w i t h such a b i n a r v d e c i s i o n - s t r a t e o y t h e number of dec i s i o n s t o be made i s m i n i m a l . However, t h e r e a r e t v o , c l o s e l y r e l a t e d , oroblems when u s i n g such an a s s u m t i c n of m i n i m a l number o f d e c i s i o n s on l o g i c a l grounds i n d e q e n d e n t o f a suhj e c t ' s r e n r e s e n t a t i o n of a s t r u c t u r e due t o ' c o g n i t i v e c o n s t r a i n t s ' (Note 1). F i r s t , such a r u l e cannot e x n l i c i t l ; ! i s n o t s e n s i t i v e f o r chanoes i n t a s k .
i n c o r c o r a t e t a s k - c o n s t r a i n t s and t h u s I n s t e a d , s e v e r a l n r o c e s s i n q models
may c o e x i s t w i t h o u t anv t o o l t o s p e c i f y i n which s i t u a t i o n a n a r t i c u l a r strateqy apnlies.
Second, a d e s c r i n t i o n o f t h e c l a s s i f i c a t i o n - n r o c e s s o f
o b j e c t s o u r e l y i n terms o f a m i n i m a l number o f d e c i s i o n s i s as such n o t a f o r m u l a t i o n i n t e r m o f t h e c o a n i t i v e c o n t e n t o f t h e c a t e g o r i e s Drocessed, i.e.
i n o r d e r t o a a t h e r i n f o r m a t i o n a b o u t t h e c o a n i t i v e c o n t e n t f r o m expe-
r i m e n t a l r e s u l t s a d d i t i o n a l i n t e r o r e t a t i o n of t h e d a t a i s needed ( c f . G e i s s l e r and P u f f e
(1983)).
Pure o o t i m a l d e c i s i o n models a r e w i t h o u t any
f u r t h e r assumption n o t s u i t e d t o d e s c r i b e t a s k - s n e c i f i c i t y o f human c l a s s i f i c a t i o n oerformance.
A model t h a t accounts f o r t a s k - s n e c i f i c aerformance
must i n c l u d e t h e o r g a n i z a t i o n o f b o t h task-deaendent r e n r e s e n t a t i o n and task-dependent D r o c e s s i n ? . G e i s s l e r and P u f f e (1983, Note 2) d e s c r i b e d a n o t h e r view, c a l l e d t h e ' g u i d e d i n f e r e n c e ' view.
I t anoeared t o account f o r task-denendence i n some expe-
r i m e n t s t h a t were designed t o t e s t c r u c i a l d i f f e r e n c e s between t h i s ' g u i d e d inference'
and t h e ' o n t i m a l d e c i s i o n ' view.
D e s p i t e t h e l i m i t e d range o f
35
Task-dependen t representation of categories
c o n d i t i o n s i n which t h e view has so f a r been s u b j e c t e d t o e x o e r i m e n t a l s c r u t i n o we b e l i e v e t h a t t h e l o q i c s o f t h e armroach may be f o r m a l i z e d .
In
t h e n r e s e n t paoer we t r y t o f o r m u l a t e t h e l o g i c s o f cruided i n f e r e n c e i n more d e t a i 1
.
2 . OPTIVAL D E C I S I O N APPLIED The b a s i c c h a r a c t e r i s t i c s o f p u i d e d i n f e r e n c e can a t b e s t be i n t r o d u c e d by d i s c u s s i n g i t s a n n l i c a t i o n w i t h i n t h e framework o f a s i n n l e e x n e r i m e n t a l ' s a r a d i a m n u b l i s h e d elsewhere ( G e i s s l e r and P u f f e (1983)) and c o n p a r i n g i t w i t h the a n p l i c a t i o n o f optimal decision.
The o a r a d i y and t h e a p o l i c a t i o n
o f o n t i m a l d e c i s i o n w i l l be d i s c u s s e d i n t h i s s e c t i o n .
2.1. THE EXPERIKNTAL ~r?RADIW I n t h e exDeriments a h i e r a r c h i c a l n a r t i t i o n o f c a t e q o r i e s was employed.
4s
an example a t h r e e - l e v e l o a r t i t i o n i s d e p i c t e d i n F i o u r e 1 where a p a r t i t i o n i n p i s symbolized by t h e growing o f t h e number o f i n d i c e s . digm c o n s i s t e d o f two t a s k s .
The p a r a -
One i s commonly known as a v e r i f i c a t i o n task
and a n o t h e r as a m u l t i n l e choice r e a c t i o n , o r c l a s s i f i c a t i o n t a s k . f i c a t i o n a c a t e g o r y - n a m , f o r i n s t a n c e , "A2",
was a i v e n and sub.iects had t o
decide whether an o b j e c t n r e s e n t e d belongs t o i t o r n o t . e d v i t b . "Yes" o r "Vo" (see a l s o F i g u r e 2 ) .
In veri-
They then respond-
I n C l a s s i f i c a t i o n s u b j e c t s had
t o respond w i t h , t h e c o r r e c t category-name o f t h e s t i m u l u s oresented.
The
category-names were s y l l a b l e s , known through p r i o r l e a r n i n g , c o r r e s p o n d i n o w i t h t h e c a p i t a l s w i t h zero, one o r two i n d i c e s i n F i g u r e 1.
I n classifica-
t i o n i t was necessary t o n r e s e n t f i r s t a cue, a l s o k n w n through p r i o r l e a r n i n g , i n d i c a t i n g t h e l e v e l , t o o medium o r bottom, on which s u b j e c t s had t o respond (see a l s o F i g u r e 2 ) .
A category could c o n s i s t o f several objects,
b u t a t t h e b - l e v e l t h e y c o n s i s t e d o f one o b j e c t o n l y .
The cases A , A*,
B
and C i n F i g u r e 1 show f o u r tynes o f a b s t r a c t f e a t u r e - d e s c r i D t i o n s o f obj e c t s used.
F i g u r e 2 shows some v i s u a l s t i m u l i t h a t f u l f i l such d e s c r i p -
t i o n s : t h e ai i n d i c a t e an o u t e r - c o n t o u r ,
t h e bi
an i n n e r - c o n t o u r and t h e
c. a f i l l i n g - i n o f t h e i n n e r - c o n t o u r . 1
T h i s e x p e r i m e n t a l paradigm guarantees a s t r i c t c o n t r o l o f t h e e f f e c t s o f t a s k - and s t i m u l u s - v a r i a t i o n a t each c a t e g o r y - l e v e l and a l l o w s f o r a unique and s i m p l e d e s c r i p t i o n o f t h e d a t a i n terms o f t h e f e a t u r e s . be discussed h e r e a r e r e a c t i o n - t i m e s o f c o r r e c t answers.
The d a t a t o
They a r e assumed
t o r e f l e c t t h e mean number of d e c i s i o n s t o be made i n each case.
It i s
36
H. Lluffurt and H.-C. Geissler
Fioure 1
A h i e r a r c h i c a l o a r t i t i o n o f c a t e g o r i e s and t h e f n u r o h j e c t comhinations used i n t h e e x n e r i m e n t s .
- -..- .
-
Correct
Cue
wrification
"RI1"
(a2blc2) ___
.
"t" classi
-
fication
"b"
( a2b2c2)
( alblcl) I
Figure 2 An example o f s t i m u l i and answers.
37
Task-dependent representation of categories
assumed t h a t t h e paralneters t o be f i t t e d i n o r d e r t o r e l a t e t h e number o f d e c i s i o n s and the r e a c t i o n - t i m e a r e inde?endent cf the t.ask a t hand.
Such
a r e q u i r e m e n t o f c r o s s - t a s k v a l i d a t i o n i s w i d e l y used i n p e r c e p t u a l r e s e a r c h (Anderson (1976); Birnbaum (1983); G e i s s l e r (1980); G e i s s l e r , K l i x and S c h e i d e r e i t e r (1978); G e i s s l e r and P u f f e (1983); G e i s s l e r , Scheidcrei'er and S t e r n ( 1 9 8 0 ) ) .
2.2. P R E D I C T I O N S BASED ON OPTIFAL D E C I S I O N ( c n j u n c t i v e feature rearesentations I n Eases A and A* each c a t e g o r v i s r e l a t e d t o a s e t o f c o n j u n c t i v e l y j o i n t features.
N i t h t h e h e l n o f F i g u r e 3a t h e o r e d i c t i o n s f o r Case A based on
an o a t i m a l d e c i s i o n view can be d e r i v e d i n accordance w i t h t h e a s s e r t i o n s mentioned i n t h e i n t r o d u c t i o n .
F i v u r e 3a r e p r e s e n t s a h i n a r v d e c i s i o n t r e e
f o r a s e r i a l e l i m i n a t i o n orocedure where each node d e s c r i b e s a o a r a l l e l dec i s i o n on two f e a t u r e s .
I n each o a t h from t o p t o b o t t o m t h e l e t t e r s i n d i -
cate the r e l e v a n t c o n j u n c t i v e l y j o i n t features.
I f i n the c l a s s i f i c a t i o n
task t h e s e r i a l t e s t i n g i s assumed t o s t o p a t t h e t-, m- o r b - l e v e l as i n &
s t o o a t the lSt, Znd
o r Jrd l e v e l i n F i g u r e 3a when s t a r t e d f r o m t h e t o n .
The number o f f e a t u r e -
d i c a t e d by t h e e x p e r i m e n t e r , t h i s corresponds w i t h
t e s t s i s t h e mean number o f nodes nassed i n o r d e r t o reach t h e a o o r o p r i a t e l e v e l , b e i n g 1, 2 o r 3 r e s n e c t i v e l y .
S e r i a l o r o c e s s i n g i n the v e r i f i c a t i o n
task i s e q u i v a l e n t t o f o l l o w i n g a ore-determined top-down a a t h o f t h e t r e e . The such p r e d i c t e d numbers o f d e c i s i o n s f o r t h e t-, m- and b - l e v e l s a r e a l s o
1, 2 and 3 r e s p e c t i v e l y .
Thus f o r each h i e r a r c h i c a l l e v e l o f t h e c a t e g o r i e s
t h e mean r e a c t i o n - t i m e s f o r c l a s s i f i c a t i o n and v e r i f i c a t i o n a r e p r e d i c t e d t o be e q u a l .
A b i n a r y d e c i s i o n i n F i a u r e 3a means t h a t a n e g a t i v e d e c i s i o n about a f e a t u r e , f o r e x a m l e ayl
automatically i m l i e s a p o s i t i v e decision about the
complemntary feature a t t h a t l e v e l
a2.
Thereqfore, t h e d i f f e r e n c e between
c l a s s i f i c a t i o n and v e r i f i c a t i o n i s minimal i n t h e sense t h a t such a b i n a r y d e c i s i o n between two f e a t u r e s i n c l a s s i f i c a t i o n amounts t o t h e same as a Yes-No d e c i s i o n on a s i n g l e f e a t u r e i n v e r i f i c a t i o n .
I n F i g u r e 3c a n o t h e r
d e c i s i o n - t r e e i s d e n i c t e d t h a t i s maximal w i t h r e s o e c t t o t h e d i f f e r e n c e between c l a s s i f i c a t i o n and v e r i f i c a t i o n .
Here the assumotion i s made t h a t
a f t e r each n e g a t i v e d e c i s i o n a second t e s t f o l l o w s c h e c k i n g t h e complementa r y f e a t u r e on a l e v e l a p a r t .
The mean number of d e c i s i o n s t o reach t h e
t-, m- o r b - l e v e l e q u a l s (1+2)/2 = 1.5,
(2+3+3+4)/4
= 3 and
H.Buffart and H . X . Ceisskr
38
b
C.
Figure 3 Decision-trees f o r c l a s s i f i c a t i o n sis.
,
based on t h e o n t i m a l d e c i s i o n hypothe-
a ) Case A w i t h m i n i m a l number o f d e c i s i o n s .
b ) Case A*.
c ) Case A
w i t h maximal number o f d e c i s i o n s .
(3+4+4+5+4+5+5+6)/e = 4.5 r e s D e c t i v e l y .
Thus f o r each h i e r a r c h i c a l l e v e l
o f t h e c a t e g o r i e s t h e r a t i o o f t h e mean r e a c t i o n - t i m e s f o r c l a s s i f i c a t i o n and v e r i f i c a t i o n i s p r e d i c t e d t o enual 1 . 5 .
39
Task-dependent representation of categories
I n accordance w i t h t h e o p t i m a l d e c i s i o n h y p o t h e s i s t h e r e a s o n i n g above i s based on t h e c o n j u n c t i v i t y n o t on t h e f e a t u r e - o u t w a r d .
This i s depicted i n
t h e s i m i l a r i t y o f t h e d e c i s i o n - t r e e s i n F i g u r e s 3a and 3b.
The l a t t e r i s
a t r e e f o r Case A* where a t each c a t e g o r y - l e v e l t h e r e l e v a n t f e a t u r e s a r e chosen t o be d i f f e r e n t f r o m any o t h e r . those f o r case A*. and A*.
The o r e d i c t i o n s a r e i d e n t i c a l w i t h
F i g u r e 4 shows some e x p e r i m e n t a l r e s u l t s f o r Cases A
The mean r e a c t i o n - t i w e t r e n d s q u a l i t a t i v e l y c o n t r a d i c t s b o t h t h e
minimal and maximal n r e d i c t i o n s .
A
(msecl
c Lassifica t ion
3400
--
3000
--
2600
--
verification, posit. predict ions
2200 --
1800
--
1400
--
1000
-t
m
b t response category Level
m
b
Fiaure 4 Exoerimental r e s u l t s ( a d a p t e d f r o m G e i s s l e r and P u f f e (1983)) and t h e o r e t i c a l p r e d i c t i ons ,
40
H. Buffart a d H.-C. Geissfer
Disjunctive feature representations The h i e r a r c h i c a l s t r u c t u r e i n Cases R and C i n F i g u r e 1 have been d e r i v e d from Case A by p e r m u t i n g f e a t u r e c o m b i n a t i o n s on t h e b - l e v e l , i .e. o b j e c t s ,
so t h a t no c o n j u n c t i v e r e o r e s e n t a t i o n on t h e m- and t - l e v e l i s p o s s i b l e . From a nure o o t i m a l d e c i s i o n view t h e b e s t s t r a t e g y f o r c l a s s i f i c a t i o n and v e r i f i c a t i o n i s , f i r s t , t o c l a s s i f y o r v e r i f y o b j e c t s by s e r i a l e l i m i n a t i o n on t h e h i g h e s t l e v e l i n v o l v i n g c o n , j u n c t i v e r e o r e s e n t a t i o n s and, second, t o determine, deDendinq on t h e r e s u l t on t h i s i n i t i a l l e v e l , t h e f u r t h e r r e Quested d e c i s i o n s t o answer t h e q u e s t i o n . F o r Cases R and C t h e i n i t i a l l e v e l , a l s o c a l l e d t h e p r i m a r y (Hoffmann (1981)) o r b a s i c (Rnsct- ( 1 9 7 5 ) ) c o n c e n t l e v e l , i s t h e b - l e v e l ( c f . F i g u r e
1 ) . T h e r e f o r e , t h e mean number o f d e c i s i o n s r e q u i r e d i n t h e f i r s t p r o c e s s i n a - s t a g e i s t h e mean number o f d e c i s i o n s f o r the b - l e v e l i n Cases A and A*. F o r t h e second p r o c e s s i n o - s t a g e s e v e r a l i m o l e m e n t a t i o n s a r e p o s s i b l e . However, responses c o n c e r n i n g t h e m- and t h e t - l e v e l a r e always o b t a i n e d through t h e b - l e v e l , so t h a t mean r e a c t i o n - t i n e s a t t h e b - l e v e l a r e n e v e r h i a h e r than those a t t h e o t h e r l e v e l s .
The m i n i m a l c l a s s i f i c a t i o n - v e r i f i -
c a t i o n r a t i o i s o b t a i n e d by assurninq t h a t t h e d e c i s i o n - t r e e i n F i g u r e 3a
i s used i n the f i r s t p r o c e s s i n q s t a a e and t h a t t h e second s t a g e s c a r c e l y influences reaction-time.
A t each l e v e l t h e r a t i o e q u a l s 3/3 = 1.
The
maximal r a t i o i s o b t a i n e d b y assuming i n t h e case o f c l a s s i f i c a t i o n t h a t t h e d e c i s i o n - t r e e i n F i a u r e 3c i s used and t h a t each t r a n s i t i o n f r o m a l o w e r t o a higher l e v e l i n the trees i n Figure 1 i s a decision-sten. and b - l e v e l t h e r a t i o e q u a l s 6.5/3 p e c t ivel y
=
2.2,
5.5/3
=
A t t h e t-, m-
1.8 and 4.5/3 = 1.5 r e s -
.
F i g u r e 5 shows r e s u l t s o f an e x n e r i m e n t f o r Cases I\, R and C . f o r Case A c o r r e s p o n d t o those o f t h e f o r m e r e x n e r i m e n t .
The r e s u l t s
The r e s u l t s f o r
Cases P and C a l s o d i s a q r e e w i t h t h e D r e d i c t i o n s c o n c e r n i n q t h e d i f f e r e n c e s between v e r i f i c a t i o n and c l a s s i f i c a t i o n .
I n a d d i t i o n , the s i g n i f i c a n t d i f -
ference between t h e t r e n d s i n t h e Cases B and C c o n t r a d i c t s t h e assumption t h a t c l a s s i f i c a t i o n and v e r i f i c a t i o n a r e i n d e p e n d e n t o f f e a t u r e c o n t e n t . N o t i c e t h a t i n Case B t h e mean r e a c t i o n - t i m e a t t h e t - l e v e l i s even much l o w e r than t h a t a t t h e b - l e v e l e n t i r e l y i n c o n t r a s t w i t h t h e c l a i m t h a t t h e b - l e v e l i s a t r a n s i t i o n a l staae f o r the t - l e v e l .
41
Task-dependent representation of Categories
A
lmsec) "i/------
3800
A
L
.-
A
0
0 clotsilication verification. posit raredictions
*---
3LOO
..
3000
..
2600
..
2200
-.
1800
.'
lL00
.-
l4
t
rn
b
t . t
rn
b
response category level
Figure 5 E x p e r i m e n t a l r e s u l t s (adaoted from G e i s s l e r and P u f f e (1983)) and t h e o r e t i c a l p r e d i c t i ons
.
3. A STRUCTURAL INFORNATION APPROACH TO GUIDED INFERENCE 3.1. GUIDED INFERENCE
In t h e i n t r o d u c t i o n we a l r e a d y discussed t h a t the o p t i m a l d e c i s i o n view cann o t account f o r d i f f e r e n c e s i n t a s k and s t i m u l u s - c o n t e n t .
We b e l i e v e t h a t
t h e disagreement above i s caused b y t h i s l a c k o f f l e x i b i l i t y .
We l i k e t o
show t h a t c l a s s i f i c a t i o n and v e r i f i c a t i o n processes may be i n t e r p r e t e d t o be determined by t h e r e p r e s e n t a t i o n s of t h e c a t e g o r i e s i n memory i n t h e sense t h a t t h e processes can be completely expressed i n terms o f r e p r e s e n t a tion-parameters.
Such a process i s s a i d t o be a guided i n f e r e n c e process
42
H. Ruffart and H.-C. Geirrler
s i n c e any n a r t i c u l a r t a s k can be c o n s i d e r e d as a oroblem s o l v i n g s i t u a t i o n wliere the s o l u t i o n i s i n f e r r e d f r o m t h e i n f o r m a t i o n c o n t a i n e d i n t h e r e p r e sen t a t i ons
.
Such a model i s suogested b y t h e most s a l i e n t d i f f e r e n c e s between D r e d i c t i o n and r e s u l t s mentioned above.
-
F o r example:
The d i f f e r e n c e observed hetween v e r i f i c a t i o n and c l a s s i f i c a t i o n can be seen as a consequence o f t a s k c o n s t r a i n t s because i n v e r i f i c a t i o n t h e c a t e g o r y t o be checked i s a l r e a d y known i n advance whereas i n c l a s s i f i c a t i o n the c o r r e c t c a t e q o r y i s t o be found from a s e t o f a l t e r n a t i v e s .
If
d e s c r i o t i o n s o f c a t e q o r i e s would be e x p l i c i t l y i n v o l v e d i n n r o c e s s i n g , v e r i f i c a t i o n s h o u l d e x n e c t t o be f a s t e r than c l a s s i f i c a t i o n .
-
I n Case 4* knowledge o f t h e b - and c - f e a t u r e s i m n l i e s a comnlete knowledqe o f t h e a- and b - f e a t u r e s t o o .
I n comparison w i t h Case A t h i s c o u l d
he used t o reduce t h e number o f f e a t u r e t e s t s necessary.
-
I n Case
B f o r a d e c i s i o n a t t h e t - l e v e l i t s u f f i c e s t o know o n l y two f e a -
t u r e s , t h e bi and c . , i n s n i t e o f t h e d i s j u n c t i v e c h a r a c t e r o f t h e r e p r e 3 s e n t a t i o n s a t t h e m - l e v e l . Thus t h e mean number o f d e c i s i o n s a t t h e t - l e v e l c o u l d be l o w e r i n s t e a d o f h i q h e r t h a n t h i s number a t t h e m - l e v e l . 3.2. ASSUMPTIONS
A g u i d e d i n f e r e n c e model r e n u i r e s two t y p e s o f assumntions.
One t y p e con-
c e r n s the r e p r e s e n t a t i o n s and t h e o t h e r t y p e t h e r e l a t i o n between r e p r e s e n t a t i o n s and d e c i s i o n s .
Ne assume t h a t t h e b a s i c i d e a s o f s t r u c t u r a l i n f o r -
m a t i o n t h e o r y ( S I T ) a b o u t r e o r e s e n t a t i o n s ( c f . B u f f a r t and L e e w e n b e r g (1983); R e s t l e (1982)) pay a l s o a o p l j h e r e .
The assumotions a b o u t t h e de-
c i s i o n s are adapted f r o m C e i s s l e r and P u f f e (1983). S I T supnoses ( 1 ) t h a t r e n r e s e n t a t i o n s d e s c r i b e i d e n t i t i e s w i t h i n and between
s t i m u l i and ( 2 ) t h a t such d e s c r i o t i o n s a r e c o n s t r a i n e d .
The c o n s t r a i n t s can
be exoressed i n terms o f a r e s t r i c t e d s e t o f grammar-rules a l l o w i n g t h e cons t r u c t i o n of sometimes i n t r i c a t e d e x o r e s s i o n s .
SIT does n o t make any assumo-
t i o n about c o n s t r a i n t s on Drocesses, b u t a r e c e n t t k e o r e t i c a l s t u d y ( B u f f a r t , Note 1) suggests t h e e x i s t e n c e o f a r e l a t i o n between r e o r e s e n t a t i o n and process c o n s t r a i n t s .
Several representations o f a stimulus are allowed.
S I T assumes t h a t s u b j e c t s p r e f e r t h e r e p r e s e n t a t i o n b e i n g most economic i n -
c l u d i n g b o t h c o n t e x t and task-demands.
T h i s economy- or m i n i m u m - p r i n c i p l e
can be seen as a formal e x n r e s s i o n o f a p e r c e o t u a l s t a b i l i t y n r i n c i p l e .
43
Task-dependent representot ion of cotegorier
A stimulus-representation i n the b r a i n i s regarded t o be l e s s s t a b l e as i t s
number o f ' f r e e ' o r non-identical elements increases.
B u f f a r t and Leewen-
bero (1983) prooosed a memory-model t h a t automatically tends t o i n f e r the more s t a b l e representation o f a stimulus.
The minimum-arinciple has been
apoeared t o be a reasonable t o o l t o c a l c u l a t e the mst stable representation. Geissler and Puffe sunnose t h a t ohiect-representations include the categoryrcrresentations belonoin? t o the task a t hand.
A renresentation o f an ob-
j e c t i s decoded i n t o a format t h a t i s consistent w i t h i t s sensory information,
Decisions are assumed t o be the comparison o f the decoded represen-
t a t i o n w i t h the sensory information.
Besides there e x i s t c o n t r o l operations
s t r u c t u r i n g the colnnarison nrocedure.
They are sunposed t o o r i q i n a t e from
the imolementation o f the categories i n the reDresentations.
Every opera-
t i o n i s assumed t o require a f i x e d time-period, so t h a t a Dart o f the reaction-time mzy be p r e d i c t e d t o be the m u l t i p l i c a t i o n o f the number o f decisions and t h i s period.
Ti-e o t h e r p a r t i s supDosed t o be caused by the en-
coding process being a constant f o r each s e t o f s t i m u l i .
Because there i s
no t h e o r e t i c a l r e l a t i o n between a renresentation and i t s encoding procedure the constant has t o be estimated from the data. 3.3. WIPED INFERENCE APPLIED TO THE DATA
The grammar-rules r e f l e c t i n g the assumtions can a t b e s t be introduced a t the hand o f examples.
Their a b s t r a c t form w i l l be discussed l a t e r on.
The
dashed l i n e s i n Figures 4 and 5 show the r e s u l t s o f the c a l c u l a t i o n s based on these r u l e s .
They coincide w i t h the r e s u l t s o f the m r e i n t u i t i v e l y
based c a l c u l a t i o n s o f Geissler and Puffe (1983).
The encoding constants and
time-neriods f o r decision are given i n Table 1. .__.I
decision ti me encoding t i m e
953
Table 1 Calculated decision and encoding t i m e i n c l a s s i f i c a t i o n and v e r i f i c a t i o n . The basic operator used i s a feature connection-oDerator, \ , as introduced i n Formulae l a and l b . I t comDoses references t o features. The f i r s t obThe operator i s j e c t i n Case A , f o r e x a m l e , can be described by al\bl\cl.
44
H. Birjfart and If.-G.Ceissler
s s s o c i a t i ve. The o a e r a t o r has t h e s o - c a l l e d d i s t r i b u t i v e p r o o e r t y as d e f i n e d i n S I T ( c f . I n . t h i s p m e r we use two s n e c i a l forms o f
P u f f a r t and L e e w e n b e r r l ( 1 9 8 3 ) ) .
t h i s n r o n e r t y , c a l l e d t h e f e a t u r e s e t p r o d u c t r u l e (Formulae l c and I d ) and t h e i n n e r p r o d u c t r u l e (Formula l e ) .
The f e a t u r e s e t n r o d u c t r u l e d e s c r i b e s
g e n e r a t i o n o f o b j e c t s b e l o n g i n a t o a c a t e g o r y b.v f u l l c o h i n a t i o n o f f e a t u res from sets o f mutually e x c l u s i v e features. bes c o m l e t e c o r r e l a t i o n o f f e a t u r e s .
The i n n e r n r o d u c t r u l e d e s c r i -
Decodinp t h e l e f t hand-side o f a F o r -
mula d e l i v e r s t h e r i g h t h a n d - s i d e .
[Y1 +th
a
*
*
Yn)\(X)
... y n l \
=L
ab
(la)
=L
abc
(Ib)
... y,)
(x)\ (y, (Y1
a\b ah\ c
[x,
-
. .. x\y,
* X\YI
(Y1)\(X)
... Xnl
*
*
a
* Y$X1
(lc)
(Y"Y ( X I
(Id)
... Yn\Xn
(le)
t h e s e r u l e s s e t s o f o b j e c t s , o r o f p a r t o f o b j e c t s can b e r e p r e s e n t e d .
Formula 2 shows a r e n r e s e n t a t i o n o f t h e s e t o f o b j e c t s i n Case A .
Formulae
3a-c show i t f o r case A* and Formula 3d r e p r e s e n t s a s e t of h i e r a r c h i c a l p a r t o f these o b j e c t s . Formula 4 .
The s e t o f o b j e c t s i n Case R i s r e D r e s e n t e d by
These r e n r e s e n t a t i o n s a r e t h e most c-ccnomic @rm i n i m a l one.
O t h e r r e p r e s e n t a t i o n s can e a s i l y be c o n s t r u c t e d . ( ( a 1 a 2 ) \ ( b 1 b2))'(C1 c2)
(2)
[(a1)(ap)I\ I(z1Hz2)1
(3.3)
z 1 = [ ( b 1 ) ( b p ) l \ L(c1 c 2 ) ( c 3 cq)1
(3b)
z2 = [ ( b 3 ) ( b 4 ) l \ [ ( c 5 c 6 ) ( c 7 c e l l
(ZC)
[ ( a 1 ) ( a p ) I \ [ ( b l bz) ( 0 3 b 4 ) I
(3d)
( [c2b2 clbll
[c2bl
clb21
I\(
[al a21 [ a 2 all
(4)
A n a i r o f b r a c k e t s { I i s used t o denote c a t e g o r i e s i n a r e n r e s e n t a t i o n .
50 {x1y21 r e p r e s e n t s a c a t e g o r y c o n s i s t i n g o f one o b j e c t w i t h f e a t u r e s x1 and y 2 and { ( x 1 x 2 ) ( y 1 y 2 ) ? r e n r e s e n t s a c a t e g o r y o f f o u r o b j e c t s d e s c r i b e d
by t h e n r o d u c t r u l e i n s i d e .
I f a c a t e g o r y c o n s i s t s o f two p a r t s connected
b y t h e o n e r a t o r \ i t s u f f i c e s t o denote t h e c a t e g o r y i n one o f b o t h p a r t s . F o r example, Ixly2)
may be denoted b y Ixll\y2.
The general r u l e i s g i v e n
45
Task-dependent rep resenta tion of ca tego ries
i n Formulae 5 a - f .
-
x\ t y } * {x\y) (xny
(5a)
tx\yl
(5b)
({XI)\!! =- { ( x ) } \ y
(5c)
* x\((y)l
(5d)
[{xll\y * {[XI n y x\ [{yll * x\ { bl I
(5e)
X\({.Y})
(5f)
The a p p l i c a t i o n o f these r u l e s and the r u l e s i n Formulae l a - e f o r the catev o r i e s "A" and "B" o f Case A can be seen i n Formulae 6a-f.
Formula 6 f shows
b o t h c a t e g o r i e s ane a one-to-one manning o n t o the resoonse-names i s possible. N o t i c e t h a t such a one-to-one maonina i s a l s o p o s s i b l e Prorr Formula 6a. Therefore, i t i s n o t necessary t o decode Formula 6a, o r i n o t h e r words f o r v e r i f i c a t i o n o r c l a s s i f i c a t i o n i t s u f f i c e s t o i n v e s t i g a t e the " a " - f e a t u r e s . However, t h e r e i s a r e s t r i c t i o n on such an i n v e s t i g a t i o n o f f e a t u r e s witkout. f u l l y decoding t h e r e p r e s e n t a t i o n .
The i n v e s t i a a t i o n can o n l y s t a r t a t the
t o p o r the bottom o f the h i e r a r c h y o f the r e p r e s e n t a t i o n . the r e p r e s e n t a t i o n i n Formula 6a i s shown i n F i g u r e 3a.
The h i e r a r c h y o f Every node i s r e l a -
t e d t o a decoding according t o one o f the Formulae l c - e i f one o f t h e arguments c o n t a i n s a category r e p r e s e n t a t i o n { I .
Thus, t h e i n v e s t i g a t i o n o f
an " a " - f e a t u r e may s t a r t w i t h the i n v e s t i g a t i o n o f a " c " - f e a t u r e (bottom) f o l l o w e d by an i n v e s t i g a t i o n o f a "b"-feature.
I t a l s o may s t a r t w i t h an
i n v e s t i g a t i o n o f an " a " - f e a t u r e ( t o o ) , b u t never w i t h an i n v e s t i g a t i o n o f a " b " - f e a t u r e (medi um)
.
(({a1)ta21)\ ( b l b2))\(C1 5 )
(6a)
W a 1 W ( b l b2) ( { a 2 } ) \ 0 1 b 2 ) ) \ ( c 1 5 )
(6b)
( { ( a p ( b l b2) { ( a , ) M J 1
(6c)
b 2 ) ) \ ( 5 c2)
5)
(6d)
b2))\(C1 5 ) )
(be)
( { ( a l ) \ ( b l b2)} {(az)\(b1 b2)})\(C1 { ( b l ) \ ( b l b 2 ) ) \ ( C 1 C2)")\(bl Ialblcl
alblc2
alb2cl
alb2c2}{a2blcl
a2blc2
a2b2cl
a2b2c2}
(6f)
I t i s assumed t h a t f o r a reDresentation given a s u b j e c t uses such a procedure t h a t a minimal number o f decisions i s r e q u i r e d .
This assumation i s c a l l e d
the minimal d e c i s i o n hypothesis and i t a p p l i e s a f t e r the minimal representat i o n has been chosen. I t i m p l i e s t h a t i n Case A s u b j e c t s s t a r t a t the topl e v e l and t h a t the r e a c t i o n - t i m e w i l l be very low.
Indeed, t h i s i s shown
H. Buffarr and H - C . Ceusler
46
i n b o t h F i a u r e s 4 and 5 .
F i g u r e 6a shows how i n the case o f c l a s s i f i c a t i o n
the mean nurnber o f d e c i s i o n s i s determined.
I f a s u b j e c t assumes c a t e g o r v
"A" he needs one comnarison i f t h e f e a t u r e i s al and two comparisons if t h e f e a t u r e appears t o he a2.
The mean number o f d e c i s i o n s e q u a l s 1 . 5 .
I n the
case o f v e r i f i c a t i o n a s u b j e c t has t o i n v e s t i a a t e w h e t h e r o r n o t the f e a t u r e belongs t o t h e c a t e a o r v p i v e n .
F o r a o o s i t i v e answer p r e c i s e l y one compari-
son i s r e q u i r e d . Formula 7a r e p r e s e n t s t h e m - l e v e l c a t e g o r i e s o f case A .
Anain, t h e comnari-
son-orocedure may be s t a r t e d a t t h e b o t t o m - o r t h e t o n - l e v e l o f t h e r e n r e sentation.
The l a t t e r nrocedure r e l u i r e s l e s s comoarisons.
comparisons f o r c l a s s i f i c a t i o n i s shown i n F i g u r e 6h.
l y assumes c a t e g o r y "Al"
The number o f
',then a s u b j e c t r i g h t -
two comoarisons, al and h l , a r e n e c e s s a r y .
t h e c a t e g o r y i s "A2"
two e x t r a comnarisons, al and b2, a r e n e c e s s a r y .
the category i s "PI"
a s u b j e c t makes two f a l s e comparisons,
o t h e r comoarisons, a2 and bl.
aim+
When [.!hen
b1 a r d two
'ahen t h e c a t e g o r y aonears t o b e "B2" a n o t h e r
two comparisons, a 2 and b2, have t o he made.
N o t i c e t h a t a comoarisov-pro-
cedure c o u l d a l s o have been s t a r t e d w i t h one o f t h e o t h e r c a t e g o r i e s .
Once
s t a r t e d the procedure follows the h i e r a r c h i c a l s t r u c t u r e o f the representation.
O t h e r orocedures c o u l d have been A2A1B1B2,
A2A1B2B1,
A1A2B2B1,
The mean number o f comparisons
R1B2A1A2, B2B1A1A2, F12B1A2Al o r B1B2A2A1. equals 4. F o r a p o s i t i v e answer i n a v e r i f i c a t i o n - t a s k two c o m a r i s o n s a r e requi red. ( ( a , a 2 ) \ ( { b p 2 m \ ( c 1 c2)
(7a)
((a1 a2)\(bl b2))\({cQ{c2})
(7b)
Formula 7b r e p r e s e n t s t h e b - l e v e l c a t e g o r i e s o f Case A .
I t makes n o d i f f e -
r e n c e f o r the number o f d e c i s i o n s w h e t h e r t h e procedure s t a r t s a t t h e t o o o r the b o t t o m o f t h e h i e r a r c h y .
F i g u r e 6 c shows a scheme o r t h e comparison-
nt-ocedure, t h a t i s analoqous t o t h e orocedure f o r t h e m - l e v e l .
The mean
number o f comparisons e q u a l s 8.5 f o r c l a s s i f i c a t i o n and 3 f o r ( P o s i t i v e ) verification.
The c a l c u l a t e d mean numbers o f comoarisons i n Cases A, A*,
9 and C a r e summarized i n T a b l e
2.
47
Task-dependent representation of c o t e p r i e s
A a
3/ = 1.5 2
x =4
16
C
Figure 6 Decision-trees f o r c l a s s i f i c a t i o n i n Case A based on the guided i n f e r e n c e hyrJothesis. a ) t - l e v e l
, ke)
m-level
, c)
b-level
.
H. Buffarr and H.-G. Geissler
48
I
classification
veri fi cation
L
C a l c u l a t e d nunher
Qf
______.
..
A* ._
B
1.5
1.5
4.0
~- C 7.5
A
too
mean number o f d e c i s i o n s
.-
-
.o
medi urn
4 .O
4.0
8.5
?!
hottom
8.5
4.5
8.5
8.5
ton
1.o
1.0
2.5
6 .O
medi urn
2 .o
2.0
3.5
4 .O
bottom 1.o __Tat12 2
1.0
. .3.0
2.0
d e c i s i o n s f o r c l a s s i f i c a t i o n and v e r i f i c a t i o n .
Formulae Sab show t h e r e D r e s e n t a t i o n s o f t h e t- and m - l e v e l c a t e g o r i e s o f Case A*.
The c a l c u l a t i o n s a r e i d e n t i c a l w i t h t h e c a l c u l a t i o n s i n Case P
(see f i g u r e s 7ab).
The r e n r e s e n t a t i o n s o f t h e c a t e g o r i e s a t t h e b - l e v e l
do n o t r e q u i r e any d i s c u s s i o n .
The comparison-procedure i s shown i n F i g u r e
7c.
Formula 9
The r e l a 2d d e c i s i o n
r e p r e s e n t s t h e t - l e v e l c a t e g o r i e s i n Case B .
scheme f o r c l a s s i f i c a t i o n i s shown i n f i g u r e 8a.
The c a l c u l a t i o n o f t h e
mean number o f d e c i s i o n s f o l l o w s f r o m t h e c o m o a r i s o n - s e r i e s : blcl
-+
A , blb2c2
+
A , blclb2blc2
+
B and blb2c2blb2cl
+
B.
The mean number
o f d e c i s i o n s f o r v e r i f i c a t i o n o f c a t e q o r y " A " f o l l o w s from t h e s e r i e s : blcl
-+
YES and blb2c2
+
YES, i m n l y i n g ( 2 + 3 ) / 2 = 2.5 d e c i s i o n s .
Formulae 9bc r e p r e s e n t the c a t e g o r i e s a t t h e m- and b - l e v e l .
As shown i n
F i g u r e s 8b ana Oc the h i e r a r c h i c a l s t r u c t u r e o f b o t h r e D r e s e n t a t i o n s i s id e n t i c a l so t h a t w i t h o u t any c a l c u l a t i o n an equal r e a c t i o n - t i m e i s p r e d i c ted.
T h i s agrees w i t h t h e d a t a (see F i g u r e 5 ) .
i s analoqous t o t h e Drocedure f o r t - l e v e l
.
The comparison-procedure
The d i f f e r e n c e between t h e m-
and b - l e v e l f o r v e r i f i c a t i o n stems f r o m t h e f a c t t h a t a c a t e g o r y a t t h e ml e v e l c o n t a i n s two o b j e c t s . cedures f o r c a t e g o r y A1,
F o r i n s t a n c e : t h e two p o s s i b l e comnarison-pro
c2clbla2
number of (4+3) = 3.5 d e c i s i o n s .
+
YES and clbla2c2
-+
YES, i m p l y a mean
49
Task-dependetlt representotiori of categories
A*
a decisiohs
decisions
2
at
: a
1
2
Y
Y
3
k
1.5
6
Figure 7 D e c i s i o n - t r e e s f o r c l a s s i f i c a t i o n i n Case A* based on t h e g u i d e d i n f e r e n c e h y p o t h e s i s . a) t - l e v e l , h ) m - l e v e l , c ) b - l e v e l .
H. fluffart and H . X . Ceisder
50
B
C
decisions
Cab,
3
ch%
cLbrQI c,b,a, ctbza, c,bp, kb,a2 C&a,
6
68 zff.5 10
II
13
{'I
/d
Fiaure 8 Decision-trees f o r classification in Case B based on the guided inference hynothesis. a ) t - l e v e l , b ) m-level, c) b-level .
Task-dependeii t representation of categories
51
Forrwlae lOah show two reoresentations of the objects used i n Case C . Both a r e minimal reoresentations. The references n . refer to a combination of 1 two features as indicated i n Formula 1Oc. I t anpeared t h a t a reasonable explanation of the data i s only nossible i f one assumes t h a t subjects reoard such a p a i r of features as one f e a t u r e . This imnlies t h a t both features are nrocessed i n p a r a l l e l . Our notation i n Formulae Nab exnresses t h i s assumntion. There are a l s o theoretical considerations t h a t suoport this chunkina hypot h e s i s . There i s a formal difference between the representations in Case C a n d the o t h e r Cases. The category reoresentations are ambicjuous as shown in Fcrrnulae l l a - c . h e r a y use the ai as well as the ni t o reoresent the categories. Another way of saying i s t h a t a t the t-level both, non-related, cateqories "A" and "B" have the s a w s e t of features even when the chunking indicated i n Formula 1Oc occurs. Ye assume t h a t t h i s causes the chunkino. I t agrees with the basic assumntion of SIT ( R u f f a r t , Note 1) t h a t elements a r e chunked when t h e i r combination anoears t o be an element i n a represent a t i o n . I t can be oroven t h a t i n s e r i a l reoresentations the d i s t r i b u t i v e r u l e s , such as the rules used i n t h i s oaper, stem from this assumption. ( ( { [ P ~ P ~ J [n3o4I 1 ) ) \ ( ( { [ a l a 2 1 1) ({[a2all 1 ) )
(1la)
( ( I{o1n2)I [ { n y 4 I l 111 ( ( [{a1a22H ) ( [{a2a111 ) )
(lib)
( [Ial!{a211 [{a2}{alll ) \ ( [{p11{n211 [ { D ~ } { D ~ ) I )
(1lc)
Anart from the chunking there i s categorical ambiouity. Islhen during a comnarison nrocedure ambiguity of eategorical reDresentations occurs, a subject has to decide which representation i s intended. Ide assume t h a t such a decision i s analogous to a feature-comparison so t h a t i t takes an equally long time-neriod. The imolementation of this assumotion in our formalism i s shown i n Formulae 12a-e and w i l l be exolained l a t e r on. Notice t h a t , due t o the d e f i n i t i o n s i n formulae lc-e the indication of a category,{}, i n a representation occurs a t the r i a h t hand-side. A t f i r s t s i g h t this i s not the case i n the representations i n Formulae 6a, 7a, 8a and 9a, b u t in these cases the r i g h t hand-side i s not involved i n the comparison-procedure.
Therefore, t h e bracl:ets
a t t h e r i p h t hand-side o f t h e Formulae l l a - c a r e
sunnosed t o r e n r e s e n t t h e c a t e o o r v i n t e n d e d . (XI\
'y?
* !4yl
(12a)
({xi)\ (iyI) * ((x)l\ { ( y ) ? [!x?l\ [(Vll
-
(12b)
1x1 I\ ( [yl
( 12c)
F o r n u l a l 2 d shows a s i t u a t i o n where a s u b j e c t has t o c o n t r o l w h i c h t h e categorical renresentation i s .
I t i s a s o e c i a l case o f Formulae l c - d when
t h e at.rurrents i n d i c a t e t h e c a t e ? o r v .
'.'hen t h e arguments a r e d i s t r i h u t e d
a c c o r d i n ? t o F o r p u l a e 1c-d one has t o keep c o n t r o l which i s t h e c a t e g o r y intended.
I t means t h a t e v e r v t i m e when such a h i e r a r c h y - n o d e i s i n v o l v e d
i n a comnarison-procedure an e x t r a d e c i s i o n has t o be t a k e n .
for e x a m l e , a t the t o n - l e v e l o f Formula l l a . i t decodes i n t o Formula 13a.
T $ i s occurs,
' W t h t h e h e l n o f Formula l c
The e x t r a d e c i s i o n , as i n d i c a t e d i n Forwula
12d, i s shown i n t h e h i e r a r c h i c a l t r e e i n F i a u r e 9 a .
The c u r v e d arrow i n -
d i c a t e s t h a t i n t h e case o f c l a s s i f i c a t i o n an e x t r a d e c i s i o n i s r e q u i r e d when a comparison-procedure t r a n s f e r s f r o m c a t e g o r v " A " i n t o c a t e g o r y "P"
o r v i c e versa.
Formula 12e i n d i c a t e s a s i t u a t i o n analooous t o Formula 12d.
( i[o1n2I ln3n41 I ) \ ( i lala21 I (
Ip1021
i( [
ln3n41 ) I\ { ( [ala21
D ~ ! [n3n41 J ~ ~ )\ ( la1a221
1 ~ ~ [n3n41 ~ ~ 2 1 ( i [a2all
?)
(13a)
) I { ( [p1p21 [o3p41 f ?\ { ( la2all
)1
( 13b)
I)
)I
({
( lapl] ) I
{ ( [P1D21 Ip3n41
(13c)
Formula 12a and t h e l e f t hand-side o f Formulae 12h-c a l s o i n d i c a t e a s i t u a t i o n where a c o n t r o l - d e c i s i o n has t o be t a k e n .
I n o r d e r t o c r e a t e a more
t r a n s n a r e n t f o r r - c l i s m such a d e c i s i o n i s o n l y a t t a c h e d t o Formula 12a, and t h e l e f t hand-side i n Formulae 12b-c i s decoded as i n d i c a t e d . i s used t o decode F o r n u l a 13a i n t o Formula 13b. c o r d i n ? t o Formula 12a i n t o Forrrula 13c.
Formula 12h
The l a t t e r i s decoded ac-
tiere a c o n t r o l - d e c i s i o n has t o he
taken as i n d i c a t e d i n F i q u r e 9 a . P . t the hand o f F i p u r e 9a t h e mean number o f d e c i s i o n s f o r c l a s s i f i c a t i o n
can be cz1ci:lated. D
~
~
~
The o o s s i b l e comDarison-series a r e plal
C A,D p1p2CDn3n4a2 P ~ ~ ~
-f
nln2a2CDn3p4CDnlD2al
+
+
A, nlalP2CDP3DqCDDlaz
R, p1n2CDD3aln4CDp1p2CDp3a2
-f
+
B,
B,
+ A,
p1p2a2
+
A,
53
Tusk-dependen t rep resentat io n of categories
and p1p22CDp D a CDp1n2CDn3n4al 3 4 2
-+
B, where CD i n d i c a t e s t h e c o n t r o l - d e c i s i o n .
C
Figure 9 D e c i s i o n - t r e e s f o r c l a s s i f i c a t i o n i n Case C based on the g u i d e d inference hypothesis.
a) t - l e v e l
, b)
m-level
, c)
b-level
.
H. Buffart and H:C. Ceirskr
54
I n v e r i f i c a t i o n s u b j e c t s looks f o r c o n f i r m a t i o n of f e a t u r e s i n a f i x e d s e t
o f objects.
The o b j e c t t o be t e s t e d i s a r b i t r a r i l y chosen, h u t f o r e v e r y
o b j e c t a control-decision ahout the category-renresentation i s necessary. The mean number o f d e c i s i o n s f o r v e r i f i c a t i o n can he d e r i v e d frollr t h e s e r i e s CDnlal
+
VES,
CDp1CDn2a2
CDolCDn2CDD3CDp4a2
-f
YES.
+
YES, CDnlCDo2CDn3al
+
YES and
The mean number o f d e c i s i o n s e q u a l s (3+5+7+9)/4=6
Because o f t h e a m h i o u i t y o f t h e c a t e g o r i c a l r e o r e s e n t a t i o n s i n Formulae l l a - c one a l s o c o u l d assume t h a t t h e comparison-procedures a r e i n accordance w i t h the 'reversed' reoresentations.
However, then i t annears t h a t t h e
mean numbers o f d e c i s i o n s i n b o t h v e r i f i c a t i o n and c l a s s i f i c a t i o n a r e l a r g e r . Our c h o i c e f o r t h e r e o r e s e n t a t i o n s s h a m i n t h e Formulae i s based on t h e assumption o f m i n i m a l number o f d e c i s i o n s . 4 . DISCGSSION
4 . 1 . A SIJ!IF?ARY OF THE ASSUPTPTInNS
A o a r t f r o m t h e r u l e s d e f i n e d , t h e assumptions t h a t u n d e r l i e t h e g u i d e d i n ference approach above can be sunnnarized as f o l l o w s . References and f e a t u r e s The b a s i c e l e t r e n t s i n a r e n r e s e n t a t i o n , c a l l e d memory-references.
refer to
f e a t u r e s d e f i n e d i n t h e e x n e r i m e n t o r t o combiriaticvs o f such f e a t u r e s . They r e o r e s e n t s t r u c t u r e s on a n o t h e r l e v e l t h a t i s n o t e x n l i c i t l y i n v o l v e d i n t h e t a s k a t hand. Reoresen t a t i o n s
-
An o b j e c t i s r e p r e s e n t e d as a c o m b i n a t i o n o f r e f e r e n c e s t o i t s f e a t u r e s . A s e t o f o b j e c t s may be r e p r e s e n t e d i n terms of r u l e s a c t i n g on o t h e r
r u l e s o r r e f e r e n c e s ; t h e decoding o f t h e r e p r e s e n t a t i o n d e l i v e r s a s e t o f r e f e r e n c e combinations u n i q u e l y r e l a t e d w i t h t h e s e t o f o b j e c t s .
A repre-
s e n t a t i o n exDresses t h e i d e n t i t y o f f e a t u r e s i n a s e t o f o b j e c t s .
-
A c a t e q o r y - r e p r e s e n t a t i o n generates t h e s e t o f o b j e c t s b e l o n g i n g t o t h e
category.
I t i s c o n s t r u c t e d f r o m a r e p r e s e n t a t i o n of a s e t o f o b j e c t s b y
i n d i c a t i n g t h e elements t h a t g e n e r a t e t h e c a t e g o r y d u r i n g decoding.
-
Vhen a l l category-renresentations o f a s e t o f o b j e c t s a r e ambiguous chunki n 9 o f references occurs i f possible.
Task-dependent representation of categories
55
Decisions
-
D u r i n a c l a s s i f i c a t i o n o r v e r i f i c a t i o n a s u b j e c t i s supposed t o compare r e f e r e n c e s and t h e r e l a t e d o h y s i c a l shares.
The number o f comparisons i s
d e t e m i n e d b y t h e s u b j e c t ' s r e n r e s e n t a t i o n and t h e e x n e r i m e n t a l t a s k , h u t n o t by t h e e x p e r i m e n t a l l y d e f i n e d f e a t u r e s . dependent
-
The r e n r e s e n t a t i o n i s t a s k -
.
The comparison-procedure f o l l o w s t h e h i e r a r c h i c a l t r e e generated by t h e categorical reDresentation. the tree.
Each decoding o f a r u l e generates a node i n
The p r o c e d u r e may s t a r t a t t h e t o p o r a t t h e b o t t o m o f t h e
tree.
-
Each coniuarison of a r e f e r e n c e i s suoposed t o l a s t a s t a n d a r d p e r i o d o f
time.
I t i s assumed t h a t references a r e s e r i a l l y processed and t h a t a l l
s t r u c t u r e s r e p r e s e n t e d by one r e f e r e n c e a r e processed i n n a r a l l e l
.
Thus
t h e answer on t h e q u e s t i o n w h e t h e r s e r i a l o r p a r a l l e l D r o c e s s i n g occurs can be d e r i v e d f r o m t h e r e p r e s e n t a t i o n s .
-
IJhen a c a t e g o r i c a l r e n r e s e n t a t i o n i s ambiguous a s u b j e c t takes a c o n t r o l d e c i s i o n e v e r y t i m e when t h e comparison-procedure nasses a node i n t h e r e l a t e d h i e r a r c h i c a l t r e e t h a t r e p r e s e n t s a decoding o f a r u l e w i t h ( d i r e c t l y ) ambiguous arguments.
The double minimum p r i n c i p l e
-
The i n f o r m a t i o n l o a d o f a r e o r e s e n t a t i o n i s d e f i n e d t o be t h e number o f references i t c o m r i s e s ,
From among a v a i l a b l e a1 t e r n a t i v e r e n r e s e n t a t i o n s
t h e ones w i t h t h e l o w e s t i n f o r m a t i o n a r e supposed t o be u s e f u l f o r a subject.
-
I f more than one m i n i m a l r e n r e s e n t a t i o n e x i s t s a s u b j e c t uses from these
t h e r e o r e s e n t a t i o n ( s ) t h a t r e q u i r e s t h e minima
number o f d e c i s i o n s i n
c l a s s i f i c a t i o n and v e r i f i c a t i o n .
4.2. A POSSIBLE GEP!ERPLIZATION The formalism o u t l i n e d oreserves t h e most e s s e n t a1 c o n s t i t u e n t s o f g u i d e d i n f e r e n c e . P r o c e s s i n g i s s D e c i f i e d i n terms o f o p e r a t i o n s a c t i n g on c a t e ? o r y r e p r e s e n t a t i o n s t h a t i n c l u d e t a s k - c o n s t r a i n t s and " r e d e f i n i t i o n o f features".
N e v e r t h e l e s s , a f u r t h e r t e s t o f t h e f o r m a l i s m and e s p e c i a l l y o f
the representation-rules
i s required.
There i s no w e l l - t e s t e d s e t o f b a s i c
r u l e s a v a i l a b l e f o r t h e d e s c r i p t i o n o f f e a t u r e combinations.
I n the e x p e r i -
H. Buffort and H.-C. Ceirsler
56
writs o f Hayes-Poth and Hayes-Roth (1977) s u b j e c t s seem t o base t h e i r de-
c i s i o n s on t h e Dower s e t o f f e a t u r e combinations a v a i l a b l e .
Our r u l e s can
he c o n s i d e r e d as examples o f dimensional o r d e r i n o as d i s c u s s e d by Garner
(1978).
I t c a n n o t be e x c l u d e d t h a t i t w i l l appear t h a t t h e f o r m u l a t i o n of
a f e n b a s i c r u l e s i s a l m o s t i m n o s s i b l e because t h e v a r i a b i l i t y o f r u l e s i s
very l a r o e . This also occurs i n S I T .
I n f a c t , t h e t h e o r y f o r m u l a t e s o n l y a chunkina-
c o n s t r a i n t t h a t r o u q h l y i m l i e s t h a t , f o r example, i n a sequence ace acf b c f t h e c o r r e l a t i o n o f a and c as we11 as t h e c o r r e l a t i o n o f c and f cannot hr e x r l i c i t l y d e s c r i b e d i n one r e a r e s e n t a t i o n .
A representation describes
e i t h e r t h e c o r r e l a t i o n o f a and c o r t h a t o f c and f. s e n t a t i o n s t h a t meet t h i s a s s u m t i o n .
S I T allows a l l reore-
I t suoposes t h a t t h e u l t i m a t e c h o i c e
of t h e r e o r e s e n t a t i o n i s based on a t v o e o f s t a h i l i t y expressed by t h e m i n i mum-nrinciole.
T h i s c h o i c e chanoes when t h e c o n t e x t o r t h e t a s k changes
( c f . t e e w e n b e r g and R u f f a r t ( 1 9 8 3 ) ) . influence.
I n f a c t , we a l s o imolemented c o n t e x t -
Because, we d e r i v e d t h e c a t e g o r y - r e o r e s e n t a t i o n s f r o m t h e whole
s e t o f o b j e c t s i n v o l v e d i n an e x o e r i m e n t .
I f t h e o r i n c i o l e s o f SIT, formu-
l a t e d f o r s e r i a l s t r u c t u r e s , a r e t r a n s f e r a b l e t o t h e domain o f conceptual l e a r n i n a the e x p e r i m e n t a l r e s e a r c h must q i v e a t t e n t i o n t o t h e f o r m u l a t i o n o f t h e i n f l u e n c e o f c n n t e x t and t a s k r a t h e r t h a n t o t h e f o r m u l a t i o n o f t h e r u l e s used b y t h e s u b j e c t s ,
The naradiqm used b y G e i s s l e r and Puffe (1983)
i s an examole o f a measurement o f c o n t e x t - and t a s k - i n f l u e n c e . H w e v e r , as we showed the m i n i m u m - o r i n c i p l e f o r m u l a t e d by SIT i s n o t s u f f i c i e n t t o e x p l a i n the experimental r e s u l t s . mum-orinciole c o n c e r n i n o t h e o r o c e s s .
'Je i n t r o d u c e d a second m i n i -
I t appeared t h a t w i t h i n t h e s c o r e
o f t h e e x p e r i m e n t a l naradiam c o n s i d e r e d i n t h i s Daper, i t i s o o s s i b l e t o derive the c o r r e c t d e s c r i o t i o n from a l a r g e s e t o f Dossible t h e o r e t i c a l descriotions.
From t h e v i e w o o i n t o f t h e double minimum c o n d i t i o n i n f e r e n c e
i s guided s i n c e f i r s t a c o n s t r a i n t i s p u t on c a t e q o r v r e p r e s e n t a t i o n and o n l y a f t e r t h i s a common optimum o f r e o r e s e n t a t i o n and o r o c e s s i n g i s s e l e c ted.
I t i s o f h i g h t h e o r e t i c a l s i g n i f i c a n c e t o s t a t e t h e c r i t e r i a f o r se-
l e c t i n g r e p r e s e n t a t i o n s and o r o c e s s i n q r u l e s i n such a way t h a t t h e y remain an i n v a r i a n t under change o f paradiam.
This i s o f p a r t i c u l a r imuortance
s i n c e i t i s d i r e c t l y l i n k e d t o t h e d i f f e r e n c e between t h e " o p t i m a l d e c i s i o n " and " g u i d e d i n f e r e n c e " approaches.
A more g e n e r a l f o r m a l d e f i n i t i o n o f t h e
double minimum c o n d i t i o n needs a o e n e r a l s p e c i f i c a t i o n o f p r o c e s s i n g l o a d .
Our w e c i f i c a t i o n i n terms o f number o f o p e r a t i o n s and t i m e - p e r i o d s i s
Task-dependen t representation of categories
57
c e r t a i n l y n o t t h e o n l y one n o s s i b l e . NOTES
1) The t e r m c o g n i t i v e c o n s t r a i n t has been adapted from R u f f a r t , and R e s t l e (1983).
Leewenbera
REFERENCES Anderson, N.A.,
I n t e y a t i o n theory, f u n c t i o n a l measurement and t h e
Dsvchoohysical law, i n H . 4 .
G e i s s l e r , and V U . ' ~ . Zabrodin (Eds.),
Advances i n Psychoohysics, R e r l i n , D.D.R.,
VER Deutscher V e r l a q d e r
W s s e n s c h a f t e n (1976). Sirnbaum, M.H.,
Scale converrience as a p r i n c i n l e f o r the s t u d y o f p e r -
c e p t i o n , i n H . 4 . c e i s s l e r , H.F.J.M.
B u f f a r t , E.L.J.
Leewenberg, and
V . S a r r i s (Eds.). flodern I s s u e s i n P e r c e p t i o n , Amsterdam, B e r l i n , D.D.R.,
N o r t h - H o l l a n d r i i h l i s h i n q Company, VEB Deutscher V e r l a g d e r
Im'issenschaften (1983). B u f f a r t , H., H.-G.
and LeeuwenbPr?, E . ,
G e i s s l e r , H.F.J.M.
(Eds.),
S I r u c t u r a l information theory, i n
B u f f a r t , E.L.J.
Leewenbero, and V . S a r r i s
Modern I s s u e s i n P e r c e p t i o n , Amsterdam, B e r l i n , D.D.R.,
North-
H o l l a n d P u b l i s h i n a Comany, VEB Deutscher V e r l a q d e r Idissenschaften
(1983) . B u f f a r t , H.,
Leewenbero, E.,
and Restle, F.,
Analysis o f ambiguity
i n v i s u a l p a t t e r n c o m a l e t i o n , Journal o f Experimental Psvchology, Human P e r c e p t i o n and Performance, 9 (1983), 980-1000. Garner, W.R., Asoects o f a s t i m u l u s : f e a t u r e s , dimensions, and c o n f i g u r a t i o n s , i n E. Rosch, and B.B. L l o y d (Eds.), r i z a t i o n , H i l l s d a l e , N.J., G e i s s l e r , H.-G.,
C o g n i t i o n and Catego-
Lawrence Erlbaum Associates (1978).
P e r c e n t u a l r e p r e s e n t a t i o n o f i n f o r m a t i o n : dynamic
frames o f r e f e r e n c e i n judgment and r e c o g n i t i o n , i n F. K l i x , and
B . Krause (Eds.), P s y c h o l o o i c a l Research, Humbolt U n i v e r s i t a t B e r l i n , 1960-1980, B e r l i n , D.D.R., VER Deutscher V e r l a o d e r Y i s s e n s c h a f t e n (1980). G e i s s l e r , H.-G.,
K l i x , F.,
and S c h e i d e r e i t e r ,
o f s e r i a l s t r u c t u r e s , i n E.L.J. (Eds.),
U.,
Visual recognition
L e e w e n b e r g , and H.F.J.M.
Buffart
Formal Theories o f V i s u a l P e r c e o t i o n , C h i c h e s t e r . New York,
W i l e y ( 1978)
.
H.Buffdrt m d H.-G. Geissler
Geissler, H.-C.,
and P u f f e ,
'I.,
The i n f e r e n t i a l b a s i s o f c l a s s i f i c a t i o n :
tl.-c.
f r o m nerceot?cal t c reclory code systems, i n R u f f a r t , E.L.J.
C e i s s l e r , H.F.J.F.
L e e w e n h e r o , and V . S a r r i s ( E d s . ) , Modern I s s u e s i n
P e r c m t i on, AFS terdam, S e r l in , 3 .D .R
. , i i o r t h - t b l 1 and
Puhl is h i ng Company,
VEB Deutscher V e r l a o d e r ' J i s s e n s c h a f t e n (1983). G e i s s l e r , H.-C..,
S c h e i d e r e i t e r , U., and S t e r n , '.I., S t r a t e g i e s o f s e r i a l
cornnarison and d e c i s i o n i n memory: i n v a r i a n t and task-denendent comnon e n t s , i n F . K l i x , and E.D.R.,
Voffmann ( E d s . ) , C o g n i t i o n and Memory, B e r l i n ,
VEB Deutscher V e r l a n d e r V i s s e n s c h a f t e n ( 1 9 8 0 ) .
Hayes-Roth,
P.,
and Ljayes-Roth,
F.,
Concent l e a r n i n r r and t h e r e c o p n i -
t i o n a d c l a s s i f i c a t i o n o f exemnlars, . l o u r n a l o f Verbal L e a r n i n g and Verbal Behaviour, 16 (19771, 321-328. H i c k , IJ.E.,
On the r a t e o f y i n o f i n f o r m a t i o n , Q u a r t e r l y J o u r n a l o f
E x n e r i w n t a l P s y c h o l a y , 4 ( 1 9 5 2 ) , 11-26. Hoffmann, J . , Concents, t h e i r f o r m a t i o n , r e n r e s e n t a t i o n and i d e n t i f i c a t i o n ( i n Geman), i n proceedinns o f t h e XXnd I n t e r n a t i o n a l Congress o f Psychology, L e i n z i o , D.D.R. Leewenberg, E.,
(19611, 97-105.
and B u f f a r t , H.,
An o u t l i n e o f c o d i n ? t h e o r y : a sum-
mary o f r e l a t e d e x n e r i m e n t s , i n H.-G. E.L.J.
k i s s l e r , H.F.J.H.
L e e w e n b e r g , and V . S a r r i s TEds.:,
Amsterdam, B e r l i n , D.D.R.,
Ruffart,
r'cdern I s s u e s i n P e r c e p t i o n ,
N o r t h - H o l l a n d P u h l i s h i n p Comnany, VEB
Deutscher V e r l an d e r 1,Iissenschaften (1983). R e s t l e , F., Codin? t h e o r y as an i n t e g r a t i o n o f qes t a l t Dsychology and i n f o r m a t i o n p r o c e s s i n p t h e o r y , i n J . Beck (Ed.
, Orpanization
R e o r e s e n t a t i o n i n P e r c e o t i o n , H i l l s d a l e , N.J.,
Lawrence Erlbaum As-
and
sociates (1982). Rosch, F.,
C o o n i t i v e r e n r e s e n t a t i o n s o f semantic c a t e o o r i e s , J o u r n a l
c f E x o e r i m e n t a l Psvchol ogy , 3 ( 1975), 382-407. REFERENCE NOTES
[ i ] B u f f a r t , H.,
Seven minus two and s t r u c t u r a l o o e r a t i o n s , I n t e r n a t i o n a l
Renort, I l n i v e r s i t ! , o f Ni.jmeoen, The N e t h e r l a n d s (1984).
[Z] Puffe, M., V i s u e l l e Erkennun? alr: an%rderunqsabhangige I n f e r e n z u b e r s e q u e n t i e l l e n Kateaorierenresentationen, Uneublished Thesis, B e r l i n , D.D.R.,
Humbolt U n i v e r s i t a t ( 1 9 6 3 ) .
TRENDS I?f IWATHEMATICAL PSYCHOLOGY E. Vegreef and J. Van Bu enhaut (editors) 0 Elsevier Science Pub&rs B . K (ivorth-Holland), 1984
59
OUTLINE OF A THEORY OF BRIGHTNESS, COLOR, AND FORM PERCEPTION Stephen Grossberg Center f o r A d a p t i v e Systems Boston U n i v e r s i t y
T h i s paper d e s c r i b e s new concepts and mechanisms from a r e a l - t i m e v i s u a l p r o c e s s i n g t h e o r y t h a t has been used t o e x p l a i n p a r a d o x i c a l d a t a about b r i g h t n e s s and form perception.
These d a t a i n c l u d e t h e C r a i k - O ' B r i e n
e f f e c t , t h e Land b r i g h t n e s s and c o l o r demonstrations, t h e f a d i n g o f s t a b i l i z e d images, neon c o l o r spreading, complementary c o l o r i n d u c t i o n , c o m p l e t i o n o f i l l u s o r y c o n t o u r s , and b i n o c u l a r r i v a l r y .
Two f u n c t i o n a l l y d i s -
t i n c t c o n t o u r processes i n t e r a c t t o generate these b r i g h t n e s s and f o r m properties i n the theory.
A Boundary Contour process i s s e n s i t i v e t o t h e
amount o f c o n t r a s t b u t n o t t o t h e d i r e c t i o n o f c o n t r a s t i n s c e n i c edges. I t i n c l u d e s a b i n o c u l a r matching stage t h a t i s s e n s i t i v e t o s p a t i a l s c a l e ,
o r i e n t a t i o n , and b i n o c u l a r d i s p a r i t y , and whose outcome t r i g g e r s a process o f monocular c o n t o u r completion.
These completed c o n t o u r s f o r m t h e boun-
d a r i e s o f monocular p e r c e p t u a l domains.
A F e a t u r e Contour process i s
s e n s i t i v e t o b o t h t h e amount o f c o n t r a s t and t o t h e d i r e c t i o n of c o n t r a s t i n s c e n i c edges.
It triggers a diffusive f i l l i n g - i n reaction o f featural
qua1 it y w i t h i n p e r c e p t u a l domains whose boundaries a r e d y n a m i c a l l y d e f i n e d by t h e completed boundary c o n t o u r s .
The d i f f u s i v e f i l l i n g - i n r e a c t i o n s
t a k e p l a c e w i t h i n s y n c y t i a o f c e l l compartments.
These preprocessed mono-
c u l a r r e p r e s e n t a t i o n s g i v e r i s e t o a p e r c e p t v i a a process o f b i n o c u l a r resonance.
The p e r c e p t t a k e s t h e f o r m o f s t a n d i n g waves o f p a t t e r n e d
a c t i v i t y among m u l t i p l e s p a t i a l s c a l e s .
The Boundary Contour process i s
hypothesized t o be analogous t o i n t e r a c t i o n s between t h e hypercolumns i n area 17 o f t h e v i s u a l c o r t e x .
The F e a t u r e Contour process i s h y p o t h e s i z e d
t o be analogous t o i n t e r a c t i o n s between t h e cytochrome oxydase s t a i n i n g b l o b s i n area 17 and p r e s t r i a t e c o r t e x i n area 18.
60
S.Crossberg
1. INTRODUCTION : PARADOXICAL VISUAL DATA AS MANIFESTATIONS OF ADAPTIVE V I S I‘PL MECHP NI S ” T T h i s c h a p t e r o u t l i n e s some o f t h e new concepts and m c h a n i s m s t h a t my c o l l e a g u e s and I have been u s i n g t o e x p l a i n p a r a d o x i c a l d a t a a b o u t b r i g h t n e s s and form p e r c e p t i o n (Cohen and Grossberg, 1983a, 1983b, 1984; Grossberg, 1981, 1983a, 1983b; Grossberg and M i n g o l l a , 1984). T h a t p a r a d o x i c a l d a t a abound i n t h e f i e l d o f v i s u a l p e r c e p t i o n does n o t seem as s u r p r i s i n g when one c o n s i d e r s t h e manner i n which v i s u a l i n f o r m a t i o n i s acquired.
L i g h t passes t h r o u g h r e t i n a l v e i n s b e f o r e i t reaches r e t i n a l
p h o t o r e c e p t o r s , and l i g h t does n o t i n f l u e n c e t h e r e t i n a l r e g i o n corresponding t o the b l i n d spot.
The p e r c e p t s o f human o b s e r v e r s a r e n o t d i s t o r t e d ,
however, by t h e i r r e t i n a l v e i n s o r b l i n d s p o t s d u r i n g normal v i e w i n g conditions.
Thus some images t h a t a r e r e t i n a l l y p r e s e n t a r e n o t p e r c e i v e d
because o u r v i s u a l processes a r e a d a p t i v e l y designed t o f r e e o u r p e r c e p t s from i m p e r f e c t i o n s o f v i s u a l scanning.
These a d a p t i v e v i s u a l processes can
a l s o g e n e r a t e p a r a d o x i c a l p e r c e p t s , as d u r i n g t h e p e r c e p t i o n o f s t a b i l i z e d imaCes (Riggs, r a t l i f f ,
Cornsweet, and Cornsweet
and t’ebb. 1960 ; P r i t c h a r d
, 1961
; Yarbus
, 1967),
, 1953
; P r i t c h a r d , Heron
and o f f i l l i n g - i n r e a c t i o n s
( G e r r i t s and de Haan, 1966; G e r r i t s and Timmerman, 1969; G e r r i t s and Vendrick, 1970). Other a d a p t i v e v i s u a l mechanisms e x i s t t h a n t h o s e which compensate f o r r e t i n a l v e i n s and t h e b l i n d s p o t .
The v i s u a l mechanisms t h a t p r e v e n t
p e r c e p t i o n of r e t i n a l v e i n s and t h e b l i n d s p o t a r e o p e r a t i v e even when o n l y one eye i s open.
Due t o t h e a c t i o n o f b i n o c u l a r v i s u a l mechanisms, some
r e t i n a l images t h a t can be m o n o c u l a r l y p e r c e i v e d may n o t be p e r c e i v e d during binocular viewing conditions.
Binocular r i v a l r y provides a c l a s s i c a l
example o f t h i s f a c t ( B l a k e and Fox, 1964; Cogan, 1982; Kaufman, 1974; K u l i k o w s k i , 1978).
The p r o p e r t i e s o f b i n o c u l a r r i v a l r y a l s o seem l e s s
p a r a d o x i c a l when one c o n s i d e r s how t h e s u p p r e s s i o n of some monocular d a t a can a c h i e v e a more c o n s i s t e n t d e p t h f u l b i n o c u l a r r e p r e s e n t a t i o n o f a scene (Kaufman. 1974; Grossberg, 1980, 1983a). The above examples i l l u s t r a t e t h a t r e t i n a l l y p r e s e n t images may be suppressed t o a c h i e v e more c o n s i s t e n t v i s u a l r e p r e s e n t a t i o n s o f t h e e x t e r n a l world.
F o r a s i m i l a r reason, o b s e r v e r s may see images t h a t a r e n o t r e t i n a l -
l y present.
The r e m a r k a b l e Land c o l o r d e m o n s t r a t i o n s (Land,
1977) a r e ,
f o r example, o f t e n i n t e r p r e t e d as consequences o f t h e v i s u a l s y s t e m ’ s a b i l i t y t o process t h e r e f l e c t a n c e 5 o f p e r c e i v e d o b j e c t s d e s p i t e t h e
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Theory ofbrigknms, color, m d f o r m perception
v a r i a b i l i t y of t h e l i g h t i n g c o n d i t i o n s under which t h e o b j e c t s a r e observed. Our theory c l a r i f i e s t h e adaptive mechanisms t h a t g i v e r i s e t o t h e f a d i n g o f s t a b i l i z e d images, f i l l i n g - i n , and t h e Land b r i g h t n e s s c o l o r demonstrations.
The same mechanisms a l s o shed l i g h t on t h e p e r c e p t i o n o f i l l u s o r y
f i g u r e s ( G e l l a t l y , 1980; Kanizsa, 1974; Kennedy, 1978, 1979, 1981; Parks, 1980; Parks and Marks, 1983; P e t r y , Harbeck, Conway, and Levey, 1983), o f neon c o l o r spreading (Redies and Spillman, 1981; van T u i j l , 1975; van T u i j l and de Weert, 1979; van T u i j l and Leeuwenberg, 1979), and o f Julesz stereograms ( J u l esz, 1971; Kaufman, 1974). Every t h e o r y o f a d a p t i v e behavior i s faced w i t h t h e problem o f s p e c i f y i n g p r e c i s e l y t h e sense i n which i t s t a r g e t e d behavior i s adaptive, w i t h o u t becoming c i r c u l a r o r f a l l i n g i n t o an i n f i n i t e regress.
This d i f f i c u l t
problem does n o t go away j u s t by c i t i n g p l a t i t u d e s about Darwinian evolution.
The p r i n c i p l e s o f o r g a n i z a t i o n and r e a l - t i m e mechanisms t h a t
i n s t a n t i a t e o u r view of adaptive behavioral processes a r e o f t e n f a r removed from t e l e o l o g i c a l concepts t h a t a r e e a s i l y expressed u s i n g concensual language.
Some o f our concepts analyse, f o r example, how t h e nervous
system, o r indeed any c e l l u l a r machine, may respond t o i t s i n p u t environment i n a s e n s i t i v e and s t a b l e f a s h i o n w i t h o u t l o s i n g i t s c a p a b i l i t y f o r r e g u l a t e d s e l f - o r g a n i z a t i o n (Grossberg, 1978, 1980, 1982).
This approach
i n d i c a t e s how behavioral p r o p e r t i e s which seem t o r e f l e c t an i n t e l l i g e n t t e l e o l o g y may a r i s e as t h e epiphenomena of general c o n s t r a i n t s on t h e p o s s i b i l i t y o f doing accurate i n f o r m a t i o n processing a t a l l . The p r e s e n t chapter does n o t d e r i v e these p r i n c i p l e s and mechanisms from f i r s t principles.
Rather i t o u t l i n e s how some o f these general t o o l s ,
once a v a i l a b l e , can be s p e c i a l i z e d f o r use i n brightness and form perception.
My focus w i l l be on processes t h a t a r e hypothesized t o occur a t a
c e n t r a l l e v e l o f neural processing, n o t a b l y i n t h e v i s u a l c o r t i c a l areas 17 and 18.
The neural i n t e r p r e t a t i o n of these processes i s not, however,
c r i t i c a l t o understanding how they work.
A l l o f t h e processes may be
i n t e r p r e t e d as formal r u l e s f o r r e a l - t i m e v i s u a l i n f o r m a t i o n processing. The paper
w i t h Professor Carpenter (Carpenter and Grossberg,
(1981 ) , by c o n t r a s t , i l l u s t r a t e s how an adaptive approach may be used t o c l a r i f y data taken from t h e most p e r i p h e r a l photoreceptive c e l l s o f t h e vertabra?e nervous system.
This work may a l s o be s t u d i e d as a p u r e l y
formal model, n o t a b l y a model concerning how a r e c e p t i v e device can r e c a l i b r a t e i t s s e n s i t i v i t y i n response t o an i n p u t source whose i n t e n s i t y
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can f l u c t u a t e o v e r a v e r y l a r g e range t h r o u g h t i m e .
2. BOUNDARY CONTOURS AND FEATURE CONTOURS. One o f t h e most i m p o r t a n t new concepts o f o u r t h e o r y i s t h a t s e v e r a l t y p e s o f c o n t o u r e x t r a c t i n g processes a r e s i m u l t a n e o u s l y o p e r a t i v e d u r i n g b r i g h t n e s s and f o r m p e r c e p t i o n (Grossberg, 1983a, 1983b).
Two o f t h e s e
c o n t o u r processes a r e s a i d t o g e n e r a t e boundary c o n t o u r s and f e a t u r e c o n t o u r s , r e s p e c t i v e l y . The v e r y f a c t t h a t more t h a n one t y p e o f c o n t o u r process i s h y p o t h e s i z e d t o e x i s t i m p l i e s t h a t c a r e must be e x e r t e d i n a n a l y s i n g what one means by a c o n t o u r , o r edge, and how i t i s computed. F o r example, n e i t h e r o f o u r c o n t o u r processes i s computed u s i n g zero c r o s s i n g s ( M a r r and H i l d r e t h , 1980), w h i c h a r e w h o l l y i n a d e q u a t e f o r o u r piirposes (Grossberg. 1983a).
T h i s c h a p t e r w i l l d e s c r i b e and use t h e s e
new c o n t o u r concepts i n an i n t u i t i v e and n o n t e c h n i c a l way. B e f o r e i l l u s t r a t i n g how boundary and f e a t u r e c o n t o u r processes h e l p t o e x p l a i n p a r a d o x i c a l v i s u a l d a t a , I w i l l d i f f e r e n t i a t e them u s i n g s e v e r a l
o f t h e i r more s a l i e n t p r o p e r t i e s .
2.1. BOUNDARY CONTOURS AND CONTOUR COMPLETION (i) C o n t r a s t : A boundary c o n t o u r i s s e n s i t i v e t o t h e amount o f c o n t r a s t ,
b u t n o t t o t h e d i r e c t i o n o f c o n t r a s t , a t an edge o f a v i s u a l scene.
Thus
a boundary c o n t o u r can be a c t i v a t e d by e i t h e r a l i g h t - d a r k edge o r a d a r k l i g h t edge a t a f i x e d s c e n i c p o s i t i o n . ( i i ) Completion : Several boundary c o n t o u r a c t i v a t i o n s t h a t c o r r e s p o n d t o c o n t r a s t d i f f e r e n c e s a t d i f f e r e n t s c e n i c p o s i t i o n s may i n t e r a c t v i a a process of c o n t o u r c o m p l e t i o n .
The c o n t o u r c o m p l e t i o n process i s c a p a b l e
o f s y n t h e s i z i n g g l o b a l v i s u a l c o n t o u r s f r o m l o c a l c o n t o u r fragments.
The
c o n t o u r s t h a t a r e p e r c e i v e d i n i l l u s o r y f i g u r e s a r e assumed t o be generated by t h i s c o n t o u r c o m p l e t i o n process.
Completed boundary c o n t o u r s a r e used
t o bound p e r c e p t u a l l y s i g n i f i c a n t domains i n t h e t h e o r y ; hence t h e name boundary c o n t o u r .
A s i m p l e d e m o n s t r a t i o n o f a c o n t o u r process w i t h t h e s e p r o p e r t i e s can be made as f o l l o w s (Grossberg and M i n g o l l a , 1984). equal r e c t a n g l e s a l o n g an i m a g i n a r y boundary. u n i f o r m shade o f g r a y .
D i v i d e a square i n t o two C o l o r one r e c t a n g l e a
C o l o r t h e o t h e r r e c t a n g l e i n shades o f g r a y t h a t
process f r o m l i g h t t o d a r k as one moves f r o m end 1 o f t h e r e c t a n g l e t o end 2 of t h e r e c t a n g l e .
C o l o r end 1 a l i g h t e r shade t h a n t h e u n i f o r m g r a y
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o f t h e o t h e r r e c t a n g l e , and c o l o r end 2 a d a r k e r shade than t h e u n i f o r m gray o f the o t h e r rectangle.
Then as one moves f r o v end 1 t o end ?,an i n t e r
mediate g r a y r e g i o n i s passed whose luminance a p p r o x i m a t e l y equals t h a t o f the uniform rectangle.
A t end 1, a l i g h t - d a r k edge e x i s t s from t h e
nonuniform r e c t a n g l e t o t h e u n i f o r m r e c t a n g l e .
A t end 2, a d a r k - l i g h t
edge e x i s t s f r o m t h e nonuniform r e c t a n g l e t o t h e u n i f o r m r e c t a n g l e .
Despite
t h i s r e v e r s a l i n t h e d i r e c t i o n of c o n t r a s t f r o m end 1 t o end 2, an o b s e r v e r can see an i l l u s o r y edge t h a t j o i n s t h e two edges o f o p p o s i t e c o n t r a s t and separates t h e i n t e r m e d i a t e r e c t a n g l e
r e g i o n s of equal luminance.
T h i s d e m o n s t r a t i o n i l l u s t r a t e s t h e e x i s t e n c e of a process o f c o m p l e t i n g c o n t o u r s between edges w i t h o p p o s i t e d i r e c t i o n s o f c o n t r a s t .
This contour
c o m p l e t i o n process i s t h u s s e n s i t i v e t o amount of c o n t r a s t , b u t n o t t o direction o f contrast. ( i i i ) B i n o c u l a r M a t c h i n g : A monocular boundary c o n t o u r can be generated when a
s i n g l e eye views a v i s u a l scene.
When two eyes view a v i s u a l
scene, p a i r s o f monocular boundary c o n t o u r s become engaged i n a process o f b i n o c u l a r matching.
T h i s b i n o c u l a r matching process i s s e n s i t i v e t o
such f a c t o r s as s p a t i a l s c a l e , b i n o c u l a r d i s p a r i t y , and edge o r i e n t a t i o n . Due t o t h e b i n o c u l a r matching process, a f u s e d b i n o c u l a r boundary c o n t o u r may be generated f r o m a p a i r o f matched monocular boundary c o n t o u r s . A l t e r n a t i v e l y , one o f t h e monocular boundary c o n t o u r s may be suppressed, o r a double image of unfused monocular boundary c o n t o u r s may o c c a s i o n a l l y be generated.
These outcomes p r o v i d e t h e l o c a l s u b s t r a t e f o r such g l o b a l
processes as b i n o c u l a r r i v a l r y and d e p t h p e r c e p t i o n i n t h e (Grossberg, 1980, 1983a).
theory
I n p a r t i c u l a r , i f monocular boundary c o n t o u r s
c o r r e s p o n d i n g t o t h e l e f t eye a r e m o m e n t a r i l y suppressed by b i n o c u l a r mismatch, t h e n these monocular boundary c o n t o u r s cannot c o n t r i b u t e t o t h e c o n t o u r c o m p l e t i o n process corresponding t o t h e l e f t v i s u a l f i e l d . p e r c e p t s t h a t can be seen d u r i n g monocular v i e w i n g may by t h i s b i n o c u l a r matching process.
Visual
t h u s be suppressed
These processes o f b i n o c u l a r matching
and c o n t o u r c o m p l e t i o n a r e i n t e r p r e t e d n e u r a l l y t o o c c u r i n t h e s t r i a t e cortex (area 17). The boundary c o n t o u r processes a r e l i k e frames w i t h o u t p i c t u r e s .
The
p i c t o r i a l d a t a themselves a r e d e r i v e d f r o m t h e f e a t u r e c o n t o u r processes. T h i s s u g g e s t i v e analogy w i t h p i c t u r e s and frames s h o u l d n o t be taken t o o l i b r a l l y , however, s i n c e t h e same v i s u a l sources i n p u t t o b o t h t h e bound a r y c o n t o u r process and t h e f e a t u r e c o n t o u r process, and t h e o u t p u t s o f
S.Crossberg
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h o t h types o f processes i n t e r a c t i n a c o n t e x - s e n s i t i v e way t o g e n e r a t e visual representations. 2 . 2 . FEATURE CONTOURS AND DIFFUSIVE FILLING-IN
( i ) C o n t r a s t : The f e a t u r e c o n t o u r process i s s e n s i t i v e t o d i r e c t i o n o f c o n t r a s t as w e l l as t o amount o f c o n t r a s t , u n l i k e t h e boundary c o n t o u r process.
F o r example, i t i s o b v i o u s l y i m p o r t a n t i n computing t h e r e l a t i v e
b r i g h t n e s s across a s c e n i c boundary t o t a k e a c c o u n t o f which s i d e of t h e s c e n i c boundary has a l a r g e r r e f l e c t a n c e .
A s i m i l a r remark h o l d s a t a
red-green s c e n i c boundary. ( i i ) Completion : Boundary c o n t o u r s a c t i v a t e a c o n t o u r c o m p l e t i o n process t h a t generates t h e g l o b a l c o n t o u r s which bound monocular p e r c e p t u a l domains. F e a t u r e c o n t o u r s a c t i v a t e a d i f f u s i v e f i l l i n g - i n process t h a t spreads f e a t u r a l q u a l i t i e s , such as b r i g h t n e s s o r c o l o r , a c r o s s t h e s e monocular p e r c e p t u a l domains.
Such a d i f f u s i v e f i l l i n g - i n r e a c t i o n i s , f o r example,
hypothesized t o prevent perception o f the b l i n d spot. T h i s d i f f u s i v e f i l l i n g - i n process i s s p a t i a l l y l i m i t e d by boundary c o n t o u r s . The b i n o c u l a r m a t c h i n g process t h a t o c c u r s between monocular boundary con. t o u r s chooses among t h e p o s s i b l e s p a t i a l s c a l e s t h a t w i l l r e c e i v e boundary contour signals.
Thus, a l t h o u g h a f e a t u r e c o n t o u r may be a c t i v a t e d a c r o s s
a l l o f these s p a t i a l s c a l e s , o n l y t h o s e s c a l e s t h a t r e c e i v e boundary cont o u r signals w i l l generate b a r r i e r s t o t h e d i f f u s i o n o f f e a t u r a l q u a l i t y . Only these s c a l e s w i l l be capable of c o n t r i b u t i n g t o t h e f i n a l p e r c e p t i n t h e t h e o r y , as t h e subsequent d i s c u s s i o n w i l l show.
(iii) Monocular F i l l i n g - I n : Because monocular boundary c o n t o u r s a r e b i n o c u l a r l y matched b e f o r e g i v i n g r i s e t o completed c o n t o u r s , each v i s u a l f i e l d ( l e f t and r i g h t ) may process d i f f e r e n t completed boundary c o n t o u r s , j u s t as i n binocular r i v a l r y .
These d i s t i n c t completed boundary c o n t o u r s can
d i f f e r e n t i a l l y r e g u l a t e t h e monocular p r o c e s s i n g o f f e a t u r a l q u a l i t y i n t h e i r respective visual f i e l d s . These concepts w i l l now be r e f i n e d u s i n g a m a c r o c i r c u i t diagram and a p p l i e d t o s e v e r a l t y p e s o f p e r c e p t u a l phenomena.
3 . M A C R O C I R C U I T OF PROCESSING STAGES. F i g u r e 1 d e s c r i b e s a m a c r o c i r c u i t of p r o c e s s i n g s t a g e s t h a t i s c a p a b l e o f s y n t h e s i z i n g g l o b a l depth, b r i g h t n e s s , and form i n f o r m a t i o n f r o m m o n o c u l a r l y
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Theory of brightness, color, and form perception
.
Figure 1 Network stages used t o e x t r a c t g l o b a l depth, b r i g h t n e s s , and f o r m i n f o r m a t i o n f r o m monocular and b i n o c u l a r images.
Table I
l i s t s t h e names o f these stages and t h e i r p o s s i b l e n e u r a l analogs.
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S. Crossberg
and binocularly viewed p a t t e r n s . Table I l i s t s the names of the abbreviated processing s t a g e s , as well as the neural s t r u c t u r e s t h a t presently seem most l i k e l y t o execute analogous processes. Table I ML
-
MR
- Right Monocular Representation
BME
-
MBDL
-
L e f t Monocular Representation (Lateral geniculate nucleus) (Lateral geniculate nucleus) Binocular Matching of Edges (Hypercolumn i n t e r a c t i o n s i n s t r i a t e cortex - Area 1 7 (Hubel and Wiesel, 1977)) Left Monocular Brightness Domain (Cytochrome oxydase s t a i n i n g blobs - Area 17 p r e s t r i a t e cortex
-
-
and/or
Area 18 (Horton and Hubel, 1981; Hendrick-
son, H u n t , and Wu, 1981; Hubel and Livingstone, 1981;
Livingstone and Hubel, 1982)) MBDR
-
Right Monocular Brightness Domain ( P r e s t r i a t e cortex - Area 18 - and/or cytochrome oxydase s t a i n i n g blobs in Area 1 7 )
FIRE - F i l l i n g - I n Resonant Exchange ( P r e s t r i a t e cortex - Area 18) The BME stage generates oriented monocular boundary contours in response
t o inputs from ML and M R , binocularly matches these monocular boundary contours, and monocularly completes the boundary contours t h a t survive t h e binocular matching process. The completed l e f t boundary contours generate s c a l e - s p e c i f i c boundary s i g n a l s t o the MBDL stage. T h e completed r i g h t boundary contours generate s c a l e - s p e c i f i c boundary s i g n a l s t o t h e MBDR stage. Monocular f e a t u r e contours a r e generated by ML
-+ MBDL s i g n a l s and MBDR s i g n a l s . Thus the monocular regions ML and MR give r i s e t o both boundary contour s i g n a l s and t o f e a t u r e contour s i g n a l s . The boundary contours undergo binocular matching and monocular contour completion before rejoining the corresponding f e a t u r e contours a t MBDL and YBDR. This divergence of f e a t u r e contour processing from boundary contour processing allows the boundary contours t o b e n e f i t from s c a l e - s p e c i f i c , o r i e n t a t i o n -
MR
+
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Theory of brigktness, color, and form perception
s p e c i f i c , and d i s p a r i t y - s p e c i f i c b i n o c u l a r s e l e c t i o n w i t h i n BME b e f o r e t h e s u r v i v i n g boundary c o n t o u r s a r e passed among t h e s p a t i a l s c a l e s o f MBOL and MBDR. 4. DIFFUSIVE FILLING-IN WITHIN MONOCULAR BRIGHTNESS AND COLOR DOMAINS. The monocular boundary c o n t o u r s i g n a l s from stage BME t o stage MBDL d e f i n e barriers t o the l a t e r a l d i f f u s i o n o f featural a c t i v i t y t h a t i s triggered by f e a t u r e boundary s i g n a l s f r o m ML t o MBOL.
This d i f f u s i v e f i l l i n g - i n
r e a c t i o n i s assumed t o o c c u r w i t h i n a s y n c y t i u m of c e l l compartments ( F i g .
2 ) . W i t h i n t h i s c e l l u l a r syncytium, t h e f e a t u r e boundary s i g n a l ' s t r i g g e r a r a p i d d i f f u s i o n , o r averaging, o f e l e c t r i c a l p o t e n t i a l across compartment boundaries, except across those compartment boundaries t h a t r e c e i v e boundary c o n t o u r s i g n a l s f r o m t h e BME s t a g e .
The f e a t u r e c o n t o u r s i g n a l s
f r o m stage ML t o stage MBDL a r e hereby smoothed w i t h i n stage MBD between L successive boundary c o n t o u r s t h a t a r e c o n s i d e r e d p e r c e p t u a l l y s i g n i f i c a n t by t h e BME s t a g e (Cohen and Grossberg, 1983b, 1984; Grossberg, 1983a, 1983b). Boundary c o n t o u r s i g n a l s from s t a g e BME t o stage MBDL a r e assumed t o p r e v e n t d i f f u s i o n across t h e i r t a r g e t compartments by i n c r e a s i n g t h e e l e c t r i c a l r e s i s t a n c e of t h e t a r g e t compartment membranes.
I t i s assumed
t h a t t h e boundary c o n t o u r s i g n a l s a c h i e v e t h i s i n c r e a s e i n c e l l r e s i s t a n c e by a c t i n g as i n h i b i t o r y s i g n a l s t o t h e i r t a r g e t compartments.
A t the
same t i m e t h a t t h e s e i n h i b i t o r y s i g n a l s i n c r e a s e membrane r e s i s t a n c e s , t h e y a l s o h e l p t o decrease, o r h y p e r p o l a r i z e , t h e t a r g e t c e l l p o t e n t i a l s . An analogous d i s c u s s i o n shows how s i g n a l s f r o m t h e BME s t a g e t o t h e MBDR stage d e f i n e t h e boundaries t h a t c o n t a i n t h e f i l l i n g - i n r e a c t i o n s which o c c u r i n response t o f e a t u r e c o n t o u r s i g n a l s from t h e MR stage t o t h e M8DR stage.
5. DOUBLE STEP BRIGHTNESS ILLUSION. Using t h e s e i n t e r a c t i o n s between boundary c o n t o u r s i g n a l s , f e a t u r e c o n t o u r s i g n a l s , and d i f f u s i v e f i l l i n g - i n ,
a l a r g e number o f p a r a d o x i c a l b r i g h t -
ness p e r c e p t s have been s i m u l a t e d (Cohen and Grossberg, 1983b, 1984). One well-known phenomenon was s t u d i e d by Arend, Buehler, and Lockhead
(1971) d u r i n g t h e i r i n v e s t i g a t i o n s o f t h e C r a i k - O ' B r i e n i l l u s i o n ( O ' B r i e n , 1958).
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68
m
m
BOUNDARY CONTOUR SIGNALS
COMP#RTMENT
FEATURE CONTOUR SIGNALS
Figure 2 A monocular b r i g h t n e s s domain i n s t a g e MBDL o r s t a g e MBDR.
The f e a t u r e c o n t o u r s i g n a l s e x c i t e c e l l compartments which permit r a p i d l a t e r a l d i f f u s i o n , o r f i l l i n g - i n ,
of a c t i v i t y
across t h e i r membranes, e x c e p t a t t h o s e membranes which r e c e i v e boundary c o n t o u r s i g n a l s f r o m t h e BME s t a g e .
These boundary
c o n t o u r s i g n a l s i n h i b i t t h e i r t a r g e t membranes and i n c r e a s e t h e i r resistance against t h e l a t e r a l f l o w of a c t i v a t i o n .
The f e a t u r e
c o n t o u r s i g n a l s a r e thus averaged w i t h i n domains whose boundaries a r e c o n s i d e r e d p e r c e p t u a l l y s i g n i f i c a n t by t h e BME stage.
Theory of brightness, color, andjorm perception
F i g u r e 3a d e s c r i b e s a l u r i n a n c e ? r o f i l e t h a t was generated by r a p i d l y r o t a t i n g a d i s k covered w i t h a p p r o p r i a t e l y c u t s e c t o r s o f b l a c k and w h i t e papers.
The luminances t o t h e l e f t and t o t h e r i g h t of t h e luminance
cusps a r e e q u a l , and t h e average luminance a c r o s s t h e cusps equals t h e background luminance.
F i g u r e 3b shows t h a t t h i s luminance p r o f i l e i s
p e r c e i v e d as (approximate) steps o f i n c r e a s i n g b r i g h t n e s s .
Figure 3 ( a ) A double C r a i k - O ' B r i e n luminance p r o f i l e i n which t h e luminances t o t h e l e f t and r i g h t of t h e luminance cusps a r e equal
, and
t h e average 1uminance across t h e cusps equals t h e
background luminance.
( b ) T h i s luminance p r o f i l e appears t o
an o b s e r v e r as two (approximate) s t e p s o f b r i g h t n e s s .
69
70
S. Crossberg
A s i m i l a r r e s u l t i s generated by o u r t h e o r y ( F i g u r e 4 ) .
I n the theory,
each o f t h e two luminance cusps i n t h e i n p u t p a t t e r n g e n e r a t e s a boundary contour t h a t r e s t r i c t s the l a t e r a l d i f f u s i o n o f f e a t u r a l a c t i v i t y .
These
two boundaries a r e d e p i c t e d by t h e reduced d i f f u s i o n c o e f f i c i e n t s i n Figure 4.
The f e a t u r e c o n t o u r process a t t e n u a t e s t h e background luminance
o f t h e i n p u t p a t t e r n and enhances t h e c o n t r a s t s o f t h e cusps.
The boundary
contour signals a l s o c o n t r i b u t e t o the t o t a l i n p u t p a t t e r n t h a t i s received by t h e c e l l compartments.
This t o t a l i n p u t p a t t e r n i s c a l l e d t h e f i l t e r e d
i n p u t p a t t e r n w i t h b o u n d a r i e s i n F i g u r e 4.
(The f l a n k s of t h i s t o t a l i n p u t
p a t t e r n were a r t i f i c i a l l y extended t o t h e l e f t and t o t h e r i g h t t o a v o i d c o n t a m i n a t i o n by boundary e f f e c t s , because o u r computer c o u l d s i m u l a t e o n l y 2500 c e l l s i n a r e a s o n a b l e amount o f t i m e ) .
When t h i s t o t a l i n p u t i s
a l l o w e d t o d i f f u s e w i t h i n t h e domains d e f i n e d by t h e boundary c o n t o u r s i g n a l s , t h e s t e p l i k e monocular preprocessed p a t t e r n o f F i g u r e 4 i s t h e result.
Appendix A d e s c r i b e s t h e network e q u a t i o n s t h a t g e n e r a t e t h e
patterns i n Figure 4.
6. NEON COLOR SPREADING AND COMPLEMENTARY COLOR INDUCTION The phenomenon o f neon c o l o r s p r e a d i n g i l l u s t r a t e s t h e e x i s t e n c e o f bound a r y c o n t o u r s and o f f e a t u r e i n a p a r t i c u l a r l y v i v i d way.
Redies and
S p i l l m a n (1981) have, f o r example, r e p o r t e d an e x p e r i m e n t u s i n g a s o l i d r e d c r o s s and an E h r e n s t e i n f i g u r e .
When t h e s o l i d r e d c r o s s i s p e r c e i v e d
i n i s o l a t i o n , i t looks q u i t e u n i n t e r e s t i n g (Figure 5a).
When an E h r e n s t e i n
f i g u r e i s p e r c e i v e d i n i s o l a t i o n , i t generates an i l l u s o r y c o n t o u r whose shape ( e . g . ,
c i r c l e or diamond) depends upon t h e v i e w i n g d i s t a n c e .
When
the red cross i s placed i n s i d e the Ehrenstein f i g u r e , the red c o l o r flows o u t o f i t s c o n t a i n i n g c o n t o u r s and tends t o f i l l t h e i l l u s o r y f i g u r e (Figure 5b). I n o u r t h e o r y , t h i s p e r c e p t i s e x p l a i n e d by showing how boundary c o n t o u r s o f t h e E h r e n s t e i n f i g u r e i n h i b i t c o n t i g u o u s houtidery c o n t o u r s o f t h e r e d c r o s s . l h i s i n h i b i t o r y process t a k e s Dlace w i t h i n t h e BME s t a p e as p a r t c f t h e c o n t o u r c o m p l e t i o n process (Grossberg and F l i n g o l l a
, 1984
).
This i n h i -
b i t o r y a c t i o n w i t h i n t h e BME does n o t n r e v e n t t h e p r o c e s s i n a o f f e a t u r e cont o u r s i g n a l s from s t a g e
'$ t o
s t a g e MBD and f r o m s t a g e MR t o s t a g e MBDR.
I.
Thus t h e r e d c o n t o u r s i g n a l s a r e r e c e i v e d by MBDL and MBDR d e s p i t e t h e f a c t t h a t some o f t h e i r c o r r e s p o n d i n g boundary c o n t o u r s i g n a l s a r e i n h i b i t e d w i t h i n t h e BME stage.
Theory of brightness, color, and form perception
71
STEP ILLUSION INPUT P A l l U N
DIFNUON
comam
l.l.loe
-l.l.lol FlLTERm INPUT PATXRN
YONQCUUR REPRO-
WITH BOUNDARIES
PATXRN
J
-8.7*13-'
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1
Figure 4 S i m u l a t i o n of t h e double s t e p b r i g h t n e s s i l l u s i o n .
The t e x t
d e s c r i b e s how t h e double C r a i k - O ' B r i e n i n p u t p a t t e r n generates boundary c o n t o u r s and f e a t u r e c o n t o u r s t h a t i n t e r a c t t o form the step-like output pattern.
S. Crossberg
Figure 5 ( a ) A r e d c r o s s seems unremarkable when p e r c e i v e d i n i s o l a t i o n .
(b) When p e r c e i v e d w i t h i n an E h r e n s t e i n f i g u r e , t h e r e d c o l o r f l o w s o u t s i d e o f t h e c r o s s and i s c o n t a i n e d by t h e i l l u s o r y c o n t o u r t h a t i s g e n e r a t e d by t h e E h r e n s t e i n f i g u r e .
Theory of bri&tness,
color, a i d
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form perception
The i n h i b i t i o n o f t h e s e boundary c o n t o u r s i g n a l s w i t h i n t h e BME s t a g e allows the r e d f e a t u r a l a c t i v i t y t o d i f f u s e outside o f the red cross.
The
i l l u s o r y boundary c o n t o u r t h a t i s induced by t h e E h r e n s t e i n f i g u r e r e s t r i c t s the d i f f u s i o n o f t h i s red-labelled activation.
T h i s example o f neon
c o l o r s p r e a d i n g t h u s p r o v i d e s a v i v i d example i n which one can "see" t h e d i f f e r e n c e between boundary c o n t o u r s and f e a t u r e c o n t o u r s , as w e l l as t h e r o l e o f i l l u s o r y boundary c o n t o u r s i n r e s t r i c t i n g t h e d i f f u s i o n o f f e a t u r a l a c t iv i ty . Redies and S p i l l m a n (1981) have s y s t e m a t i c a l l y v a r i e d t h e d i s t a n c e o f t h e r e d cross from t h e Ehrenstein contour, t h e i r r e l a t i v e o r i e n t a t i o n s , t h e i r r e l a t i v e s i z e s , e t c . t o s t u d y how t h e s t r e n g t h o f t h e s p r e a d i n g e f f e c t changes w i t h s c e n i c parameters. "thin
O f p a r t i c u l a r i n t e r e s t i s the f a c t t h a t
[ r e d ] f l a n k s r u n n i n g a l o n g s i d e t h e r e d c o n n e c t i n g l i n e s " (p. 679)
can o c c u r i f t h e E h r e n s t e i n f i g u r e i s s l i g h t l y separated f r o m t h e c r o s s . I n o u r t h e o r y , s e p a r a t i n g t h e E h r e n s t e i n f i g u r e from t h e c r o s s weakens t h e i n h i b i t o r y e f f e c t o f E h r e n s t e i n boundary c o n t o u r s upon t h e boundary cont o u r s o f the cross.
When t h e boundary c o n t o u r s of t h e c r o s s a r e p a r t i a l l y
weakened, t h e y can s t i l l p a r t i a l l y r e s t r i c t t h e d i f f u s i o n of r e d - l a b e l l e d activation.
Then t h e r e d c o l o r can o n l y "bleed" o u t s i d e t h e c o n t o u r s o f
the cross. One m i g h t o b j e c t t o t h i s e x p l a n a t i o n by q u e s t i o n i n g t h e a b i l i t y o f t h e E h r e n s t e i n boundary c o n t o u r s t o i n h i b i t t h e boundary c o n t o u r s o f t h e c r o s s . Why do n o t t h e E h r e n s t e i n boundary c o n t o u r s a l s o i n h i b i t c o n t i g u o u s E h r e n s t e i n boundary c o n t o u r s ? If t h e y do, t h e n how do any boundary cont o u r s s u r v i v e t h i s process o f mutual i n h i b i t i o n ? I f t h e y do n o t , t h e n d o e s n ' t t h i s e x p l a n a t i o n o f neon c o l o r s p r e a d i n g c o l l a p s e ? A t t h i s p o i n t , l e t us r e c a l l t h a t t h e boundary c o n t o u r process i s s e n s i t i v e t o t h e amount o f c o n t r a s t , even though i t i s i n s e n s i t i v e t o t h e d i r e c t i o n o f contrast.
I c l a i m t h a t t h e E h r e n s t e i n boundary c o n t o u r s do m u t u a l l y
i n h i b i t one another, b u t t h a t t h i s
i n h i b i t i o n i s a type o f shunting l a t e r a l
i n h i b i t i o n such t h a t e q u a l l y stro'ng i n h i b i t o r y c o n t o u r s i g n a l s can remain p o s i t i v e and balanced (Grossberg, 1973. 1983a).
I f , however, t h e E h r e n s t e i n
boundary s i g n a l s a r e s t r o n g e r t h a n t h e boundary
s i g n a l s o f t h e c r o s s by
a s u f f i c i e n t amount, t h e n t h e l a t t e r s i g n a l s can be s i g n i f i c a n t l y i n h i b i t e d .
f h i s formal p r o p e r t y of t h e c o n t o u r c o m p l e t i o n process p r o v i d e s an explanat i o n o f t h e e m p i r i c a l f a c t t h a t neon c o l o r spreading occurs o n l y i f t h e
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S. Crossberg
c o n t r a s t o f a f i g u r e (e.g.,
t h e c r o s s ) r e l a t i v e t o t h e background i l l u m i n a -
t i o n i s l e s s t h a n t h e c o n t r a s t o f t h e bounding c o n t o u r s (e.g.,
t h e Ehren-
s t e i n f i g u r e ) r e l a t i v e t o t h e background i l l u m i n a t i o n (van T u i j l and de Weert, 1979). A s i m i l a r argument h e l p s t o e x p l a i n t h e complementary c o l o r i n d u c t i o n t h a t
van T u i j l (1975) r e p o r t e d i n h i s o r i g i n a l a r t i c l e a b o u t t h e neon e f f e c t . Uraw on w h i t e paper a r e g u l a r g r i d o f h o r i z o n t a l and v e r t i c a l b l a c k l i n e s t h a t form 5 m . squares.
Replace a s u b s e t o f b l a c k l i n e s by b l u e l i n e s .
L e t t h i s subset o f l i n e s be r e p l a c e d , say, f r o m t h e s m a l l e s t i m a g i n a r y diamond shape t h a t i n c l u d e s complete v e r t i c a l o r h o r i z o n t a l l i n e segments o f the g r i d .
When an o b s e r v e r i n s p e c t s t h i s p a t t e r n , t h e b l u e c o l o r o f
t h e l i n e s appears t o spread around t h e b l u e l i n e elements u n t i l i t reaches t h e s u b j e c t i v e c o n t o u r s o f t h e diamond shape.
T h i s p e r c e p t has t h e same
e x p l a n a t i o n as t h e p e r c e p t i n F i g u r e 5b. Next r e p l a c e t h e b l a c k l i n e s by b l u e l i n e s and t h e b l u e l i n e s by b l a c k lines.
Then t h e i l l u s o r y diamond l o o k s y e l l o w r a t h e r t h a n b l u e .
L e t us
suppose t h a t t h e y e l l o w c o l o r i n t h e diamond i s induced by t h e b l u e l i n e s i n t h e background m a t r i x .
Then why i n t h e p r e v i o u s d i s p l a y i s n o t a
y e l l o w c o l o r i n t h e background induced by t h e b l u e l i n e s i n t h e diamond ? Why i s t h e complementary c o l o r y e l l o w p e r c e i v e d when t h e background cont a i n s b l u e l i n e s , whereas t h e o r i g i n a l c o l o r b l u e i s p e r c e i v e d when t h e diamond c o n t a i n s b l u e l i n e s ? What i s t h e reason f o r t h i s asymmetry ?
I suggest t h a t t h e reason i s as f o l l o w s .
When t h e diamond i s composed
of b l u e l i n e s , t h e n opponent c o l o r p r o c e s s i n g enables t h e b l u e l i n e s t o induce c o n t i g u o u s y e l l o w c o n t o u r s i g n a l s i n t h e background.
However these
y e l l o w c o n t o u r s i g n a l s a r e c o n s t r a i n e d by t h e boundary c o n t o u r s i g n a l s o f t h e b l a c k l i n e s t o remain w i t h i n t h e s p a t i a l domain o f t h e f e a t u r e contour o f the black lines. ground.
Thus t h e y e l l o w c o l o r i s n o t seen i n t h e back-
By c o n t r a s t , t h e boundary c o n t o u r s i g n a l s o f t h e b l a c k l i n e s i n
t h e background i n h i b i t t h e c o n t i g u o u s boundary c o n t o u r s i g n a l s of t h e b l u e l i n e s i n t h e diamond.
Hence t h e b l u e f e a t u r e c o n t o u r s i g n a l s o f t h e b l u e
lines can f l o w w i t h i n t h e diamond. When b l u e l i n e s form t h e background, t h e y have two e f f e c t s on t h e diamond. They i n d u c e y e l l o w f e a t u r e c o n t o u r s v i a opponent p r o c e s s i n g .
They a l s o
i n h i b i t t h e boundary c o n t o u r s i g n a l s o f t h e c o n t i g u o u s b l a c k l i n e s . t h e y e l l o w c o l o r can f l o w w i t h i n t h e diamond.
Hence
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Theory of brightness, color, and form perception
To c a r r y o u t t h i s e x p l a n a t i o n q u a n t . i t a t i v e l y , we need t o s t u d y how opponent c o l o r processes (red-green, ye1 l o w - b l ue) preprocess t h e f e a t u r e c o n t o u r s i g n a l s f r o m s t a g e ML t o stage MBDL and from stage MR t o stage MBDR. Opponent c o l o r processes w i t h t h e r e q u i s i t e p r o p e r t i e s a r e d e f i n e d by gated d i p o l e f i e l d s (Grossberg, 1980).
We a l s o need t o q u a n t i t a t i v e l y
s t u d y how t h e c o n t o u r c o m p l e t i o n process responds t o complex s p a t i a l p a t t e r n s such as g r i d s and E h r e n s t e i n f i g u r e s (Grossberg and M i n g o l l a , 1984). 7 . FILLING-IN DUE TO STABILIZED IMAGES
The d i s t i n c t i o n between boundary c o n t o u r s and f e a t u r e c o n t o u r s a1 so h e l p s t o e x p l a i n p e r c e p t s t h a t a r e induced by s t a b i l i z e d images (Yarbus, 1967). Suppose, f o r example, t h a t a l a r g e c i r c l e i s surrounded by a u n i f o r m r e d background ( F i g u r e 6 ) . vertical line.
L e t t h e c i r c l e be d i v i d e d down t h e m i d d l e by a
W i t h i n each h e m i c i r c l e , p l a c e a small c i r c l e f i l l e d w i t h
t h e same r e d c o l o r as t h e background.
C o l o r t h e remainder o f t h e l e f t
h e m i f i e l d b l a c k and t h e remainder o f t h e r i g h t h e m i f i e l d w h i t e . Now s t a b i l i z e t h e r e t i n a l images o f t h e l a r g e c i r c u l a r c o n t o u r and i t s v e r t i c a l edge.
The background r e d i s t h e n seen t o f l o w o v e r t h e b l a c k and
white hemifields.
Since t h e background r e d c o l o r i s t h e same c o l o r as
t h e r e d c o l o r t h a t f i l l s t h e small c i r c l e s , one m i g h t have supposed t h a t t h e e n t i r e f i g u r e would l o o k u n i f o r m l y red.
I t does not..
The r e d c i r c l e
t h a t l i e s w i t h i n t h e p r e v i o u s l y b l a c k h e m i f i e l d l o o k s b r i g h t e r , whereas the red c i r c l e t h a t l i e s w i t h i n the previously white hemifield looks darker.
To e x p l a i n t h i s p e r c e p t , l e t us suppose t h a t s t a b i l i z i n g a c o n t o u r causes i t s boundary c o n t o u r s i g n a l s t o decay (Grossberg, 1983b).
Then t h e r e d
f e a t u r e c o n t o u r s t h a t a r e e x t e r i o r t o t h e l a r g e c i r c l e can f r e e l y f l o w i n t o t h e i n t e r i o r o f t h e c i r c l e , even as t h e b l a c k and w h i t e f e a t u r a l c o n t o u r s o f t h e i n t e r i o r o f t h e c i r c l e i n t e r m i x w i t h one a n o t h e r and t h e r e d background.
T h i s f i l l i n g - i n process does n o t , however, e n t e r t h e
small r e d c i r c l e s , because t h e i r s c e n i c c o n t o u r s a r e n o t r e t i n a l l y stabilized. The r e l a t i v e c o n t r a s t o f t h e r e d f e a t u r e c o n t o u r w i t h i n t h e l e f t r e d c i r c l e i s g r e a t e r t h a n t h e r e l a t i v e c o n t r a s t of t h e r e d f e a t u r e c o n t o u r w i t h i n the r i g h t red c i r c l e .
These r e d f e a t u r e c o n t o u r s a r e c o n t a i n e d by t h e
u n i n h i b i t e d boundary c o n t o u r s i g n a l s around each small c i r c l e .
When t h e
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s. Grossberg
Figure 6 When the boundary of the large c i r c l e and the vertical l i n e are stabilized on the r e t i n a , the red color outside the large c i r c l e f i l l s - i n the c i r c l e except a t the small red c i r c l e s . The l e f t small c i r c l e appears as a brighter red, whereas the right small c i r c l e appears as a darker red than the f i l l e d - i n red color, even though a l l the red regions are colored using the same red shade.
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Theory of brightness, color, ond form perception
f i l l i n g - i n process t a k e s p l a c e , t h e s e r e l a t i v e c o n t r a s t s a r e t h e r e f o r e p r e s e r v e d as t h e t o t a l l e v e l of f e a t u r a l a c t i v i t y becomes more u n i f o r m l y d i s t r i b u t e d around them.
The n e t e f f e c t i s t h e o b t a i n e d p e r c e p t .
8. THE LAND COLOR AXIOMS. Land (1977) has developed a formal model t o e x p l a i n t h e remarkable b r i g h t ness and c o l o r e f f e c t s t h a t he has demonstrated u s i n g McCann Mondrians. T h i s model computes t h e c u m u l a t i v e r e l a t i v e c o n t r a s t changes t h a t o c c u r when one s e r i a l l y t r a v e r s e s a p a t h a c r o s s a p i c t u r e , and t h e n averages t h e outcomes across many such paths. shades o f g r a y need t o be considered.
For brightness predictions, only F o r c o l o r p r e d i c t i o n s , t h e computa-
t i o n s a r e done w i t h r e s p e c t t o more t h a n one wavelength.
T h i s procedure
l e a d s t o e x c e l l e n t p r e d i c t i o n s about t h e p e r c e i v e d b r i g h t n e s s e s and c o l o r s o f many v i s u a l d i s p l a y s . D e s p i t e these f o r m a l successes, i t i s d i f f i c u l t t o p h y s i c a l l y i n t e r p r e t L a n d ' s s e r i a l scanning and s e r i a l a v e r a g i n g p r o cesses.
Moreover, h i s formal procedure a l s o f a i l s t o p r e d i c t t h e c o r r e c t
r e s u l t i n a number o f i m o r t a n t cases.
A s i m p l e example t h a t i l l u s t r a t e s
t h e b a s i c n a t u r e o f t h e s e f a i l u r e s i s shown i n F i g u r e 7.
Figure 7 The f a c t t h a t t h e i l l u s o r y c i r c l e w i t h i n t h e d a r k l i n e s i s b r i g h t e r t h a n i t s surround cannot be e x p l a i n e d by t h e Land (1977) r e t i n e x t h e o r y .
The p r e s e n t t h e o r y can e x p l a i n i l l u s o r y
b r i g h t n e s s d i f f e r e n c e s and p r o v i d e s a p h y s i c a l i n t e r p r e t a t i o n o f t h e formal operations o f Land's r e t i n e x theory.
78
S.Grossberg
I n F i g u r e 7 , an i l l u s o r y c i r c l e i s c l e a r l y seen a t t h e i n n e r t i p s of t h e radial lines. brightness.
T h i s i l l u s o r y c i r c l e e n c l o s e s a r e g i o n o f enhanced i l l u s o r y
No m a t t e r how many r a d i a l l y o r i e n t e d s e r i a l scans a r e made
between t h e r a d i a l l i n e s , t h e i r average c o n t r a s t change c a n n o t p r e d i c t t h i s i l l u s o r y b r i g h t n e s s d i f f e r e n c e , because t h e c o n t r a s t does n o t change a t a l l d u r i n g these scans.
I n c l u d i n g t h e d a r k r a d i a l l i n e s w i t h i n t h e scans
does n o t improve t h i s i n c o r r e c t p r e d i c t i o n , because t h e i l l u s o r y b r i g h t n e s s d i f f e r e n c e does n o t depend on t h e number, l e n g t h , o r c o n t r a s t o f t h e s e dark l i n e s i n a s i m p l e way.
F o r example, t h e s t r e n g t h of t h e i l l u s o r y
b r i g h t n e s s e f f e c t i s reduced as t h e a n g l e formed by t h e l i n e s w i t h t h e i l l u s o r y c i r c l e decreases (Kennedy, 1978).
T h i s happens even if t h e
number, l e n g t h , and c o n t r a s t o f t h e l i n e s remain unchanged. T h i s e m p i r i c a l p r o p e r t y c a n n o t be e x p l a i n e d by any t h e o r y t h a t depends o n l y on averages o f l o c a l c o n t r a s t changes. configuration o f the lines.
The e f f e c t s a r e c l e a r l y due t o t h e g l o b a l I n a s i m i l a r f a s h i o n , t h e Land (1977) r e t i n e x
t h e o r y cannot e x p l a i n why r e a r r a n g i n g t h e o r i e n t a t i o n of t h e l i n e s i n F i g u r e 7 can change t h e shape of t h e r e g i o n where i l l u s o r y b r i g h t n e s s o c c u r s (Kennedy, 1975), o r why a s i n g l e l i n e causes no i l l u s o r y b r i g h t n e s s change whereas even f o u r such l i n e s , i f p r o p e r l y c o n f i g u r e d , can cause a s i g n i f i c a n t i l l u s o r y b r i g h t n e s s change. The p r e s e n t t h e o r y can d e a l w i t h t h e i l l u s o r y e f f e c t s by showing how t h e c o n t o u r c o m p l e t i o n process can g e n e r a t e i l l u s o r y c o n t o u r s , and how f e a t u r e c o n t o u r s can i n t e r a c t w i t h i l l u s o r y boundary c o n t o u r s t o g e n e r a t e i l l u s o r y b r i g h t n e s s changes (Grossberg and M i n g o l l a , 1984).
Our t h e o r y a l s o
c l a r i f i e s why L a n d ' s formal o p e r a t i o n s o f t e n work so w e l l by g i v i n g these operations a physical i n t e r p r e t a t i o n .
S e r i a l computations o f c o n t r a s t
changes a l o n g m u l t i p l e p a t h s a r e r e p l a c e d by p a r a l l e l c o m p u t a t i o n s o f c o n t r a s t - s e n s i t i v e feature contours.
Averages a c r o s s s e r i a l scans a r e
r e p l a c e d by d i f f u s i v e (hence, a v e r a g i n g ) f i l l i n g - i n w i t h i n domains d e f i n e d by boundary c o n t o u r s .
To q u a n t i t a t i v e l y c a r r y o u t t h i s t y p e o f e x p l a n a t i o n
f o r c o l o r p e r c e p t s , we a l s o need t o p r e p r o c e s s t h e f e a t u r e c o n t o u r s i g n a l s from s t a g e ML t o s t a g e MBDL and from s t a g e M R t o s t a g e MBDR u s i n g opponent c o l o r processes ( r e d - g r e e n , y e l l o w - b l u e ) t h a t a r e d e f i n e d by g a t e d d i p o l e f i e l d s (Grossberg, 1980).
79
Theory ofbrklitness, color. and form perception
9. FILLING-IN RESONANT EXCHANGE.
A l l o f t h e p r o c e s s i n g stages d e s c r i b e d above a r e assumed t o be pre-perceptual.
Only a c t i v i t y i n t h e FIRE stage of F i g u r e 1 i s assumed t o e n t e r
conscious perception.
S i g n a l s form s t a g e MBDL and/or MBDR t h a t a r e capable
o f a c t i v a t i n g t h e FIRE s t a g e a r e s a i d t o " l i f t " t h e preprocessed monocular p a t t e r n s i n t o t h e p e r c e p t u a l domain.
I assume t h a t on)y c o n t o u r s of t h e
MBDL and MBDR a c t i v i t y p a t t e r n s i n p u t t o t h e FIRE stage.
These c o n t o u r s
s h o u l d n o t be confused w i t h t h e edges o f a v i s u a l scene.
They a r e non-
l i n e a r s t a t i s t i c a l c o n s t r u c t s of a r a t h e r h i g h o r d e r by t h e t i m e t h e y g e n e r a t e s i g n a l s from t h e MBDL stage o r t h e MBDR stage t o t h e FIRE stage. One needs o n l y t o remember t h a t t h e c o n t o u r c o m p l e t i o n process w i t h i n t h e BME stage g i v e s r i s e t o t h e c o n t o u r s w i t h i n
MBDL and MBDR t o a p p r e c i a t e
this fact. The s c a l e - s p e c i f i c c o n t o u r s i g n a l s f r o m t h e MBDL and MBDR stages undergo a b i n o c u l a r matching process i n t h e FIRE stage t h a t i s analogous t o t h e matching process whereby boundary c o n t o u r s from t h e ML and MR matched a t t h e BME stage.
stages a r e
Only c o n t o u r s t h a t s u r v i v e t h i s b i n o c u l a r
matching process can send feedback s i g n a l s back t o t h e MBDL and/or MBDR stages.
These l o c a l feedback s i g n a l s i n i t i a t e r e s o n a n t exchanges between
t h e FIRE s t a g e and t h e MBD stages.
These r e s o n a n t exchanges s e l e c t i v e l y
l i f t some, b u t n o t a l l , o f t h e preprocessed monocular subdomains i n t o
conscious perception. Once such a f i l l i n g - i n r e s o n a n t exchange i s i n i t i a t e d by a feedback exchange of matched c o n t o u r s , i t can r a p i d l y propagate across thousands of r e c e p t i v e f i e l d s w i t h i n i t s s p a t i a l scale.
Each such r e s o n a n t exchange
generates a s t a n d i n g wave o f p a t t e r n e d a c t i v i t y t h e FIRE stage.
across a subdomain o f
The ensemble o f a l l such s t a n d i n g waves c o n s t i t u t e s t h e
network's percept. The p r o p e r t i e s o f t h e s e f i l l i n g - i n r e s o n a n t exchanges l i e beyond t h e scope o f t h i s chapter.
The concept o f a FIRE stage was i n t r o d u c e d i n Grossberg
(1981, 1983a) and e x t e n s i v e l y analysed i n Cohen and Grossberg (1983a, 1984). 10. BINOCULAR RIVALRY, STABILIZED IMAGES AND THE GANZFELD. Three q u a l i t a t i v e p r o p e r t i e s of t h e s e resonances a r e , however, e a s i l y noted.
These p r o p e r t i e s f u r t h e r c l a r i f y how b i n o c u l a r r i v a l r y and t h e
80
S. Crossbeit
f a d i n g o f s t a b i l i z e d images and o f a g a n z f e l d o c c u r w i t h i n t h e network o f F i g u r e 1. Suppose t h a t , due t o b i n o c u l a r matching, some l e f t - m o n o c u l a r boundary cont o u r s a r e suppressed w i t h i n t h e BME stage. Then t h e s e boundary c o n t o u r s cannot send boundary c o n t o u r s i g n a l s t o t h e c o r r e s p o n d i n g r e g i o n o f stage MBDL. F e a t u r a l a c t i v i t y t h e n q u i c k l y d i f f u s e s a c r o s s t h e network p o s i t i o n s
c o r r e s p o n d i n g t o t h e s e suppressed c o n t o u r s ( G e r r i t s and V e n d r i c k , 1 9 7 0 ) . Consequently, no c o n t o u r s i g n a l s can be e m i t t e d from these p o s i t i o n s w i t h i n No edge matches w i t h i n t h e FIRE stage
t h e MBDL s t a g e t o t h e FIRE s t a g e .
can o c c u r a t t h e s e p o s i t i o n s , and no feedback s i g n a l s a r e r e t u r n e d from t h e FIRE stage t o t h e MBDL s t a g e t o l i f t t h e c o r r e s p o n d i n g monocular subdomain i n t o p e r c e p t i o n .
Thus t h e subdomains whose boundary c o n t o u r s
a r e suppressed w i t h i n t h e BME s t a g e a r e no l o n g e r p e r c e i v e d .
As soon as
t h e c o r r e s p o n d i n g boundary c o n t o u r s w i n t h e BME b i n o c u l a r c o m p e t i t i o n , however, these subdomains can a g a i n r a p i d l y s u p p o r t t h e r e s o n a n t l i f t i n g o f t h e i r subdomain i n t o p e r c e p t i o n a t t h e FIRE stage. A s i m i l a r argument h o l d s f o r t h e n o n p e r c e p t i o n o f a subdomain whether i t s
boundary edges a r e suppressed by b i n o c u l a r r i v a l r y w i t h i n t h e BME stage, by image s t a b i l i z a t i o n , o r do n o t e x i s t , as i n a g a n z f e l d . REFERENCES. [ 11
Arend. L.E.,
B u e h l e r , J.N.,
and Lockhead, C.R.,
Difference information
i n b r i g h t n e s s p e r c e o t i o n , P e r c e o t i o n and Psychor)hysics, 9 (1971),
367-370. 121
Blake, R, and Fox, P . , R i n o c u l a r r i v a l r y s u n n r e s s i o n : i n s e n s i t i v e t o s n a t i a l frequency and o r i e n t a t i o n change, V i s i o n Research, 14 (1974)
687-692. I31
Carpenter, G.A.,
and Crossher!,
v e r t e b r a t e DhotoreceDtors,
5., A d a o t i o n and t r a n s m i t t e r g a t i n g i n
,'ournal o f T h e o r e t i c a l N e u r o b i o l o a y , 1
(19819, '-42.
141
Cogan, A . I . ,
15 1
Cohen, M.A.,
tronocular s e n s i t i v i t y d u r i n g b i q o c u l a r v i e w i n g , V i s i o n
Research, 2 2 (1982), 1-16. and Grossbera, S.,
Some q l o b a l p r o r J e r t i e s of b i n o c u l a r re
sonances: d i s p a r i t y m t c h i n g , f i l l i n g - i n , in
P.
and f i g u r e - g r o u n d s y n t h e s i s ,
Dodwell and T. C a e l l i ( E d s . ) , F i g u r a l s y n t h e s i s , H i l l s d a l e , NJ.,
Erltiaum A s s o c i a t e s (1983a).
Tlieory of brightness, colnr, andforni perception
[71-
rn1 191
1101 [111
[121
[131
U41
[15 1
U6 1
81
Cohen, W . P . , and Crossbero, q., Neural dynamics of binocular form Derceotion, Society f o r Neuroscience Abstracts, 13th Annual Fleeting, Poston, 353.8 (1983b). Cohen, M . A . , and Grosshero, S., The dynamics of briohtness nerceDtion: a nsychological and a neural theory, Submitted f o r nublication (1984). G e l l a t l y , A . R . H . , Percention of an i l l u s o r y t r i a n a l e w i t h masked i n d u cincl f i g u r e , Percention, (1 (1980), 599-602. G e r r i t s , H.J.P., de t'aan, R . , and Vendrick, A.J.H., Experiments with r e t i n a l s t a h i l i z e d imacres: r e l a t i o n s between the observations and neural d a t a , Vision Research, 6 (1566), 427-440. G e r r i t s , H.J.!'., and Timwrman, J.G.Y.E.N., The f i l l i n a - i n qrocess in n a t i e n t s w i t h r e t i n a l scotomata, Vision qesearch, 9 (1969), 439-442. C e r r i t s , H.J.P., and Vendrick, A . J . H . , Simultaneous c o n t r a s t , f i l l i n a i n nrocess and information processing i n man's visual system, ExDerimental Brain Research, 11 (1970), 411-430. Grossberq, S . , Contour enhancement, short-term memory, and constancies i n reverberating neural networks, Studies i n Pnnlied rlathematics, 52 (1973), 217-257. Crossbera, S . , How does a brain build a coonitive code?, Psvchological Review, 87 ( l ? : , O ) , 1-51. Grossberg, S . , Adaptive resonance i n develonment, nerception, and cognit i o n , i n 5 . Grossbera ( E d . ) , r'athematical Dsycholoqy and psychophysiolooy, Providence, R I , American Mathematical Society (1981). Grossberg, S . , Studies of mind and brain: neural n r i n c i o l e s of l e a r n i n a , perception, develonment, coqnition, and motor c o n t r o l , Boston, Reidel Press (1982). Grossberg, S . , The quantized ceometry of visual snace: the coherent computation of denth, form, and l i g h t n e s s , The Behavioral and Brain Sciences, i n press (1983a). Grossberq, S . , Neural s u b s t r a t e s of binocular form Derception: f i l t e r i n g , matchinq, d i f f u s i o n , and resonance, i n E. Easar, H . Flohr, H . Haken, and A.J. Mandell (Eds.), qynergetics of the brain, New York, Springer Verl ag ( 1983b) . Grossberg, S . , and Mingolla, E . , Neural dynamics of form Derception: i l l u s o r y figures and neon color soreading, submitted f o r publication (1984). Hendrickson, A.E., H u n t , S.P., and Wu, J.-Y., Immunocytochemical l o c a l i zation of glutamic acid decarboxylase i n monkey s t r i a t e cortex, Nature,
82
S. Crossberg
292 (1981), 605-607.
1201
Horton, J . C . , and Huhel, D.H.,
Regular p a t c h y d i s t r i b u t i o n o f c y t o -
chrome o x i d a s e s t a i n i n g i n n r i m a r y v i s u a l c o r t e x o f macaque monkev, Nature, 292 (1961), 762-764. Hubel
, D.H.,
and L i v i n g s t o n e , t4.S.,
Reaions o f o o o r o r i e n t a t i o n t u -
n i n g c o i n c i d e w i t h natches o f cytochrome o x i d a s e s t a i n i n g i n monkey s t r i a t e c o r t e x , S o c i e t y f o r Neuroscience P b s t r a c t , 1 1 t h Annual Meetina,
Los Angeles, 118.12 ( 1 9 8 1 ) . , D.H., and V i e s e l , T .N., F u n c t i o n a l a r c h i t e c t u r e of macaque mon-
Hubel
key v i s u a l c o r t e x , Proceedings o f the Royal S o c i e t y o f London ( B ) , 198
( 1977). 1-59. Foundations o f cyclonean p e r c e o t i o n , Chicaoo, U n i v e r s i t v
Julesz, R . ,
o f Chicago Press ( 1 3 7 1 ) . Kanizsa, C . ,
Contours w i t h o u t v r a d i e n t s o r c o C n i t i v e c o n t o u r s ? ,
l i a n J o u r n a l o f Psycholoay, Kaufman, L.,
Ita-
1 (1974). 93-113.
S i g h t and mind: an i n t r o d u c t i o n t o v i s u a l o e r c e p t i o n ,
New York, O x f o r d U n i v e r s i t y Press ( 1974). Kennedy, J.H.,
Denth a t an edge, c o p l a n a r i t y , s l a n t deoth, change i n
d i r e c t i o n , and chanoe i n b r i g h t n e s s i n the o r o d u c t i o n o f s u b j e c t i v e con t o u r s , I t a l ian J o u r n a l of Psycholoay , 2 ( 1975) Kennedy, J.M.,
,
107-123.
ill us or,^ c o n t o u r s and t h e ends o f l i n e s , P e r c e n t i o n , 7
(1978), 605-607. Kennedy, J.M.,
S u b j e c t i v e c o n t o u r s , c o n t r a s t , and a s s i m i l a t i o n , i n
C.F. Nodiqe, and D.F. F i s h e r ( E d s . ) , P e r c e o t i o n and o i c t o r i a l r e p r e sen t a t i on, New York , Praeger ( 1979)
.
Kennedy, J.M., I l l u s o r y b r i g h t c e s s and the ends of n e t a l s : change i n brightness w i t h o u t a i d o f s t r a t i f i c a t i o n o r a s s i m i l a t i o n e f f e c t s , P e r c e p t i o n , 10 (1981), 503-585. K u l i k o w s k i , J . J . , L i m i t o f s i n q l e v i s i o n i n s t e r e o o s i s denends on cont o u r sharnness, N a t u r e , 275 (1973), 126-127. Land,
E.H., The r e t i n e x t h e o r y o f c o l o r v i s i o n , S c i e n t i f i c American,
237 (1977), 108-128. L i v i n g s t o n e , F . S . , and H u b e l , D.H.,
Thalarnic i n o u t s t o cytochrome o x i -
d a s e - r i c h r e g i o n s i n monkey v i s u a l c o r t e x , Proceedings o f t h e N a t i o n a l Academy o f Sciences, 79 (1982), 6098-6101.
1331
Marr, D.,
and H i l d r e t h ,
E., Theory o f edge d e t e c t i o n , Proceedings o f
t h e Royal S o c i e t y o f London ( B ) ,
207 (1980), 187-217.
83
Theory ofbrightness, colur, a n d j o r m perception
O'Brien,
V., Contour n e r c e o t i o n , i l l u s i o n , and r e a l i t y . J o u r n a l o f t h e
b t i c a l S o c i e t y o f America, 48 ( 1 9 5 8 ) , 112-119. Parks, T.E.,
S u b j e c t i v e f i g u r e s : some unusual c o n c o m i t a n t b r i g h t n e s s
e f f e c t s , P e r c e n t i o n , 9 (1980), 239-241. Parks, T.E.,
and Marks,
(I.,
Sharo-edges vs. d i f f u s e i l l u s o r y c i r c l e s :
t h e e f f e c t s o f v a r y i n a luminance, P e r c e n t i o n and Psychonhysics, 33
( 1 9 8 3 ) , 172-176. P e t r y , S.,
Harbeck, A . , Conwav, J., and Levey, J., S t i m u l u s d e t e r m i -
n a n t s c f b r i g h t n e s s and d i s t i n c t n e s s of s u b j e c t i v e contours, Perceat i o n and Psychophysics, 34 (1983) , 169-174. P r i t c h a r d , R.M.,
S t a b i l i z e d imanes on t h e r e t i n a , S c i e n t i f i c American,
204 ( 1 9 6 1 ) , 72-78. P r i t c h a r d , R.V.,
Heron, I d . ,
and Hebb, D.O.,
V i s u a l D e r c e o t i o n aoproach-
e d by t h e method o f s t a b i l i z e d images, Canadian J o u r n a l o f Psychology,
14 (1960), 67-77. I401
Redies, C.,
and S n i l l m a n , L . The neon c o l o r e f f e c t i n t h e E h r e n s t e i n
i l l u s i o n , P e r c e p t i o n , 10 (1981). 667-681.
[411
Riggs, L.A.,
E a t l i f f , F.,
Cornsweet, J.C.,
and Cornsweet, T.N.,
The
disaFpearance o f s t e a d i l y f i x a t e d v i s u a l t e s t o b j e c t s , J o u r n a l o f t h e O D t i c a l S o c i e t y o f America, 43 (1953), 495-501.
I421
van T u i j l , H.F.J.M.,
A new v i s u a l i l l u s i o n : n e o n l i k e c o l o r s p r e a d i n o
and complementary c o l o r i n d u c t i o n between s u b j e c t i v e c o n t o u r s , A c t a P s y c h o l o g i ca, 39 ( 1 9 7 5 ) , 441-445. van T u i j l , H.F.J.If,,
and de Weert, C.M.M.,
Sensory c o n d i t i o n s f o r t h e
occurrence o f t h e neon s p r e e d i n g illu s i o n , PerceDtion , 8 (1979), 211-
215. I441
van T u i j l , H.F.J.W.,
and Leewenberg, E.L.J.,
Neon c o l o r s n r e a d i n g
and s t r u c t u r a l i n f o r m a t i o n measures, P e r c e o t i o n and PsychoDhysics, 25
( 1979) , 269-284. Eye movements and v i s i o n , New York, Plenum Press ( 1 9 6 7 ) .
Yarbus, A.L.,
84
S. Cressberg
APPENDIX
T h i s a m e n d i x d e s c r i b e s t h e c e u r a l network t h a t Cohen and Grossbera (1984) use t o s i m u l a t e b r i a h t n e s s o e r c e o t s .
The f o l l o w i n o s i m u l a t i o n s were done
on one-dimensional f i e l d s o f c e l l s .
The i n n u t o a t t e r n ( I , ,
i s transformed i n t o t h e o u t n u t D a t t e r n
( z l , z2,
...,
12,
...,
In)
zn) v i a t h e f o l l o w i n o
equations. FF'
1 ._CWTOURS
...,
The i n o u t o a t t e r n ( I 1 , 1 2 ,
I n ) i s transformed i n t o f e a t u r e c o n t o u r s
v i a a f e e d f o r w a r d o n - c e n t e r o f f - s u r r o u n d network o f c e l l s u n d e r g o i n g shimt i n r , o r membrane c o w t i o n , i n t e r a c t i o n s .
The a c t i v i t y , o r D o t e n t i a l
>i,
c f t h e ith c e l l < n a feature contour o a t t e r n i s d a xi
+
= -Pxi
-
(€3
n 7 IkCki
xi)
-
(xi
t
k=l
n D! I IkEki
.
C=l
(1)
and t h e o f f - s u r r o u n d c o e f f i c i e n t s Fki
Both t h e o n - c e n t e r c o e f f i c i e n t s Cki
are Gaussian f u n c t i o n s o f i n t e r c e l l u l a r d i s t a n c e ; v i z . , , , 2 Cki = C e x o [ - u ( i - k ) L 1 and Eki = E e x n [ - v ( i - k ) 1 , where p
>
V.
System (1)
i s assumed t o r e a c t m c h m r e o u i c k l y than t h e d i f f u s i v e f i l l i n g - i n Drocess.
Hence, we assume t h a t each xi
respect t o the i n o u t o a t t e r n . n
x. = 1
The a c t i v i t y p a t t e r n (xl,
I: (PCki k=l n
xi),
i s i n aporoximate e n u i l i b r i u m w i t h d xi = 0 and
A t equilibrium,
...,
-
DEki)Ik
xn) i s s e n s i t i v e t o b o t h t h e amount and
t h e d i r e c t i o n o f c o n t r a s t i n edges o f t h e i n o u t D a t t e r n (Grossberg ( 1 9 8 3 a ) ) . These f e a t u r e c o n t o u r a c t i v i t i e s a e n e r a t e i n p u t s o f t h e f o r m
t o the d i f f u s i v e f i l l i n o - i n orocess.
The i n h i b i t o r y t e r m Si
i s d e f i n e d b;)
the boundary c o n t o u r nrocess i n e q u a t i o n ( 7 ) below. BqUNDARy CONTOURS
The i n p u t o a t t e r n
(I1, 12,
...,
In) a l s o a c t i v a t e s t h e boundary c o n t o u r
process, which we r e D r e s e n t as a feedforward o n - c e n t e r o f f - s u r r o u n d n e t w o r k i n a underqoing s h u n t i n g i n t e r a c t i o n s .
T h i s s i m p l i f i e d view o f t h e
as
Tlieory ofbrightness, color, andjornt perception
boundary c o n t o u r nrocess i s n e r m i s s i b l e i n t h e n r e s e n t s i m u l a t i o n s because t h e s i m u l a t i o n s , b e i n g one-dimensional and monocular, do n o t need t o account f o r o r i e n t a t i o n a l t u n i n a o r b i n o c u l a r matchinq.
Since t h e s i m u l a t i o n s do
n o t orobe t h e dynamics o f i l l u s o r y c o n t o u r formation,
t h e c o n t o u r comple-
t i o n Drocess can a l s o he i q n o r e d .
As i n e n u a t i o n ( Z ) , t h e i n p u t n a t t e r n r a p i d l y l i v e s r i s e t o an a c t i v i t y nattern
__
n
c
-
(SCki
--
DEki)Ik
k=l Yi =
-
P
n
+c
(Cki
(4) + Eki)Ik
k=l where Cki
-
= C exp [ - p ( i - k )
ficients V,
and
<
2
1 and Eki
-
= E exn[-v(i-k)
2
1
.
The Caussian c o e f -
are, however, l a r ? e r than t h e Gaussian c o e f f i c i e n t s 1.1 and
because boundary c o n t o u r s a r e n a r r o w e r than f e a t u r e c o n t o u r s .
The a c t i v i t y o a t t e r n (yl,
y2,
..., y n )
i s s e n s i t i v e both t o the d i r e c t i o n
and anc.unt o f c o n t r a s t i n the i n n u t o a t t e r n ( I 1 ,
12,
...,
In).
The s e n s i -
t i v i t . y t o the d i r e c t i o n o f contrast i s progressively eliminated by the f o l lowing onerations.
L e t t h e o u t o u t z i g n a l f r o m W E t o FqRD t h a t i s e l i c i t e d
b y a c t i v i t y yi equal f ( y i ) , p a r t o f yi;
where f ( w ) i s a s i a m i d s i a n a l b f t h e r e c t i f i e d
viz.,
The n o t a t i o n [w]+ = rax(w,O) and y > 1.
The o u t p u t s i g n a l s f ( y i )
a r e spa-
t i a l l y d i s t r i b u t e d b e f o r e i n f l u e n c i n q c e l l comnartments o f t h e c e l l s y ~ c y tium.
The t o t a l s i a n a l t o t h e ith c e l l c o r p a r t m e n t due t o t h e a c t i v i t y
o a t t e r n (yl,
y2,
..., Y,)
is n
where Gik viz.,
Gik
i s a Gaussian f u n c t i o n o f i n t e r c e l l u l a r d i s t a n c e ; 2 = G e x p [ - w ( i - k ) 1 . T h i s Gaussian f a l l - o f f ( w ) i s l e s s narrow
t h a n t h e boundary c o n t o u r nrocess ( f i , G ) ,
b u t more narrow t h a n t h e f e a t u r e
c o n t o u r Drocess ( p . ~ ) , DIFFUSIVE FILLING-IN The a c t i v i t y zi o f t h e ith c e l l u l a r compartment o f t h e c e l l u l a r s y n c y t i u m
86
S. Crossberg
obeys t h e n o n l i n e a r d i f f u s i o n e q u a t i o n
d zi
+ Ji+l,i(Zi+l
= -Hzi
~ ti J ) i-lYi(Zi-1
i s defined h y equation ( 3 ) .
where t h e i n n u t Fi
J i. + ~ , and i 'i-1,i
-
-
Zi)
+ Fi,
(8)
The d i f f u s i o n c o e f f i c i e n t s
a r e determined b y boundary c o n t o u r s i n n a l s a c c o r d i n g t o
equations o f the form
and
where t h e t h r e s h o l d
r
>
0.
Thus an i n c r e a s e i n t h e boundary s i g n a l Si
creases b o t h d i f f u s i o n c o e f f i c i e n t s J i tl,i s i n n a l Fi
and Ji-l,i.
The f e a t u r e c o n t o u r
a l s o decreases when t h e boundary s i g n a l Si
e q u a t i o n s ( 3 ) , (Q), and
(lo!,
de-
increases.
I n a l l the
t h e i n h i b i t o r y e f f e c t s o f boundary s i g n a l s Si
on c e l l compartment memhranes a c t v i a s h u n t i n g i n h i b i t i o n .
A o o s i t i v e thres-
h o l d r occurs i n ( 9 ) and ( 1 0 ) . b u t n o t i n ( 3 ) , hecause we assume t h a t t h e i n t e r c o m n a r t m e n t a l membranes t h a t r e g u l a t e d i f f u s i o n o f a c t i v i t y between comnartments a r e l e s s a c c e s s i b l e t o t h e s i g n a l s Si
than are t h e e x t e r i o r
s u r f a c e mmbranes t h a t bound t h e c e l l u l a r s y n c y t i u n .
Sunnorted i n n a r t b y t h e A i r Force D f f i c e o f S c i e n t i f i c Research (AFOSK i-r-Cl48).
TRENDS I N hfATHEMATK4L PSYCHOLOGY E. Degreefond J . Van Bu enhaut (editors) 0 Elsevier Science h b l i s g r s B. V. (North-Holland), 1984
87
ATTENTION I N DETECTION THEORY
H. C. Yicko I n s t i t u t e o f Psycholoay U n i v e r s i t v o f Technology Brunswi ck, Germany
Detection theory takes account o f the c o n d i t i o n a l h i t and f a l s e alarm o r o h a b i l i t i e s , given s t i m u l i have been insnected and decided upon.
From t h a t
exact formulation a theory i s d e r i v e d which oerm i t s , under adequate exDerimenta1 c o n d i t i o n s , the separate es ti ma ti on o f a t t e n t i on, d i s c r i m i n a t i o n and b i a s parameters.
1. DEFINITIONS Detection theory (see e.9. Rreen. and Swets (1966)) i s expressed i n most general form by the ? a i r o f equations p ( Y / s ) = fs(d,b
,... ),
~ ( Y l n )= fn(d,b
,...) ,
(1)
w i t h s denoting the event o f a s i g n a l , n t h a t o f a nonsignal o r noise s t i mulus b e i n g presented t o an observer 0. the stimulus, s o r n, as a s i g n a l .
Y denotes the event o f 0 r e p o r t i n g
I n o t h e r words, D ( Y [ s ) and p ( Y [ n ) deno-
t e h i t and false alarm p r o b a b i l i t i e s t o be estimated by emFirica1 h i t and f a l s e alarm r a t e s .
~ ( Y l s )and p ( Y ] n ) are r e l a t e d by f u n c t i o n s fS resp.
fn
t o some d i s c r i m i n a t i o n oarameter, d, and some d e c i s i o n b i a s parameter, b. Sometimes a d d i t i o n a l Darameters are considered, e.0. s i g n a l t o n o i s e variance r a t i o o r c r i t e r i o n variance.
Ye s h a l l leave them n u t o f account i n order
t o s i m p l i f y arguments and n o t a t i o n . I n subsequent equations we s h a l l consider an observation event, i.e. the event o f 0 making a perceptual decision about whether o r n o t t o r e p o r t a signal.
'le designate t h a t event by o
F i n a l l y , we l e t
7
and
8
denote the
events complementary t o Y and 0 , i . e . the events o f 0 n o t r e p o r t i n g resp. n o t observing the s t i m u l u s presented.
Nonobservation may be due t o the
f a c t t h a t the stimulus does n o t a r r i v e a t 0's perceptual system o r t h a t i t
88
H.C. Micko
i s n o t orocessed up t o the response s e l e c t i o n stacre. S o e c i f i c d e t e c t i o n , choice and t h r e s h o l d models d i f f e r by the n a i r o f funct i o n s fS, fn oostulated, which may r e f l e c t d i f f e r e n t numerical representat i o n s and/or i n t e r p r e t a t i o n s o f d i s c r i m i n a t i o n and b i a s .
For convenience
we d e f i n e f S and fn t o be nondecreasino functions of b y l a r a e r values of which reoresent s t r o n q e r b i a s towards r e p o r t i n g s t i v u l i as s i g n a l s . ascume f,
2 fp
bk
f o r any b and l e t o r e a t e r values o f d be associated w i t h
l a r o e r excess o f fS over fn f o r any p a r t i c u l a r b (see F i g u r e 1). Thus,e.g., b i s o o s i t i v e l y r e l a t e d t o ouessinp o r o b a b i l i t y i n h i o h t h r e s h o l d models,
n e q a t i v e l y t o b e t a o r C i n c l a s s i c a l d e t e c t i o n models t h a t n o s t u l a t e Gauss i a n d i s t r i b u t i o n s f o r f s , fn; d i s p o s i t i v e l y r e l a t e d t o d' i n Gaussian I-, i n Luce's (1963) choice nodel b u t n e g a t i v e l y b t o I-,.When c o n s i d e r i n g Gaussian d e t e c t i o n models we denote by -mN(O,l)
d e t e c t i o n models and -104
the value o f t h e standard normal d i s t r i b u t i o n f u n c t i o n a t h.
Fiaure 1 C l a s s i c a l (Gaussian) d e t e c t i o n t h e o r y . L e f t : Decision f u n c t i o n s ( r e p o r t p r o b a b i l i t y f o r s i g n a l s (f,) n o i s e (f,)
as f u n c t i o n o f b i a s ( b ) ) .
and
D i s c r i m i n a b i l i t y , d=2u.
Right: ROC-Curve o f the u n c o n d i t i o n a l Gaussian d e t e c t i o n model, d=23.
2. A T A C I T ASSUMPTION OF DETECTION THEORY The usual i n t e r p r e t a t i o n o f the parameters d and b r e s t s h e a v i l y on an i m p l i c i t b u t fundamental assumption o f d e t e c t i o n theory: d e t e c t i o n i s viewed as a perceptual d e c i s i o n task, i.e. responses Y and the outcome o f a d e c i s i o n about a perceptual event. q u i r e d i n o r d e r t o j u s t i f y inferences about
@Is
'i
are assumed t o r e f l e c t
This assumption i s r e -
perceptual d i s c r i m i n a t i o n
89
Attention m detection theory
and d e c i s i o n b i a s f r o m h i s h i t and f a l s e a l a r m r a t e s .
ble make t h a t assumn-
t i o n e x o l i c i t by r e w r i t i n g t h e fundamental e q u a t i o n s (1) o f d e t e c t i o n theory :
~ ( Y l o h s )= f s ( d , b ) ,
n(Y1ohn) = fn(d,b)
Errs. ( 2 ) s t a t e , more n r e c i s e l y than eqs. (1). t h a t d e t e c t i o n t h e o r y takes a c c o m t o f t h e n r o b a b i l i t i e s o f a h i t and f a l s e alarm, c i v e n t h e i n s o e c t i o n o f and d e c i s i o n a b o u t t h e s t i r u l u s Dresented.
I n f a c t , h i t s and f a l s e
--
alarms may r e s u l t f r o m a r b i t r a r y , n e r c e p t i o n - i n d e n e n d e n t guesses i n t h e ab~
sence o f any i n s p e c t i o n o f t h e s t i m u l u s t o be judged, apnarent misses and c o r r e c t r e j e c t i o n s may r e f l e c t j u s t l a c k o f o b s e r v a t i o n . resnonses Y and
I n b o t h cases t h e
a r e n o t outcomes o f a p e r c e o t u a l d e c i s i o n and, t h e r e f o r e ,
As a
cannot p r o p e r l y be taken account o f by c l a s s i c a l d e t e c t i o n t h e o r y .
consequence, h i t and f a l s e a l a r m r a t e s t h a t a r e a f f e c t e d by a r b i t r a r y guess i n g o r l a c k o f a t t e n t i o n a r e u n s u i t a b l e f o r a n a l y s i s by d e t e c t i o n t h e o r y and s h o u l d he p r e c l u d e d i f m i s l e a d i n g parameter e s t i m a t e s a r e t o be a v o i d e d ( i t m i q h t be mentioned t h a t p e r c e n t i o n - i n d e p e n d e n t guessing d i f f e r s f r o m t h e k i n d o f guessing assumed i n t h r e s h o l d models, t h e p r o b a b i l i t y o f w h i c h e x p l i c i t e l y depends on 0 ' s p e r c e o t u a l s t a t e " D e t e c t " resD. " N o n - d e t e c t " ) . I n c l a s s i c a l d e t e c t i o n experiments i t was assumed t k a t a l l s t i m u l i present e d were i n f a c t i n s p e c t e d and j u d g e d as r e q u i r e d by t h e t h e o r y .
T h a t as-
sumption i s n o t unreasonable s i n c e 0 ' s were i n f o r m e d about t i m e and p l a c e o f occurrence o f the s t i m u l i ,
Ide may a l s o hope t h a t s u f f i c i e n t c a r e has
been taken t o secure t h e h i g h morale necessary t o immunize observers, who made thousands o f judqements, a g a i n s t t h e t e m n t a t i o n t o nroducp responses w i t h o u t b o t h e r i n g about t h e s t i m u l i .
The above assumption, however, cannot
be unheld, i f d e t e c t i o n t h e o r y i s a p o l i e d t o d a t a o f a t t e n t i o n experiments, because i t i s t h e v e r y purpose o f a t t e n t i o n paradigms t o i n v e s t i g a t e behav i o r under c o n d i t i o n s t h a t make s t i m u l i l i k e l y t o escape 0 ' s n o t i c e .
It
has been r e c o g n i z e d t h a t b, when e s t i m a t e d f r o m eqs. (1) i n s t e a d o f eqs. ( Z ) , measures t h e confounded e f f e c t s of a t t e n t i o n and j u d g e m n t b i a s ( J e r i s o n , P i c k e t t & Stenson (1965); Swets & K r i s t o f f e r s e n (1970); Broadbent (1971),
p . 93f.).
S i m i l a r l y , n e r c e p t i o n - i n d e p e n d e n t guessing may reduce d w i t h o u t
t h e D e r c e p t u a l system b e i n g i m n a i r e d o r s t i m u l u s d i s c r i m i n a b i l i ty b e i n g r e duced.
Perception-independent quessing, however, may o c c u r i n a t t e n t i o n ex-
p e r i m e n t s i f no
s p e c i a l p r e c a u t i o n s a r e taken.
I t i s bound t o o c c u r i f 0's
a r e f o r c e d t o respond t o e v e r y s t i m u l u s presented, whether observed o r unnoticed.
I t i s t h e purpose o f t h e p r e s e n t paper t o d e r i v e a more a p p r o p r i a t e
90
H.C. Micko
f o r m u l a t i o n o f d e t e c t i o n theory from eqs. ( 2 ) which p e r m i t s , under adequate exDerimenta1 c o n d i t i o n s , the seaerate estirnation o f a t t e n t i o n , d i s c r i m i n a t i o n and decision parameters. 3. THEORY !?F CONDTTIPNPL DETECTION The c o n d i t i o n a l h i t and f a l s e alarm p r o b a b i l i t i e s given i n eqs. ( 2 ) are the aoDroDriate p r o b a b i l i t i e s from which t o determine d i s c r i m i n a t i o n and b i a s narameters.
As a r u l e , however, the r e s p e c t i v e c o n d i t i o n a l h i t and false
alarm r a t e s cannot be obtained e x p e r i m e n t a l l y , because w i t h o u t s p e c i a l q u e s t i o n i n g o f 0 ' s we do n o t know whether o r n o t a response, Y resp. r e s u l t e d from a perceotual d e c i s i o n .
v,
I n most cases, the a v a i l a b l e data w i l l
be the u n c o n d i i i o n a l h i t and f a l s e alarm r a t e s which estimate o ( Y l s ) and Therefore eqs. ( 2 ) i s o f l i t t l e p r a c t i c a l value so f a r and we p(Y1n). have t o d e r i v e a p p r o n r i a t e exnressions f o r the u n c o n d i t i o n a l n r o b a b i l i t i e s . We assume t h a t the s e t of c o r r e c t o r f a l s e r e p o r t s o f a s i g n a l can be subd i v i d e d w i t h o u t remainder o r o v e r l a p i n t o a s e t o f r e o o r t s based on perceot u a l decisions (althouqh s o s s i b l y d e r i v e d from weak o r f a u l t y evidence) and a s e t o f r e p o r t s based on oerception-independent cuessing, i .e. {YAs) = { Y A o A s ~ u {YA%s~ iYAn) = {YAoAn]
u iYAoAn1.
From t h a t assumption we o b t a i n f o r the r e s p e c t i v e n r o b a b i l i t i e s o f occurrence : D ( Y A S )= D ( Y A O A S ) + ~ ( Y A ~ A S ) n(YAn) = o(YAoAn)
+
p(YAh),
and d i v i d i n g by p ( s ) , resD. o ( n ) , y i e l d s the f o l l o w i n g expression f o r the unconditional h i t and f a l s e alarm p r o b a b i l i t i e s :
Applying the elementary theorem on c o n d i t i o n a l p r o b a b i l i t i e s , P(AAR\C) = ~ ( B ( C ) - ~ ( A / B A Ct )o, t h e p r o b a b i l i t i e s on the r i g h t sides o f eqs. ( 3 ) we o b t a i n p ( y l S ) = D(0lS) p ( Y \ n ) = p(o1n)
-
D(Yl0AS) + p ( 0 l S ) p(Y1ohn)
+
p(iiln)
-
p(Y(oAS),
*
p(Yl6An).
E q u i v a l e n t equations were discussed by Wykoff (1963) i n the c o n t e x t o f
(4)
91
Attention in detection theory
discrimination learning. I n eqs. ( 4 ) a l l p r o b a b i l i t i e s which a r e o f r e l e v a n c e t o d e t e c t i o n and a t t e n t i o n do occur.
~ ( Y ~ o A s and ) n(Y1oAn) a r e t h e f a m i l i a r c o n d i t i o n a l pro-
b a b i l i t i e s n f eqs. ( 2 ) which a r e taken account o f by c l a s s i c a l d e t e c t i o n theory.
P ( Y l k ) and p ( Y l o h n ) , a r e t h e c o n d i t i o n a l h i t and f a l s e a l a r m o r o b a b i l i t i e s , g i v e n t h e s t i m u l u s has n o t been observed.
They a r e t h e o r o b a b i l i t i e s
of r e p o r t i n g a s i g n a l on t h e b a s i s of an a r b i t r a r y guess.
S i n c e n o percep-
t i o n i s i n v o l v e d , t h e r e p o r t o f a s i g n a l i s as l i k e l y as t h a t o f a n o i s e s t i m u l u s and t h e p r o b a b i l i t i e s a r e o b v i o u s l y independent o f d.
They may be
a f u n c t i o n o f b, however, because under c o n d i t i o n s t h a t make 0 more l i k e l y t o judge a p e r c e i v e d s t i m u l u s as a s i g n a l he m i g h t a l s o be m r e l i k e l y t o guess t h a t an unobserved s t i m u l u s i s a s i g n a l . p(Yl&Is)
Hence we assume
= n(Yl8An) = f,(b),
(5)
w i t h f ( b ) d e n o t i n g sow nondecreasing f u n c t i o n o f b . The assumption does 9 n o t p r e c l u d e t h e s p e c i a l case f ( b ) E const., o r even f,(b) 5 0. 9 ~ ( 0 1 s )and p(o1n) a r e o b s e r v a t i o n o r o h a b i l i t i e s , i .e. t h e p r o b a b i l i t i e s o f
~ ( 6 1 =~ )l - p ( o l s )
0 n o t i c i n c r and d e c i d i n g upon t h e s t i m u l u s p r e s e n t e d .
and
p ( 8 l n ) = l - p ( o l ~ ) a r e t h e r e s o e c t i v e complements, i.e. t h e p r o b a b i l i t i e s o f o v e r l o o k i n g t h e s t i m u l u s o r l o s i n g i t b e f o r e i t i s Drocessed
UD
t o the res-
ponse s e l e c t i o n s t a g e .
These a r e t h e p r o b a b i l i t i e s o f i n t e r e s t t o t h e
scholar o f attention.
IJe may t r e a t them as parameters o f t h e t h e o r y o r
d e r i v e them f r o m a s e o a r a t e model, f o l l o w i n g J e r i s o n , P i c k e t t & Stenson
(1967) who argued t h a t d e t e c t i o n t h e o r y s h o u l d be a p p l i e d t o o b s e r v a t i o n p r o b a b i l i t i e s r a t h e r t h a n t o judgement p robabilities. non-inspection,
i,
I n s o e c t i o n , 0, and
o f a s t i m u l u s p r e s e n t e d may be r e g a r d e d as outcomes o f
a d e c i s i o n j u s t as a r e p o r t , Y, and n o n - r e p o r t ,
7.
Hence, we assume, i n
analogy t o eqs. (1) n ( o l s ) = f;(c,a),
p ( o l n ) = f;(c,a)
(6)
w i t h f u n c t i o n s f; and f; n o t n e c e s s a r i l y equal t o , b u t h a v i n g t h e same p r o p e r t i e s as fs and fn ( s e e D e f i n i t i o n s ) , t o d and b.
c and a as parameters e q u i v a l e n t
The parameter c may be i n t e r p r e t e d as a measure of r e l a t i v e
c o n s p i c u i t y o f s o v e r n , i . e . as a measure o f p r e a t t e n t i o n a l d i s c r i m i n a b i lity.
The parameter a i s a b i a s parameter l i k e b.
a t t e n t i o n b i a s towards t h e s t i m u l i p r e s e n t e d
-
However, i t measures
as opnosed t o d e c i s i o n b i a s .
92
H.C. Micko
and f;
B y f;
o b s e r v a t i o n p r o b a b i l i t y i n c r e a s e s as a t t e n t i o n towards t h e s t i -
mulus source i n c r e a s e s and n ( o l s ) exceeds p ( o / n ) p r o a r e s s i v e l v w i t h i n c r e a sing signal consoicuity. I n s e r t i n g eqs. ( 2 ) . ( 5 ) and ( 6 ) i n t o eqs. ( 4 ) we o b t a i n t h e g e n e r a l equat i o n s o f a theory o f c o n d i t i o n a l detection:
*
o ( Y l s ) = f (d,b,c,a)
= f;(c,a)
S
*
p(Ylrl) = fn(d,h,c,a)
. fs(d,b)
t
(1-f;(c,a))
fn(d,b)
t
(1-f,',(c,a))
= fi(c.a)
. f,(b)
-
(7) fg(b).
E n ~ . ( 7 ) t u r n o u t t o r e o r e s e n t a two s t e p d e t e c t i o n t h e o r y supplemented by c o r r e c t i o n f o r guessinq.
The f i r s t s t e o concerns t h e d e c i s i o n w h e t h e r o r
n o t i n s o e c t t h e s t i m u l u s , t h e second s t e p concerns t h e d e c i s i o n o f whether o r not renort i t .
C l a s s i c a l d e t e c t i o n t h e o r y ( e q s . 1) i s a s o e c i a l case
o b t a i n e d by l e t t i n o a
=m
o r e o u i v a l e n t l y ~ ( 0 1 s )= n ( o l n ) = 1 (see eqs. 7a).
C o n d i t i o n a l d e t e c t i o n t h e o r y may b e r e g a r d e d as an i d e n t i f i a b l e and t e s t a b l e f o r m u l a t i o n o f i d e a s advanced b y J e r i s o n , P i c k e r t & Stenson ( 1 9 6 ) who post u l a t e d three states o f a t t e n t i o n " a l e r t " ,
" b l u r r e d " and " d i s t r a c t e d " , each
o f which o u g h t t o be taken a c c o u n t o f by a d e t e c t i o n model o f i t s ow?. T h p i r " d i s t r a c t e d " s t a t e i s about e q u i v a l e n t t o o u r n o n - o b s e r v a t i o n s t a t e
6.
" A l e r t n e s s " vs " b l u r r e d n e s s " i s t r e a t e d h e r e as a c o n t i n u o u s v a r i a b l e
t o be measured b y t h e s e n s i t i v i t y oarameter d. I n subsequent s e c t i o n s we s h a l l e x n r e s s eqs. ( 7 ) a l s o i n t h e l e s s e x o l i c i t fom D ( Y l s ) = f*(d,b,D(ols))
= ~ ( 0 1 s ) * f(d,b)
t
(l-o(ols))
~ ( y ~ n= )f * ( d , b , p ( o ~ n ) )
= o(oIn)
+
(l-p(oIn))
f(d,b)
i n o r d e r t o make arouments more a n o r e h e n s i b l e .
fg(b),
-
(7a) fg(b),
I n eqs. ( 7 a ) t h e observa-
t i o n p r o b a b i l i t i e s a r e t r e a t e d as oarameters which may o r may n o t be f u r t h e r e x p l a i n e d by an a t t e n t i o n model o f t h e t y p e suggested above (eqs. 6 ) .
f;l and f ' have g' t o be s p e c i f i e d i n a d d i t i o n t o fS and fn, w i t h f; = fS, f,', = f b e i n g an n obvious c a n d i d a t e and a h i g h t h r e s h o l d model f; = c + (1-c)a. f ' = a, ann I n o r d e r t o a p p l y t h e t h e o r y t o data, t h r e e f u n c t i o n s , f;,
other.
Two parameters, c and a, have t o be e s t i m a t e d i n a d d i t i o n t o t h e
t r a d i t i o n a l parameters d , b e t c .
T h e r e f o r e , parameter e s t i m a t i o n and good-
ness o f f i t t e s t i n g r e q u i r e t h e e m o i r i c a l d e t e r m i n a t i o n o f s e v e r a l p o i n t s o f t h e ROC-curve.
T h a t may b e awesome b u t s u r e l y i s o f e d u c a t i o n a l value,
93
Attention in detection theory
because i t p r e v e n t s t h e s i m p l e c o m u t a t i o n o f d and b from a s i n g l e p a i r
o f h i t and f a l s e a l a r m r a t e s w i t h o u t a p p l y i n g a goodness o f f i t t e s t t o t h e model emoloyed.
The r e d u c t i o n of parsimony i n comparison w i t h uncondi t i o -
n a l models may be accented i n view o f t h e i n c r e a s e d D r e c i s i o n o f i n t e r p r e t a t i o n and t h e avoidance o f ccnfounding a t t e n t i o n a l e f f e c t s on h i t and f a l s e a l a r m r a t e s w i t h p e r c e n t i o n a l and judqemental e f f e c t s .
Ploreover, n o t much
parsimony can be l o s t , c o n s i d e r i n q t h a t a l m o s t any e m D i r i c a l ROC-curve can be f i t t e d by one o f t h e many u n c o n d i t i o n a l models, (Laming , ( 1973) )
fS,
fn, a1 ready proposed
.
The i n t e r D r e t a t i o n of c o n d i t i o n a l d e t e c t i o n t h e o r y and i t s parameters i s t e s t a b l e too, s i n c e a t t e n t i o n and judgement b i a s can be m a n i p u l a t e d i n d e oenckntly.
Any t e s t , a d m i t t e d l y , r e s t s on t h e assumption t h a t e f f e c t s o f
attention-,
d i s c r i m i n a t i o n - and d e c i s i o n - v a r i a b l e s on h i t and f a l s e a l a r m
r a t e s do n o t i n t e r a c t .
I t may be o r e d i c t e d , e.a.,
t h a t confidence r a t i n q s
r e p r e s e n t d e c i s i o n b i a s and do n o t v a r y w i t h s h i f t s o f a t t e n t i o n .
On t h e
o t h e r hand, d i s t r a c t i o n s o u q h t t o reduce a t t e n t i o n b i a s b u t s h o u l d n o t i n fluence decision c r i t e r i a .
Poth k i n d s o f b i a s s h o u l d a l s o be a f f e c t e d i n -
denendently b y a p p r o p r i a t e m a n i o u l a t i o n s o f t h e p a y - o f f m a t r i x , d e c i s i o n b i a s by v a r y i n g t h e r a t i o o f p a y - o f f s f o r r e p o r t s versus r e j e c t i o n s , a t t e n t i o n b i a s by p r o p o r t i o n a l i n c r e a s e s o f a l l f o u r e n t r i e s o f t h e p a y - o f f matrix.
The s e p a r a t e m a n i o u l a t i o n o f c o n s p i c u i t y and d i s c r i m i n a b i l i t y i s
more d i f f i c u l t . I n a d i s c r i m i n a t i o n paradigm t h e d i f f e r e n c e between t h e two c l a s s e s o f s t i m u l i t o be d i s c r i m i n a t e d may be maniDulated on a v a r i a b l e which i s u n r e l a t e d t o c o n s p i c u i t y .
E.g.,s
and n may be tones o f d i f f e r e n t
frequency b u t equal i n t e n s i t y .
Thus, d v a r i e s w i t h frequency d i f f e r e n c e
whereas c = 0 remains c o n s t a n t .
I n s i g n a l d e t e c t i o n and a t t e n t i o n p a r a -
digms, however, c and d w i l l u s u a l l y be confounded. may be a reasonable a p r i o r i a s s u m t i o n .
I n t h a t case, c = d
I n any case, we have t o assume
c 4 d, because we carlnot e x p e c t p r e a t t e n t i o n a l d i s c r i m i n a t i o n t o be superi o r t o overall discriminability.
I f n o i s e s t i m u l i a r e more conspicuous
than s i g n a l s , c may t a k e n e g a t i v e values, r e s u l t i n a i n ~ ( 0 1 s )
p(o1n).
By assuming c = 0 o r c = d, t h e number o f parameters o f a c o n d i t i o n a l de-
t e c t i o n model may be reduced b y one t o a minimum of t h r e e .
4. MODELS WITH NON-CONSTANT GUESSING FUNCTIONS adopThe f o r m o f an ROC-curve i s determined b y t h e d e t e c t i o n model fS, f, T h a t o f an ROC-curve p r e d i c t e d f r o m a model o f c o n d i t i o n a l d e t e c tion ted.
94
I€. C. Micko
depends h e a v i l y , a l s o , on t h e auessinq f u n c t i o n f o assumed
-
particularly
w i t h r e s n e c t t o those p r o o e r t i e s by which t h e anoGooriateness o f a c o n d i t i e n a l model can be e v a l u a t e d .
The n r o b a b i l i t y p ( Y l 8 A s ) = n(YloAn) o f r e o o r t -
i n ? a s t i m u l u s on t h e b a s i s o f a o u r e guess may o r r a y n o t c o - v a r y w i t h 0 ' s I n t h e former case f ( b ) i s a monotoneously i n c r e a s i n g 9 c o n s t . I n t h i s s e c t i o n we c o n s i d e r t h e
decision hias b.
f u n c t i o n o f b , i n t h e l a t t e r f,(b)former case.
Many, i fn o t mGst d e t e c t i o n models, can be w r i t t e n i n t h e form fs(d,b,m)
= f(b+m-d),
fn(d,b,m)
= f(b-(l-m).d),
Osm
(8)
i .e. fS and fn d i f f e r o n l y b y a t r a n s l a t i o n f r o m each o t h e r on t h e ( a p p r o n r i a t e l y transformed) decision a x i s .
E q u i v a l e n t l y , we may say, fS and fn
d i f f e r by a t r a n s l a t i o n f r o m some r e f e r e n c e f u n c t i o n f r ( b ) which i s s i t u a h t e d somewhere between fS and fn ( e . g . Gaussian d e t e c t i o n t h e o r y : fr=--N(0,l), -b+m*dM(n,l\, -m
f 5-
Ln
=s-:1-
fr = e x o t b ) / l + e x o ( b ) ,
' .'PI -m
( 0 ,1) ; 1o g i s t i c c h o i ce the0 r y :
f5 = ex~(btm.d)/ltexn(htm.dl.
fn = e x r l f b - f I - m ) * d \ / l + e x p l b - ( l - m ) . d ) : exnonential d e t e c t i o n t h r o r y : b b+m* d b - ( l-m)d fr = l / e x n ( k ) , f s = l / e x o ( k ) , fn = l / e x p ( k 1. The d i s t a n c e between fS and fn i s m.d+(l-m).d
= d.
The a d d i t i o n a l Darameter
s ' f n , b u t determines t h e t - e s p c c t i v e d i s tances o f fs and fn f r o m t h e so f a r a r b i t r a r i l y chosen r e f e r e n c e f u n c t i o n
m i s n o t a oarameter o f t h e model f
F o r s i m p l i c i t y o f p r e s e n t a t i o n , we d i s r e g a r d models t h a t p r e d i c t funcfr* tions fS, fn o f d i f f e r e n t s t e e m e s s . The r e a d e r w i l l r e a l i z e , however, t h a t t h e f o l l o w i n g arguments and c o n c l u s i o n s h o l d more o r l e s s f o r a l l models w i t h s i m i l a r and o v e r l a p o i n q f u n c t i o n s fS, fn ( s e e F i p u r e 2 ) .
I f fS and fn d i f f e r b y t r a n s l a t i o n o n l y , i t i s most n a t u r a l t o assume t h a t t h e guessing f u n c t i o n fc, too, d i f f e r s f r o m fS and fn b y t r a n s l a t i o n o n l y , and t h a t i t i s l o c a t e d iomewhere between fs and fn. !.re l e t
f (b) = fr(b) = f ( b ) , (9) 9 t h e r e b y s i v i n g t h e r e f e r e n c e f u n c t i o n f r ( b ) an e m p i r i c a l meaning. Now t h e Darameter m r e p r e s e n t s t h e r e l a t i v e closeness o f f assumed t o e q u a l a p r i o r i s i a n a l D r o b a h i l i t y .
9
t o fS and fn and may he
I n s e r t i n g eqs. (8) and ( 9 ) i n eqs. ( 7 a ) , we o b t a i n t h e g e n e r a l e x p r e s s i o n f o r c o n d i t i o n a l d e t e c t i o n models w i t h f u n c t i o n s fS = fn = f
9
= f e q u a l up
95
Attention in detection theory
t o trans ation. D Y Is) = f:(
d,b ,m,n(o I s ) ) = n(ols).f(b+m.d)t( d ,b ,m,p( 01 n ) ) =
P ( Y r ) = (:f
1-p(o I s ) ) * f ( b ) ,
n( o I n ) f ( b+( 1-m) . d ) t ( l - o ( o I n ) ) - f ( b) .
(10)
We want t o compare these c o n d i t i o n a l models w i t h t h e f o l l o w i n g uncondi t i o n a l d e t e c t i o n models: ~ ( Y l s ) = <(do,b) D(Yln) = f:(do,b)
= flb+pfols).m-d),
(11)
= f(b-p(oln).(l-m)-d).
F o r e v e r y c o n d i t i c n a l model o f t h e f o r m (10) t h e r e e x i s t s an u n c o n d i t i o n a l model o f t h e f o r m (11) w i t h d i s c r i m i n a b i l i t y parameter do = D(ols)-m.d f:
+
p(oln)*(l-m)*d d d (!!)
and f,* a r e o b t a i n e d by computing f o r each argument b t h e w e i g h t e d mean
of f s ( b ) and f ( b ) , r e s p . fq(b) and fn(b), w i t h o b s e r v a t i o n and guessing 9 p r o b a b i l i t i e s as w e i g h t s . and f: a r e o b t a i n e d by t r a n s l a t i o n f r o m f , 9 i . e . more p r e c i s e l y by, t a k i n g f o r each p a i r o f equal values o f t h e funct i o n s fS and f
r e s o . f and fn, t h e w e i g h t e d mean o f t h e i r r e s p e c t i v e 9' q arguments ( s e e F i g u r e 2 ) . Thus b o t h , f:
and f!,
a r e w e i g h t e d averages o f f
and f
f: and f: averag' fn because a v e r a g i n g across
8
and f: # ges o f f and fn. As a r u l e , f: # 9 o r d i n a t e s and across abscissas does n o t y i e l d t h e same f u n c t i o n . f:
as w e l l as f:
and b y f
However,
i s bounded, o v e r t h e whole range o f b, by fS f r o m above
f r o m below, both approach
fS f o r p ( o / s ) + l and f f o r p(o\s)-.O. 9 9 I n view o f t h e l a r g e o v e r l a p o f fS and fn i n u s u a l d e t e c t i o n s i t u a t i o n s and,
we conclude t h a t , fS and f g' By t h e same argument, f,* and fo n are simi-
as a consequence, t h e s t i l l l a r g e r o v e r l a p o f f:
d i f f e r s o n l y a l i t t l e from f!.
l a r functions.
Consequently t h e c o n d i t i o n a l d e t e c t i o n model f:
d i s c r i m i n a b i l i t y parameter d e t e c t i o n model
f,
, f:
with
d d i f f e r s only a l i t t l e from the unconditional
f: w i t h t h e d i s c r i m i n a b i l i t y parameter do ,< d g i v e n i n
Thus, i f f
i s o f t h e same, o r s i m i l a r , form as f S and fn, nong o b s e r v a t i o n combined w i t h guessing has an e f f e c t s i m i l a r t o t h a t o f reduced
eq. ( 1 2 ) .
d i s c r i m i n a b i l i ty.
C o n s i d e r i n g t h e c o m p a r a t i v e l y l a r g e samnling e r r o r s o f
e m p i r i c a l h i t and f a l s e a l a r m r a t e s , we conclude t h a t t h e ROC-curve o f a c o n d i t i o n a l d e t e c t i o n model w i l l be p r a c t i c a l l y i n d i s t i n g u i s h a b l e from t h e ROC-curve o f some c l a s s i c a l d e t e c t i o n model w i t h s m a l l e r d i s c r i m i n a t i o n narameter. -
H.C. Micko
96
Fioure 2 C o n d i t i o n a l d e t e c t i o n t h e o r y w i t h Gaussian d e c i s i o n and guessing f u n c t i o n s . Left:
( S o l i d l i n e s ) Renort o r o b a b i l i t y f o r s i g n a l (f:)
and n o i s e (f:)
as
weighted mean o f the Gaussian d e c i s i o n f u n c t i o n fS reso. fn o f F i aure 1 and a Gaussian guessing f u n c t i o n f o e o u i d i s t a n t from f S and fn* f:(b)
= .67 f s ( b ) t . 3 3 f,(b)y
f i ( b ) = .5 fn(b)+.5
fo(b).
(Broken l i n e s ) Decision f u n c t i o n s of an u n c o n d i t i o n a l Gaussian detec-
(c,
t i o n model
f:)
w i t h d i s c r i m i n a b i l i t y parameter do = 1.16
( = .58-d = .67-.5-d + . 5 - . 5 - d y see eq. ( 1 2 ) ) . Right: ( S o l i d l i n e ) ROC-curve f o r the c o n d i t i o n a l d e t e c t i o n model ( f z , f:). (Broken l i n e ) ROC-curve f o r the u n c o n d i t i o n a l d e t e c t i o n model 0 (fS'
0
fJ.
A c o n d i t i o n a l h i g h t h r e s h o l d model cannot be d i s t i n g u i s h e d even t h e o r e t i c a l l y from an u n c o n d i t i o n a l h i g h t h r e s h o l d model w i t h s m a l l e r d i s c r i m i n a b i l i t y parameter.
Consider e.q. t h e f o l l o w i n g f u n c t i o n s : = d, f
= f = 0, n 9 fS = d+( 1-d) .b ,
f
f o r b i 0,
S
fn = b , f f
? s
} for 0 < b <
= by
= f = f =1, n a
for b
The r e s p e c t i v e c o n d i t i o n a l model w i t h m = 1
2
1.
1,
Attention in detectiun theory
p(Y I S ) = f:(d,b
,p(ols))
= p(0
IS).(
97
d + j l - d ) * b ) + ( l - o ( o Is)).b O < b < l ,
p(Y1n) = f:(d,b,n(oln))
= D ( o l n ) . b + ( l - p ( o l n ) ) - b = b,
i s i d e n t i c a l w i t h t h e u n c o n d i t i o n a l model ~ ( Y l s )= f ~ ( ~ ( o l s ) . d . b ) = ~ ( ~ ] ~ ) * d + ( l - p ( ~ l ~ ) * d ) * b , O < b < l , p(Y I n ) = f i ( o ( o l s ) . d , b ) = b w i t h d i s c r i m i n a b i l i t y parameter n ( o l s ) . d 6 d. O b v i o u s l y t h e r e l a t i v e frequency o f guessing can be a s c e r t a i n e d o n l y i f f d i f f e r s c l e a r l y f r o m fs and fn.
Even so t h e c o n t r i b u t i o n s o f p e r c e p t u a l
g
d e c i s i o n s and guesses t o h i t and f a l s e a l a r m r a t e s may n o t be easy t o sepaConsider e.g.
rate.
sten function f
i.e.
a model t h a t assumes gaussian f u n c t i o n s
fS,
fn and a
9
u n n o t i c e d s t i m u l i w i l l be r e o o r t e d i f and o n l y i f b exceeds
some v a l u e
I n o r d e r t o a v o i d t h e i n t r o d u c t i o n o f an a d d i t i o n a l f r e e parameter, bo. we may r e a s o n a b l y p o s t u l a t e t h a t bo corresponds t o p a y - o f f c o n d i t i o n s which y i e l d a z e r o (or m i n i m a l s a t i s f a c t o r y ) e x p e c t e d q a i n f o r r e p o r t s under a given a p r i o r i s i g n a l p r o b a b i l i t y .
The r e s u l t i n g f u n c t i o n s o f t h e c o n d i t i o -
n a l d e t e c t i o n model
r e p r e s e n t t o t h e l e f t and r i g h t o f bo, o g i v e s which a r e reduced by m u l t i p l i c a t i o n w i t h t h e r e s n e c t i v e i n s p e c t i o n p r o b a b i l i t i e s (see F i g u r e 3).
A t bo
b o t h f u n c t i o n s show a jump upwards o f s t e p s i z e equal t o t h e r e s p e c t i v e non observation p r o b a b i l i t y .
The ROC-curve o f t h a t model has t h r e e d i s t i n c t
The upper and l o w e r end a r e determined by fs and fn, whereas t h e The d i f f e r e n c e middle p a r t i s a s t r a i g h t l i n e o f slope l - p ( o l s ) / l - o ( p ( n ) . parts.
98
H.C. Micko
between t h e p ( Y ( s ) - v a l u e s o f t h e end n o i n t s o f t h e s t r a i n h t p a r t e q u a l s l - n ( o l s ) , the d i f f e r e n c e between t h e i r o ( Y 1 n ) - v a l u e s e q u a l s l - n ( o \ n ) . ( I n
two a d d i t i o n a l ROC-curves a r e shown f o r i l l u s t r a t i v e ourposes.
Fioure 3
They d i f f e r f r o m t h e main one w i t h r c s o e c t t o t1.e o b s e r v a t i o n p r o b a b i l i t i e s ~ ( o l s ) , o(o1n) w h i c h were d e t e r m i n e d f r o m t h e a t t e n t i o n model f;ic,a)
=
a+. 2 2 -m
= a-*22N(0,1) _m
N ( 0 , 1 ) , f,',(c,a!
by v a r v i n g t h e a t t e n t i o n o a r a -
meter a.)
Fioure 3 C o n d i t i o n a l d e t e c t i o n t h e o r y w i t h C'aussian d e c i s i o n f u n c t i o n s and a s t e n f u n c t i o n f o r quessing o r o b a b i l i t y . Left:
R e n o r t n r o b a b i l i t y f o r s i y a l (f:)
of t h e Gaussian d e c i s i o n f u n c t i o n f steo function f
a
f:(b) Right:
r e s o . fn o f F i a u r e 1 and a
b i bo
+ .33,
b
>
bo
.5 fn(b).
b
s
bo
.5 f n ( b ) + .5,
b
> bo
.67 f s ( b )
S
as w e i g h t e d mean
for ouessing p r o b a b i l i t y
.67 f s ( b ) . fr(b) =
and n o i s e (f:)
=
( S o l i d l i n e ) ROC-curve o f t h e c o n d i t i o n a l d e t e c t i o n model (f:,
*
fn).
(Broken l i n e s ) Examples o f ROC-curves o f t h e same model w i t h h i g h e r and l o w e r o b s e r v a t i o n p r o b a b i l i t i e s .
( F o r values o f ~ ( 0 : s )and
a ( o l n ) , see F i g u r e 4 ) . ROC-curves o f c o n d i t i o n a l d e t e c t i o n models w i t h s t e p f u n c t i o n s f resemble t h e ROC-curves o f u n c o n d i t i o n a l two t h r e s h o l d models.
somewhat 9 I f , however,
99
Attention in detection theory
a model w i t h smooth f u n c t i o n s fs, fn can be assumed
-
and models o f t h a t
k i n d s t i l l have most e m n i r i c a l s u o p o r t (Laming (1973), n. 1 5 9 f f )
-
then a
model o f c o n d i t i o n a l d e t e c t i o n i s i d e n t i f i a b l e i n D r i n c i o l e and may be id e n t i f i a b l e i n p r a c t i c e i n s o i t e o f exoerinental e r r o r , provided a s u f f i c i e n t number o f ROC-Points, w e l l d i s t r i b u t e d o v e r t h e whole range o f b , has been determined e m p i r i c a l l y .
I n a subsequent s e c t i o n we c o n s i d e r e x o e r i -
mental c o n d i t i o n s which favor t h e a p p l i c a b i l i t y o f t h e model g i v e n i n eqs.
(13). 5. MODELS VITH CONSTANT GUESSING FUNCTIONS
Here, we c o n s i d e r models w i t h c o n s t a n t p r o b a b i l i t i e s o f r e p o r t i n g u n n o t i c e d s t i m u l i as s i g n a l s .
I-lhen assuming p ( Y l 0 A s ) = p(Yl6hn) = p
eqs. ( 7 a ) t a k e t h e f o r m
9
f ( b ) z const.,
5
9
~ ( ~ 1 s= ) f:(d,b,gfl.p(oIs))
= n(oIs).fs(d,h)
+
(~-O(OIS))*P~,
P ( Y l n ) = fn(d,b,pfl,o(oln))
= n(oln).fn(dsb)
+
(l-o(oln))*pg.
*
(14)
I n view o f t h e r a t h e r r e s t r i c t e d p l a u s i b i l i t y o f such a model w i t h O < p g < 1,
*
e x a m l e s o f t h e f u n c t i o n s f,, f: and ROC-curves w i l l be shown o n l y f o r t h e s o e c i a l case p = 0 ( s e e F i g u r e 4 ) . I n t h e general case t h e f,, fz-ROC-curve 9 i s a reduced v e r s i o n o f t h e fS, fn-curue which b e g i n s a t t h e o o i n t
*
(l-o(oln))*pg, (l-o(ols)).po) (l-n(ols))-(l-pg).
and ends a t ( I - n ( o l n ) ) . ( l - D
9
),
. The v a l i e s o f p ( Y l s ) a r e reduced b y t h e f a c t o r ~ ( 0 1 s ) .
t h e v a l u e s o f p ( Y l n ) b y p(o1n). The most i n t u i t i v e c o n s t a n t D r o b a b i l i t y o f r e o o r t i n g a s i g n a l on t h e b a s i s o f a r b i t r a r y guessing i s
i . e . i t i s assumed t h a t u n n o t i c e d s t i m u l i w i l l n e v e r be r e p o r t e d as s i g n a l s . T h a t assumption i s most l i k e l y t o be m e t by e x p e r i m e n t a l d a t a i f s i g n a l s a r e i n f r e q u e n t such as e.g. i n v i g i l a n c e t a s k s . duce
I n t h a t case eqs (14) r e -
to
o(Y Is) = f:(d,b,c,a)
= p(o(s)*fs(d,b)
= f;(c,a)-fs(d,b),
p ( Y l n ) = fz(d,b,c,a)
= o(oln).fn(d,b)
= f;l(c,a)*fn(d,b),
(15)
*
*
t h e f u n c t i o n fs i n c r e a s e s f r o m z e r o t o ~ ( 0 1 s )and fn from zero t o p ( o l n ) , t h e ROC-curve f r o m 0,O t o p ( o l n ) , ~(01s). I f s i g n a l and n o i s e s t i m u l i a r e
100
H.C. Micko
e q u a l l y consoicuous, i . e . i f c = 0 and c o n s e q u e n t l y n ( o l s ) = n ( o l n ) , t h e RDC-curve ends a t t h e main d i a g o n a l .
Figure 4 Conditional detection theory w i t h o u t guessing. Left:
Reoort o r o h a b i l i ty f o r s i g n a l (f:)
r e s o . n o i s e (f:)
as w e i g h t e d
mean o f t h e Gaussian d e c i s i o n f u n c t i o n s f S r e s n . fn o f F i g u r e 1 and t h e c o n s t a n t guessing f u n c t i o n f ( b ) : 0. 9 = .67 f s ( b ) + O , f i ( b ) = .5 fn(b)+O.
f:(b) Right:
( S o l i d l i n e ) RVC-curve o f t h e c o n d i t i o n a l d e t e c t i o n model (f:, (broken l i n e s )
f:).
Examoles o f RnC-curves o f t h e same model w i t h h i g h e r
and l o w e r o b s e r v a t i o n Drobabi lit i e s . The
lowermost
model (f;,
f;)
s o l i d c u r v e i s t h e ROC-curve o f a Gaussian a t t e n t i o n w i t h c o n s n i c u i t y parameter c = .44 f r o m which t h e
t h r e e p a i r s o f o b s e r v a t i o n o r o b a b i l i t i e s p ( o I s ) , p ( o l n ) have been computed ( ~ ( 0 1 s ) = . 3 4 , .67, .83, p ( o l n ) = .2, I n F i g u r e 4, t h r e e ROC-curves a r e shown.
.5,
.7).
They a r e d e r i v e d f r o m t h e Gaussian
d e c i s i o n model w i t h d i s c r i m i n a t i o n parameter d = 2:
n(Y!ohs) = fs(d,b)
=
b+ 1
--h(O,l),
p ( Y l o h n ) = fn(d,b)
=
b:LN(O,l)l
combined
w i t h t h e Gaussian a t t e n t i o n model w i t h r e l a t i v e c o n s n i c u i t y parameter c = 0.44:
~ ( 0 1 s ) = f;(c,a)
= a + . 2 2 ~ ( ~ , ~o)( o, l n ) = fA(c,a)
They d i f f e r w i t h r e s p e c t t o t h e a t t e n t i o n parameter a.
=
a-.22 -m
N(0,1).
T h e i r endpoints l i e
on t h e ROC-curve which corresponds t o t h e a t t e n t i o n model adonted.
Eqs. ( 1 5 ) have been f i t t e d w i t h some success t o v i g i l a n c e d a t a (Tergan (1972);
101
Attnition in detectioii tlieory
Zimolong & Meyer ( 1 9 7 4 ) ) .
The model i s i d e n t i f i a b l e , because, a g a i n , f
i s c l e a r l y d i f f e r e n t f r o m fS and fn. T h a t would be t r u e a l s o f o r
9
f a ( b ) : oq = 1, an analoguous assumotion a o D l i c a b l e t o c o n d i t i o n s o f h i g h a ' p r i o r i si!nal Eqs. ( 1 4 ) w i t h p
( 1 3 ) w i t h bo =+-,
p r o b a b i l i t y o r v e r y low c o s t o f f a l s e alarms. 9
= 0 and nq = 1 can be regarded as sDeci a1 cases o f eqs.
r e s n . bo =--.
I n f a c t , t h a t model t u r n s o u t t o be t h e
o n l y i d e n t i f i a b l e c o n d i t i o n a l d e t e c t i o n model.
G. CONSEQUENCES FOR DETECTION AND PTTENTION PARADIGtlS The o b j e c t i v e o f d e t e c t i o n t h e o r y i s t h e s e o a r a t i o n and s e o a r a t e measurement o f s e n s i t i v i t y and d e c i s i o n b i a s .
I f imoerfect a t t e n t i o n , too, i s t o
be taken i n t o account and t o be measured, e s t i m a t e s o f a t t e n t i o n - , d i s c r i m i n a t i o n - and d e c i s i o n Darameters can be o b t a i n e d f r o m a model o f c o n d i t i o n -
a l d e t e c t i o n , p r o v i d e d a r b i t r a r y quesses can be d i s t i n g u i s h e d f r o m percept u a l decisions.
I n o r d e r t o make them distinguishable,experimental
and
p a y - o f f c o n d i t i o n s s h o u l d f a v o r , s i n c e t h e y cannot e n f o r c e , a p a r t i c u l a r response i n t h e case o f unobserved s t i m u l i .
A t f i r s t s i a h t an "undecided"-
resoonse a s s o c i a t e d w i t h n e i t h e r n e g a t i v e n o r p o s i t i ve D a y - o f f appears most natural.
I n f a c t , i f t h e c o n f i d e n c e r a t i n g method i s employed and s i g n a l
p r o b a b i l i t y i s .5, t h e u t i l i z a t i o n o f a n e u t r a l c a t e g o r y i s i n d i c a t e d . T h a t c a t e g o r y w i l l be used f o r resnonses based on d e c i s i o n s as w e l l as guesses, b u t o t h e r c a t e g o r i e s a r e l i k e l y t o be chosen o n l y a f t e r i n s p e c t i o n o f t h e s t i m u l u s t o be judged.
The frequency o f n e u t r a l choices a t t r i b u t a b l e
t o D e r c e n t i o n a l d e c i s i o n s can be e s t i m a t e d by d e t e r m i n i n g t h e b e s t f i t t i n g ROC-curve o f an u n c o n d i t i o n a l d e t e c t i o n model f,,
fn, g i v e n t h e response
f r e q u e n c i e s i n t h e n o n - n e u t r a l c a t e g o r i e s and an a r b i t r a r y p o r t i o n o f t h e frequency o f n e u t r a l c h o i c e t o be e s t i m a t e d as a f r e e parameter.
The r e -
m a i n i n g n e u t r a l choices can be a t t r i b u t e d t o DerceDtion indeoendent guessing. T h e i r r e l a t i v e frequency i s an e s t i m a t e o f p ( E l s ) resp. D ( o l n ) . s i g n a l frequency i s l a r g e r o r s m a l l e r than .5,
If a priori
the i d e n t i f i c a t i o n o f a
"guessing"-category w i l l be much more d i f f i c u l t b u t may n o t be i m p o s s i b l e a1 t o g e t h e r .
A n e u t r a l c a t e g o r y cannot be used, however, i n a Yes-No procedure because, i n o r d e r t o t r a c e t h e whole course o f an ROC-curve, s t r o n g n o s i t i v e ( r e s o . n e g a t i v e ) d e c i s i o n b i a s s e s must be induced.
They r e q u i r e p a y - o f f m a t r i c e s
t h a t y i e l d s u f f i c i e n t l y n o s i t i v e exDected p a y - o f f s f o r r e o o r t s , ( r e s p . r e j e c t i o n s ) g i v e n an a o r i o r i s i g n a l D r o b a b i l i t y .
Under such n a y - o f f
102
1I.C. Micko
c o n d i t i o n s , t i s more p r o f i t a b l e t o r e o o r t a s i onal ( r e s o . t o r e j e c t t h e s t i m u l u s ) a t v e n t u r e r a t h e r t h a n t o n i v e a n e u t r a l resoonse o r t o suspense the decision.
Thus, Yes- and No-resnonses based on o e r c e n t i o n indeoendent
quessing cannot b e o r e c l u d e d by p e r m i t t i n g 0 t o a v o i d a n o s i t i v e o r n e g a t i v e decision.
A way o u t o f t h e d i l e m a i s t h e s e p a r a t e e m p i r i c a l d e t e r m i n a t i o n o f t h e lowF o r t h e l o w e r p a r t we mav emoloy pav-
e r and uoper D a r t o f t h e ROC-curve.
o f f m a t r i c e s t h a t p r e s c r i b e q a i n s f o r h i t s , losses f o r f a l s e alarms and zero n a y - o f f s f o r misses and c o r r e c t r e j e c t i o n s .
D e c i s i o n b i a s may be mani-
nu:ated by v a r y i n g t h e g a i n / l o s s r a t i o , s u b j e c t t o t h e r e s t r i c t i o n t h a t exp e c t e d p a y - o f f f o r r e p o r t i n g unobserved s t i m u l i i s ( s u f f i c i e n t l y ) n e g a t i v e . llnder t h a t c o n d i t i o n s 0 ' s can be e x n e c t e d n o t t o r e n o r t a s i f l n a l on t h e b a s i s o f a p u r e Guess.
The RnC-points thus determined w i l l r a r e l y s u f f i c e
f o r f i t t i n g t h e t h e o r e t i c a l ROC-curve a f an eqs. ( 1 5 ) - t v o e o f model.
As a
r u l e t h e e s t i m a t e s o f t h e c o o r d i n a t e s ~ ( 0 1 s ) . p ( o 1 n ) o f t h e uoner end o f t h e ROC-curve w i l l be u n r e l i a b l e t o o ifo n l y e m n i r i c a l n o i n t s o f t h e l o w e r p a r t o f t h e c u r v e a r e a v a i l a b l e f o r t h e f i t t i n g procedure.
T h e r e f o r e we
have t o determine t h e unner D a r t o f t h e ROC-curve by emnioying p a y - o f f mat r i c e s t h a t o r e s c r i b e g a i n s f o r c o r r e c t r e j e c t i o n s , l o s s e s f o r misses and zero n a y - o f f s f o r h i t s and f a l s e alarms. s u b j e c t t o t h e r e s t r i c t i o n t h a t exp e c t e d D a y - o f f f o r a s t i m u l u s r e j e c t i o n i s n e g a t i v e under t h e g i v e n a p r i o r i signal probability.
Under t h e s e Day-off
c o n d i t i o n s o u r e guesses can be ex-
A c o m b i n a t i o n o f t h e two nrocedures
o e c t e d t o r e s u l t i n Yes-resnonses.
y i e l d s the upoermost and lowermost branch o f
a ROC-curve o f t h e t y p e shown
i n Figure 3. By f i t t i n g a model o f t h e t y n e g i v e n i n eqs. ( 1 3 ) w i t h b > bo t o t h e upDer,
and b
t o t c t h e l o w e r s e t o f d a t a n o i n t s we o b t a i n t h e wanted e s t i m a t e s
o f t h e a t t e n t i o n and d e t e c t i o n parameters.
Here again. t h e d e c i s i v e a s p e c t
o f t h e nrocedure i s t h e c o n c e n t r a t i o n of guessing resnonses on one response class, e i t h e r reports o r rejections.
If guesses r e s u l t i n Yes- as w e l l as
No-responses, parameter i d e n t i f i c a t i o n w i l l he v i r t u a l l y i m p o s s i b l e . REFERENCES 111
Broadbent, D.E.,
121
Green, D.M.,
D e c i s i o n and s t r e s s , London, Academic Press (1971).
and Swets, J.A.,
New York, bli l e y ( 1966)
.
S i g n a l d e t e c t i o n t h e o r y and Dsychophysics,
103
Attention in detection theory
[31
J e r i s o n , H.J.,
P i c k e t t , R.M.,
and Stenson, H.H.,
The e l i c i t e d obser-
v i n g r a t e and d e c i s i o n nrocesses i n v i g i l a n c e , Human F a c t o r s 7 ( l S f S ) ,
107- 128.
[41 [51
Laming, D.,
Mathematical Psychology, London, Academic Press (1973).
Luce, R.D..
D e t e c t i o n and r e c o g n i t i o n , i n Luce, R.D.,
Galanter, E . , Ililey
[61
Handbook o f mathematical psycholooy,
Bush, R.R.,
and
v o l , l . , New York,
(1963).
Swets, J.A.,
and K r i s t o f f e r s o n , A.B.,
A t t e n t i o n , Annual Review o f Psy-
chology, 2 1 (1070). 333-366.
171
Tergan, S.O.,
V i g i l a n z und S i g n a l e n t d e c k u n g s t h e o r i e ( V i g i l a n c e and
s i g n a l detection theory)
, UnDublished
diploma t h e s i s , Marburq U n i v e r s i -
ty, r i r m a n y (1972).
[81
Plykoff, L.B., ing,
The r o l e o f o b s e r v i n g responses i n d i s c r i m i n a t i o n l e a r n -
i n Luce, R.D.,
Bush, R.R.,
and G a l a n t e r , E . , Readings i n mathe-
m a t i c a l psycholoay, v o l . 1, New York, b l i l e y (1963).
[91
Zimolong,
B.
, and
Meyer, F.,
Die Anwendung d e r Siqnalentdeckungstheo-
r i e a u f V i g i l a n z d a t e n (The a m l i c a t i o n o f s i a n a l d e t e c t i o n t h e o r y t o v i g i l a n c e d a t a ) , B e r i c h t u b e r den 28. KongreB d e r Deutschen G e s e l l s c h a f t fiir P s y c h o l o g i e i n S a a r b r k k e n , 1972, G'dttingen, Hogrefe (1974).
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TRENDS IN MATHEMATICAL PSYCHOLOGY E . Degreefond J . Van Bufenkaut (editors) 0 Elsevier Science Publis err B . V. (North-Holland), 1984
IMPOSSIBLE OBJECTS AND INCONSISTENT INTERPRETATIONS
E. Terouanne Uni v e r s i t e Paul V a l e r y t n e l 1 i e r , France
Since Penrose t r i a n g l e and Escher drawinrls, a v a r i e t y o f " i m n o s s i b l e o b j e c t s " o r " i m p o s s i b l e f i g u r e s " annear e d , as w e l l i n p r a n b i c s r t s as i n v i s u a l p e r c e p t i c n studies.
Several a t t e m o t s have been made t o modelize
such s i t u a t i o n s .
The n r e s e n t paaer groposes a new
a n a l y s i s i n terms o f a mathematical r o d e l o f t h e cons i s t e n c y o f an i n t e r a r e t a t i o n .
The nremise f o r i t i s
t h a t t h e i m n o s s i b i l i t y , i f any, i s n o t t o be searched i n t h e drawing i t s e l f , b u t r a t h e r i n t h e i n t e r a r e t a t i o n t h a t an o b s e r v e r Cives o f i t .
1. INTRODUCTION The concent " i m n o s s i b l e o b j e c t " was i n t r o d u c e d by L.S.
and R. Penrose as
"a s p e c i a l t y p e o f i l l u s i o n " , w i t h r e f e r e n c e t o drawings o f M.C.
Escher.
E v e r s i n c e , numerous examnles o f "imDossible o b j e c t s " have been p u b l i s h e d , and s e v e r a l a u t h o r s have made v a r i o u s a t t e m p t s t o analyse and modelize t h i s conceDt (Cowan, Draner, Gregory, Huffman, Suaihara, Terouanne, among o t h e r s ) . The common c h a r a c t e r i s t i c o f these s o - c a l l e d " i m c o s s i b l e o b j e c t s " i s t o be two-dimensional drawinps conveying t h e i m p r e s s i o n o f q u i t e f a m i l i a r t h r e e dimensional o b j e c t s a t a f i r s t glance, and g i v i n g t h e observer, p o s s i b l y a f t e r a delay, a s t r o n g f e e l i n g o f " i m n o s s i b i l i t y " o r " f a l s e n e s s " .
I n fact,
i t l o o k s d i f f i c u l t t o D r o v i d e any more f o r m a l d e f i n i t i o n o f " i m o o s s i b l e
o b j e c t s " o r " i m n o s s i b l e f i a u r e s " (how c o u l d one d e f i n e something which c o u l d not exist?
.., ) .
The f i r s t aim o f t h e D r e s e n t Daper i s t o c l a i m t h a t , i f one i n t e n d s t o formal i z e such a s i t u a t i o n f o r t h e sake o f s c i e n t i f i c a n a l y s i s , i t i s b e t t e r t o r e n l a c e t h e concepts " i m o o s s i b l e something" b y t h a t o f " i n c o n s i s t e n t i n t e r Dretation" o f a figure.
Iblhat we rear: here by "an i n t e r n r e t a t i o n o f a f i o u r e " i s any s e t o f assert i o n s about the elements o f the f i ? u r e , d e a l i n o w i t h : (i)
the surfaces o f the v i r t u a l o b j e c t which would be represented by such
and such an area i n tlie f i o u r e , and t h e i r q u a l i t y ( n l a n a r , o r convex, o r concave,.
..) ,
( i i ) the i n t e r s e c t i o n l i n e s o r m i n t s between these surfaces which would be represented by such and such a l i n e o r p o i n t i n the f i p u r e , and t h e i r concavi ty o r convexity. ( i i i ) t h e occlusions, i . e . the l i n e s i n the f i g u r e along which one surface o f the v i r t u a l o b j e c t would h i d e another one, ( i v ) and o o s s i b l y a d d i t i o n a l n r o o e r t i e s such as n a r a l l e l i s m between some faces, o r c o l l i n e a r i t y between edges, and s o on
... is
I n these t e r n , the s i t u a t i o n u s u a l l y described as "imoossihle o b j e c t "
t h a t o f an observer who, havina s t a t e d h i s own i n t e r n r e t a t i o n o f a f i g u r e , detects a c o n t r a d i c t i o n between some o f h i s a s s e r t i o n s o r some o f t h e i r consequences.
Then w s h s l l soeak o f " i n c o n s i s t e n t i n t e r p r e t a t i o n " o f the f i -
nllre.
Thus the oremise f o r the Present work i s t h a t the " i m n o s s i b i l i t y " ,
i f any,
i s somethinp oerceived by the s u b j e c t , and n o t an i n t r i n s i c Drooerty o f the fiwre.
The f o l l o w i n g w i l l e x a m n l i f y t h i s main n o i n t .
F i r s t , i t i s w e l l k n w n , f o r i n s t a n c e since Necker's cube ( f i p . 1 - a ) , twcj
that.
observers o f a f i g u r e , o r the same observer a t two d i f f e r e n t times, can
give d i f f e r e n t i n t e r o r e t a t i o n s o f i t .
Another examnle i s f i g . 1-b which
can be i n t e r p r e t e d e i t h e r as a h o l l o w cube o r as a square-based t r u n c a t e d c y r a r i d b u i l t on the t o p r f a cube (unoer views). Secondly, l e t us consider a v a r i a n t of Penrose's t r i a n g l e ( f i g . 1 - c ) .
The
imnression o f impossibi1it.y occurs when i n t e r p r e t i n g i t as the image o f an o b j e c t made o f three o a r a l l e l e p i n e d i c bars , i o i n i n a by n a i r s ( t h a t i s what we c a l l a " m u l t i b a r ob.ject").
Nevertheless, a l t e r n a t i v e i n t e r p r e t a t i o n s ex-
i s t which are r e a l i z a b l e as three-dimensional o b j e c t s (Gregory, Terouanne) ,
Moreover, some sub,jects do n o t f e e l any impression o f i m p o s s i b i l i t y w i t h t h i s fiaure,
s i n c e they sDontaneously i n t e r n r e t i t as an imaqe o f a puzzle,
o r of an assembly o f paner s t r i n s , and they can a c t u a l l y b u i l d such o b j e c t s . F i n a l l y , a geometrical argument (developed below) allows us t o orove t h a t f i g . 1-d i s a c t u a l l y the image o f a m u l t i b a r ob.ject, when most observers f i n d
Impossible objects and inconsistent interpretations
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107
108
E. Teboudnne
SGr-ethinq " i m p o s s i b l e " . o r a t l e a s t "odd" i n i t . On t h e o t h e r hand, a s i m n l e o e o m e t r i c a l argument Droves t h a t f i g . I - e cannot he t h e imaoe o f a m u l t i h a r o b j e c t ( i t s t h r e e e x t e r n a l f o l d s s h o u l d meet), when most o h s e r v e r s - ' v t e r n r e t i t as such.
Here aoain, an a l t e r n a t i v e , and g e o m e t r i c a l l y c o n s i s t e n t , i n -
t e r n r e t a t i o n o f t h e same f i o u r e 1-e can be given, f o r i n s t a n c e , by assuminn t h a t the l e f t c o r n e r o f t h e v i r t u a l t r i h a r has been t r u n c a t e d . !I'e w i l l now d e v e l o n a mathematical node1 f o r t h e i n t e r p r e t a t i o n s o f a " p o l y hec'ral ficrure".
'.'hen s n e a k i n g o f a " a o l y h e d r a l f i g u r e " , we assume t h a t t h e
f i v u r e i s s x c l u s i v e l v v a d e of s t r a i c l h t l i n e s and t h a t t h e o h s e r v e r i n t e r n r e t s i t as t h e image o f a n o l y h e d r a l scene, t h a t i s a scene e x c l u s i v e l y made o f
obiects w i t h nlanar faces.
Althouoh i t does n o t t a k e i n t o account a l l t h e
s i t u a t i o n s usually described as"ivnossib1e o b j e c t s " , the f a m i l y o f polyhed r a l f i c u r e s c o n t a i n s most o f t h e known examoles o f them.
2 . POCYHFDRAL SCENF Consider a o o l y h e d r a l scene and an o h s e r v e r o f i t : f o r i n s t a n c e i n f i g . 2-a t h e scene i s composed o f a t r i a n g u l a r o y r a v i d ( w i t h faces P , R , C ) t h e qround 7.
b u i l t on
Fach face o f t h e scene i s s u p n o r t e d by a n l a n e o f t h e space.
L e t F be the s e t o f these n l a n e s ( i n o u r example F =
(A,R,C,DI).
Each v i -
s u a l r a y o f t h e o b s e r v e r i s r e p r e s e n t e d b y an o r i e n t e d h a l f - l i n e i s s u i n g f r o m a q i v e n o o i n t O and g o i n q through t h e scene, much l i k e X-rays.
Then
w i t h each v i s u a l r a y R WP can a s s o c i a t e a b i n a r y r e l a t i o n upon t h e s e t F, e x o r e s s i n g t h e o r d e r i n which t h e elements o f F a r e met b y R.
ii o u r example,
R
For instance
meets A and 8 t o o e t h e r a t a n o i n t o f t h e i r i n t e r s e c t i o n ,
then i t crosses t h e n v r a m i d and meets t h e t h i r d s i d e C b e f o r e t h e ground we say t h a t A=B > C > D a l o n g R .
On t h e o t h e r hand, t h e r a v S crosses t h e
same Dlanes i n t h e o r d e r : A > D > R > C .
r:
Impossible objects and inconsistent inferpretations
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Figure 2
109
110
E. T&nmine
3 . POLYHEDRAL FIGURE
L e t us now 90 back t o a n o l v h e d r a l f i o u r e and an o b s e r v e r o f i t . The obs e r v e r i s no l o n g e r r e o r e s e n t e d i n t h e f i q u r e s i n c e he i s o u t o f i t s s u o n o r t the nresent reader f o r f i a . 2-b.
and i s , t y p i c a l l y ,
Each o o i n t i n t h e sun-
n o r t o f t h e f i q u r e r e n r e s e n t s one o f t h e o b s e r v e r ' s v i s i i a l r a y s towards t h e fioure,
and hence one o f h i s v i s u a l r a y s across t h e snace o f t h e v i r t u a l
scene.
I4henever t h e o h s e r v e r i n t e r q r e t s a l i n e o f t h e f i g u r e as t h e i n t e r -
s e c t i o n o f two f a c e s , he s n l i t s t h e s u o o o r t o f t h e f i n u r e i n t o t h r e e p a r t s a c c o r d i n g t o t h e o r d e r i n w h i c h t h e c o r r e s p o n d i n o r z y s k o u l d c u t t h e planes
For i n s t a n c e , i f y o u i n t e r n r e t t h e l i n e P i n f i p .
s u o v o r t i n q these f a c e s .
2 - 0 as t h e s u o o o r t o f t h e convex i n t e r s e c t i o n o f faces A and B, y o u mean t h a t a t every o o i n t o f
r,
A = 9 h o l d s , whence a t
any p o i n t on t h e r i g h t
s i d e o f f', A < R F o l d s and a t any o o i n t on t h e l e f t s i d e o f
P,
A > B holds.
S i i v i l a r l y , whenever tke o b s e r v e r i n t e r n r e t s a seflment, say s , as an o c c l u s i o n o f a f a c e , sav R , o v e r a n o t h e r face, 0 i n o u r examole, he assumes t h a t a t e v e r y o o i n t o f s, B >
D 'lolds.
4 . INTFRPRETATION OF A POLYHEDRAL FIGURE L e t C be t h e s e t
Jf
segments, h a l f - l i n e s o r l i n e s which cornnose a g i v e n
DO-
l y h e d r a l f i g u r e , and P t h e s e t o f n o l v g o n a l areas t h e y determine i n t h e suno o r t o f the f i g u r e . DEFINITION I
An i n t e r m e t a t i o n o f a n o l v h e d r a l f i q u r e (S,L) c o n s i s t s o f : (i)
the s e t F o f " f a c e s " i n t h e i n t e r o r e t a t i o n .
a oart o f P.
Each f a c e i s r e l a t e d t o
I f t h i s n a r t i s ernnty, t h e f a c e i s a " h i d d e n " f a c e .
( i i ) t h e s e t IE o f "edges" i n t h e i n t e r o r e t a t i o n . one o r s e v e r a l c o l i n e a r serlments o r l i n e s .
Each edge i s r e l a t e d t o
Each o f these segments o r l i n e s
e i t h e r belongs t o S o r i s added by t h e o b s e r v e r w i t h d o t s and i s t h e n c a l l e d a " h i Aden edge". ( i i i ) t h e s e t A c F x E d o f " a d j a c e n c i e s " i n t h e i n t e m r e t a t i o n . (A,e,B) F A means t h a t e i s i n t e r o r e t e d as a common edge between A and B and t h a t t h e o b s e r v e r has d e c i d e d ( f o r i n s t a n c e b y c o n s i d e r i n g c o n v e x i t y o r c o n c a v i t y )
on which s i d e o f e , A
>
B i s valid.
( i v ) the s e t m x s b o f "occlusions" i n the i n t e r o r e t a t i o n . (A,s,B) means t h a t s belongs t o an edge o f A a l o n g which A occludes B .
E 0
111
Impossible objects and nlconsistent interpretations
U[T t r a n s l a t i n g p o s s i b l e assump( v ) a s e t o f e q u i v a l e n c e r e l a t i o n s on tions o f parallelism, colinearity o r coolanarity.
DEFINITION 2 The "adjacency granh" o f an i n t e r n r e t a t i o n i s t h e graoh i n which v e r t i c e s a r e elements o f F , t h e edges a r e elements o f v e r t i c e s A and B i f and o n l y i f (A,e;R)
IE,
w i t h t h e edge e j o i n i n g t h e
€A.
DEFINITION 3 The " o r d e r i n g araph" o f an i n t e r n r e t a t i o n a t a n o i n t M o f t h e f i g u r e i s t h e o r i e n t e d graoh o b t a i n e d i n t h e . f o l l o w i n g way f r o m t h e adjacency graph:
( i ) i f (A,e,B) E /Pa and M i s on t h e l i n e s w p o r t i n g e, then t h e edge e i s removed f r o m t h e granh and i t s v e r t i c e s a r e merged. ( i i ) i f (A,e,B)
E RI and !Ii s n o t on t h e l i n e s u o n o r t i n g e, then t h e edge e
B ( o r R towards A) i f t h e i n t e r n r e t a t i o n o f
i s d i r e c t e d f r o m A towards (A,e,B)
suDposes t h a t A
( i i i ) i f (A,s,B)
E
>
F
( o r D < A) a t ! I .
0 and M i s on s , then an o r i e n t e d edge i s added f r o m A
towards R .
5. LOCAL CONSISTENCY OF AN INTERPRETATION DEFINITION 4
A n o i n t i n t h e s u p p o r t of a n o l y h e d r a l f i g u r e i s a " c r i t i c a l o o i n t " f o r a o i v e n i n t e r p r e t a t i o n o f t h i s f i g u r e i f and o n l y i f t h e o r d e r i n g graph o f t h e i n t e r n r e t a t i o i a t t h i s n o i n t contains a c i r c u i t , DEFINITION 5 An i n t e r n r e t a t i o n o f a n o l y h e d r a l f i g u r e i s " l o c a l l y c o n s i s t e n t " i f and o n l y i f i t allows no c r i t i c a l n o i n t . F o r i n s t a n c e , f i g . 3-b i s t h e adjacency graoh o f an i n t e r n r e t a t i o n o f f i g . 3-a which i n v o l v e s o n l y t h r e e v i s i b l e faces A, B and C, and t h e i r p a i r e d convex a d j a c e n c i e s a, b and c.
T h i s i n t e r p r e t a t i o n Droves t o be l o c a l l y
c o n s i s t e n t : as a m a t t e r o f f a c t , t h e l i n e s s u o p o r t i n g a, b and c s p l i t t h e s u p o o r t o f t h e f i g u r e i n t o s i x areas, s i x h a l f - l i n e s and one o o i n t , and t h e o r d e r i n g graphs i n each o f these t h i r t e e n r e g i o n s c o n t a i n n o c i r c u i t : f o r instance, f i g u r e
3-c ( o r 3-d) i s t h e o r d e r i n g aranh a t p o i n t
M ( o r N) b u t
a l s o a t any o t h e r p o i n t o f t h e same area d e l i n e a t e d b y t h e l i n e s s u p p o r t i n g
112
I
Q
I I
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Impossible objects and inconsistent interpretations
a, b and c . The n o l y h e d r a l f i g u r e i n f i p . 3-e i s t h e same as i n f i n . 3-a, b u t we now c o n s i d e r i t s i n t e r n r e t a t i o n i n terms of a t r u n c a t e d pyramid: two hidden f a ces D and E and a hidden edae between them aopear. i n v o l v e s f i v e faces and n i n e edges.
The adjacency graph
POW
The p a i r e d i n t e r s e c t i o n s b y d and e
between faces B y C and D a r e n o t c o n c u r r e n t , t h u s t h e i n t e r n r e t a t i o n leads t o a contradiction.
As a m a t t e r o f f a c t ,
t h e o r d e r i n n granh a t p o i n t P con-
t a i n s t h e c i r c u i t R>C>D>B ( f i o . 3 - f ) and P i s a c r i t i c a l p o i n t f o r t h i s i n ternretation. F i n a l l y , f i g . 3-a pronoses a t h i r d i n t e r p r e t a t i o n o f t h e same p o l y h e d r a l f i g u r e , w i t h an a d d i t i o n a l t r u n c a t u r e o f t h e v i r t u a l nyramid a l o n g a hidden face
F.
And t h i s l a t t e r i n t e r p r e t a t i o n i s l o c a l l v c o n s i s t e n t .
I n o r d e r C.c check t h e l o c a l c o n s i s t e n c v o f an i n t e r n r e t a t i o n , i t i s n o t ne-1 cessarv t o compute t h e o r d e r i n n qraoh a t e v e r v D o i n t o f t h e s u p o o r t , n o r a t e v e r y element of t h e d i s s e c t i o n o f t h e s u m o r t by t h e l i n e s s u D p o r t i n a Consider f o r examnle t h e Dentabar o f f i g .
t h e edges o f t h e i n t e r p r e t a t i o n . 4-a,
and an i n t e r p r e t a t i o n of i t r e s t r i c t e d t o i t s f i v e v i s i b l e faces and
t h e i r n a i r e d i n t e r s e c t i o n s alonrl t h e c e n t r a l edces o f each o f t h e f i v e "bars".
The adjacency aranh c o n t a i n s b u t one c y c l e , which c o u l d D r o v i d e a t
most two o p p o s i t e c i r c u i t s .
The l i n e s a t o e have been o r i e n t e d f o r t h e con-
venience o f t h e demonstration: i n o r d e r t o make t h e c i r c u i t o f f i a . 4-b f r o m t h e adjacency graph, we must s i t u a t e o u r s e l v e s i n the h a l f - g l a n e t o t h e l e f t o f a, because a i s convex and we want A
S i m i l a r l y , we must s i t u a t e o u r -
s e l v e s t o t h e l e f t o f b y t o t h e r i o h t o f c, t o t h e l e f t o f d and t o t h e r i n h t o f e.
T h i s i n t e r s e c t i o n i s empty.
half-olanes.
So i s t h e i n t e r s e c t i o n o f t h e oDposin9
T h i s i n t e r n r e t a t i o n i s thus l o c a l l y c o n s i s t e n t .
The same n r o -
cedure a n o l i e s t o any i n t e r p r e t a t i o n a f t e r h a v i n a l i s t e d a l l t h e p o s s i b l e c i r c u i t s , and i n p a r t i c u l a r a l l t h e c y c l e s i n t h e adjacency graoh. B u t t h e l o c a l c o n s i s t e n c y o f an i n t e r p r e t a t i o n i s n o t a s u f f i c i e n t c o n d i t i o n f o r i t s f e a s i b i l i t y : f o r i n s t a n c e , t h e Penrose " i m p o s s i b l e s t a i r c a s e " o f f i a . 5-a, o r t h e Huffman " i n t r i c a t e d D a r a l l e l e D i o e d s " o f f i g . 5-b a r e l o c a l l y c o n s i s t e n t , when t h e i r "common sense" i n t e r p r e t a t i o n s c l e a r l y c o n t a i n con-
t r a d i c t i o n s and a r e i n f e a s i b l e .
On t h e o t h e r hand, t h e " i n t r i c a t e d card-
boards" i n t e r p r e t a t i o n o f f i q . 5-c l o o k s f a i r a t a f i r s t glance, b u t t h e " t r i r a t i o r u l e " i n t r o d u c e d below w i l l nrove i t s i n f e a s i b i l i t y .
I n order t o
deal w i t h t h i s more e v o l v e d n o t i o n o f c o n s i s t e n c y , we need some new t o o l s .
114
E . Tirouanne
a
b
Fioure 4
115
Impossible objects and inconsistent ititerpretatiuns
6 . J O I N OPERATIONS Consider t h e f o l l o w i n o o n e r a t i o n s d e f i n e d i n the E u c l i d e a n Dlane: i f M and N a r e two a r b i t r a r y p o i n t s i n t h e n l a n e , t h e " j o i n " FIN o f I' and f4 i s t h e
oaen segment between F7 and N, and t h e " e x t e n s i o n " o f
from N i s the h a l f -
l i n e f l / N i s s u i n g f r o m 11 and e x t e n d i n g t h e segment MN.
These o o e r a t i o n s a r e
d e f i n e d f o r t h e p a r t s o f t h e a l a n e b y union.
They f u r n i s h i t w i t h a s t r u c -
t u r e o f " j o i n geometrv" o r " j o i n snace", a f t e r P r e n o w i t z .
Now, l e t A and B be two faces o f an i n t e r p r e t a t i o n o f a a n o l y h e d r a l f i g u r e , and FI and N be two n o i n t s i n t h e s u a o o r t o f t h e f i q u r e . I n o r d e r f o r t h e i n t e r n r e t a t i o n t o be f e a s i b l e , t h e o r d e r i n g s between A and R a t
M, a t N and
a t some o t h e r p o i n t s o f t h e l i n e determined by F1 and N s h o u l d f u l f i l c e r t a i n c o n s i s t e n c y r e l a t i o n s : f o r examole, i f A>B h o l d s a t b o t h M and N one s h o u l d n o t f i n d any p o i n t a t which A
S a t e v e r v p o i n t o f W N .
O r i f A>B a t M and And so on.
A l l these
r e l a t i o n s a r e summarized i n t h e f o l l o w i n g t a b l e s , where t h e empty squares corresnond t o t h e cases i n which t h e o r d e r i n g s a t 11 and N do n o t s u f f i c e t o determine t h e o r d e r i n q a t t h e p o i n t considered.
Tables o f t h e o r d e r i n ? between two faces A and B a t an a r b i t r a r y p o i n t o f PIN ( l e f t ) o r F?/N ( r i g h t ) as a f u n c t i o n o f t h e i r o r d e r i n g s a t M and N.
7. TRANSITIVITY Another f e a t u r e o f o r d e r i n g r e l a t i o n s which i s c l e a r l y d e s i r a b l e i n o r d e r t o ensure t h e c o n s i s t e n c y o f an i n t e r o r e t a t i o n i s t h e t r a n s i t i v i t y : whenever A>B and B>C h o l d , one s h o u l d a l s o have A>C.
8. COMPREHENSIVE CONSISTENCY Now we s h a l l complete e v e r y l o c a l o r d e r i n g r e l a t i o n by t r a n s i t i v i t y and by a p D l y i n g t h e c o n s i s t e n c y r u l e s a t t a c h e d t o t h e j o i n and e x t e n s i o n o p e r a t i o n s .
I16
DEFINITInPI 6 The "comnrehensive o r d e r i n q r e l a t i o n s " o f an i n t e r n r e t a t i o n o f a o o l y h e d r a l f i q u r e a t e v e r y p o i n t o f i t s s u o o o r t a r e o b t a i n e d from t h e l o c a l o r d e r i n a r e l a t i o n s a t each p o i n t hy a l t e r n a t e l y a p p l y i n o t h e two f o l l o w i n g procedures: ( i l t r a n s i t i v i t y : f o r e v e r y n o i n t '1 and e v e r y I\.B,C and R z C a t M, we add A>C a t 11.
S such t h a t A>B a t PI
(ii,c l n s u r e o z t h e o r w r a t i o n s o f j o i n and e x t e n s i o n : f o r any A , R S ,
l e t us
denote t h e s e t I A > R , A = R , A 4 1 o f t h e o o s s i b l e o r d e r i n q s between A and B a t F n o i n t as n(A,B).
fiqure,
Then f o r e v e r y o a i r M, N o f p o i n t s i n t h e s u p p o r t o f t h e
i n which t h e i n t e r s e c t i o n s o f Q(A,B) w i t h t h e c u r r e n t o r d e r i n g
oranhs a t M and N a r e non emnty, we determine, a c c o r d i n g t o t h e t a b l e s , which e l e m e n t o f O(A,Q),
i f anv, s h o u l d be added t o t h e o r d e r i n g r e l a t i o n s
a t e v e r y p o i n t o f IlN, V/PI and N / U . Because a l l t h e s e t s and r e l a t i o n s i n use h e r e a r e f i n i t e ones, we can p r o v e t h a t t h e above orocedure n e c e s s a r i l y s t a b j l i 7 e s ? t t h e epd o f a f i n i t e numb e r (oresumably s m a l l ) o f r e n e t i t i o n s . NFINITIOPI 7 The n o t i o n s o f "comnrehensive c r i t i c a l p o i n t " o r "comorehensive c o n s i s t e n c y " o f an i n t e r o r e t a t i o n f o l l o w i m m e d i a t e l y . p o i n t " and " c o n s i s t e n c y " ,
We s h a l l e v e n t u a l l y use " c r i t i c a l
f o r s h o r t , and s i m p l y w r i t e "A>B a t M" f o r "A>B
belongs t o t h e comprehensive o r d e r i n g r e l a t i o n a t M". Here i s f o r i n s t a n c e t h e d e m o n s t r a t i o n of t h e i n c o n s i s t e n c y o f t h e i n t e r o r e t a t i o n o f f i g . 5-h i n terms o f t h e i m b r i c a t i o n o f two a r c h e d D a r a l l e l e n i p e d s : (A,a,R)
i s i n t e r o r e t e d as a convex adjacency and (B,KL,C)
I t f o l l o w s t h a t A>R>C a t t i . o f MP.
S i m i l a r l y A>E>C a t
B u t PI E MP and C>D>A a t N.
as an o c c l u s i o n .
P. Hence, A>C a t e v e r y p o i n t
Thus A>C>A a t N and N i s a c r i t i c a l n o i n t
The " s t a i r c a s e " i n t e r n r e t a t i o n o f f i o . 5-a assumes t h a t a l l t h e " h o r i z o n t a l " D a r t s o f t h e " s t a i r s " , l i k e X,Y
and 2, a r e p a r a l l e l .
B u t whenever t v o F a r a l -
l e l faces v e r i f y X > Y a t a q i v e n n o i n t , t h e j o i n r u l e s imply t h a t X>Y h o l d s a t e v e r y o o i n t o f t h e s u n o o r t o f t h e f i g u r e ( f o r i n s t a n c e because we assume that
X=V
holds a t
some i n f i n i t e l y remote p o i n t s ) .
i n c o n s i s t e n c y f o l 1ows immedi a t e l y
.
The d e m o n s t r a t i o n o f t h e
hipossible objects and inconsistent interpretations
b
C
117
On tbe o t h e r hand, when a n i n t e r v r e t a t i o n of a nolyhedral f i g u r e i s consis-
t e n t , i t often allows the observer t o construct the point o r l i n e i n t e r s e c tions of some t r i p l e s o r n a i r s of nlanes even thoucrh they h a d n o common m i n t i n the f i g u r e i t s e l f . For i n s t a n c e , a dotted l i n e i n f i s . 3-e shows t b p i n t p r s e c t i o n l i n e between faces C a n d D . And n o i n t p renresents t h e i n t e r s e c t i o n between B , C a n d D. PEFIhITInY E An i n t e r p r e t a t i o n i s "comnletable" when i t i s c o n s i s t e n t a n d the comorehen-
s i v e ordering procedure oermits the construction o f the l i n e ( o r p o i n t ) int e r s e c t i o n of any p a i r ( o r t r i p l e ) of i t s f a c e s . 9 . PARALLEL OR CONCURRENT INTEYECTIONS
The comoletahility of an i n t e r p r e t a t i o n i s not a s u f f i c i e n t condition f o r i t s f e a s i b i l i t y i n the snecial case, i l l u s t r a t e d in f i n . 5 - c , where a l l the naired i n t e r s e c t i o n s hetween i t s faces a r e concurrent o r n a r a l l e l . Then we w i l l need an additional nronerty, r e l a t e d t o the notion of " t r i r a t i o " introduced by the fcllowinq r e s u l t . TtIFnREI' 1 I f ( D i ) (1:i:6)
i s a family of concurrent o r o a r a l l e l l i n e s i n a nlane, the
quantity
[D1 ,I!2 , D 3 ,@ 4 ,I!5 ,@ 61 i s the same f o r every familv ( A i ) c f
m 6 -% -F .m%Tm c o l i n e a r point; i n which
=
*
AiEDi
(l~i~6).
DEFINITInb4 9
For any family ( D i ) ( l $ i < 6 ) of p a r a l l e l o r concurrent l i n e s i n a plane, we s h a l l c a l l the " t r i r a t i o " of the (Di), the q u a n t i t y ID1,D2,D3,D4,D5,D61 defined by theorem 1. THEORE!!
2
Let ( D i ) ( l ~ i s 6 )be a family of p a r a l l e l or concurrent l i n e s in the plane. TI-cn the Di are the orojections of the p a i r e d i n t e r s e c t i o n s FOnF1, FOIF3, Fo;*F2, F p F 3 , F2nF1, F nF of four planes F0,F1,F?,F3 i f and only i f t h e i r 3 1 t r i r a t i o i s -1. I n t h i s case we say t h a t the l i n e s ( D i ) f u l f i l t h e " t r i r a ti0 rule".
119
Impossible objects and inconsistent interpretations
10. FF.'SrBILITY OF AN INTERPQETATION DEFINITInN 10 An i n t e r n r e t a t i o n o f a n o l y h e d r a l f i r l u r e i s " f e a s i b l e " when t h e r e e x i s t a p o l y h e d r a l scene and a b i j e c t i v e correspondence between t h e faces o f t h e i n t e r o r e t a t i o n and a s e t o f faces o f t h e scene, and a b i j e c t i v e correspondence between the r a y s i n t h e scene snace and t h e o o i n t s i n the f i g u r e suno o r t , so t h a t the o r d e r i n r l of any two faces o f t h e i n t e r p r e t a t i o n a t an a r b i t r a r y n o i n t c o i n c i d e s w i t h t h a t o f t h e c o r r e s o o n d i n p faces o f t h e scene along the corresponding ray. Now we can s t a t e t h e f o l l o w i n g r e s u l t . THEOREM 3
A s u f f i c i e n t c o n d i t i o n f o r an i n t e r o r e t a t i o n t o be f e a s i b l e i s t h a t i t i s c o m l e t a b l e and, i f a l l t h e p a i r e d i n t e r s e c t i o n s of i t s faces a r e concurr e n t o r o a r a l l e l , t h a t they f u l f i l the t r i r a t i o r u l e .
A more complete account o f t h e n r e s e n t m a t t e r , w i t h t h e o r o o f s o f a l l t h e r e s u l t s , i s now a v a i l a b l e as a p r e n r i n t p a p e r . REFERENCES
[l] Cowan, T.M., The t h e o r y o f b r a i d s and t h e a n a l y s i s o f i m n o s s i b l e f i g u r e s , J o u r n a l o f Elathematical Psychology, 6 (1977). 41-56. [2]
Draoer, S.Y.,
The Penrose t r i a n a l e and a f a m i l y o f r e l a t e d f i g u r e s ,
P e r c e p t i o n , 7 (1978), 283-296. [3]
Escher, M . C . , The w o r l d o f p1.C.
[4]
Gregory, R.L., (Eds.),
[5]
Escher, New York, Abrams
(1971).
The confounded eye, I n R.L. Gregory, and E.H.
Gombrich
I l l u s i o n i n n a t u r e and a r t , London, Duckworth ( 1 9 7 3 ) .
Huffman, D . A . ,
I m o s s i b l e o b j e c t s as nonsense sentences, I n
B. M e t z l e r
and D. M i c h i e (Eds.), Machine i n t e l l i a e n c e , New York, H a l s t e d , 6 (1971), 295-323. [6]
Necker, L.A.,
Observations on some remarkable o n t i c a l nhenomena seen i n
S w i t z e r l a n d , and on an o n t i c a l nhenomenon which occurs on v i e w i n g a f i g u r e of a c r y s t a l o r p e o w t r i c s o l i d , London, Edinborough P h i l o s o p h i c a l Magazine, 1 (1832), 329-337. [7]
Penrose, L.S.,
and Penrose, R.,
Impossible object: a special type o f
i l l u s i o n s , B r i t i s h J o u r n a l o f Psychology, 49 (1958), 31-33.
120
IC]
E . Tbrouanne
Prenowitz, '*I.,
and J a n t o s c i a k , J . , J o i n geometries: a t h e o r y o f convex
s e t s and l i n e a r geometry, New York, S p r i n g e r V e r l a n ( 1 3 7 9 ) . 131
Sugihara,
, Classification
I:.
o f i l r p o s s i b l e o h j e c t s , P e r c e p t i o n , 11
( 1982) , (55-74.
[lo1
Terouanne, E . , On a c l a s s o f " i m n o s s i b l e " f i q u r e s : a new languaae f o r a new a n a l y s i s , J o u r n a l o f V a t h e m a t i c a l Psycholoqy, 22 ( 1 3 3 0 ) , 24-47.
[I11
Terouanne, E . ,
" I m n o s s i b l e f i n u r e s " and i n t e r p r e t a t i o n s o f D o l y h e d r a l
f i q u r e s , h l u l t i g r a n h i e U.E.R. "ontnellier
(1933).
Hathematiqws, L h i v e r s i t P Pzul Valery,
TREA'DS lh' MATHEMATICAL PSYCHOLOGY E. Degreefand]. Van Bu mhaut (editors) 0 Ekevier Science Publb& B. V. (North-Holland), 1984
121
I N SEARCHING OF GENERAL REGULARITIES OF ADAPTATION DYNAMICS ON THE TRANSFORMATION LEARNING THEORY V.F.
Venda
USSR Academy o f Science, I n s t i t u t e o f Psychology
The s y s t e m a t i c s t u d y o f c h a r a c t e r i z a t i o n e q u a t i o n s , curves o f s t r a t e g i e s and o f e v o l u t i o n s and t r a n s f o r mations o f s t r a t e g i e s d u r i n g t h e l e a r n i n g t a s k a r e t h e fundamentals o f a new t r a n s f o r m a t i o n t h e o r y o f l e a r n i n g processes.
The t h e o r y has a t h e o r e t i c a l
and p r a c t i c a l s i g n i f i c a n c e , p a r t i c u l a r l y , f o r t h e s o l v i n g o f t h e problem o f t h e o p t i m a l c o n t r o l o f t h e l e a r n i n g process, and f o r more p e r f e c t e x p e r i m e n t a l and a n a l y t i c a l models o f t h e s e processes.
The
i n d i v i d u a l i z a t i o n o f t h e methods and means of t h e o p e r a t o r ' s l e a r n i n g , which becomes wide-spread nowadays, makes q u i t e a c t u a l t h e development o f a "micro-approach"
t o t h e s t u d y o f l e a r n i n g processes,
t o g e t h e r w i t h t h e t r a d i t i o n a l "macro-approach".
1. THE HISTORY OF THE DEVELOPMENT OF THE LEARNING THEORY AND THE CRITICAL ANALYSIS OF ITS PRESENT STATE T r a d i t i o n a l l y , t h e problem o f l e a r n i n g i s one o f t h e m a j o r problems of psychological science.
S t u d i e s i n t h i s f i e l d were commenced by H. Ebbinghaus
i n 1885; he used t h e m a t e r i a l o f f o r g e t t i n g and remembering words.
A great
number o f experiments gave him a p o s s i b i l i t y t o f i n d a monotonous exponent i a l r e g u l a r i t y f o r t h e d e c r e a s i n g o f number o f words one can s t o r e i n h i s memory 1171
.
Then, well-known E.L. T h o r n d i k e ' s experiments i n t h e sphere o f a s s o c i a t i o n s and P a v l o v ' s s t u d i e s aimed a t t h e forming and d e s t r o y i n g s o - c a l l e d " c o n d i t i o n e d r e f l e x e s " i n animals, gave t h e analogous r e s u l t s .
The g r e a t e s t r o l e i n development of monotonous a s y m p t o t i c models of l e a r n i n g has been p l a y e d by C.L. H u l l [ i ] .
He assumed t h a t t h e h i g h e s t ( a s y m p t o t i c )
l e v e l o f s k i l l i s d e t e r m i n e d b y i n t e n s i t y o f need c o n f i r m a t i o n , c o n d i t i o n e d s t i m u l u s and a l s o by t h e temporal c h a r a c t e r i s t i c s o f s t i m u l i and r e a c t i o n s . He i n v e n t e d t h e p o s t u l a t e s a b o u t t h e g e n e r a l i z a t i o n s o f s t i m u l u s and i t ' s d y i n g away on t h e base o f t h e s e p o s t u l a t e s , and a l s o o f p a r t i c u l a r theorems on t h e l e a r n i n g t o d i f f e r t o s t r u c t u r e s t i m u l i , t o r e c o n s t r u c t i t t i o n a l l y , and so on C.L.
voli-
H u l l has gone t o t h e main f o r m u l a t o c a l c u l a t e
" t h e power o f s k i l l " . L a t e r , a f t e r W.K.
Estes', R.C.
A t k i n s o n ' s , G.H.
Bower'5,J.G.
Greeno's,
R . D . L u c e ' s r e s e a r c h , t h e monotonous a s y m p t o t i c models o f l e a r n i n g processes became c l a s s i c a l and well-known.
The f i r s t s t e p s i n t h e b r a n c h o f l e a r n i n g
psychology were f u n d a m e n t a l l y reviewed by G.W. "Experimental Psychology" 1151 ,[201
K l i n g i n Woodworth's
.
S i g n i f i c a n t l y more t h a n a h a l f o f q u a n t i t a t i v e e x p e r i m e n t a l d a t a a b o u t t h e l e a r n i n g processes were t a k e n e i t h e r f r o m a n i m a l s o r f r o m p e o p l e , b u t u s i n g extremely simple tasks.
Experimental s i t u a t i o n s w i t h a1 t e r n a t i v e r e a c t i o n s
between 2 t o 3 s i g n a l s were most t y p i c a l . The l a r g e f a c t u a l m a t e r i a l a b o u t t h e l e a r n i n g processes gave a p o s s i b i l i t y f o r R . Bush and F. M o s t e l l e r
141 t o b u i l d c o m p a r a t i v e l y s i m p l e s t o c h a s t i c
models o f l e a r n i n g , which were based on t h e assumption t h a t i t ' s p o s s i b l e t o d e s c r i b e t h e s e processes as M a r k o v i a n ones. E x p o n e n t i a l models o f l e a r n i n g processes found t h e i r way t o a p p l i e d psychology.
They appeared t o be u s e f u l f o r c a l c u l a t i n g t h e l o n g i t u d e of e x p e r i -
mental s e r i e s , i n w h i c h t h e s u b j e c t a c h i e v e s e f f e c t i v e s o l v i n g frr each from t h e f i n i t e s e t o f problems; meanwhile t h i s e f f e c t i v i t y
i s comparable
w i t h the l e v e l o f s k i l l e d s p e c i a l i s t s ( b u t i f i t i s a p r a c t i c a l l y i n f i n i t e set o f possible real situations).
This question i s extremely important
f o r t h e problem o f t r a n s f e r i n g l a b o r a t o r y d a t a o n t o r e a l c o n d i t i o n s o f work. The way o f p r o g r a m i n g o f e n g i n e e r i n g p s y c h o l o g i c a l experiments founded on u s i n g l e a r n i n g models w h i c h were proposed by W.K.
Estes, R.C.
R . Bush, F. M o s t e l l e r i s d e s c r i b e d i n our books [25, 311.
Atkinson,
Shortly, t h i s
I23
Transformation learning theuy
v a y reduces t o t h e f o l locrina t h i n g s : b e g i n n i n g assumntions d u r i n g t h e des i g n and w o r k i n g as f o l l m s : 1) o n e r a t i v e problems chosen a r e s o l v e d w i t h t h e h e l o o f one c e r t a i n method, i . e . a l l t h e ways and s o l u t i o n s a r e r e l a t e d w i t h an unique s t r a t e g y ; 2 ) t h e l e a r n i n g Drocess w i t h i n t h e f r o n t i e r s o f t h i s s t r a t e g y i s d e s c r i b e d as a P a r k o v i a n one. $1. B r u n e r t a l k s a b o u t t h e s t r a t e g y o f d e c i s i o n makino as on t h e c e r t a i n way o f a c q u i r i n g , s t o r i n g and u s i n g i n f o r m a t i o n which serves f o r a c h i e v i n a c e r t a i n aims ( r e s u l t s ) [31. The s o l u t i o n o f problems by a suh.ject means t h a t h i s r e a l s t r a t e g y SR i s id e n t i c a l t o an a o r i o r i s t r a t e g y S a y and t h e l a s t one must be a s s i m i l a t e d b y t h e s u b j e c t SR = S a y e l s e S k # Sa. IJhen l o o k i n g a t t h e sequence o f experiments as a t t h e elements o f a Parkov process, we can see t h e f o l l o w i n a n r o b a b i j i t i e s o f t r a n s i t i o n s :
k a n w h i l e , i t i s necessary t o assume t h a t r i g h t and wrong s o l u t i o n s cannot e x i s t s i m u l t a n e o u s l y i n one exaeriment, and t h e sum o f i t s o r o h a b i l i t i e s equals 1; i n p a r t i c u l a r , f o r b e g i n n i n g D r o b a b i l i t i e s ( b e f o r e l e a r n i n g ) : P o ( s R = s a ) t P o ( s ' R # s a ) = 1. F o r t h e complete d e f i n i n g o f t h e l e a r n i n g processas a Harkovian one i t i s necessary t o f i n d , f r o m exoeriments, f o l l o w i n a meanings: Po(SR=Sa),
P(S R =S a ] S' tR #C a ) and F(SoR#SaISR=Sa); t h e o t h e r meanings o f t r a n s i t i o n p r o b a b i l i t i e s a r e easy t o f i n d f r o m these ones. We w i 11 symbolize P(SR=SalS IR#Sa) as a and P(SR=SaISR=Sa)-P(SR=SalS'R#Sa) as 6; then t h e p r o b a b i l i t y o f t h e f a c t t h a t i n t h e t r a i n i n g e x n e r i m e n t number n, t h e s u b j e c t would use t h e r i g h t s t r a t e g y SR=Sa can be expressed as: Pn(SR=Sa)= BnPo(sR=s,)+( 1-6 n )P,(SR=Sa), where Pm(SR=Sa)esuals
&
P (SR=sa I s ' RPSa) = P ( S ' R#sa I SR=Sa)+P ( 5 R=Sa[ 5 ' RfSa)
-
Thus, i f t h e p r o b a b i l i t y o f a r i g h t s o l u t i o n o f each problem f r o m t h e g i v e n
s e t b y t h e e x p e r i e n c e d o o e r a t o r s , who were i n v i t e d t o t a k e p a r t i n t h e experiments, P ( S =S )=P O P a
* , then
f r o m t h e above formulas, we can f i n d t h e
V . F . Venda
124
number o f exrJeriment n , i n w h i c h a Civen s u b j e c t who has c e r t a i n i n i t i a l t r a i n i n g Po(SR=Sa) and a b i l i t y t o a c n u i r e and s t o r e s k i l l s o f r i g h t Droblem s o l v i n g , P(S'R#TaISR=Sa) and P ( 5 R = S a l S ' R # S a ) , would a c h i e v e t h e s t a b l e l e v e l
of r e s u l t s ( t h e D r o b a b i l i t v o f r i r r h t d e c i s i o n which i s normal f o r s k i l l e d o n e r a t o r s : pn = P * ) .
I n c o n n e c t i o n w i t h R u s h - t l o s t e l l e r ' s l e a r n i n o model, i n t h e case o f -1<$<1 and n-,
t h e v a l u e o f Pn(SR=Sa) ooes t o t h e a s y m n t o t i c l i n e P(Sq=Sa), which
r e f l e c t s some f i n i t e c o n s t a n t l e v e l , t h e s o - c a l l e d " l e a r n i n q p l a t e a u " . The f u r t h e r develonment o f t h e l e a r n i n g processes m d e l was done by Q.C.
A t k i n s o n , C.H.
Rower, and E . C . C r o t h e r s , i n " I n t r o d u c t i o n t o mathema-
t i c a l theory o f l e a r n i n g " . An i n o r t a n t a c h i e v e w n t of R . C .
A t k i n s o n e t a l . , was t + e c o n s t r u c t i o n o f
t h e model o f w h o l i s t l e a r n i n q Drocess: t h e l e a r n i n n
lpwl
i n n was e x p r e s -
sed r a t h e r f r o m dynamics o f l e a r n i n o , which was c h a r a c t e r i z e d b y t h e v a r i a t i v e v e l o c i t y , than f r o m t h e s t a t e d u r i n g t h e e x n e r i m e n t n - 1 and c o n s t a n t t r a n s i t i o n p r o b a b i l i t i e s f r o m one s t a t e t o a n o t h e r , as i t was i n B u s h ' s S e q u e n t i a l l e a r n i n c f o f t w o - k i n d r e a c t i o n s i s o r e s e n t e d by R.C. n-1 and A t k i n s o n as two e x n o n e n t i a l curves: P(S'R#Sa) = ( 1 - c ) n-1 D(Sp=S'a) = ( 1 - q c ) , where ( 1 - c ) i s tl7e l o q a r i t h m i c a l decrement o f d y i n p
models.
away o f w o n a r o l u t i o n s p r o b a b i l i t y ; ( I - q c ) i s t h e l e s s decrement o f d y i n o away ( q z l ) of tt-e p r o b a b i l i t y o f s t r a t e g y a c q u i r e d i n b e g i n n i n q Sa, which would be s u b s t i t u t e d by s t r a t e a y S",.
The D r o b a b i l i t y o f t h e f a c t , t h a t i n
e x n e r i m e n t number n w o u l d be observed t h e s t r a t e p y S'IR=S'la i s g i v e n by
P ( s " R=s a) = 1-P ( s 1 a fS R ) - P ( s R=s' a ) . '1
I n the b e g i n n i n g P(SR=S',)
i s n o t h i g h , then, when we a r e l e a r n i n g , i t i n -
creases, achieves a maximum and then b e g i n s t o be s u b s t i t u t e d b y t h e second k i n d o f reactions
-
hy t h e s t r a t e g y S"R=S"a; i n a m e a n t i r e P(S"R=S"a) i n -
creases i n c o n n e c t i o n w i t h t h e above men ti oned f o r m u l a. U s i n q t h e c o n t r o l e x p e r i m e n t a l d a t a R.C.
A t k i n s o n had shown t h e dependence
o f t h e r e l a t i v e q u a n t i t y o f dops who a v o i d e d t h e e l e c t r i c a l c u r r e n t shock i n experiments conducted b y Solomon-Uinn.
These f a c t were used b y A t k i n s o n
125
Tromfomotion learning theory
t o prove c o i n c i d e n c e between h i s mathematical models and e x p e r i m e n t a l results. A l t h o u g h R.C.
A t k i n s o n and c o l . g e n e r a l i z e d i n t h e i r mathematical models
p r a c t i c a l l y e x c l u s i v e l y t h e d a t a o f s i m p l e experiments w i t h animals, t h e s e models r e f l e c t e d e x t r e m e l y i m p o r t a n t aspects o f l e a r n i n g process f r o m one t y p e o f a c q u i r e d r e a c t i o n s t o a n o t h e r
-
-
the t r a n s i t i o n
and marked t h e
development o f non-monotonous mathematical l e a r n i n g models. Non-coincidence between r e a l l e a r n i n g processes and monotonous exponents was shown i n t h e wide range o f experiments, b u t d i d n ' t f i n d enough t h e o r e t i c a l proofs.
On t h e base o f t h e g r e a t number o f e x p e r i m e n t a l
d a t a H. Ebbinghaus and l a t e r R. Woodworth d i f f e r e d i n i t i a l , i n t e r m e d i a t e and f i n a l p l a t e a u s i n t h e l e a r n i n g curves. An i n t e r m e d i a t e p l a t e a u was f i r s t l y found e x p e r i m e n t a l l y by B r i e n and H a r t e r i n 1896 when s t u d y i n g processes w i t h t e l e g r a p h i s t s .
And i t was
i n t e r e s t i n g t h a t t h e l e a r n i n g c u r v e f o r t h e s i m p l e s t o p e r a t i o n s was monotonous and h a d n ' t an i n t e r m e d i a t e p l a t e a u , which was b e a u t i f u l l y seen i n r e c e i v i n g c a b l e messages l e a r n i n g curves [61. B r i e n and H a r t e r e x p l a i n e d t h e appearance o f i n t e r m e d i a t e p l a t e a u s d u r i n g t h e l e a r n i n g , by t h e presence of forming elementary s k i l l s needed f o r f u r t h e r development, more complex and p e r f e c t s k i l l s ( f i g . 1 ) .
Also they
r e g i s t r a t e d t h e f a c t t h a t t h e r e c e p t i o n d i f f e r s f r o m t r a n s l a t i o n , because i n t h e f i r s t case a t e l e g r a p h i s t can n o t v a r y t h e temporal c h a r a c t e r i s t i c s o f h i s work, because t h e y a r e g i v e n by t h e o b j e c t i v e c h a r a c t e r i s t i c s o f the external world.
A plateau i s a comparatively long delay i n increasing
t h e p r o f e s s i o n a l r e s u l t s , connected w i t h t h e a c q u i r i n g o f wide-spread v e r b a l l o r e , and t h e n t h e process o f l e a r n i n g makes a b i g l e a p .
Brien
and H a r t e r showed i n 1899 t h a t t h e p l a t e a u i s s t i l l more c l e a r when l e a r n i n g more complete s k i l l s , f o r i n s t a n c e , when l e a r n i n g a r e c e p t l o n connected t e x t ( f i g . 1, c u r v e 3 ) , meanwhile when l e a r n i n g a r e c e p t i o n words i t ' s expressed l e s s c l e a r , and when l e a r n i n g a r e c e p t i o n independent l e t t e r s t h e r e i s n ' t any p l a t e a u i n g e n e r a l ( c u r v e 1). l e a r n i n g process has a monotonous c h a r a c t e r .
I n t h e l a s t case t h e I n 1906 S w i f t has p u b l i s h e d
t h e d a t a about t h e presence o f an i n t e r m e d i a t e p l a t e a u when s t u d y i n g f o r e i g n languages.
Figure 1 The c u r v e s o f l e a r n i n g o f t e l e g r a p h i s t s (by Rrier; and H a r t e r , 1899).
Betson i n 1916 and Kow i n 1937 s u p p o r t e d t h e c o n c e p t i o n t h a t t h e f o r m a t i o n of an i n t e r m e d i a t e p l a t e a u i n l e a r n i n g c u r v e s i s more p r o b a b l e w i t h complex o p e r a t i o n s t h a n w i t h s i m p l e .
F u r t h e r m o r e , t h e y showed t h a t
complex o p e r a t i o n s , a c q u i r i n g as a whole, d i d n o t g i v e a p l a t e a u i m p e r a t i v e l y ; t h e y met i t u s u a l l y i n t h o s e cases, when t h e s t u d e n t f i x e d h i s a t t e n t i o n on t h e s i n g l e elements of a complex o p e r a t i o n , one a f t e r a n o t h e r . If e a r l i e r monotonous l e a r n i n g processes appeared more o f t e n i n a n i m a l s
and r e s u l t i v e l y had analogous processes when l e a r n i n g p e o p l e t o f u l f i l s i m p l e t a s k s , t h e n l a t e r on i t was made a s u c c e s s f u l t r i a l t o r e p r o d u c t non-monotonous l e a r n i n g processes i n a n i m a l s by means o f c o m p l i c a t i n g t h e i r
127
Transfomiation learning theory
b e h a v i o u r and l e a r n i n g c o n d i t i o n s .
F o r example, Hanter (1929) found a
p l a t e a u when l e a r n i n g r a t s t o search a way w i t h i n t h e l a b y r i n t h . G e n e r a l i z i n g t h e r e s u l t s o f a1 1 t h e e x p e r i m e n t a l s t u d i e s where i n t e r m e d i a t e p l a t e a u s were found,
R. Woodworth e x p l a i n s t h i s phenomenon by means o f t h e
changing o f f u l f i l l i n g t a s k method.
K.K.
The analogous change was found by
P l a t o n o v and L.M. Schwarz (1949), when t h e y worked o u t motor s k i l l s ,
and t h e s u b j e c t t u r n e d from s u r p l u s , bed r e g u l a t e d motions i n t o more l a c o n i c p u r p o s e f u l motions.
D u r i n g t h e t r a i n i n g two p l a t e a u s were observed:
1 5 t h t o 1 8 t h and 2 6 t h t o 2 8 t h days o f s h o t ' s t r a i n i n g .
Among o t h e r causes
o f p l a t e a u s r i s e , t h e y emphasize t h e i n f l u e n c e o f changing s k i l l s t r u c t u r e , t u r n i n g i n t o t h e s k i l l w i t h more p e r f e c t s t r u c t u r e .
For a l l t h a t , i t ' s
p o s s i b l e a p r o v i s i o n a l change f o r t h e worse of t h e indexes which f o l l o w s plateau. B.F.
Lomov 1211 i n h i s book has shown t h e c o n v i n c i n g d a t a about t h e non-
monotonous c h a r a c t e r o f l e a r n i n g processes and i t s dependence on t h e l e v e l o f complexity o f s k i l l s acquired ( f i g . 2). The p u b l i c a t i o n o f such d a t a was e x t r e m e l y i n t e r e s t i n g , because t h e g e n e r a l l y accepted p s y c h o l o g i c a l and mathematical l e a r n i n g t h e o r i e s d i c t a t e d a l m o s t o b l i g a t o r y an a p p r o x i m a t i o n o f e x p e r i m e n t a l d a t a by means o f a monotonous a s y m p t o t i c l e a r n i n g curve.
F o r i n s t a n c e , i n t h e well-known
handbook o f A. Chapanis, devoted t o methods and techniques o f e n g i n e e r i n g p s y c h o l o g i c a l experiments, namely t h i s a p p r o x i m a t i o n o f e x p e r i m e n t a l d a t a by an a s y m p t o t i c l e a r n i n g c u r v e was recomnended.
Fig. 3 taken from t h i s
book, shows an e x p e r i m e n t a l p o i n t and "average c u r v e " (unbroken l i n e ) , proposed by Chapanis.
I t ' s p o s s i b l e , however, t o p u t f o r w a r d an a l t e r n a t i v e
assumption as t o t h e more complex c h a r a c t e r o f a l e a r n i n g process ( a d d i t i o n a l touch l i n e a t f i g . 3).
I t ' s a l s o i n t e r e s t i n g t o note t h a t i n
a n o t h e r c h a p t e r o f t h e handbook, A. Chapanis p r e s e n t s
non-approximated
d a t a and, however, d o e s n ' t g i v e an e x p l a n a t i o n of t h e complex c h a r a c t e r o f t h i s process I161.
30
\ , -
lo
Figure 2
Examples of non-monotonous curves o f learning o f operators (by Lomov, 1966).
129
Transformiation fearnhig theory
t,sec
10
3
9
I
w
7
.*
1
2
'
4
3
5
L . .
6
7
8
9
10
11 T, days
Figure 3 Experimental d o t s and t h e i r a p p r o x i m a t i o n by an e x p o n e n t i a l l e a r n i n g c u r v e ( b y Chapanis, 1959). t : average k e y i n g time;
T : days o f t e s t ;
.
approximation l i n e ;
: e x p e r i m e n t a l dots;
--- .
-
: Chapanis'
a d d i t i o n a l approximation . l i n e
suggested by us.
An a p p r o x i m a t i o n o f t h e l e a r n i n g processes by t h e monotonous e x p o n e n t i a l c u r v e became t h e most s t a b l e t r a d i t i o n i n some a p p l i e d branches o f psychol o g y and i n ergonomics.
F o r example, i n t h e competent work " I n t r o d u c t i o n
t o ergonomics" (Moscow, 1977) mathematical methods o f computation c u r v e s , p r o c e e d i n g f r o m t h r e e o r even f r o m o n l y two p o i n t s a r e reviewed. showed t h e monotonous dependence of t h e o p e r a t o r ' s j o b e f f e c t i v i t y
Authors upon
t h e volume o f i n f o r m a t i o n I used and t h e c o r r e s p o n d i n g monotonous l e a r n i n g c u r v e a s y m p t o t i c a l l y approaching t o some "maximal" l e v e l of e f f e c t i v i t y ( F i g . 4 ) , i f t h e number o f l e a r n i n g t r i a l s n i s b i g enough.
130
Q Q
inax
n
I
8,
a) Figure 4
E f f e c t i v e n e s s Q as a f u n c t i o n o f i n f o r m a t i o n volume I ( a ) and number o f t r i a l s n ( b ) .
( b y Zarakovsky e t a l . , 1977).
Because a n a l y s i s methods o f l e a r n i n g processes p r e s e n t e d above r e f l e c t a modern s t a t e o f l e a r n i n g t h e o r y and a r e proposed by a u t h o r s f o r t h e aims o f wide p r a c t i c a l a p p l i c a t i o n , l e t ' s d i s c u s s i n d e t a i l t h e s e m a t e r i a l s . The a u t h o r s proceed f r o m t h e s o - c a l l e d "macro-approach",
which i s based
on a g e n e r a l i z e d a n a l y s i s o f a g r e a t number o f e x p e r i m e n t a l d a t a and t h e i r
mean q u a n t i t i e s .
No d o u b t t h a t such approach g i v e s a p o s s i b i l i t y t o show
sometimes t h e most g e n e r a l i z e d r e g u l a r i t i e s , b u t i n t h i s case i t may l o s e t h e v e r y s i g n i f i c a n t t r a i t s o f l e a r n i n g process. f a c t by t h e f o l l o w i n g example.
Let's illustrate this
Our e x p e r i m e n t a l d a t a (Venda, 1969) about
t h e l e a r n i n g s u b j e c t i n compensatory t r a c k i n g o f dynamic s i g n a l s a r e shown i n F i g . 5 and an i d e n t i c a l f l o w o f watched s i g n a l s , were o u t p u t t o d i f f e r e n t numbers o f apparata (n=1+6! d h i c h he c o u l d deal w i t h s i m u l t a n e o u s l y and send
131
Transformation learning theory
t8,8 C 40
35 30 25
20
15
10
1
2
3
4
5
6
7
8 Ttuays
Figure 5 Experimental data on learning process with compensatory tracking : t : average time o f tracking o f one s i g n a l ; T : days o f learning; n : number o f simultaniously perceived s i g n a l s ( n = 1; 2 ; 3; 4; 5; 6 ) . (by Venda, 1969).
w i t h t h e h e l p o f c o r r e s p o n d i n g channels t h e r e g u l a t i v e s i g n a l s The process o f l e a r n i n g appears t o have o p e n l y a non-monotonous c h a r a c t e r when any number o f a p p a r a t a e x i s t s ; i t i n c l u d e s d i f f e r e n t p l a t f o r m s and even i n some p e r i o d shows t h e change of w a t c h i n g indexes t o t h e worse.
If
a l l t h e s e complex c u r v e s a r e s u m a r i z e d a r i t h m e t i c a l l y , i t would be q u i t e t h e t r a d i t i o n a l monotonous average c u r v e (see F i g . 6 ) .
t,
ao c
35 30 25 20
15 1
2
3
4
5
6
7
T,
days
Figure 6 T o t a l summarized and averaged d a t a o f F i g u r e 5. F o r a l l t h a t , however, we l o s e t h e p r e s e n t a t i o n about t h e d i f f e r e n c e s d u r i n g
t h e l e a r n i n g w i t h t h e d i f f e r e n t q u a n t i t i e s o f apparata, i n t e r m e d i a t e and f i n a l e f f e c t i v i t y o f w a t c h i n g , depending on t h e number o f s i m u l t a n e o u s l y receiving signals.
A f t e r a l l , when u s i n g "macro-approach",
we c a n n o t
a n a l y s e t h e i n f l u e n c e o f t h e i n f o r m a t i o n a l f a c t o r on t h e l e a r n i n g , c o n t r o l t h e l e a r n i n g . s o l v e t h e problem o f a d a p t a t i o n o f i n f o r m a t i o n a l f l o w t o o p e r a t o r by means o f r e g u l a t i n g a number o f s i g n a l s which a r e t o be r e c e i v e d
133
lranrfomation learning theory
by him simul t a n e o u s l y . Yet a t t h e b e g i n n i n g o f o u r c e n t u r y , t h e s c a l e o f r e l a t i v e t i m e w i t h t h e r e g i s t r a t i o n o f i n d i v i d u a l d i f f e r e n c e s i n l e a r n i n g speed and t h e c o r r e s ponding s t r e t c h i n g and compression o f t i m e a x i s d u r i n g t h e j o i n t r e f l e c t i o n o f d a t a i n t h e experiments w i t h t h e d i f f e r e n t p e o p l e o r under d i f f e r e n t circumstances o f t h e i r l e a r n i n g , aimed a t a p p r o x i m a t i o n o f curves, was proposed.
The s c a l e was named a f t e r i t s a u t h o r , S.B. V i n c e n t , who used
s u c c e s s f u l l y t h i s s c a l e when g e n e r a l i z i n g d i f f e r e n t d a t a about t h e l e a r n i n g ; however, t h e n t h e s c a l e a p p l i e d v e r y seldom. I f we would use S.B. V i n c e n t ' s s c a l e t o f i n d mean q u a n t i t i e s o f t h e same data, m a r k i n g a t t h e a x i s o f r e l a t i v e t i m e Trel,,
the co-ordinates o f the
d i f f e r e n t phases of t h e l e a r n i n g process, we would have a non-monotonous c u r v e ( F i g . 7 ) , i n which f o r each s t r a t e g y , a c q u i r e d i n succession, i t i s p o s s i b l e t o watch f o r t h e f o l l o w i n g phases : t h e b e g i n n i n g o f s t r a t e g y a c q u i r i n g (1, 2 ) , s w i f t improvement of indexes (2, 3 ) , p l a t e a u ( 3 , 4 ) , t u r n i n g f r o m t h e p r e c e d i n g s t r a t e g y i n t o t h e f o l l o w i n g (4, 5 ) . An a n a l y s i s o f such a c u r v e g i v e s us a p o s s i b i l i t y t o ask a q u e s t i o n , n o t o n l y about i s o l a t e d s t u d i e s of t h e processes o f human l e a r n i n g t o each s t r a t e g y as such (as i t was done by B r i e n and H a r t e r ) , b u t a l s o a b o u t t h e s t u d y i n g of t u r n i n g one s t r a t e g y i n t o another.
I t g i v e s us a r i g h t t o
r e s e a r c h n o t o n l y t h e e v o l u t i o n of s i n g l e s t r a t e g y , b u t a l s o t h e t r a n s formations o f s t r a t e g i e s . Up t o t h e t i m e such t r a n s f o r m a t i o n s interesting
-
-
as i t seems t o me, t h e most
o f t e n were o u t o f mind.
Authors o f " I n t r o d u c t i o n t o ergonomics" k i n d l y p r e s e n t e d t o me t h e i r r e a l d a t a about some complex processes o f l e a r n i n g which were n o t i n c l u d e d i n t h e i r book, because t h e y were c o n s i d e r e d t o be " a r t e f a c t s " and " b l u n d e r s " . I n one case, t h i s changing o f t h e number o f misses a d m i t t e d by o p e r a t o r s when f r a c t i o n e d u s e f u l s i g n a l s c o n c e r n i n g t h e n o i s e s (see F i g . 8 ) . F o r a l l t h a t , d u r i n g t h e s i x t h hour of l e a r n i n g , i t was proposed t o t h e s u b j e c t s t o c o n t i n u e t h e i r work independently, w i t h o u t i n s t r u c t i o n s . The s t u d e n t t u r n e d i n t o t h e independent w o r k i n g o u t and d e f i n i n g t h e c r i t e r i a
134
35 30 25
20 15
1 2
5-1 2
3 4
3 4 5-1 2
Trelativ
Figure 7 The same e x p e r i m e n t a l d a t a (of F i g . 5 ) summarized and averaged w i t h s c a l e by Vincent,(l912).
1
2
3
4
5
6
7
8
9T
hours
Figure 8 Real e x p e r i m e n t a l d a t a (compare w i t h t h e o r e t i c a l c u r v e s o f t h e same a u t h o r s a t t h e F i g . 4 ) . ( b y Zarakovsky e t a l . ,
1977).
135
Transfonnatiun learning theory
o f the fractioning useful signals.
These q u i t e numerous d a t a cannot be
approximated by monotonous curves f o r enough reasons. I n R. A t k i n s o n ' s e t a l . book " I n t r o d u c t i o n i n mathematical t h e o r y o f l e a r n i n g " i s p r e s e n t e d t h e dependence o f l a t e n t p e r i o d o f s u b j e c t s responses t upon t h e q u a n t i t y o f p r e l i m i n a r y remembered o b j e c t s d; and f o r a l l t h a t
t h e segments f o r " s m a l l " and " b i g " number o f o b j e c t s a r e taken, b u t t h e i n t e r m e d i a t e numbers a r e e l i m i n a t e d f r o m t h e c o n s i d e r a t i o n (see F i g . 9 ) .
t,
m aec
8 50
7 50
6 50 5 50 450
0
5
10
15
20
25
30 d
Figure 9 by A t k i n s o n e t a l . (1971). The dependence o f t on d has a c l e a r l i n e a r c h a r a c t e r .
I t ' s easy t o f i n d
a monotonous a s y m p t o t i c c h a r a c t e r o f t h e l e a r n i n g , i f we would assume r e g u l a r monotonous r e l a t i o n between t h e c h a r a c t e r i s t i c s o f c o m p l e x i t y a r d t h e indexes o f s u c c e s s f u l n e s s , c o m p l e x i t y o f i t ' s f u l f i l l i n g i n each moment o f 1e a r n i n g . We must e s p e c i a l l y remark t h e q u e s t i o n about i n t e r r e l a t i o n s between t h e f a c t o r s ( i n i t i a l c o n d i t i o n s ) and indexes ( r e s u l t s ) o f f u l f i l l i n g tasks.
V.F. Vendo
136
For instance, A.N.
Leontyev and E.P.
K r i n c h i k (1963) showed a g r e a t s e t of
d a t a about l i n e a r r e l a t i o n between t h e i n f o r m a t i o n a l c a p a c i t y o f s i g n a l s (as p s y c h o l o g i c a l f a c t o r o f t a s k f u l f i l l i n g c o m p l e x i t y ) and t i m e o f r e a c t i o n t o s i g n a l s (as t h e i n d e x , c r i t e r i o n o f t a s k f u l f i l l i n g c o m p l e x i t y ) . An analogous l i n e a r r e l a t i o n
the task f u l f i l l i n g
-
-
the less information i n signals, the easier
c a n n o t be observed, i f we watch f o r r e a l a c t i v i t y
when a c h i e v i n g c o n c r e t e aims. F o r example, when s o l v i n g some t h i n k i n g o r senso-motor problem, a human
o p e r a t o r must have c e r t a i n volume o f i n f o r m a t i o n .
When t h e volume has an
o p t i m a l meaning, t h e problem i s s o l v e d e a s i l y , b u t i f t h e volume o f i n f o r m a t i o n i s more o r l e s s t h a n o p t i m a l , t h e n t h e c o m p l e x i t y o f s o l v i n g would increase. F i g . 10 shows an e x p e r i m e n t a l c u r v e of t h e dependence of t h e meantime o f t h e compensatory w a t c h i n g f o r s i g n a l s i n t h e case, i n which t h e s i g n a l s composing t h e same f l o w a r e r e f l e c t e d i n 1-6 a p p a r a t a by t u r n s ( n = 1 - 6 ) . T h i s c u r v e i s b u i l t i n c o n n e c t i o n w i t h t h e d a t a o f Venda, 1969. ( s e e f i g . 1 0 ) F i g . 5 c l e a r l y shows an o p t i m a l number o f apparata, w h i c h g i v e s a p o s s i b i l i t y f o r meantime o f t h e w a t c h i n g o f s i g n a l s t o be minimal t ( n = 3 ) = l l sec., 11 =3. On t h e l e f t ( n c 3 ) and on t h e r i g h t (n>3), t h e c u r v e i s a p p r o x i opt. 2 mated by t h e d i f f e r e n t r e g r e s s i o n e q u a t i o n s : tl(nc3)=9+(4,5-n) sec; 2 t (n>3)=ll+(n-2,5) sec. 2
The analogous c u r v e s w i t h t h e c l e a r l y expressed optimum f o r many cases of c o n n e c t i o n s between t h e d i f f e r e n t p s y c h o l o g i c a l f a c t o r s and c r i t e r i a o f t h i n k i n g , p e r c e p t i v e and senso-motor problems s o l v i n g c o m p l e x i t y a r e found.
For a l l t h i s , t o each problem s o l v i n g s t r a t e g y . corresponds a c e r t a i n k i n d o f c u r v e and c e r t a i n o p t i m a l meaning of t h e p s y c h o l o g i c a l c o m p l e x i t y factor.
For s i m p l i c i t y ,
l e t ' s r e v i e w a monofactor dependence; t h e methods
of f i n d i n g o u t of t h e system o f p s y c h o l o g i c a l f a c t o r s o f c o m p l e x i t y and t h e i r i n f l u e n c e o n t o c r i t e r i a o f c o m p l e x i t y chosen a r e d i s c u s s e d i n d e t a i l i n o u r monograph " E n g i n e e r i n g psychology and s y n t h e s i s o f i n f o r m a t i o n d i s p l a y systems'' (Moscow,( 1975); 2nd ed. -( 1981)).
Tramjurmatiun learning theory
137
F i g u r e 10 C h a r a c t e r i z a t i o n c u r v e o f s t r a t e g y i n compensatory t r a c k i n g ( b y Venda, 1969).
t : average t r a c k i n g time;
n : number o f p e r c e i v e d s i g n a l s .
Thus, t h e r e i s a s p e c i a l aspect i n t h e problem of l e a r n i n g t o be discussed, and t h i s aspect i s c l o s e l y connected w i t h t h e emphasizing n o n - l i n e a r c h a r a c t e r o f c o n n e c t i o n between t h e f a c t o r s ( c o n d i t i o n s ) and c r i t e r i a o f t h e s u c c e s s f u l n e s s o f human t a s k f u l f i l l e d d u r i n g t h e l e a r n i n g and a l s o o f t h e presence o f an o p t i m a l f a c t o r i n d e x f o r each t a s k when f u l f i l l i n g a r e a l s t r a t e g y i n each experiment. Although y e t i n 1896 t h e complex n o n - l i n e a r components i n l e a r n i n g dynamics were found (and p a r t i c u l a r l y , i n t e r m e d i a t e p l a t e a u s were observed) and t h e d i f f e r e n c e s between p o s s i b l e s t r a t e g i e s o f f u l f i l l i n g e d u c a t i o n a l t a s k s were proved, t h e r e a r e p r a c t i c a l l y o n l y monotonous models o f l e a r n i n g i n
contemporary p s y c h o l o g i c a l and mathematical l e a r n i n g t h e o r i e s . Psychologists-mathematicians selected from the experimental-psychological body o f f a c t s about l e a r n i n g t h e most s u i t a b l e ones f o r a p p r o x i m a t i o n by t h e s i m p l e a n a l y t i c a l f u n c t i o n s - M a r k o v i a n c h a i n s , e x p o n e n t i a l and power e q u a t i o n s , p o l y n o m i a l s and o t h e r mathematical models, w h i c h a r e q u i t e understandable and a c q u i r a b l e f o r a l l t h e members of r e s e a r c h o f t h e complex problem o f l e a r n i n g , and p a r t i c u l a r l y , f o r e x p e r i m e n t a l p s y c h o l o gists.
Such a m a t e r i a l i n c l u d e s i n main t h e processes of animal l e a r n i n g
b u t a so t h e processes o f a c q u i r i n g e x c l u s i v e l y s i m p l e s k i l l s by humans. Then, t h e e x p e r i m e n t a l p s y c h o l o g i s t s who s t u d i e d more complex l e a r n i n g proce ses, o f t e n tended t o r e p r e s e n t t h e r e s u l t s o f t h e i r r e s e a r c h e s by well-known a n a l y t i c a l f u n c t i o n s .
F o r a l l t h i s , t h e d a t a w h i c h were
n o t consonant w i t h monotonous a s y m p t o t i c a l and o t h e r analogous mathematlcal models, were e l i r i n a t e d as " a r t e f a c t s " . The smoothing and " s t a n d a r d i z a t i o n " o f l e a r n i n g curves were caused a l s o by a v e r a g i n g o f e x p e r i m e n t a l d a t a i n c o n n e c t i o n w i t h t h e g r e a t number o f s u b j e c t s and under t h e d i f f e r e n t c i r c u m s t a n c e s o f l e a r n i n g . The monotonous c h a r a c t e r o f l e a r n i n g c u r v e s was determined a l s o by t h e assumption about t h e monotonous (usua l y , l i n e a r ) c h a r a c t e r o f dependence between t h e c o n d i t i o n s ( f a c t o r s ) and complexity
he c r i t e r i a o f t a s k f u l f lment
.
When s t u d y i n g t h e l e a r n i n g processes
n e n g i n e e r i n g psychology and ergono-
mics, t h e s c i e n t i s t s c o n f i n e d t h e i r s e l v e s o n l y w i t h t h e "macro-approach'' and d i d n ' t p e n e t r a t e t o t h e s t r u c t u r e o f processes, d i d n ' t f i n d o u t t h e leading psychological factors.
L a s t ones d e t e r m i n e t h e r e g u l a r i t i e s o f
l e a r n i n g and t h e c h a r a c t e r i s t i c s of t h e l e a r n i n g processes which consequentl y make themselves f a m i l i a r t o d e s t r o y , change by more adequate ones and p e r s p e c t i v e s o f t h e problem s o l v i n g . The f u r t h e r development of t h e l e a r n i n g t h e o r y i s needed t o be connected w i t h t h e s t u d y i n g n o t o n l y of t h e s t r a t e g i e s e v o l u t i o n processes, b u t a l s o o f t h e processes o f i t s t r a n s f o r m a t i o n , t u r n i n g f r o m one s t r a t e g y t o
139
Transformution learning theory
a n o t h e r under t h e i n f l u e n c e o f l e a r n i n g methods, i n d i v i d u a l d i f f e r e n c e s between t h e s t u d e n t s , c o n d i t i o n s and means o f f u l f i l l i n g e d u c a t i o n a l t a s k s .
2. SOME WAYS OF FURTHER DEVELOPMENT OF THE LEARNING THEORY (INTRODUCTION TO THE TRANSFORMATION THEORY OF LEARNING) An a n a l y s i s o f dependence between t h e p s y c h o l o g i c a l f a c t o r and t h e c r i t e r i o n o f t a s k f u l f i l l i n g c o m p l e x i t y , c h a r a c t e r i s t i c f o r each s t r a t e g y , and a l s o an e x a m i n a t i o n o f o p t i m a l i n d e x o f t h i s f a c t o r , g i v e s us a p o s s i b i l i t y t o d e f i n e t o p p o s s i b i l i t i e s o f t h e p e r f e c t i o n f o r each s t r a t e g y and t o d e t e r mine t h e c h a r a c t e r o f s t r a t e g i e s t r a n s f o r m a t i o n s , i.e. o f t h e t u r n i n g from one i n t o a n o t h e r . L e t ' s r e v i e w an example o f e x p e r i m e n t a l d a t a about t h e dependence between t h e p s y c h o l o g i c a l f a c t o r and t h e c o m p l e x i t y c r i t e r i o n when f u l f i l l i n g t a s k s and u s i n g t h e d i f f e r e n t s t r a t e g i e s o f i t s f u l f i l m e n t . D e t a i l e d d e s c r i p t i o n s o f such experiments a r e g i v e n i n [22, 25, 311. F i g . 11 shows on t h e l e f t t h e dependences o f t h e o p e r a t i v e problems s o l v i n g t i m e t upon t h e number o f s i n g l e i n f o r m a t i o n a l s i g n a l s n, r e c e i v e d by t h e o p e r a t o r , when s o l v i n g problems i n mnemoscheme.
S t r a t e g y A which begins t h e
l e a r n i n g has a c h a r a c t e r o f element-to-element p e r c e p t i o n o f i n f o r m a t i o n f r o m t h e mnemoscheme. The minimal s o l v i n g t i m e when u s i n g a s t r a t e g y A A tAmin, we have i n t h e case o f noopt.. During the f o l l o w i n g stage o f learning, the operator acquires a strategy
B,
i n which t h e elements p e r c e i v e d a r e j o i n e d i n t o w h o l i s t f u n c t i o n a l groups :
B
tBmin in t h e case o f nopt..
The o p e r a t o r achieves t h e b e s t indexes when l e a r n i n g t o p e r c e i v e a l l t h e mnemoscheme s i m u l t a n e o u s l y and t o i d e n t i f y t h e s t a t e o f o b j e c t as a whole : C tcrnin bopt= 1 ) . I t ' s necessary t o draw a t t e n t i o n t o t h e s p e c i a l p o i n t s on t h e l e f t i n t h e curves o f f i g . 11,the comnon p o i n t AB o f s t r a t e g i e s A and B and t h e common p o i n t BC o f t h e s t r a t e g i e s B and C.
Co-ordinates o f these p o i n t s correspond
t, sec
IA
tAO
t~~ tAmi n
tB C
tBmi n
%min
c no p t
nsc
nB opt
AB nA
ont
TAH
TAoDt
TRoqt
TBC
F i g u r e 11 C h a r a c t e r i z a t i o n c u r v e s o f s t r a t e g i e s A , B, C ( l e f t ) and l e a r n i n g c u r v e ( r i g h t ) f o r c r i t e r i o n o f t i m e o f d e c i s i o n making.
T
141
'liansformatwn learning tlieory
t o i d e n t i c a l indexes when u s i n g two d i f f e r e n t s t r a t e g i e s .
These p o i n t s
can be examined as t h e t u r n i n g f r o m one s t r a t e g y i n t o a n o t h e r , more p e r f e c t , when l e a r n i n g ( f r o m A i n t o
B and from 8 i n t o C ) .
T h i s phenomenon can be
p a r t i a l y e x p l a i n e d on t h e base o f t h e d a t a about t h e t r a n s f e r o f s k i l l s . Now t h e n a t u r a l q u e s t i o n a r i s e s : how t h e strategy-change g i v e s us a A B C p o s s i b i i t y t o decrease an o p t i m a l l e v e l o f c o m p l e x i t y tmin> tmin> tmin, maxi ma 1 y a c h i e v a b l e when u s i n g each s t r a t e g y . I t ' s necessary t o make a r e s e r v a t i o n t h a t i n t h e g i v e n case we mean one o f f o u r l e v e l s o f e x p e r i m e n t a l d a t a a v e r a g i n g : t o t a l ( f o r a l l men), c o n t i n g e n t a l ( f o r c e r t a i n mainland o f s t u d e n t s ) , group ( f o r group of people from t h e s t u d e n t s c o n t i n g e n t who a r e u n i t e d by some comnon c h a r a c t e r i s t i c s ) and t h e n t h e i n d i v i d u a l l e v e l o f t h e a n a l y s i s o f e x p e r i m e n t a l d a t a conc e r n i n g t h e s t u d y i n g c o n c r e t e man.
We have t h e r e t h e main l e v e l s u s u a l l y
shown when we t a l k about t h e m u l t i - l e v e l mutual a d a p t a t i o n o f man and machine i n t h e man-machine systems [21, 311. main components o f such an a d a p t a t i o n .
The l e a r n i n g i s between t h e
F o r a l l t h a t , when r e v i e w i n g
e v o l u t i o n a r y and t r a n s f o r m a t i o n a l processes i n t h e f i e l d o f l e a r n i n g s t r a t e g i e s we a l s o draw o u r a t t e n t i o n t o t h e mutual a d a p t a t i o n o f man and l e a r n i n g c o n d i t i o n s , means, i n s t r u c t i o n s .
P a r t i c u l a r l y , the s e l e c t i o n o f
o p t i m a l , i n d i v i d u a l l y adapted t o t h e s t u d e n t means of t h e f u l f i l l i n g e d u c a t i o n a l t a s k s , g i v e s us a p o s s i b i l i t y t o make t h e t r a n s f o r m a t i o n f r o m one s t r a t e g y i n t o a n o t h e r , s i m p l e and t o a c q u i r e them t i l l t h e l e v e l demanded. So, l e t ' s r e t u r n t o t h e q u e s t i o n about t h e n a t u r e of p e r f e c t i o n a c t i v i t y i n d i c e s when l e a r n i n g and t u r n i n g f r o m one s t r a t e g y i n t o another. L e t ' s c o n s i d e r a v e r y s i m p l e example.
We t e a c h a s u b j e c t t o i d e n t i f y
(make a d i a g n o s i s ) 16 d i f f e r e n t s t a t e s o f some o b j e c t .
An o b j e c t i s r e f l e c t e d by means o f 30 i n f o r m a t i o n a l elements and each one can have two
d i f f e r e n t s t a t e s : parameter d i s p l a y i n g n o r m a l l y and p a t o l o g i c a l l y . A t t h e f i r s t stage o f l e a r n i n g t h e s u b j e c t e v a l u a t e s t h e o b j e c t ' s s t a t e l o o k i n g o v e r a l l 30 elements.
The i n f o r m a t i o n a l c o n t e n t s o f s i g n a l , i . e .
o f c o n c r e t e p e r c e i v i n g s t a t e o f parameter complex, Ia=3010g22=30 b i t s .
142
V.F. Venda
A t t h e second s t a g e o f l e a r n i n g t h e s u b j e c t groups 30 b i t s i n t o f i v e f u n c t i o n a l groups and each one possesses f o u r d i f f e r e n t s t a t e s .
An
i n f o r m a t i o n a l c o n t e n t s o f an analogous s i g n a l , c o n c r e t e s t a t e o f i n f o r m a t i o n d i s p l a y system
(DIS), Ib=510g24=10 b i t s .
F i n a l l y , a t t h e t h i r d stage t h e s u b j e c t l e a r n s t o i d e n t i f y t h e s t a t e of a l l t h e combinations f r o m 30 i n f o r m a t i o n a l elements as a whole, and f o r a l l t h a t t h e g e n e r a l number o f t h e s t a t e s of t h i s c o m b i n a t i o n , s i m u l t a n e o u s l y distinguished, i s 16. I,=log
Then t h e i n f o r m a t i o n a l c o n t e n t s of s i g n a l
2 i6 = 4 b i t s .
When comparing w i t h t h e w h o l i s t e s t i m a t i o n o f each f r o m 1 6 s t a t e s , t h e abundance o f i n f o r m a t i o n a l c o n t e n t s o f t h e s i g n a l a t t h e f i r s t s t a g e IaiIC=7,5,
and a t t h e second s t a g e I b / I c = 2 , 5 .
The s t r a t e g y o f t h e i d e n t i f i c a t i o n o f t h e o b j e c t ' s s t a t e s , from t h e i n f o r m a t i o n a l complex s t a t e , used by t h e s u b j e c t a t t h e t h i r d s t a g e o f l e a r n i n g i s more s p e c i a l i z e d
-
i t ' s more adapted f o r t h e i d e n t i f i c a t i o n
Qc=16 c o n c r e t e s t a t e s b y means o f t h e i r d i f f e r e n t i a t i o n . The s t r a t e g y t h a t i s used a t t h e second s t a g e i s more u n i v e r s a l t h a n t h e s t r a t e g y e x i s t i n g a t t h e t h i r d s t a g e : i t h e l p s t o d i f f e r Qb=2lo s t a t e s of object.
B u t t h e second s t r a t e g y i s more s p e c i a l i z e d t h a n t h e f i r s t : i t
h e l p s t o d i f f e r 220 t i m e s l e s s s t a t e s t h a n by means o f t h e f i r s t s t r a t e g y
Q
=z3'.
One can see from i t t h a t f o r r e l a t i v e l y minimal decrease o f t h e a s i t u a t i o n s i d e n t i f i c a t i o n t i m e we pay a s i g n i f i c a n t l i m i t a t i o n o f freedom degrees number : t a / t b = l , 3 3 ; Qa/Qb=220; ta/tc=2,35; Qa/Qc=226 .
And now t h e i m p l i c a t i o n f o l l o w s t h a t each f o l l o w i n g s t r a t e g y as a r u l e ,
improves t h e d e c r e a s i n g o f f u l f i l l i n g t a s k s c o m p l e x i t y a t t h e expense o f i t s more s p e c i a l i z a t i o n as compared w i t h t h e p r e c e d i n g s t r a t e g y . :f we r e t u r n t o t h e d a t a a b o u t t h e s o l v i n g problems i n mnemoscheme, t h e n
g r a p h i c a l l y t h i s phenomenon can be expressed by t h e f a c t t h a t t h e w i d t h of t h e B s t r a t e g y c u r v e (an i n t e r v a l of v a l u e s of n o r o t h e r p s y c h o l o g i c a l f a c t o r s o f problem s o l v i n g c o m p l e x i t y (KJR) i n which t h e B s t r a t e g y h e l p s t o manage w i t h t h e problem) a t t h e F i g . 11 i s more narrow t h a n t h e w i d t h o f
143
Tronsfonnatiori leamirig tlieory
the A strategy.
A c c o r d i n g l y , t h e i n t e r v a l o f i n d i c e s n, i n which t h e
s t r a t e g y C can be r e a l i z e d , i s l e s s t h a n t h e i n t e r v a l s f o r t h e s t r a t e g i e s A and B.
S t a t e d o t h e r w i s e , a l t h o u g h t h e s t r a t e g y A has an optimum which i s worse b t h a n t h o s e o f t h e B and C s t r a t e g i e s (tiin> tmin> tkin), and s i m u l t a n e o u s l y t h e B s t r a t e g y can, even w i t h d i f f i c u l t y , be r e a l i z e d when n moves a s i d e s i g n i f i c a n t l y f r o m nA
-
an o p t i m a l i n d e x o f t h i s f a c t o r (under t h e
I n experiments on t h e s o l v i n g o p e r a t i v e problems i n mnemoscheme, i s found t h a t t h i s i n t e r v a l f o r t h e A s t r a t e g y i s AnA and C A f o r t h e C ' s t r a t e g y i s An << An The curves on t h e l e f t s i d e we name
circumstances o f t>t$;.).
.
c h a r a c t e r i z a t i o n curves o f s t r a t e g i e s . On t h e r i g h t s i d e o f F i g . 11 i s shown t h e c o n s t r u c t i n g of t h e l e a r n i n g ci'rve, i.e.
o f t h e change o f t h e meantime o f t h e s o l v i n g problems t i n mnemoscheme
i n the c u r r e n t o f the time o f l e a r n i n g
T.
We t r a n s f e r i n t o t h e r i g h t s i d e of Fig. 11 t h e i n d i c e s o f tan( t h e t i m e o f s o l v i n g a t t h e b e T i n n i n g o f l e a r n i n g ) , tAmin, t
~
t~
~ y t~~ mine'
The number o f t h e i n f o r m a t i o n a l elements i n t h e g i v e n case i s l o o k e d t h r o u g h as a g e n e r a l i z e d p s y c h o l o g i c a l f a c t o r o f t h e c o m p l e x i t y [21, 311.
n is a
is its r e a l index o f t h e psychological f a c t o r o f complexity, n opt t h e o r e t i c a l l y o p t i m a l i n d e x f o r t h e c o n c r e t e s t r a t e g y ( i = A , B, C ) ;
n""
#are
nn
and
t h e i n d i c e s o f t h e p s y c h o l o g i c a l f a c t o r o f c o m p l e x i t y which a r e
common f o r b o t h s t r a t e g i e s .
P o i n t e r s i n t h e l e f t s i d e o f F i g . 11 show t h e
d i r e c t i o n of t h e t ( K . ) change d u r i n g t h e l e a r n i n g . T u r n i n g f r o m one JP BC B s t r a t e g y i n t o a n o t h e r i n t h e g i v e n case occurs when nAB< ntpt and n < nopt, i . e . when t h e r e a l i n d e x o f t h e p s y c h o l o g i c a l f a c t o r o f c o m p l e x i t y decreases as compared w i t h i t s o p t i m a l i n d e x f o r g i v e n s t r a t e g y . t A B > tA
and t B C > tB
Consequently,
T h i s f a c t does e x p l a i n t h e emergency o f humps
when t u r n i n g f r o m t h e A s t r a t e g y i n t o t h e B s t r a t e g y and from t h e B s t r a t e g y i n t o t h e C s t r a t e g y , w i t h t h e maximum a t t h e moments TAB and TBC. I f we would t a k e as t h e c r i t e r i o n of c o m p l e x i t y r a t h e r a n o t h e r i n d e x t h a n t h e time, f o r i n s t a n c e , t h e e f f e c t i v i t y o r t h e speed o f t h e problem s o l v i n g which i s adverse t o t h e s o l u t i o n t i m e , t h e n t h e g r a p h i c s Q(K ) and Q ( T ) jR
144
U
V.F. Venda
145
Transformation kaniing theory
would l o o k as shown i n F i g . 12, where t h e case when K . i n c r e a s e s i n t h e JR process o f s t r a t e g y e v o l u t i o n and t r a n s f o r m a t i o n d u r i n g t h e l e a r n i n g , i s presented. The d i f f e r e n t r e l a t i v e p o s i t i o n s o f c h a r a c t e r i z a t i o n curves o f s t r a t e g i e s g i v e us d i f f e r e n t t r a n s f o r m a t i o n phases on t h e l e a r n i n g curve. o f t h e A and B s t r a t e g i e s (see F i g . 11, 12
-
Optimums
l e f t ) a r e f a r each f r o m
a n o t h e r and t h e i r comnon p o i n t AB i s v e r y low.
That i s because we f i n d
d o w n h i l l t h e phase o f A t o B t r a n s f o r m a t i o n a t t h e l e a r n i n g c u r v e (see F i g . 11, 12
-
r i g h t ) ; B and C s t r a t e g i e s a r e c l o s e r and t r a n s f o r m a t i o n
phase has o n l y one p l a t e a u .
L a t e r we s h a l l show p a r t i c u l a r cases where
t r a n s f o r m a t i o n s may be almost e x p o n e n t i a l monotonous. I t ' s w e l l known t h a t t h e s o l v i n g o f t h e complex problems, p a r t i c u l a r l y , connected w i t h t h e o p e r a t i v e c o n t r o l o f o b j e c t s , i s c h a r a c t e r i z e d by t h e presence o f a wide range o f v a r i a n t s .
There can be, t h e o r e t i c a l l y , an
i n f i n i t e number o f s o l v i n g s t r a t e g i e s . F i g . 13 shows a c e r t a i n s e t of r e a l l y p o s s i b l e s t r a t e g i e s ( t h i s s e t i s p r a c t i c a l l y 1 i m i t e d by t h e t e c h n i c a l d e v i c e s , a b i l i t i e s o f d i f f e r e n t p e o p l e t o r e l a t e one t o a n o t h e r , t o t r a n s l a t e t h e e x p e r i e n c e , t o s t a n d a r d i z e i t and so
On).
I f we draw o u r a t t e n t i o n t o t h e d e c r e a s i n g o f t h e l e a r n i n g t h e n i n s p i t e o f t u r n i n g t h e s t r a t e g y A i n t o B-C-D-E-F,
we would r e a l i z e f a s t e r t u r n i n g
o f t h e A s t r a t e g y d i r e c t l y i n t o D-F, a l t h o u g h , i t ' s connected w i t h g r e a t p r o v i s i o n a l d i f f i c u l t i e s and w i t h t h e d e c r e a s i n g o f Q e f f e c t i v i t y d u r i n g t h e p e r i o d s TT\D and T;F.
When t h i s t u r n i n g e x i s t s , t h e l e v e l o f t h e i n d e x
o f e f f e c t i v i t y c r i t e r i o n QFopt would a c h i e v e TFopt before t h e s e q u e n t i a l t u r n i n g from A i n t o
B- ...-F.
The a n a l y s i s shows t h a t t h e l e a r n i n g process, which i s r e f l e c t e d by t h e c u r v e Q ( T ) may proceed monotonously, w i t h o u t "humps", i f t h e c u r v e o f t h e f o l l o w i n g s t r a t e g y qi(K. ) crosses t h e c u r v e of t h e p r e c e d i n g s t r a t e g y JP Q. ( K . ) near by i t s optimum. The g r a d u a l "peaceful g r o w i n g " o f t h e new 1-1 JP s t r a t e g y i n t h e o l d one i s proceeding.
Figure 13 Constructing of d i f f e r e n t learning processes with d i f f e r e n t transformation of s t r a t e g i e s .
Transformation learning theory
147
The e d u c a t i o n a l experiments aimed a t t h e e x p e r i m e n t a l v e r i f i c a t i o n o f t h e h y p o t h e s i s about t h e r i s e o f "humps" d u r i n g t h e t u r n i n g f r o m one s t r a t e g y i n t o a n o t h e r and about t h e dependence o f t h e "humps" h e i g h t upon t h e d i s t a n c e between t h e s t r a t e g i e s , namely, between t h e o p t i m a l q u a l i t i e s o f t h e l e a d i n g f a c t o r s o f c o m p l e x i t y were c a r r i e d o u t and t h e d i f f e r e n t t y p e s o f problems and t h e t e c h n i c a l l e a r n i n g devices were used i n i t . J o i n t l y w i t h Y.M.
Magazanik and M.V.
Rybalchenko t h e comparison o f t h e
l e a r n i n g processes i n o p e r a t o r s o f power p l a n t s u s i n g usual mnemoschemes ( 1 ) , mnemoschemes w i t h t h e e l u c i d a t i o n of t h e a c t u a l c o n t o u r s connected w i t h t h e problems ( 2 ) and t h e command and i n f o r m a t i o n a l mnemoschemes ( 3 ) , which r e f l e c t v i s u a l l y t h e a l g o r i t h m o f t h e problem s o l v i n g were a l s o c a r r i e d o u t [?I]
The experiments have shown t h a t t h e r e g u l a t i o n o f t h e
v i s u a l m a t e r i a l a t t h e stage o f mnemoscheme d e s i g n b r i n g s t h e number o f t h e s t r a t e g i e s s u c c e s s i v e l y a c q u i r e d by t h e s t u d e n t s down, b u t when d i s p l a y i n g t h e a l g o r i t h m s , o n l y t h e e v o l u t i o n of t h e unique s t r a t e g y C, s t r i c t l y determined by t h e s t r u c t u r e o f t h e i n f o r m a t i o n a l system occurs, and t h a t ' s why t h e c u r v e 3 i n F i g . 14 has a monotonous a s y m p t o t i c c h a r a c t e r . Curve 2 r e f l e c t s t h e s e q u e n t i a l a c q u i r i n g o f two s t r a t e g i e s B2 and C2, and c u r v e 1,
o f a l l t h e main s t r a t e g i e s A1,
B1 and C1.
A1 i n c l u d e s t h e r e v i e w i n g o f a l l t h e i n f o r m a t i o n a l f i e l d and t h e d i s t i n g u i s h i n g o f t h e p a r t o f i n f o r m a t i o n , which r e l a t e d t o t h e c o n c r e t e problem B1 p e r m i t s t o b e g i n t h e p r o c e s s i n g of t h e r e l e v a n t i n f o r m a t i o n a t once,
because t h e c o n t o u r s a c t u a l f o r t h e problem a r e e l u c i d a t e d i n t h e mnemoscheme.
S t r a t e g y C3 i s d i r e c t l y connected w i t h t h e forming o f t h e s o l v i n g .
The a n a l y s i s o f non-monotonous curves of l e a r n i n g shows t h a t a b s o l u t e i n d i c e s o f the complexity c r i t e r i o n ( f o r instance, the solving time) i s n o t enough t o e v a l u a t e t h e l e v e l o f man's l e a r n i n g t o s o l v e problems.
Let's
propose t h a t t h e s t u d e n t o p e r a t o r when working w i t h t h e usual mnemoscheme ( F i g . 14, c u r v e 1 ) has shown i n t h e experiment t=132 sec.
L e t ' s a l s o draw
i n F i g . 14 a h o r i z o n t a l l i n e , which corresponds t o t h i s i n d e x and t h e n we conclude t h a t such an index of s o l v i n g t i m e may be achieved under t h e circumstances o f f i v e q u a l i t a t i v e l y d i f f e r e n t ways of s o l v i n g .
The p o i n t a
corresponds t o t h e A s t r a t e g y p l a t e a u where t h e s u b j e c t s a f e l y possesses t h i s strategy.
The experiments have shown t h a t if t h e s t u d e n t i s a t t h e
I46
V . F . Vmda
F i a u r e 14
Exnerimental learning curves.
Transformation learning theory
149
o p t i n i l l l e v e l i n one of t h e s t r a t e g i e s ( i n t h e g i v e n case o f t h e A s t r a t e g y ) , t h e n t h e s t r u c t u r e and i n d i c e s of h i s s o l v i n g o p e r a t i v e problems, b e l o n g i n g t o a l l t y p e s researched ( t h i n k i n g , p e r c e p t u a l and senso-motor)
, are
m a x i m a l l y s t a b l e t o t h e i n f l u e n c e of s t r e s s f a c t o r s and m i n i m a l l y dependent on t h e human's p s y c h o p h y s i o l o g i c a l s t a t e . The p o i n t b i n F i g . 14 corresponds t o t h e i d e n t i c a l index t=132 sec.,
but
i t l i e s a t t h e b e g i n n i n g o f t h e t u r n i n g from t h e A1 s t r a t e g y i n t o t h e B1
s t r a t e g y , and t h i s f a c t i s connected w i t h t h e d e v i a t i o n f r o m t h e s o l v i n g s t r u c t u r e o p t i m a l f o r t h e A s t r a t e g y and w i t h t h e p a r t i a l t r a n s f o r m a t i o n o f t h i s structure.
The d i s p e r s i o n o f t h e s o l v i n g i n d i c e s i n t h i s p o i n t ,
as a l s o i n t h e C p o i n t ,
-
i n t h e a b r u p t l o s s o f t h e B1 s t r a t e g y p e r f e c t i o n
-
i s more t h a n i n t h e p o i n t a. The i n d i c e s i n t h e p o i n t s b and c a r e more exposed t o t h e i n f l u e n c e o f s t r e s s f a c t o r s which a f f e c t t h e man when he s o l v e s t h e problem ( t h e c o n d i t i o n s o f exams were used as such f a c t o r s i n experiments w i t h t h e students). The p o i n t s d and e l i e a t t h e r i d g e of t u r n i n g from t h e Bl s t r a t e g y i n t o the
C1 s t r a t e g y .
There i s a maximal d i s p e r s i o n o f i n d i c e s .
The a n a l y s i s o f t h e l e a r n i n g c u r v e 1 i n F i g . 14 shows t h a t t h e drawing a t t e n t i o n a t t h e a b s o l u t e i n d e x o f c o m p l e x i t y c r i t e r i o n i s n o t enough t o a s s e r t about t h e l e v e l of t h e s u b j e c t ' s t r a i n i n g ; i t ' s a necessary q u a l i t a t i v e and q u a n t i t a t i v e a n a l y s i s o f t h e s t r a t e g y w i t h t h e h e l p o f which t h i s i n d e x o f c r i t e r i o n i s achieved. Under t h e circumstances o f t=132 sec.,
t h e l e a r n i n g can be stopped o n l y
i n one f r o m f i v e p o i n t s , which corresponds i n c u r v e 1 t o t h i s index, namely, i n t h e p o i n t a, when t h e s t u d e n t a c q u i r e s s a f e l y t h e A1 s t r a t e g y , a l t h o u g h i t ' s m i n i m a l l y p e r s p e c t i v e f r o m t h e p o s s i b l e s e t A1,
B1,
C1.
To b r a k e t h e l e a r n i n g i n t h e p o i n t c i s n ' t an e x p e d i e n t t h i n g , because t h e n t h e small i n t e r v a l of l e a r n i n g ( n e a r l y two days) i s enough f o r s i g n i f i c a n t d e c r e a s i n g and s t a b i l i z a t i o n o f t h e t i n d e x .
I f the learning i n b o r s t i l l
worse i n p o i n t d brakes, t h e s t u d e n t appears t o be i n t h e v e r y d i f f i c u l t
150
I’.F. Venda
s i t u a t i o n : he does s a f e l y possess no one s t r a t e g y . We should account t h e f a c t t h a t t a = t b = t c = t d = t e n o t o n l y d u r i n g t h e o r g a n i z a t i o n of t h e l e a r n i n g p r o c e s s , b u t a l s o d u r i n g t h e s o l v i n g o f t h e problem of t h e t r a n s f e r o f t h e l a b o r a t o r y r e s u l t s i n t o t h e p r a c t i c e o f t h e r e a l working a c t i v i t y organization. The method o f p l a n n i n g experiments based on t h e models of R . Bush-
F. M o s t e l l e r , i s d e s c r i b e d some pages l a t e r .
Its
s i g n i f i c a n t element
was t h e comparison between t h e s o l v i n g i n d i c e s o f i d e n t i c a l problems i n e x p e r i e n c e d o p e r a t o r s ( i n t h e c o n t r o l e x p e r i m e n t w i t h one o f t h e t y p e s o f t o o l s o r o f t h e j o b c o n d i t i o n s ) and i n s u b j e c t s who g i v e t h e i n d i c e s f o r comparative e v a l u a t i o n o f d i f f e r e n t types.
9 e t u r n i n g t o F i g . 14 and
p r o p o s i n g t h a t t h e e x p e r i e n c e d o p e r a t o r s show t h a t d u r i n g t h e problem s o l v i n g t o p = 132 sec. and i t c o i n c i d e d w i t h t h e i n d i c e s o f s u b j e c t s tsub:132 sec.,
t h e n from t h i s c o i n c i d e n c e top=tsub there i s not t o
conclude t h e correspondence between t h e s o l v i n g i d e n t i c a l problems by t h e s u b j e c t s and e x p e r i e n c e d o p e r a t o r s , because t h e s t r a t e g i e s o f s o l v i n g may be s i g n i f i c a n t l y d i f f e r e n t . Thus, a l s o , d u r i n g t h e o r g a n i z a t i o n of a l e a r n i n g process, choosing o f i t s l o n g i t u d e , f i n i s h i n g l e v e l o f i n d i c e s i s needed, and d u r i n g t h e s o l v i n g q u e s t i o n about t h e p r a c t i c a l u s e f u l n e s s of t h e p s y c h o l o g i c a l experiments t o g e t h e r w i t h t h e a b s o l u t e i n d i c e s must be compared t h e s t r a t e g i e s o f s o l v i n g w i t h t h e h e l p o f which t h e i n d i c e s g i v e n a r e achieved. The main a t t e n t i o n must be drawn a t t h e a n a l y s i s of t h e s t r a t e g y t r a n s f o r m a t i o n as t h e most complex and unsteady zones d u r i n g t h e l e a r n i n g . I t ’ s necessary t o say t h a t t h e concepts o f s t a b i l i t y and u n s t a b i l i t y a r e c e r t a i n l y r e l a t i v e ; t h e y must be added by t h e q u a n t i t a t i v e v a l u e s .
In
t h e experiments d e s c r i b e d , i t i s found t h a t a t t h e analogous phases o f a c q u i r i n g d i f f e r e n t s t r a t e g i e s , t h e d i s p e r s i o n i s l e s s when u s i n g a more p e r f e c t e d ( p r e c i s e l y , more s p e c i a l i z e d ) s t r a t e g y . Moreover, t h e s i g n i f i c a n t unevenness o f s o l v i n g s t a b i l i t y i n d i c e s when a c q u i r i n g one s t r a t e g y i s found.
F o r i n s t a n c e , when l e a r n i n g t o s o l v e
151
Transformation learning theory
problems i n t h e command and i n f o r m a t i o n a l mnemoscheme, and t h i s l e a r n i n g i s c h a r a c t e r i z e d by t h e s t r a t e g y s t a b i l i t y , t h e r e a r e some zones w i t h t h e decreasing dispersion o f indices. I t g i v e s us a p o s s i b i l i t y t o make a p r o p o s a l about few l e v e l s o f d e t a l i t y
o f l e a r n i n g processes research.
When we s t u d y t h e process o f t h e p e r f e c t i o n
t h e ways o f a c h i e v i n g t h e c e r t a i n c l a s s o f aims and o f t h e s o l v i n g t h e c o r r e s p o n d i n g problems a t a l l , t h e p o s s i b l e s t r a t e g i e s may be u n i t e d by t h e more general concept of s o l v i n g p o l i c y .
A t t h i s l e v e l of a n a l y s i s
o f t h e p o l i c y , t h e t r a n s f o r m a t i o n s between t h e s t r a t e g i e s may be reviewed as monotonous.
A c c o r d i n g l y , namely, t h i s i s proposed by t h e a u t h o r s o f
“ I n t r o d u c t i o n t o ergonomics” who work o u t a “macro-approach” t o t h e l e a r n i n g processes a n a l y s i s . From t h e o t h e r s i d e , n o t o n l y t h e r e s e a r c h o f t h e s o l v i n g p o l i c i e s , b u t a l s o of s t r a t e g i e s must become d e t a l i z e d when r e s e a r c h i n g i n d i v i d u a l l y adapted methods o f l e a r n i n g , c o n t r o l l i n g t h e experiments and so on. t h i s case, t h e s t r a t e g y must be segregated i n t o t a c t s
-
In
more narrow
c l a s s e s o f t h e s o l v i n g ways.
In t h i s case, i t ’ s p o s s i b l e sometimes t o f i n d t h e ‘‘humps’’ even i n such smooth, monotonous c u r v e s as C3 i n F i g . 14. I n t h e i r t u r n , t a c t i c s a r e d i v i d e d i n t o c o n c r e t e ways and a c t s of s o l v i n g and a l s o may be presented as t h e curves w i t h humps.
I t seems t o be e x p l a i n e d by t h e general
r e g u l a r i t y o f t h e o s c i l l a t i o n s of t h e s o l v i n g processes as such. U
-
o i d curves analogous t o t h e curves i n t h e l e f t p a r t o f F i g . 13,
q u a l i t a t i v e l y r e f l e c t t h e dynamics of p o l i c i e s i n a whole, s t r a t e g i e s , If Q
t a c t i c s , c o n c r e t e s o l v i n g and even t h e s i n g l e s t e p i n t h e s o l v i n g .
i s i n t e r p r e t e d as t h e success p r o b a b i l i t y i n each s t e p of l e a r n i n g , t h e n t h e c u r v e a t t h e r i g h t s i d e would show t h e t r a c k o f o s c i l l a t i o n s s t e p s around t h e maximal success p r o b a b i l i t y .
-
search
Such a t r a c k would show
t h e r e g u l a r ( b u t n o t t r i a l and e r r o r ) search o f s o l v i n g based on t h e u s i n g
o f p o l i c i e s , s t r a t e g i e s , t a c t i c s , ways o f s o l v i n g worked o u t by t h e g i v e n i n d i v i d u a l and h i s predecessors and analogous t o met i n t h e g i v e n case. Each s t e p o f s o l v i n g may be connected w i t h t h e o p t i m a l index of a p a r t i c u l a r psychological f a c t o r o f complexity.
I t ’ s necessary t o emphasize t h e
c o n v e n t i o n a l i t y o f t h e p r e s e n t a t i o n s o f dependences t ( K j p ) ,
Q(Kjp) a t t h e
p l a n e ; everywhere we must l o o k a t t h e few g r a n h i e s i n c o r n e c t i o n w i t h t h e numher o f o s y c h o l o o i c a l f a c t o r s o f comnlexi t y s i m u l t a n e o u s l v . we reviewed 15 d i f f e r e n t nroblems s o l v i n p .
K
in
I n 122,311
i n the case o f t h e a n a l v s i s o f o o e r a t i v e
I t i s necessary t o c r e a t e s p e c i a l methods o f t h e g r a p h i -
c a l d i s n l a y i n q o f such m u l t i - f a c t o r nrocesses o f t h e o n t i m i z a t i o n o f t h e i n f o r m a t i o n a l base o f s o l v i n g , i n which t h e s o l v i n o becomes a c c e s s i b l e t o t h e c o n c r e t e i n d i v i d u a l i n t h e c o n c r e t e exDerience (when we r e a l i z e t h e i n d i v i d u a l l e v e l o f a n a l v s i s ) , t o tk c c n t i n a e n t o f s n e c i a l i s t s o r t o t h e "mean" man (when we r e a l i z e t h e t o t a l l e v e l o f a t l a l y s i s and e x n e r i m e n t a l data o e n e r a l i z a t i o n ) .
As we see i n F i a . C , t h e r e s e a r c h o f t h e i n f l u e n c e
o f t b e t e c h n i c a l d e v i c e s t v n e s t o t h e Drocesses o f e v o l u t i o n and t r a n s f o r m a t i o n o f t h e s t r a t e q i e s t a k e s an i m o r t a n t D a r t f o r t h e l e a r n i n g nrocesses ontimization. I n t h e e x n e r i m e n t s h e l d by o u r l a b o r a t o r y t h e l e a r n i n o o f t h r e e groups o f o o e r a t o r s was s t u d i e d ( s e e F i p . 1 5 ) .
The f i r s t n r o u n ( c u r v e 1) s o l v e d t h e
nroblerns i n t h e t e x t i n s t r u c t i o n s , t h e second qroua i n t h e command and i n f o r m a t i o n a l mnemoschemes ( c u r v e 2 ) .
The t h i r d groun a f t e r few lessons,
connected w i t h t h e t e x t i n s t r u c t i o n s , t u r n e d i n t o t h e s o l v i n a w i t h t h e h e l n of cornrend and i n f o r m a t i o n a l mnemoschemes.
A1 thouqh t h e s o l v i n o o f opera-
t i v e problems w i t h t h e h e l n o f such mnemoschemes i s s i o n i f i c a n t l y e a s i e r than i n the t e x t i n s t r u c t i o n s , t h e t u r n i n a f r o m i n s t r u c t i o n s i n t o mnemoschemes i s connected w i t h s i g n i f i c a n t o r o v i s i o n a l i n c r e a s i n o o f t h e s o l v i n a comnlexi ty, and t h e n t h i s c o m n l e x i t y r a p i d l y decreases (see F i g . 15, c u r v e
3). Y . A . Cvozdev's e x p e r i m e n t connected w i t h t e a c h i n p t h e groun o f s t u d e n t s t o
r e a d t h e o o e r a t i v e i n f o r m a t i o n n u i c k l y u s i n o t h e metronome w h i c h s e t t h e r'lytliti o f r e a d i n g and nacer, which h e l o e d t o r e o u l a t e t h e sDeed o f t h e transference o f t e x t n r c j e c t e d t o the screen, c o n f i r w d the r e g u l a r i t i e s o f e v o l u t i o n and t r a n s f o r m a t i o n found (see F i a . 1 6 ) . I t i s necessary t o o o i n t o u t , t h a t t h e Dhenomena analoaous t o such which
a r e observed when t r a n s f o r m i n a s t r a t e g i e s d u r i n g t h e l e a r n i n g o f t h e man t o t h e o p e r a t i v e o e r c e n t i o n and o r o c e s s i n g o f t h e i n f o r m a t i o n , a r e c h a r a c t e r i s t i c a l f o r a g r e a t number o f o t h e r nrocesses o f human's a d a p t a t i o n t o t h e external conditions
.
153
360
300 240
180
120
60 1
2
3
4
5
6
7
8
9
N aaye
Figure 15 Experimental l e a r n i n g curves on t h e o p e r a t i v e problem s o l v i n g w i t h mnemonic schemes as t h e d i s p l a y s .
154
Figure 16 Experimental learning curve on the speedreading. C : read words per one minute N : days o f learning
I55
Transformation learning theory
Thus, f o r i n s t a n c e , B.F. Lomov showed t h e process of d a r k a d a p t a t i o n o f t h e human's v i s u a l a n a l i z e r when t h e c o l o u r c h a r a c t e r i s t i c s o f l i g h t a r e different.
F. S n e i d e r and N. Pronco t a u g h t men t o group game cards
a c c o r d i n g l y w i t h t h e d i f f e r e n t s i g n s and p u t on s u b j e c t s g l a s s e s r e v e r s i n g t h e image d u r i n g t h e p e r i o d from t h e 1 7 t h t o 4 5 t h experiment.
A f t e r the
g l a s s e s were t a k e n o f f , t h e temporal d e c r e a s i n g o f i n d i c e s was observed, i d e n t i c a l l y t o t h e f a c t o f s i g n i f i c a n t d e c r e a s i n g o f o p e r a t o r ' s work when he t u r n s from t h e complex t e x t i n s t r u c t i o n i n t o more s i m p l e comnand and i n f o r m a t i o n a l mnemoschemes.
3. CONCLUSIONS Thus, t h e s y s t e m a t i c s t u d i e s o f c h a r a c t e r i z a t i o n e q u a t i o n s , curves o f s t r a t e g i e s and o f e v o l u t i o n s and t r a n s f o r m a t i o n s o f s t r a t e g i e s d u r i n g t h e l e a r n i n g were t h e fundamentals o f a new t r a n s f o r m a t i o n t h e o r y o f l e a r n i n g processes.
The t h e o r y has a t h e o r e t i c a l and p r a c t i c a l s i g n i f i c a n c e ,
p a r t i c u l a r l y , f o r t h e s o l v i n g o f t h e problem o f t h e o p t i m a l c o n t r o l o f t h e l e a r n i n g process, and f o r more p e r f e c t e x p e r i m e n t a l and a n a l y t i c a l models o f t h i s process b u i l d i n g . The process of l e a r n i n g may be d e s c r i b e d by t h e t r a d i t i o n a l monotonous models o n l y in p a r t i c u l a r cases, a t t h e s t a g e o f t h e c o n c r e t e s t r a t e g i e s e v o l u t i o n o r when t h e s u b j e c t consequently a c q u i r e s n e a r l y l y i n g s t r a t e g i e s and a l s o when i t ' s a v e r a g i n g t h e g r e a t a r r a y s o f e x p e r i m e n t a l data. The i n d i v i d u a l i z a t i o n o f t h e methods and means o f t h e o p e r a t o r ' s l e a r n i n g which becomes wide-spread nowadays makes q u i t e a c t u a l t h e development o f a "micro-approach" t o t h e s t u d y o f l e a r n i n g processes t o g e t h e r w i t h t h e t r a d it i on a 1 "ma c r o -a p p r o ac h " .
Monotonous and non-monotonous ( w i t h i n t e r m e d i a t e p l a t e a u s and "humps") forms o f curves a r e s i g n i f i c a n t l y dependent on t h e l e v e l o f p s y c h o l o g i c a l a n a l y s i s o f t h e l e a r n i n g processes.
P r a c t i c a l l y , i t i s always p o s s i b l e
t o f i n d a q u i t e h i g h l e v e l o f g e n e r a l i z a t i o n , i n which t h e l e a r n i n g process would be approximated by t h e smooth monotonous curve.
From t h e o t h e r s i d e ,
i t ' s p o s s i b l e t o achieve such a l e v e l o f a n a l y s i s and such d e t a l i z a t i o n degree as t o t h e l e a r n i n g processes dynamics, i n which t h e t r a n s i t i o n a l
V.F. I'enda
156
nrncessec which n;Psen+ a c n e c i a l i n t e r e s t , would he found. F o r t h e a n a l y s i s o f t h e staqes o f s t r a t e a i e s t r a n s f o r m a t i o n , o f models, p a r t i c u l a r l y , n r a n h o - a n a l : i t i c a l
t h e new tynes
ncdels must be worked o u t .
The
main ?nwada!ts i s t h e a c c u m u l a t i o n o f t h e e x o e r i m e n t a l p a t e r i a l w i t h o u t t h e ancient
t r a d i t i o n s o f "smoothing" w h i c h mask t h e most i n t e r e s t i n g psycho-
l o o i c a l n e c u l a r i t i e s o f t h e l e a r n i n o orocesses.
The u s i n q o f t h e system
aonroach oronoses t o s t u d y i t i n t h e d i f f e r e n t l e v e l s : frnm t h e t o t a l uo t o i n d i v i d u a l and o p e r a t i v e , f r o m t h e o o l i c y
UD
t o s i n o l e sten w i t h i n the
c o n c r e t e a c t o f s o l v i n n , and a l s o f r o m t h e v i e w n o i n t s o f c o m n l e x i t y , s t a -
b i 1 it y and e f f e c t i v i t v o f t h e s o l u t i o n s
.
The develo?ment of t h e "micro-anproach" t o t h e a n a l y s i s o f t h e l e a r n i n g nrocesses and t r a n s f o m a t i o n t h e o r y o f l e a r n i n a g i v e us a a o s s i b i l i t y t o use them i n a b r o a d e r c l a s s o f c o g n i t i v e processes as
R
r e t h o 6 of a n a l y s i s
of c o q n i t i v e dynamics i n t h e s c i e n t i f i c and t e c h n i c a l c r e a t i v i t y , t e c h n o l o r r i c a l nrocesses and so on. PEFE RE NCES A t k i n s o n , R.C., Crothers, E -3.. A comparison o f o a i r e d - a s s o c i a t e l e a r n i n ? models h a v i n o d i f f e r e n t a c n u i s i t i o n and r e t e n t i o n axioms, J . o f V a t h e m a t i c a l Psycholooy, Bower, C.H.,
1 ( 1 9 6 4 ) , 285-315.
A o o l i c a t i o n o f a model t o p a i r e d - a s s o c i a t e l e a r n i n g ,
Psychometrika, 26 (1961), 255-280. Bruner, J . S . , Goodnow, J.J.,
Austin, G.A.,
A s t u d y o f t h i n k i n a , :'ew
York. l f il e y ( 1 9 5 6 ) . bush, R.R., lili l e y
V ' o s t e l l e r , F., S t o c h a s t i c models f o r l e a r n i n g , New York,
( 1955 \
Estes, IJ.K.,
. Toward a s t a t i s t i c a l t h e o r y o f l e a r n i n q , P s y c h o l o g i c a l
Review, 57 (1950), 97-107. E s t e s , \.I.K,, The s t a t i s t i c a l anoroach :c
(Ed.),
l e a r n i n n t h e o r y , i n S. Koch
Psycholoqy: a s t u d y o f a s c i e n c e (V.2),
New York, McGraw-Hill,
(lsw). Greeno, J.G..
P a i r e d - a s s o c i a t e l e a r n i n o w i t h s h o r t - t e r m r e t e n t i o n : Ma-
t h e m a t i c a l a n a l y s i s and d a t a r e q a r d i n q i d e n t i f i c a t i o n o f Darameters,
J . o f Mathematical Psycholoay, 4 (1967). 430-472.
157
Transformation learning theory
H u l l , C.L.,
P r i n c i o l e s o f b e h a v i o r : an i n t r o d u c t i o n t o b e h a v i o r tl.Eory,
New York, A o p l e t o n - C e n t u r y - C r o f t s ( 1 9 4 3 ) . I-uce, R.D.,
I n d i v i d u a l c h o i c e b e h a v i o r : a t h e o r e t i c a l a n a l y s i s , New
York , IJi l e y ( 1 9 5 9 ) . I101
postman, L.,
tindenwad, 3.J.. C r i t i c a l i s s u e s
Vemory and coani ti on , 1 (1973)
1111
R e s t l e , F.,
S k i n n e r , B.F., New York
interference theory,
The s e l e c t i o n o f s t r a t e g i e s i n cue l e a r n i n g , P s y c h o l o g i c a l
Review, 69 (1962) 1121
ir,
, 19-40.
, 329-343.
The b e h a v i o r o f organisms: an e x n e r i m e n t a l a n a l y s i s ,
, Anpleton-Century-Crofts
Thorndike, E.L.,
(1938).
The psychology o f a r i t h m e t i c , New York, P!acMillan,
(1922). Purnosive b e h a v i o r i n animals and men, New York, Centurv
Tolman, E.C., (1922).
, Exnerimental
I,Jood,qorth, R.S.
nsycholoay, New York, H o l t (1938).
Chapanis, A,,
Research t e c h n i q u e s i n human e n g i n e e r i n g , Ral t i m o r e (1959).
Ebbinghaus , H ,
, Ueber
Thorndike, E.L.,
das Gedachtnis , Leip-Duncker (1285).
Animal i n t e l l i g e n c e : an exDerimenta1 s t u d y o f t h e as-
s o c i a t i v e processes i n animals, F s y c h o l . Monogr.. Vincent, S.B.,
V.2,
NO8 ( 1 8 9 8 ) .
The f u n c t i o n o f t h e v i b r i s s a l i n t h e b e h a v i o r o f t h e
w h i t e r a t s , Behavior. Monosr. V.5 ( 1 9 1 2 ) . Woodworth and Schosbera's Experimental Psycholoqy, ed. Hol t, R i n e h a r t and !dinston, New York. L o m v , B .F.,
Plan and t e c h n i q u e , !loscow ( 1 9 6 6 ) .
Lomov, B.F.,
Venda, V.F.,
Human f a c t o r s : problems o f a d a o t i n g systems
f o r the i n t e r a c t i o n o f information t o the i n d i v i d u a l : the theory o f h y b r i d i n t e l l i g e n c e , i n Proc. o f t h e Human F a c t o r s S o c i e t y , 2 1 s t Annual Meeting, San F r a n c i s c o ( 1 9 7 7 ) . LOMV, R.F.,
Venda, V.F.
, Methodological
p r i n c i n l e s o f synthesis c f
h y b r i d i n t e l l i g e n c e systems: P r o c . o f ' t h e I n t e r n . Conf. on C y b e r n e t i c s and S o c i e t y , Tokyo (197E). Sheridan, T.B.,
F e r r e l l , l*f.R,.
, Man-Machine
systems: i n f o r m a t i o n , con-
t r o l and d e c i s i o n models o f human performance, The '1IT Press (1974). Venda, V.F.,
I n f o r m a t i o n d i s p l a y systems: Erconomics and d e s i g n aspects,
Moscow ( 1969)
.
( F u s s i an)
Venda, V.F., The n e r s p e c t i v e s o f development o f t h e n s y c h o l o g i c a l t h e o r y o f l e a r n i n g , P s y c h o l o a i c a l Magazine, 4 ( 1 9 8 0 ) . (Russian)
158
1271
V.F. Venda
Venda, V.F.,
P s y c h o l o n i c a l a n a l y s i s o f l e a r n i n ? processes dynamics,
Paper on USP/I'C:l!
apd IISSR/Finland symoosi urns on n s y c h o l o y o f a c t i -
v i t v . Yoscow-Helsinki ( 1 9 8 0 ) . [28]
Venda, V.F.,
t'oies n o u v e l l e s p o u r une t h @ o r i e de l ' a n n r e n t i s s a q e .
PrA-
s e n t e t f u t u r de l a n s v c h o l o a i e du t r a v a i l , P a r i s (1980). 507-517. 1291
Venda, V.F.,
svstems. 1301
Lolnov, R.F.,
Human f a c t o r s l e a d i n p t o e n g i n e e r i n g s a f e t v
Hazard P r e v e n t i o n , \'.15,
Oshanin, D.A.,
Venda, V.F.,
-, Los
r n o e l e s (19FO).
"eqe z u r E r h o l u n g des A r b e i t s e f f e k t s dcs
O w r a t e u r s i n Svstemen "Mensch-Automat", Prohleme und E r q e b n i s s e d e r Psycholopie, E e r l i n (1962).
1311
Venda, V.F.,
E n a i n e e r i n n osycholoqy and s y n t h e s i s o f i n f o r m a t i o n d i s -
p l a v systems, Yoscow ( 2 n d ed. ( 1 9 8 1 ) ) ( 1 9 7 5 ) . ( R u s s i a n )
1321
Venda, V.F.,
Er?onomic nroblems o f i n d i v i d u a l a d a p t a t i o n o f o ? e r a t . o r ' s
work means, Erponomics, 6 (1976). (Paper o r e s e n t e d i n t h e V I Congress
o f the I E A ) .
T R E D S IN MA THEMATICAL PS YCHOLOG Y E. Depejatid]. Vun Bu enlruut (editors) 0 Elsevier Science Publisgrs B.V. (h'ortli-lfohd), 1984
159
A PROBABILISTIC CHOICE MODEL ADAPTED FOF. Tt!E F M L F PERCEPTION EXPERIHEKT N.D.
Verhelst
State U n i v e r s i t y o f Utrecht The Netherlands
A c r o h a b i l i s t i c model i s developed t h a t i s adapted t o t h e d a t a c o l l e c t i o n method known as " p i c k any o u t of n " , under t h e c o n d i t i o n t h a t s t a t i s t i c a l r e p l i c a t i o n s o f t h e choices a r e p o s s i b l e .
The osycho-
p h y s i c a l aroblem o f s y s t e m a t i c e r r o r s i n a n g l e rep r o d u c t i o n i s chosen as a t y p i c a l example.
Further,
i t i s shown under which c o n d i t i o n s t h e model i s n o t
i d e n t i f i a b l e and i t i s p o i n t e d o u t t h a t t h e m d e l i n i t s most g e n e r a l form, a l t h o u g h i d e n t i f i a b l e , may be ill c o n d i t i o n e d because t h e parameter e s t i m a t i o n i s a k i n t o t h e nroblem o f e s t i m a t i n g sums o f exDonentia1 f u n c t i o n s ,
I.
INTRODUCTION
I n t h e p r o b l e m a r e a o f s c a l e c o n s t r u c t i o n , an i m v o r t a n t d i s t i n c t i o n can be made between s c a l e s c o n s t r u c t e d f r o m d a t a which r e f l e c t a dominance r e l a t i o n between i n d i v i d u a l and s t i m u l u s on t h e one hand, and s c a l e s c o n s t r u c t e d f r o m d a t a r e f l e c t i n g a p m x i m i t y r e l a t i o n between i n d i v i d u a l and s t i m u l u s .
This
d i s t i n c t i o n corresponds t o Coombs' d i s t i n c t i o n between Q I I a and Q I I b d a t a r e s p e c t i v e l y (Coombs (1964)). I n t h e framework o f ' l a t e n t t r a i t t h e o r y , i t can be argued t h a t these two k i n d s o f d a t a sidered
-
-
as f a r as b i n a r y d a t a a r e con-
a r e r e f l e c t e d i n models w i t h monotone and single-Deaked i t e m cha-
r a c t e r i s t i c f u n c t i o n s , a d i s t i n c t i o n l a b e l e d by Torqerson (1958) as models w i t h monotone vs. nonmonotone iterns.
I n t h i s paner we w i 11 be concerned
w i t h t h e c o n s t r u c t i o n o f a p r o b a b i l i s t i c model f o r s c a l i n g b i n a r y p r o x i m i t y data.
A t y p i c a l a p p l i c a t i o n i s t h e s c a l i n g o f o b j e c t s f r o m f r e e binar.y c h o i c e data, i.e.,
t h e s u b j e c t i s f r e e t o choose any o u t o f a number of o b j e c t s ( t h e
160
S . D . VerlieL1
s o - c a l l e d n i c k any/n d a t a r a t h e r i n c s t r a t e o y ) .
I n manv s c a l i n q a n p l i c a t i o n s however, e s o e c i a l l y wl-bere a r l t i tudes, a t t i t u d e s o r o r e f e r e n c e s a r e s t u d i e d , a douhle Droblem u s u a l l y must be s o l v e d : t h e s t i m u l i as w e l l as t h e i n d i v i d u a l s must be s c a l e d .
I n the context o f nre-
ference research, t h e s c a l e v a l u e a t t a c h e d t o t h e i n d i v i d u a l i s c a l l e d h i s i d e a l m i n t , ir t e r m which we w i l l use g e n e r i c a l l y i n a b r o a d e r c o n t e x t .
I f t h e r e i s o n l y one resnonse n e r s t i m u l u s - p e r s o n c o m b i n a t i o n a v a i l a b l e , t h e a n y r e o a t i o n o f d a t a ( o v e r nersons f o r examnle) i s made d i f f i c u l t , cause each i n d i v i d u a l w i l l e n t e r t h e model w i t h a new p a r 2 m t e r noint
-
-
he-
his ideal
and these n a r a r r t e r s w i l l a c t as nuisance oarameters i n t h e s t i m a -
t i o n o f t h e s c a l e values o f t h e s t i l r u l i . t e n t t r a i t theory (Fisclier (1974)).
T h i s n r o b l e m i s w e l l known i n l a -
Therefore i t i s i n t e r e s t i n g t o consider
s i t u a t i o n s where t h e r e s e a r c 9 e r has t h e o p o o r t u n i t y t o ? c c u l r u l a t e i n f o r m a t i o n w i t h o u t t h e i n t r o d u c t i o n o f e x t r a nuisance narameters, e . ~ . , by h a v i n n nure r e o l i c a t i o n s o f the data. d a t a we bave
Therefore, we w i l l assume t h a t t h e b a s i c
e t c u r d i s n o s a l c o n s i s t of an i n d i v i d u a l x s t i m u l u s x r e n l i c a -
t i o n a r r a y o f b i n a r v data, and t h a t i n t h e model t o he develoned, parameters a r e a s s o c i a t e d w i t h s t i m u l i and i n d i v i d u a l s , h u t n o t w i t h t h e r e p l i c a t i o n s . I n o r d e r t o make t h e c h o i c e o f t h e model n l a u s i b l e . we w i l l develop i t i n t h e l i q h t o f a n s y c h o n h y s i c a l nroblem t o be i n t r o d u c e d i n t h e n e x t s e c t i o n ,
2 . A PSVCYOPHYSICAL PRORLCY An o l d , h u t uq t o now u n s o l v e d p r o b l e m i s g i v e n by t h e o b s e r v a t i o n t h a t , when s u b j e c t s a r e asked t o renroduce an a c u t e a n o l e
-
e i t h e r b y memory o r
bv simultaneous v i s i o n o f t h e s t a n d a r d and h i s own r e a r o d u c t i o n
-
by adding
a l i n e t o a q i v e n h o r i z o n t a l l i n e , t h e reproduced a n g l e t e n d s t o be w i d e r than the s t a n d a r d , whereas t h e r e v e r s e e f f e c t was o b t a i n e d w i t h obtuse s t a n dards ( J a s t r o w ( 1 8 9 2 ) ) .
The suggested e x o l a n a t i o n o f a n e r c e v t u a l over- o r
underestimate o f t h e s t a n d a r d r e s t s on a l o g i c a l f l a w i n t h e a r q u m n t , b e cause, as was m i n t e d o u t hy F i s h e r (1969),
t h e same p e r c e n t u a l d i s t o r t i o n
stlnttld a o p l y t o t h e renroduced anqle, so t h a t t h e r e i s n o c o m n e l l i n g reason why t h e y s h o u l d d i f f e r .
HacRae and Loh (1981) t r i e d t o e v a l u a t e t h e p o s s i -
h l e e f f e c t s o f t h e h o r i z o n t a l - v e r t i c a l i l l u s i o n ( v e r t i c a l d i s t a n c e s anpear h i q n e r t h a n h o r i z o n t a l ones), t h e d i r e c t i o n o f t h e i n i t i a l a d j u s t m e n t s ( t h e y used an a d j u s t a b l e a n g l e w i t h f i x e d b i s e c t o r ; i t s s t a r t i n q p o s i t i o n was e i t h e r much smal l e r o r much b i ager t h a n t h e w i d t h o f t h e s t a n d a r d ) ,
161
Probabilistic choice model
and o o s s i b l y a s t a t i s t i c a l a r t i f a c t ( t h e d i s t r i b u t i o n o f reproduced angles b e i n p skew i n o n p o s i t e d i r e c t i o n s f o r acute and obtuse angles; t h e use o f t h e a r i t h m e t i c mean as sam?le s t a t i s t i c m i g h t e x o l a i n the n r o b l e m ) .
Their
f i n d i n u s were g e n e r a l l y n e g a t i v e : c o n t r o l o v e r t h e t h r e e l i k e l y b i a s s e s c o u l d n o t t a k e away t h e s y s t e m a t i c e f f e c t i n t h e renroduced anqles.
Thus,
i t aooears t h a t t h e orohlem i s r a t h e r tough and c o m p l i c a t e d and t h a t t h e
e f f e c t m i g h t b e caused by l n u l t i n l e sources, e.?. n e r c e n t u a l , motor and dec i s i o n a l orocesses.
In o r d e r t o d i s e n t a n g l e t h e nroblem i t seems w i s e t o
do exneriments where t h e motor component i s excluded, and t o c o n s t r u c t a formal model such t h a t , in combination w i t h good e x p e r i m e n t a t i o n , t h e n e r c e n t u a l and d e c i s i o n a l c o m o n e n t can be separated, much l i k e i n s i g n a l det e c t i o n t h e o r y (Green and Swets ( 1 9 6 6 ) ) . The t y n i c a l e x o e r i m e n t we have i n mind i s a s i m n l e b i n a r y c l a s s i f i c a t i o n task, where t h e s u h . j e c t i s asked t o judge i f an anqle ( c a l l e d ' v a r i a b l e ' a n g l e ) i s equal o r n o t t o a o i v e n a n a l e ( c a l l e d t h e ' s t a n d a r d ' ) . exoerimental session
-
or Dart o f i t
-
I n one
t h e s t a n d a r d i s f i x e d and v i s i b l e
throughout, and a s e r i e s o f v a r i a b l e angles i s o r e s e n t e d t o t h e s u b j e c t . The whole e x o e r i m e n t c o n s i s t o f a number o f these subexperiments, each w i t h a d i f f e r e n t standard.
I t i s h y n o t h e s i z e d t h a t t h e n e r c e o t u a l nrocesses a r e
c o n t r o l l e d by t h e p h y s i c a l c h a r a c t e r i s t i c s o f t h e s t i m u l i and t h e general v i s u a l c o n t e x t o f t h e exneriment, whereas t h e d e c i s i o n nrocesses i n v o l v e d a r e under t h e c o n t r o l o f i n s t r u c t i o n s , reward, e t c . I t m i n h t be suggested t h a t one c o u l d e q u a l l y w e l l ask t h e s u b j e c t t o judpe
t h e d i r e c t i o n o f t h e u n e q u a l i t y , so t h a t w i t h t h e same e x n e r i m e n t a l e f f o r t , much more i n f o r m a t i o n can be o b t a i n e d .
Undoubtedlv. t h i s i s t r u e , b u t o u r
main aim i s t o c o n s t r u c t a formal model f o r cases where such d i r e c t i o n a l information i s unavailable.
I n t h e case o f t h e a n g l e problem a t hand, one
c o u l d use t h i s r i c h e r i n f o r m a t i o n t o t e s t t h e more general model.
Within
t h e framework o f t h i s oaper however, we w i l l n o t c o n s i d e r t h i s n o s s i b i 1 i t . y i n detail. To keen t h e use o f concents c l e a r , t h e correspondence between t h e g e n e r i c
concepts used i n t h e i n t r o d u c t i o n and t h e t e r m i n o l o g y used i n t h e d e s c r i p t l o n o f t h e model is l i s t e d i n t a b l e 1.
N.D. Verlrekf
exoerimen t a l
i
stimuli
v a r i a b l e anoles
in d i v i dua 1s
s t a n d a r d angles
renlicatioqs
one o r s e v e r a l nersons a t one o r more o c c a s i o n ( s ) Table 1
Corresnondence between q e n e r i c concents and t e r m i n o l o c y used i n the e x n e r i m e n t .
3 . TFF iinnEL ‘.le w i l l use the f o l l o w i n a c o n v e n t i o n s f o r t h e n o t a t i o n .
s :
nunher o f s t a n d a r d anoles used i n t h e e x p e r i m e n t .
j :
aeneral index t o i n d i c a t e standard ansles.
k :
number o f v a r i a b l e anrrles i n case t h i s number i s c o n s t a n t f o r a l l standards.
k: J’
numher o f v a r i a b l e an?les i n t h e subexperiment w i t h t h e j t h s t a n d a r d .
1 :
general index t o i n d i c a t e v a r i a b l e angles.
n :
number o f r e n l i c a t i o n s o e r s t a n d a r d - v a r i a b l e c o m h i n a t i o n .
I t i s as-
sumed t h a t t h i s nunher i s c o n s t a n t t h r o u g h o u t . v :
qeneral index o f a r e o l i c a t i o n .
x :
w i d t h o f an a n c l e , expressed i n r a d i a n s .
The node1 we w i l l d i s c u s s i s i n t e n d e d t o b e v a l i d f o r a l l angles i n a c e r t a i n onen i n t e r v a l (a,b),
e.a.
(0, n/2),
The range o f t b e v e l i r ‘ i t y i s an
e m i r i c a l Droblem. The resnonse o f t h e s u b j e c t i n the e x n e r i m e n t w i l l be c o n s i d e r e d as a r a n -
dolr v a r i a b l e , d e f i n e d by A
..
=
1
‘lJ
i f a t r e o l i c a t i o n v v a r i a b l e a n g l e i i s j u d o e d t o be equal t o standard anqle j.
0
otherwise.
And we d e f i n e fvi The axioms o f t h e model a r e :
= Proh(Avi
= 1).
163
hobabilisric choice model
A X I O Y 1 : Independence
F o r a l l v, i and j t h e random v a r i a b l e s Avij
a r e m u t u a l l y indeoendent.
T h i s axiom amounts t o t h e a s s u m t i o n o f e x p e r i m e n t a l indeoendence between sub.iects and t h e s o - c a l l e d l o c a l s t o c h a s t i c independence w i t h i n s u b j e c t s . F o r t h e n e x t a x i o m , we assume t h e e x i s t e n c e o f an unbounded monotone i n c r e a s i n g f u n c t i o n u from ( a & )
onto the r e a l s : u : (a,b)
+R
and such t h a t l i m o(x) = x+a
and l i m u ( x ) =
-m
m .
X-h
For conciseness i n t h e n o t a t i o n we w i l l w r i t e f c r a l l angles xh, uh i n s t e a d The range o f t h e f u n c t i o n u w i l l be c a l l e d t h e ( u r i d i m e n s i o n a l )
o f u(x,).
l a t e n t continuum.
A l l t h e f o l l o w i n g axioms w i l l be exoressed i n terms o f
t h i s l a t e n t continuum.
The f u n c t i o n u w i l l be c a l l e d t h e Dsychophysical
function. AXIOF' 2 : Acceptance r e g i o n As a consequence o f t h e e x o e r i m e n t a l nrocedure and t h e i n s t r u c t i o n s , an i n t e r v a l (r,
) i s c r e a t e d on t h e l a t e n t continuum.
F,
j' j
T h i s i n t e r v a l w i l l serve as t h e c r i t e r i o n used b y t h e s u b j e c t i n makina h i s judgments o f e q u a l i t y o r u n e q u a l i t y .
I t i s hypothesized t h a t the w i d t h o f
t h i s i n t e r v a l can be m a n i o u l a t e d by i n s t r u c t i o n s and rewards.
N o t i c e how-
ever, t h a t the v a l i d i t y o f t h i s osychological hypothesis i s n o t c r i t i c a l f o r t h e v a l i d i t y o f t h e model as a f o r m a l d e s c r i p t i o n o f o b s e r v a t i o n s , g i v i n a room t o t e s t hypotheses about p s y c h o l o g i c a l processes.
AXIOil 3 : Accevtance Drocesses Upon t h e o r e s e n t a t i o n o f a v a r i a b l e s t i m u l u s i,two independent ( l a t e n t ) processes t a k e Dlace, which we w i l l denote as t h e l e f t h a n d nrocess and t h e r i g h t h a n d Drocess.
Roughly spoken, t h e l e f t h a n d Drocess checks whether t h e
v a r i a b l e a n g l e i s n o t t o o s m a l l t o he acceptable, and t h e r i g h t h a n d process checks whether i t i s n o t t o o b i a . v a r i a b l e s A(!).
V1 J
A(<), = 'lJ
Ye t h e r e f o r e d e f i n e two l a t e n t random
and AbYA:
1
i f v a r i a b l e angle i i s accepted by s u b j e c t v
0
otherwise.
'
i n t h e l e f t h a n d l a t e n t process.
.v.D. r'erlielrt
164
A(!-!
Vl
.I
i s defined analonously for the righthand l a t e n t nrocess.
!,Je d e f i r e f w t h e r : f ' .2 \. Vi
J
I!Okl,
the axiom on the accentance v r o c e s s e r can he s t a t e d as f o l l o w s :
3.a
Pvi( P I and A(:).
3.0
f ( r.! . i.s a monotone i n c r e a s i n ? f u n c t i o n o f E,
3.c
a r e s t o c h a s t i c a l l y indenendent
V1 J
V1 J
fvij (.el i' s a monotone d e c r e a s i n ? f u n c t i o n o f
j
"i
-
0. 1
- 0
i'
!t s h o u l d he s t r e s s e d t h a t Axiov 3 i s concerned w i t h l a t e n t nrocessps, which
are unobservable, and t h a t i t does n o t f o l l o l r from t h e axiom t h a t always two indeoendent D s y c h o l o o i c a l orocesses a r e assumed.
T h i s w i l l be c l e a r f r o m
t h e examole f o l l o w i n g axiom 4 . A Y I n f l 4 : Response axiom The observable response v a r i a b l e A . . i s d e f i n e d t o he V1.I
/Iv =iA ( P. I. Avi( r ) V1 J
.
Recause o f axiom 3a i t f o l l o w s i m m e d i a t e l y t h a t f .. = VlJ
fw. f k ) . V1 J
VlJ
.
Now, as an example c o n s i d e r t h e D s y c h o l o g i c a l t h e o r v which says t h a t t h e two l a t e n t nrocesses a r e n o t indenendent, b u t t h a t t h e s u b j e c t i n i t i a t e s one l a t e n t nrocess, e . g . the l e f t h a n d one, and t h a t he o n l y i n i t i a t e s t h e r i g h t hand one i f necessarv, i . e . i f t h e v a r i a b l e a n g l e i s n o t j u d c e d t o o s m a l l . Furthermore,
t h e r e i s some o r o b a b i l i t y g t h a t t h e nrocess he b e g i n s w i t h i s
t h e l e f t h a n d one.
I f we denote t h e n r o b a b i l i t y t h a t t h e a n ? l e i s n o t judged
t o o s m a l l by t h e l e f t h a n d nrocess by n, and t h e n r o b a b i l i t y t h a t t h e a n a l e i s n o t judged t o o b i g by t h e r i g h t h a n d process b y q, and assuming t h a t an a n q l e t h a t i s n e i t h e r judsed t o o s m a l l n o r t o o b i g , i s j u d o e d t o be e q u a l , then prob(equa1 judgment) = g p q + ( 1
-
g)p q
= P q which i s f o r m a l l y t h e same as s t a t e d i m n e d i a t e l y f o l l o w i n g axiom 4 .
That
means t h a t , as f a r as t h e o b s e r v a b l e b i n a r y r e m o n s e s a r e concerned, t h e D s y c h o l o g i c a l t h e o r y s k e t c h e d i s f o r m a l l y e q u i v a l e n t w i t h axioms 3 and 4.
165
Probabilistic ckoice model
The model i s n o t c o m o l e t e l y s o e c i f i e d by t h e f o r e g o i n g axioms; e s p e c i a l ? y the class o f functions
f$!
and ftri( has i n f i n i t e l y many twmbers.
ble choose
t h e l o g i s t i c f u n c t i o n because i t has nroven t o be v e r y useful i n o t h e r contexts.
Thus we d e f i n e t h e mcrdel i n i t s most general f c r m as
However, t h e model as s n e c i f i e d i n (1) and ( 2 ) i s n o t i d e n t i f i a b l e because w i t h s standards and k v a r i a b l e angles, and assuming t h a t a l l combinations a r e t e s t e d , t h e model has
sk t k t ?s
parameters w h i l e t h e r e a r e o n l y sk
c e l l s w i t h i n f o r m a t i o n about t h e Darameters s i n c e t h e r e p l i c a t i o n s a r e s t a t i s t i c a l l y equivalent. t h e Darameters.
T h e r e f o r e we need f u r t h e r s t r u c t u r a l c o n s t r a i n t s on
To s i m l i f y t h e e x n o s i t i o n , we w i l l assume t h a t
for a l l i , (i=l,
...,k ) ,
aij
= a j and R~~ = B j
(j=l,,..,s).
I t i s c l e a r t h a t o t h e r c o n s t r a i n t s c o u l d have been chosen and t h i s problem
w i l l be discussed f u r t h e r i n t h e s e c t i o n a b o u t i d e n t i f i a h i l i t y .
With t h i s
s i m p l i f i c a t i o n t h e model can be s t a t e d as:
I t i s c l e a r t h a t i n ( 3 ) and ( 4 ) a m u l t i p l i c a t i o n o f a by a D o s i t i v e c o n s t a n t
can be n e u t r a l i s e d by t h e m u l t i p l i c a t i o n o f 8 and t h e d i v i s i o n o f 5, 5 and u b y t h i s same c o n s t a n t ,
Analogously, t h e a d d i t i o n o f an a r b i t r a r y c o n s t a n t
t o u can be n e u t r a l i s e d b y t h e s u b t r a c t i o n o f t h e same c o n s t a n t f r o m 5 and
5.
I n o t h e r words, we a r e f r e e t o choose t h e u n i t and o r i g i n o f t h e l a t e n t
continuum. As t o t h e i n t e r p r e t a t i o n o f t h e modelparameters, i t may be observed t h a t when a and B b o t h t e n d t o i n f i n i t y , t h e p r o d u c t o f ( 3 ) and ( 4 ) has a s t e p f u n c t i o n as i t l i m i t , whose v a l u e i s 1 i n t h e Dpen i n t e r v a l ( c , ~ ) and z e r o o u t s i d e i t . T h a t means t h a t once t h e boundaries o f t h e acceptance r e g i o n
166
N.D. Verhekt
a r e s e t , a l l u n c e r t a i n t y a b o u t t h e resoonses has disaoneared. a
and a become s m a l l , fvij i s a smooth s i n g l e neaked f u n c t i o n ,
mum value i s l e s s than u n i t y .
If however, whose maxi-
T h a t means t h a t t h e d e c i s i o n c r i t e r i a become
l e s s sharo, and i t i s h v n o t h e s i z e d t h a t t h i s i s a consequence o f n o i s e i n t h e o e r c e o t u a l processes.
However, i t s h o u l d be s t r e s s e d a o a i n , t h a t t h e
corresnondence between n e r c e o t u a l Drocesses and t h e v a l u e o f a and 3 on t h e one hand, and between d e c i s i o n a l orocesses and t h e w i d t h of t h e acceptance r e g i o n on the o t h e r hand c o n s t i t u t e s an e m p i r i c a l nroblem.
The model d e f i -
ned hy ( 3 ) and ( 4 ) o f f e r s a f o r m a l m a n s t o d i s t i n g u i s h between them.
It
may be u s e f u l t o observe t h a t t h e model does n o t o a r a m e t r i z e t h e i d e a l point,
i . e . , t h e r e i s no s c a l e v a l u e a s s o c i a t e d d i r e c t l y t o t h e s t a n d a r d angle x . . J The s c a l e v a l u e o f x . may be e s t i m a t e d i n d i r e c t l y by u s i n q a v a r i a b l e s t i J mulus xi such t h a t xi = x . . J Suopose we observe t h e same ohenomenon i n t h e c l a s s i f i c a t i o n e x o e r i m e n t as d i d J a s t r o w i n t h e r e n r o d u c t i o n e x o e r i m e n t , i .e., w i t h an a c u t e s t a n d a r d angle x , t h e v a r i a b l e a n a l e x t x ' ( x ' > O ) i s j u d ? e d more o f t e n equal t o t h e s t a n d a r d than t h e v a r i a b l e a n q l e x
-
XI,
then there are a t l e a s t three
d i f f e r e n t reasons p o s s i b l e t o e x o l a i n t h i s ohenomenon.
1.k w i l l c o n s i d e r
them i n t u r n . Case 1. a
= E~ and(Ej
-
n.) = (uj
J I n t h i s case t h e f u n c t i o n fvij, j
5
cj).
c o n s i d e r e d as a f u n c t i o n o f a(xi)
m e t r i c w i t h regard t o the midooint function
(c.
t
i s sym-
~ . ) / 2 . But the psychophysical J
J may b e n o n l i n e a r i n such a manner t h a t
That m a n s t h a t t h e f u n c t i o n fviju(xi) become asymmetrical.
c o n s i d e r e d as a f u n c t i o n o f xi
has
So t h e reason f o r t h e asymmetry i n t h e d a t a i s e x p l a i -
ned by the n o n - l i n e a r i t y o f t h e Dsychoohysical f u n c t i o n n . Case 2 .
= 3 . and t h e n s y c h o p h y s i c a l f u n c t i o n i s l i n e a r . 3 J The observed a s y m t r y i n t h e d a t a can be e x p l a i n e d by t h e i n e q u a l i t y a.
- u . ) > ( u . - Cj). J J J If t h i s i s t h e case, t h i s asymmetric acceptance r e g i o n may be made symmetric (E.
by m a n i p u l a t i o n o f t h e i n s t r u c t i o n and/or rewards.
167
Probabilistic choice model
sj -
-
5 . and the ?svchoniiysical f u n c t i o n i s l i n e a r . j ~ Here t h e observed asymmetry i s due t o t h e d i f f e r e n c e between a and 6 . and j J may r e f l e c t genuine v e r c e v t u a l d i f f e r e n c e s .
Case 3.
u
j
= u
I t can e a s i l y be shown t h a t t h e o r e d i c t e d asymmetry occurs when aj < p j .
4. PARWETER ESTIPATION AND IDENTIFIABILITY We w i l l f i r s t c o n s i d e r an e x o e r i m e n t w i t h one standard. h e n c e f o r t h r e f e r r e d t o as t h e s t a n d a r d e x n e r i m n t . To have a c o n v e n i e n t f o r m f o r t h e estimates, we d e f i n e ( d r o p p i n g t h e now s u o e r f l uous i n d i c e s v and j) Q = 1 < = @ -U
e
= ci
(i=l,
...,k )
e‘ = e. The i t e m - c h a r a c t e r i s t i c f u n c t i o n , i . e . t h e D r o d u c t o f ( 3 ) and ( 4 ) can now be w r i t t e n as a
’
‘i
f. =
N o t i c e t h a t if5
+
(1
--,t h e
t
(i=l,. t eEi)
€:)(I
.. ,k).
(5)
second f a c t o r i n t h e denominator o f ( 5 ) becomes
u n i t y , and t h e r e s u l t i n g r o d e l i s t h e s i m p l e 2-parameter l o a i s t i c model
( B i rnbaum ( 1968) )
.
I t can r e a d i l y be seen t h a t t h e l i k e l i h o o d o f t h e d a t a i s a i v e n b y
k
L =
n
t. fi’(l
- fi)
i
n-ti
where t . = c a vi ’ v
.
(7)
T h a t means t h a t t h e s i m p l e t o t a l scores a r e t h e mini,mal s u f f i c i e n t s t a t i s t i c s f o r t h e whole s e t o f oarameters. N o w c o n s i d e r a t r a n s f o r m a t i o n t o a e t t h e l i k e l i h o o d f u n c t i o n ( 6 ) i n some s t a n d a r d form. Define zi yi
=
=
The l i k e l i h o o d can now be w r i t t e n as
Ea
i
eEi
.
.V. D. Ve rhe Lc t
168
L
=
c
exo c (t.lnzi+(n-t.)ln(ltyityizi)). 1 i 1
((1+zi)(lt.vi)\-"
i
(10)
Now d e f i n e = l n ( z i / ( l t y . + y1. z 1. ) ) .1
qi
(11)
5 u h s t i t u t i n g (11) i n (10) nives L = : : ( e 'i i
t
l)-n exn 1 t.' 1 i'
(12)
T h a t means ( s e e Andersen (1980)) t h a t t h e model belonas t n t h e e x p o n e n t i a l family, w i t h k c s n n n i c a l narameters oi.
I t follows from standard theory
t h a t t h e maximum l i k e l i h o o d e s t i m a t e s , i f t h e y e x i s t , a r e unique.
Further-
more, t h e e s t i m a t e s o f t h e c a n o n i c a l parameters a r e e x t r e m e l y s i m p l e : d i f f e r e n t i a t i n p (12) w i t h r e s o e c t t o ni
and s e t t i n g z e r o c i v e s t h e l i k e l i h o o d
equations exo( ni )
1t
exn(ni)
= ti/n
(i=l,
...,k )
w i t h s o l u t i on fi.1 = I n ( t i / ( n - t i ) )
(i=l,...,k)
(14)
so t h a t i t i s r e a d i l y seen t h a t a s o l u t i o n e x i s t s i f and o n l y i f 0 < ti < n .
As t o t h e o r i g i n a l narameters, we must say t h a t t h e r e a r e i n f i n i t e l y many s o l u t i o n s , because t h e r e a r e kt2 parameters whereas t h e r e a r e o n l y k s u f ficient statistics.
So, i n o r d e r t h e model t o be i d e n t i f i a b l e a t l e a s t two
c o n s t r a i n t s must be n u t on t h e o a r a n e t e r space, f o r examole by f i x i n g two parameters a t a g i v e n v a l u e .
Even then, however, t h e n r o b l e m i s n o t s o l v e d
corn?letely, t h e reason b e i n g t h a t t h e c a n o n i c a l narameters a r e n o n m n o t o n e t r a n s f o r m a t i o n s o f t h e o r i g i n a l model oarameters.
To understand t h e i m p l i -
c a t i o n o f t h i s , c o n s i d e r t h e f o l l o w i n g example: suppose we l o o k f o r t h e sol u t i o n s e t under t h e r e s t r i c t i o n
(14) t h a t
I. I t
E l-
1 fir
a =
t BEi
f o l l o w s f r o m ( 8 ) . (9), (11) and
- q = o t ec;
equivalently,
where gi = t i / ( n - t i ) .
I69
Probabilistic choice model
By a s u i t a b l e choice o f e , t h e r e a r e two s o l u t i o n s f o r each
t i v e reason f o r t h i s i s t h a t ( 5 ) ,
E.
as a f u n c t i o n o f t h e s c a l e v a l u e o f t h e
v a r i a b l e angle, i s s i n o l e neaked, so t h a t f o r each v a l u e o f i s a n o t h e r value,
E',
The i n t u i -
f o r which ( 5 ) takes t h e same v a l u e .
E,
say E , t h e r e
Since t h e l i k e l i -
hood f u n c t i o n i s a o r o d u c t o f f u n c t i o n s l i k e (5), i t i s c l e a r t h a t t h e v a l u e k of L a t a l o c a l maximum, say ( E l ,... ,ii, ...,i k )can , he a t t a i n e d a t a l l 2 p o i n t s , where any o f t h e i i s r e n l a c e d by t h e c o r r e s n o n d i n p a l l those p o i n t s corresoond t o l o c a l maxipa.
E'.
and t h a t
Since t h e c a n o n i c a l parame-
t e r s a r e unique, 1"e can s t a t e t h a t , when a and 8 a r e s u i t a b l y f i x e d , t h e r e k can he UD 2 l o c a l paxima. The wav o u t o f t h i s i n d e t e r m i n a t e n e s s c o n s i s t s o f f i x i n o t h e o r d e r o f t h e E'S
on t h e l a t e n t continuum: chanoina an i i n t o t h e corresponding
E'
neces-
s a r i l y changes the o r d e r between two o r m r e parameters, e x c e o t f o r t h e one i n the 'middle',
i.e.
t h e one w i t h t h e h i g h e s t f - v a l u e (see ( 5 ) ) .
I n case
o f t h e p r e s e n t e x p e r i m e n t where t h e n s y c h o p h y s i c a l f u n c t i o n i s m n o t o n i c , t h e o r d e r o f t h e E-narameters c o n s t i t u t e s no oroblem, b u t i n t h e more gener a l case e.g. t h e p u r c h a s i n a o f goods, f i n d i n g t b e o r d e r on t h e l a t e n t continuum i s one o f t h e m a j o r goals o f c o l l e c t i n g t h e d a t a . I n summary then, two problems can be d i s c e r n e d i n c o n n e c t i o n w i t h t h e s t a n d a r d experiment: (i)
t h e oroblem o f o v e r u a r a m e t r i z a t i o n , t h e r e b e i n g more parameters than
s u f f i c i e n t s t a t i s t i c s , and ( i i ) as an immediate consquence o f t h e use o f a non-monotonic i t e m - c h a r a c t e r i s t i c f u n c t i o n , t h e i n d e t e r m i n a t e n e s s of any s o l u t i o n u n l e s s s u f f i c i e n t o r d i n a l i n f o r m a t i o n on the parameters i s a v a i l a h l e . The way o u t f r o m these two oroblems c o n s i s t s o f a combination o f s e v e r a l s t a n d a r d experiments and c a p i t a l i z i n s on t h e assumption t h a t the psychop h y s i c a l f u n c t i o n i s indeoendent o f t h e e x p e r i m e n t a l c o n t e x t .
1.e. p e r -
f o r m s s t a n d a r d experiments, each w i t h k . v a r i a b l e s t i m u l i , b u t chosen i n J such a way t h a t t h e number o f p h y s i c a l l y d i f f e r e n t s t i m u l i , say m y i s l e s s t h a n Ck
j* I n such a case t h e number o f m i n i m a l l y s u f f i c i e n t s t a t i s t i c s , i . e . t h e num-
b e r o f c a n o n i c a l parameters i s c k . w h i l e t h e number o f model parameters J' can be made s m a l l e r . We can then c o n s i d e r two main e x t e n s i o n s o f t h e model, one c o n c e r n i n g t h e d i s c r i m i n a t i o n parameters a and 8, and one c o n c e r n i n g t h e w i d t h o f t h e
I70
N.D. VerheLt
F o r t h e d i s c r i m i n a t i o n oarameters c o n s i d e r t h e f o l l o -
acceptance i n t e r v a l .
w< ng h i e r a r c h i c a l l y o r d e r e d s ubmode 1s . (1)
(2)
a. =...= cx and B 1 =...= 5 . = . . . = 6, = 1. J S J T h a t m a n s t h a t one d i s c r i m i n a t i o n p a r a w t e r has t o be e s t i m a t e d . The a's a r e n o t c o n s t r a i n e d t o be e q u a l , b u t t h e 3 ' s a r e , o r t h e cona1
=...=
verse.
I n t h i s case s n a r a n e t e r s a r e t o be e s t i m a t e d , t h e common cx
o r ? t a k e n as u n i t y .
(3)
P!o c o n s t r a i n t s on t h e d i s c r i m i n a t i o n p a r a w t e r s .
7s
-
1 oarameters
must be e s t i m a t e d , one t a k e n as u n i t y . Crossed w i t h t h e f o r e g o i n g cases, we can d i s c e r n two submodels, c o n c e r n i n 9 t h e w i d t h o f t h e acceotance i n t e r v a l s .
(P,)
The w i d t h o f a l l i n t e r v a l s i s c o n s t a n t and unity, l e a v i n g s parameters t o be e s t i m a t e d , e.o.
(8)
the lower boundaries 5
j*
No c o n s t r a i n t on t h e w i d t h o f t h e i n t e r v a l s , l e a v i n g 2s - 1 oarameters t o he e s t i m a t e d .
To be i d e n t i f i a b l e , z k . must n o t b e l e s s t h a n t h e number o f f r e e parameters J as shown i n t a b l e 2 f o r t h e 6 cases j u s t mentioned.
l p
- .. - ... . . .... ..
IW3s-1
B m2s
~ 4 s - 2
1~3s-1
Table 2 Number o f f r e e Darameters f o r s e v e r a l submodels. The v a r i a b l e s t i m u l i always g i v i n q m f r e e oarameters. Since t h e models a r e h i e r a r c h i c a l l y o r d e r e d
-
a l l t h e models b e i n g s p e c i a l
cases o f a l l o t h e r models a o o e a r i n a t o t h e r i g h t and/or below t h e i r c e l l i n table 2
-
t h e y can be t e s t e d a q a i n s t each o t h e r e.g. by u s i n ? t h e l i k e l i -
hood r a t i o . The oroblem o f i d e n t i f i a b q l i t y however, i s n o t n e c e s s a r i l y s o l v e d because i t i s n o t c e r t a i n t h a t by addincr mure c a n o n i c a l parameters w h i l e k e e p i n g
t h e number o f modelparameters w i t h i n t h e l i m i t s g i v e n i n t a b l e 2, t h e o r d e r o f t h e s c a l e values o f t h e v a r i a b l e angles w i l l be unique.
To see
t h i s , c o n s i d e r t h e s i t u a t i o n where f o r an e x p e r i m e n t w i t h 2 s t a n d a r d s and 5 common v a r i a b l e angles, l a b e l e d A , R , C,
D and E r e s p e c t i v e l y , a l o c a l
maximum o f t h e l i k e l i h o o d f u n c t i o n has been found.
Then, f o r a l l v a r i a b l e
171
Probabilistic choice model
anqles, t h e p r o b a b i l i t y o f b e i n g j u d g e d equal ( t h e n r o d u c t o f ( 3 ) and ( 4 ) ) can he c a l c u l a t e d , m d these f u n c t i o n values can be o r d e r e d f o r each o f t h e two s t a n d a r d s , thus g i v i n a 2 I - s c a l e s as d e f i n e d i n u n f o l d i n g t h e o r y .
As
an e x a m l e c o n s i d e r t h e two I-SCaleS t o be I-scale 1 : F C A G E I-scale 2 : C R D A E , Tben t h e o o s s i b l e candidates f o r t h e o r d e r o f t h e s t i m u l i on the l a t e n t continuum a r e a s u b s e t o f t h e J - s c a l e s c o m p a t i b l e w i t h t h e two I - s c a l e s I n t h i s case these J - s c a l e s a r e A B C
D E and E A B C D.
The s o l u t i o n o f t h i s oroblem o f course i s t o add more s t a n d a r d s i n such a way t h a t t h e I - s c a l e s d e r i v e d determine u n i q u e l y t h e J - s c a l e , and g e n e r a l l y t h i s w i l l b e accomplished b y choosina standards w i t h s c a l e values w h i c h a r e w e l l s c a t t e r e d o v e r t h e l a t e n t continuum, an i d e a which i s v e r y c e n t r a l i n u n f o l d i n g t h e o r y (Coombs, c h a o t e r 5 (1964)). I t s h o u l d be c l e a r t h a t t h e f a m i l y o f models, as summarized i n t h e margins
o f t a b l e 2, does n o t exhaust t h e model: as was i n d i c a t e d i n s e c t i o n 3, i t i s p o s s i b l e t o conceive o f t h e d i s c r i m i n a t i o n parameters a and 6 , as b e i n g a f u n c t i o n o f t h e v a r i a b l e anqles, o r o f some s o e c i f i e d combination o f two parameters, one dependent on t h e v a r i a b l e angle, t h e o t h e r on t h e s t a n d a r d . A l t h o u c h t h e model as i t i s d e f i n e d now, i s v e r y f l e x i b l e and can be adaot e d and extended i n a a r e a t number o f ways
-
e s D e c i a l l y i n cases where t h e
c o n t r o l o v e r s t i m u l i , standards as w e l l as v a r i a b l e , causes no problem and where t h e r e p l i c a t i o n s a r e e a s i l y o b t a i n e d
-
one s h o u l d n o t be t o o o p t i m i s -
t i c a b o u t t h e p o s s i b i l i t i e s o f t e s t i n q t h e o r i e s o f p e r c e o t i o n and d e c i s i o n making on one and t h e same d a t a s e t , b y c o n t i n u o u s l y imposing r e s t r i c t i o n on t h e parameter space u n t i l t h e model c r a c k s down.
The reason f o r t h i s i s
t h a t g e n e r a l l y l i t t l e i s known about t h e power o f t h e t e s t s used. case a t hand, however, t h e power o f some t e s t s
-
I n the
e s o e c i a l l y those concer-
n i n q t h e e q u a l i t y o f t h e d i s c r i m i n a t i o n parameters
- will
g e n e r a l l y be v e r y
l o w as may be seen b y t h e f o l l o w i n g s i m p l i f i c a t i o n o f t h e general model f o r the standard experiment.
Some evidence i s a v a i l a b l e i n t h e l i t e r a t u r e t h a t
t h e Dsychophysical f u n c t i o n f o r angles i s l i n e a r (Stevens and G a l a n t e r
(1957)).
So i t m i g h t seem reasonable t o p e r f o r m t h e oarameter e s t i m a t i o n
o f a , 6 , r, and 5 w i t h t h e s c a l e values as known c o n s t a n t s , i.e. ui = xi.
17:
gi
= t . ( n - t . ) w i t h t . = I: a . 1
1
1
V l
and assuming f o r t h e moment t h a t t h e r e were 4 v a r i a h l e s t i m u l i i n t h e exoer i m e n t , then t h e maximum l i k e l i h o o d e s t i m a t e s a r e a i v e n by t h e s o l u t i o n o f t h e n o n - l i n e a r system -1 nX. = T
ax. 1
-EXi
' t f
-Ski
(15)
(i=1,2,3,4).
8
t T
F u t ( 1 5 ) i s a sum o f e x n o n e n t i a l f u n c t i o n s and i t i s w e l l known (Lanczos
( 1 9 6 6 ) ) t h a t t h e s o l u t i o n s a r e v e r y d i f f i c u l t t o f i n d , because t h e f u n c t i o n s are n o t o r t h o q o n a l t o each c t h e r .
T h a t means t h a t i n g e n e r a l
-
i .e. w i t h
more s t a n d a r d s and more v a r i a b l e s t i m u l i , t h e m a t r i x o f second d e r i v a t i v e s o f the l i k e l i h o o d f u n c t i o n w i l l be a l m o s t s i n g u l a r , c a u s i n g n u m e r i c a l p r o blem
ill
the e s t i m a t i o n process, b u t a l s o y i v i n a l a r o e s t a n d a r d e r r o r s f o r
t h e e s t i m a t e s , thus makina t h e s t a t i s t i c a l t e s t s o f t h e model a l m o s t powerless.
5 . CONCLUSION h D r o b a b i l i s t i c model i s develooed t h a t i s adapted t o t h e d a t a c o l f l e c t i o n method known as ' p i c k any o u t o f n ' , under t h e c o n d i t i o n t h a t s t a t i s t i c a l r e p l i c a t i o n s o f t h e choices a r e n o s s i h l e .
The n s y c h o n h y s i c a l Droblem o f
s y s t e m a t i c e r r o r s i n a n q l e r e o r o d u c t i o n i s chosen as a t y p i c a l example. The b a s i c e x n e r i m n t c o n s i s t s o f a c l a s s i f i c a t i o n task, where human s u b j e c t s , s e r v i n a as r e p l i c a t e s , judqe f o r each o f a s e r i e s o f angles ( c a l l e d v a r i a b l e a n g l e s ) i f t h e y a r e equal o r n o t t o a p i v e n s t a n d a r d . bability ment
-
-
i n terms o f t h e e x o e r i m e n t t h e p r o b a b i l i t y o f an ' e q u a l ' j u d g -
as a f u n c t i o n o f t h e s c a l e v a l u e s o f t h e v a r i a b l e angles i s conside-
r e d t o b e a o r o d u c t o f two l o q i s t i c f u n c t i o n s .
two parameters
-
A p a r t f r o m t h e s c a l e values
t h e b o u n d a r i e s o f t h e acceDtance r e g i o n
w i t h the i d e a l p o i n t , which i t s e l f i s n o t p a r a m e t r i z e d . %rs
-
The c h o i c e p r o -
t h e d i s c r i m i n a t i o n oarameters
-
-
are associated
Two o t h e r parame-
can be i n t e r p r e t e d as t h e sharpness
o f the boundaries. F u r t h e r , i t i s shown under w h i c h c o n d i t i o n s t h e model i s n o t i d e n t i f i a b l e . There a r e e s s e n t i a l l y two reasons: o v e r p a r a m e t r i z a t i o n and u n c e r t a i n t y ab o u t t h e o r d e r o f t h e s c a l e v a l u e s , t h e l a t t e r b e i n q due t o t h e s i n g l e - p e a kedness o f t h e c h o i c e f u n c t i o n .
173
Probabilistic choice model
F i n a l l y i t i s p o i n t e d o u t t h a t the nodel i n i t s most oeneral form, a l t h o u g h i d e n t i f i a b l e , may be ill c o n d i t i o n e d because t h e parameter e s t i m a t i o n i s a k i n t o t h e problem o f e s t i m a t i n ? sums o f e x o o n e n t i a l f u n c t i o n s .
Therefore,
i t may t u r n o u t t h a t o n l y s i m p l i f i e d v e r s i o n s o f t h e model a r e u s e f u l i n
practice
.
REFERENCES Andersen, E.B.,
D i s c r e t e s t a t i s t i c a l models w i t h s o c i a l s c i e n c e a o p l i -
c a t i o n s , Amst,erdam, N o r t h H o l l a n d (1980). Birnbaum, A.,
Some l a t e n t t r a i t models and t h e i r use i n i n f e r r i n a an
examinee's a b i l i t v , i n F.M. L o r d and M.R.
Novick, S t a t i s t i c a l t h e o r i e s
o f mental t e s t scores, Readino, Addison-Wesley (1968).
[31 141
Coombs, C.H., F i s c h e r , G.H.,
A t h e o r y o f d a t a , New York, h l e y (1964). E i n f i i h r u n g i n d e r T h e o r i e P s y c h o l o a i s c h e r Tests, Bern,
Huber (1974).
[51
F i s c h e r . G.H.,
An e x p e r i m e n t a l s t u d y of a n g u l a r subtension, Q u a r t e r l y
Journal o f Experimental Psychology, 21, (1969), 356-366.
[6I
Green, D.M.,
[71
Jastrow, J., On t h e judgment o f angles and o o s i t i o n s o f l i n e s , A. On
and Swets, J.A.,
S i g n a l d e t e c t i o n t h e o r y , New Vork, b l i l e y
(1966). t h e judgment o f angles, American J o u r n a l o f Psychology, 5 (1892),
214-217. [81 Lanczos, C., A n o l i e d a n a l y s i s , Englewood C l i f f s , P r e n t i c e H a l l (1956). 191 FlacRae, A.W., and Loh, P.K., Constant e r r o r s o c c u r i n matched r e p r o c!uction o f angles even i f l i k e l y b i a s e s a r e e l i m i n a t e d , P e r c e p t i o n and Psychophysics, 30 (1981). 341-346.
[lo1
Stevens, S . S . , and G a l a n t e r , E.H.,
R a t i o s c a l e s and c a t e g o r y s c a l e s
f o r a dozen perceDtua1 c o n t i n u a , J o u r n a l o f Exoerimental Psychology,
54 (1957), 377-411. [111 Torqerson, W.S., Theory and methods o f s c a l i n a , New York, N i l e y (1958).
This Page Intentionally Left Blank
PART 11 ORDER AND MEASUREMENT
This Page Intentionally Left Blank
TREh'DS Ih' MATHEMATICAL PSYCHOLOGY E . Degreef and J. Van Bu enhuut (editors) Olilseuier Science Publisf&s B. V, (fiortll-Holland),1984
ABOUT THE P6YWETRIES OF SIMILARITY JUDWENTS: AN ORDINAL POINT OF V I E Y Jean-Pierre BarthelGmy Ecole N a t i o n a l e S u p e r i e u r e des TC?lC?cnnmunic a t i o n s P a r i s , France
'Je oropose t o r e v i s i t t h e n o t i o n o f asymmetry o f s i m i l a r i t v judgments, w i t h a p u r e l y o r d i n a l m i n t o f view.
'Je w i l l use t h e m a j o r i t y r u l e , comoute
median o r d e r s and t h e n comnare them i n o r d e r t o d e t e c t ( o r d i n a l ) asymmetries.
An e x n e r i m e n t w i t h
m i crocomnuters i s analyzed.
1. INTRODUCTION The a s y m t r i c a l r e l a t i o n s h i n s i n t h e c o n t e x t o f s i m i l a r i t y judgments p l a y some D a r t i n a t t r i b u t i o n t h e o r y ( c f . S . Duval and V.H.
Duval ( 1 9 8 3 ) ) .
In-
deed i t f o l l o w s f r o m T v e r s k y ' s paper "Features c f s i m i l a r i t y " (1977) t h a t a s t r o n g c o n n e c t i o n e x i s t s between t h e asymmetry o f a r e l a t i o n , t h e s a l i e n c e
o f a s t i m u l u s and t h e n o t i o n o f n r o t o t y p i c a l i t y . T h i s paner s e t s up a t e c h n i c a l r e f l e x i o n about t h i s n o t i o n o f asymmetry coming f r o m t h e f o l l o w i n o remark: assume t h a t o e r c e i v e d s i m i l a r i t i e s between f o u r s t i m u l i a,b,c,d i n two e x n e r i m e n t s :
S(a,b)
are r a t e d
( t h e s i m i l a r i t y o f a t o b ) and S(c,d) a r e r a t e d
i n e x n e r i m e n t 1; S(b,a) and S(d,c) i n e x p e r i m e n t 2 .
A n u m e r i c a l asymmetrv i s observed when, f o r i n s t a n c e , S(a,b)
# S(b,a) (no-
ticeably).
Pssume now t h a t t h e e x n e r i m e n t 1 l e a d s t o t h e i n e q u a l i t y S(a,b)
< S(c,d)
( t h e n e r c e i v e d s i m i l a r i t y o f c t o d exceeds t h e p e r c e i v e d s i m i l a r i t y o f a to b). (i)
Then f r o m e x n e r i m e n t 2 we can g e t e i t h e r
S(b,a)
< S(d,c),
or
( i i ) S(d,c) s S(b,a). I n t h e f i r s t case t h e a c t i o n o f asymmetry remains o u r e l y l o c a l ; i t j u s t i n v o l v e s t h e p a i r s o f s t i m u l i , w i t h o u t d i s t u r b i n g i n deeoth t h e o e r c e i v e d
similarity itself.
I n t h e second case, t h a t n o t i o n o f s i m i l a r i t y ( v i e w e d
as a r a n k i n ? o n e r a t o r on t 9 e n a i r s o f s t i m u l i ) i s d i s r u n t e d . floreover, we know f r o m t r a d i t i o n s borrowed from s c a l i n o and w a s u r e m e n t and from t h e common sense as w e l l , t h a t s i m i l a r i t y analyses a r e o f t e n founded on S i m i l a r i t y o r d e r s on n a i r s . F o r a l l those reasons we nronose, i n t h i s Daner, t o r e v i s i t t h e n o t i o n o f asymmetry w i t h a n u r e l y o r d i n a l n o i n t o f view. p o i n t s sketched i n t h i s i n t r o d u c t i o n .
The f i r s t a a r t d e v e l r r e s
The second P a r t n r e s e n t s an e x o e r i -
ment analyzed i n the t h i r d a a r t .
2. F40!1 NUMERICAL IrSV’lWTPv T O nRDINAL ASYIWETRY 2 . 1 . TVERSKY’S APPRPACH A s y m t r . y i n t h e c o n t e x t o f s i m i l a r i t y judgments has been s t u d i e d by Tversky
(1977) ( c . f . a l s o Tversky and G a t i ( 1 9 7 8 ) ) .
I n T v e r s k y ’ s o a w r asynrnetries
a r e observed w i t h t h e h e l n o f c l a s s i c a l s t a t i s t i c s ( S t u d e n t t ) .
lhe invol-
ved s i m i l a r i t i e s a r e d i r e c t i o n a l : t h e y a r e r a t e d on o r d e r e d n a i r s ( a , b ) , w i t h one s u b j e c t a and one r e f e r e n t b .
Tversky’s hypothesis i s t h a t the d i -
r e c t i o n of a s y m t . r , y i s determined by t h e r e l a t i v e s a l i e n c e o f t h e s t i m u i : t h e v a r i a n t ( s u b j e c t ) i s more s i m i l a r t o t h e o r o t o t y p e ( r e f e r e n t ) than v ce versa.
Tversky r e l a t e s s e v e r a l experiments s u r m o r t i n g t h i s h y o o t h e s i s .
tioreover, he Dronoses a model i n v o l v i n g those asymmetries: a and b b e i n a two s t i m u l i , t h e s i m i l a r i t y S(a,b)
o f a t o b i s expressed as a f u n c t i o n o f t h r e e
arguments :
A
B t h e f e a t u r e s which a r e c o m n t o b o t h a and b ;
A \ 3 t h e f e a t u r e s t h a t b e l o n g t o a b u t n o t t o b; B \ A t h e f e a t u r e s t h a t b e l o n g t o b b u t n o t t o a. Ifloreover, S(a,h)
<
S ( c,d) whenever
A n Q C C n D , A \
S >- C \
D,R\
A > D \
C.
Hence Tversky‘s general model may be expressed as’ f o l l o w s : S(a,b)
= F(A n B , A \ 6 , 8 \ A ) , F b e i n g monotonic i n t h e above sense.
I n t r o d u c i n g a d d i t i o n a l assumDtions about t h e s i m i l a r i t y o r d e r i n g , Tversky o b t a i n s a r e o r e s e n t a t i o n o f t h e g e n e r a l model by t h e c o n t r a s t model: S*(a,b)
= uf(A n
B)
-
hf(P \ B)
-
uf(B \ A ) .
I n a d i r e c t i o n a l task,
u may d i f f e r and c o n s e q u e n t l y asymmetry may n o t h o l d .
x
and
179
Asymmerries ofsimilarity judgments
Indeed, t h e o b t e n t i o n o f t h a t model denends uoon the o r d i n a l n r o n e r t i e s of s i m i l a r i t i e s : t h e values S(a,b)
, cornDuted
from t h e o b s e r v a t i o n s , j u s t p l a y
t h e r o l e o f a medium f o r t h e o b t e n t i o n o f t h e comnlete n r e o r d e r ,< on t h e s e t o f pairs o f stimuli: (a,b)
i f f S(a,b)
,< (c,d)
That preorder i s reoresented by the m d e l .
,< S(c,d).
Moreover, i t c o n t a i n s c l e a r l y
t h e most r e l i a b l e i n f o r m a t i o n about t h e s i m i l a r i t i e s , t h e numerical d a t a S(a,b)
being Dossibly noised.
N e v e r t h e l e s s , i t i s i n those numerical d a t a
t h a t Tversky l o o k s f o r asymmetry.
!f one has t h e d e s i r e t o a v o i d t h e p r o -
blem l i n k e d t o t h e n o i s e o r o b a h l y o r e s e n t i n t h e data, one c o u l d prooose a n o t i o n o f asymmetrv based o n l y uoon t h e p r e o r d e r 6. Rut t h i s o r d i n a l n o i n t o f view i s m a i n l y s u n n o r t e d by t h e deeply o r d i n a l asoect o f a t y p e of asymmetry as emohasized D r e v i o u s l y i n t h e i n t r o d u c t i o n .
2.2. AN APPROACH RASED UPON THE AVERAGE O R D E R I N G I n T v e r s k y ' s paper, S(a,b) i s t h e mean v a l u e o f i n d i v i d u a l r a t i n g s .
First
keen t h a t way t o o b t a i n s i m i l a r i t i e s . Consider t h e f o l l o w i n o s i t u a t i o n : two groups a r e l u e s t i o n n e d f o r each o a i r a,b.
The grouo G1 works on j u d a i n o t h e s i m i l a r i t y o f a t o b, t h e group G2
works on t h e s i m i l a r i t y o f b t o a. cornouted f o r each group: S(a,b)
The mean values S(a,b)
f o r Gl, S(b,a)
f o r G2.
and S(b,a)
are
Therefore, t o each
group corresponds a comolete n r e o r d e r i n d u c e d b y S . ble say t h a t t h e two arouDs a r e i n a symmetrical judament when t h e two comn l e t e Dreorders c o i n c i d e .
The comnarison o f those o r e o r d e r s w i l l g i v e
us f n f o r r a t i o n nhout t h e imoortance and t h e l o c a t i o n o f asymmetries.
Un-
d e r t h a t name o f asymmetry, we understand here, any d e v i a t i o n between these two o r d e r i n g s
.
We n o t i c e some weak n o i n t s a t t h a t a m r o a c h ; l e t ' s mention two o f them. The f i r s t w i l l l e a d t o sow c a r e i n use.
The second w i l l sugaest a method
s u i t a b l e w i t h o u r o r d i n a l D o i n t o f view.
2.3.
P
DRAWBACK: THE INFLUENCE OF THE DISTRIBUTION OF THE P A I R S
OF STIFlULI
F o l l o w i n g a g a i n T v e r s k y ' s l i n e , l e t ' s imagine t h a t t h e d i s t r i b u t i o n between
G I and G2, o f t h e o r d e r e d n a i r s o f s t i m u l i i s such t h a t : s u b j e c t s i n GI meet o n l y t h e o r d e r e d o a i r s (a,b)
where b i s "more s a l i e n t " t h a n a, s u b j e c t s
i n G2 m e e t i n g t h e i n v e r t e d s i t u a t i o n .
I80
J . - P Barthehny
Then we can e x p e c t t h a t t h e S(a,b)
( i n GI) would be g r e a t e r t h a n the S(b,a)
R u t t h e two o r d e r i n g would oerhaps c o i n c i d e ( i n t h a t case t h e
( i n G?). observed asymmetries w o u l d be o n l y n u m e r i c a l b u t n o t o r d i n a l ) . a n o t h e r d i s t r i b u t i o n o f t h e o r d e r e d p a i r s between
cl
However,
and c2, r e t a i n i n g t h e
same n u m e r i c a l a s y v w t r i e s , would l e a d p o s s i h l y t o d s t i n c t o r d e r i n g s .
I n o r d e r t o observe a l l n o s s i b l e o r d i n a l asymmetries the 2n-1 D o s s i b l e d i s t r i b u t i o n s o f o u r n p a i r s .
we need t o l o o k a t
For l a c k o f m e e t i n g such
a design, i t seems d e s i r a b l e t o d e a l w i t h an e x n e r i m e n t f i t t i n o t h e two f o l lowinq qrooerties (a\
.
A n r i o r i : t h e o r d e r e d p a i r s (a,h) where b i s s a l i e n t , a r e equita'-l!/
d i s t r i h u t e d between t h e two prouns
.
( b ) A o o s t e r i o r i : ift h o s e a D r i o r i s a l i e n c e s do n o t l e a d t o asymmetries ( a c c o r d i n g t o T v e r s k v ' s i n t e r n r e t a t i o n ) one c o u l d however w i s h t o o b t a i n , as a way o f c o n t r o l , an m u i t a b l e d i s t r i b u t i o n o f t h e o r d e r e d n a i r s (a,b)
so t h a t s(a,b)
( a f t e r havinq standardized the v a r i a b l e S i n t o s )
s(b,a)
between t h e two s r o u m
.
2.4. ANOTHER D R A W A C K ANn I T S PE'lEDv Consider now t h e o b t e n t i o n o f t h e two o r d e r i n s s . t h e mean values S(a,h). (individual level).
They a r e c o n s t r u c t e d f r o m
Hence we r e s t on t h e r a t e s g i v e n by each Derson
Those r a t i n a s a r e " f o r g o t t e n " a t t h e c o l l e c t i v e l e v e l
a t the b e n e f i t o f en o r d e r .
Under t h e assumption t h a t those n u m e r i c a l
values a r e n o i s e d , some looseness i s t o be f a r e d on t h e o r d e r s themselves. I n t h i s r e s p e c t i t seems more c o n v e n i e n t t o found t h e c o l l e c t i v e o r d e r i n a over i n d i v i d u a l orderincjs.
Q u t t h i s aooroach r a i s e s a d i f f i c u l t p r o b l e m
( i l l u s t r a t e d f o r i n s t a n c e , b y t h e famous A r r o w ' s theorem).
Ue nropose h e r e
under t h e name o f condorcean a n a l y s i s , t o use t h e m a j o r i t y r u l e and t o comDute median o r d e r s and t h e n t o cornnare them i n o r d e r t o d e t e c t ( o r d i n a l ) asvmtries. To each qroun i s a s s o c i a t e d a l i n e a r o r d e r on t h e p a i r s b u i l d e d up f r o m t h e o r d e r s i n d u c e d by t h e r a t i n g g i v e n by each i n d i v i d u a l .
T h a t i s t o say
t h a t t h e drawback mentioned above i n 2.3 remains, and some orudence w i l l command a t t e n t i o n .
181
Asymmetries ofsimilnrity judgments
3. THE EXPERIMENTAL DATA 3.1.
THE BOOKLETS
Fourteen p a i r s o f m i c r o c o v p u t e r s have been s e t t l e d ( w i t h t w e n t y machines). They have been chosen w i t h t h e h e l p o f a f i r s t i n v e s t i q a t i o n b e s i d e e x p e r t s i n microcomputing ( a c c o r d i n n t o those e x p e r t s , t h e n a i r s do n o t p r e s e n t any drastic dissimilarity feature).
Poreover, t e n i17no~qti-em a r e such t h a t a
s t i m u l u s (i .e. a microcomauter) i s more " d a z z l i n g " , than t h e o t h e r .
"known",
"standard",
. ..
Isle say t h a t such a p a i r admits an a p r i o r i s a l i e n c e . The
f o u r o t h e r D a i r s a r e s a i d t o be a p r i o r i n e u t r a l . Each p a i r a,b produces two o r d e r e d o a i r s (a,b) and (b,a).
When a,b admits
an a p r i o r i s a l i e n c e , we say, when b i s more s a l i e n t than a, t h a t (a,b) i s i n (-,+)
s i t u a t i o n , w h i i e (h,a)
i s i n (+,-) s i t u a t i o n .
The t w e n t y e i g h t o r d e r e d n a i r s s o formed a r e d i v i d e d i n t o two b o o k l e t s (one green, one b l u e ) so t h a t :
-
each o a i r occurs one and o n l y one t i m e i n each b o o k l e t ; each b o o k l e t c o n t a i n s e x a c t l y f i v e o r d e r e d D a i r s i n (-,+)
s i t u a t i o n and
f i v e o r d e r e d p a i r s i n (+,-) s i t u a t i o n . The o r d e r f o r i n t r o d u c i n g t h e n a i r s has been randomized i n each b o o k l e t . T o r t h e r e r s o n s i n t h e tun grou@s, t h e i n s t r u c t i o n i s , f o r each o r d e r e d o a i r (a,b):
" g i v e a mark i n d i c a t i n g t h e degree o f s i m i l a r i t y of a t o b " .
I n o r d e r t o do so, t h e s t i m u l i a and b a r e c o n s e c u t i v e l y i n t r o d u c e d w i t h some d e s c r i p t i v e f e a t u r e s : c o n s t r u c t o r , c o u n t r y , comnosi t i o n o f t h e c e n t r a l u n i t (ROM,
RAM, p r o c e s s o r ) , a p p l i c a t i o n s and languages.
It i s well noti-
c i n g t h a t those f e a t u r e s correspond t o a b a s i c o r o d u c t ( n o t i n v o l v i n g any extension p o s s i b i l i t y ) . 3 . 2 . GROUPS AND SUBJECTS I n o r d e r t o t e k e c a r e c f t h e i n t e r i n d i v i d u a l v a r i a b i l i t y we have adooted a match group procedure by c o n t r o l l i n ? t h e D r o f e s s i o n s and t h e l e v e l s o f knowledge i n i n f o r m a t i c s . The s u b j e c t s a r e d i v i d e d i n t o two aroups. l e t , t h e second t h e green b o o k l e t . cond 41.
The f i r s t r e c e i v e t h e b l u e book-
The f i r s t c o n t a i n s 31 nersons, t h e se-
3 . 3 . IION INFLLIEVCE OF A PRIOR1 SALIENCES !lN TtlF NUrlERICPL ASYIVSFTRIES F o r each p a i r a,h t h e f o l l o w i n a numbers have heen comouted: S(a,h)
and S(h,a):
t h e mean values, i n t o each groun, of t h e deqree o f s i m i -
l a r i t y of one s t i m u l u s t o t h e o t h e r . 9,- ( a , b ) : t h e mean v a l u e o f t h e o r d e r e d p a i r i n s i t u a t i o n (+,-)
from a,b:(S+,-
= S(a,b)
(a,h)
o r S(b,a)h
S-,+
issued
( a & ) the value f c r the order-
ed c s i r i n s i t u a t i o n (-,+).
A l l those v a r i a b l e s a r e s t a n d a r d i z e d i n t o s ; s + , - : s - , + . The e x a m i n a t i o n o f t h e values o f S ( c . f .
apnendix) i n d i c a t e s t h a t , among
the t e n n a i r s w i t h an a n r i o r i s a l i e n c e , t h e d i f f e r e n c e
5-,+ ( a \ b )
-
S+,-
( a l b ) i s o o s i t i v e o n l y on f o u r n a i r s .
T h i s remark may
n o t be n i c e l y i n t e r n r e t a t e d ( t h e averages o f t h e two qrouns b e i n g d i f f e r e n t ) . Otherwise, u s i n g t h e s t a n d a r d i z e d v a r i a b l e s , t h i s tendency r e v e r s e s . F o r each o f t h e 7 2 b o o k l e t s , t h e average f o r t h e f i v e o r d e r e d o a i r s i n (-,+) s i t u a t i o n has been computed as w e l l as t h e average f o r t h e f i v e p a i r s i n
(+,-) s i t u a t i o n , heen o b t a i n e d .
So two v a r i a b l e s Ti(-,+)
and Ti(+'-)
( 1 6 i 5 72) have
T h e i r mean values are: IT
~
1
(-,+) = (+,-)
=
. -
Z Ti (-,+) i=l
A?
Ti
(+,-)
= 10,730 = 10,647.
i= 1 T h i s l e a d s t o ( S t u d e n t t e s t ) t = 0,186; f e r s i g n i f i c a n t l y from
IT
p = .67.
So m (-,+)
toes n o t d i f -
(+ ,-).
I t i s w o r t h n o t i c i n ? t h a t t h i s r e s u l t does n o t c o n t r a d i c t T v e r s k y ' s i n t e r -
pretation.
Indeed, t h e r o l e s o f s u b j e c t and o f r e f e r e n t a r e n o t v e r y c l e a r
Although i t i s asked t o t h e Dersons t o compare a t o b, t h e y d i s c o v e r
here.
(and " l e a r n " ) a t f i r s t a , t h e n b ( i f t h e paqes o f t h e b o o k l e t s a r e r e g u l a r l y r e a d f r o m t h e t o n t o t h e hottom..
.. ) .
?.4. THE DISTRIBUTION OF THE ORDERED P A I R S AND THE MF\XII.?AL FIUVERICAL
ASYIWETRY Amono t h e f o u r t e e n o r d e r e d p a i r s (a,h) s(a,b)
> s(b,a).
i n each b o o k l e t , seven a r e so t h a t
A n o s t e r i o r i the ordered o a i r s are, concerning the d i r e c -
t i o n o f t h e asymmetry, f a i r l y d i s t r i b u t e d and t h e demand ( b ) o f 2.3 i s satisfied.
183
Asymmetries of similarity judgments
so t h a t
L e t ' s comoute, f o r each b o o k l e t t h e mean v a l u e o f t h e seven S(a,b) s(a,b)
>
s(b,a).
from t h e seven S(a,b) so t h a t s(a,b) 72 1/72 I i=l
72 1/72
Ui
1
The v a r i a h l e Vi
IJe o b t a i n so t h e v a r i a b l e Ui.
= 11,242;
i=l
I t becomes
< s(b,a).
V . = 10,523:
'
i s defined
t = 1,769;
p = .038.
This
g r o u n i n q corresponds t o t h e "maximal n u m e r i c a l asymmetry" (remember t h a t one has t o t a k e t h e v a l u e of P w i t h g r e a t prudence, k e e n i n g i n mind t h a t f r o m t h e a o o s t e r i o r i s e l e c t i o n o f U and V, p i s o n l y a l o w e r bound).
4. CONDORCEAN ANALYSIS 4.1.
FROM CONDORCET TO THE MEDIAN PROCEDURE
...,
Consider a s e t G = {l, o } o f i n d i v i d u a l s and a s e t X o f o b j e c t s . t h a t each i n d i v i d u a l i c a r r i e s o u t a n a i r e d comnarison. t y p i f i e d b y a b i n a r v r e l a t i o n Ri on X.
I n o r d e r t o n u t f l e s h on t h e bone,
as "y i s n r e f e r r e d t o x by t h e i n d i v i d u a l i".Consi-
we can i n t e r p r e t xRiy d e r i n g a l l t h e Ri
, we
o b t a i n t h e o r o f i l e n = ( R1,.
. .,RD).
\Je denote b y
~ ( x , y ) t h e number of i n d i v i d u a l s o r e f e r r i n g y t o x: n ( x , y ) = l { i '!hen n(x,y)
Assume
T h a t comparison i s
2 P/2, y i s s a i d t o be m a j o r i t a r y p r e f e r r e d t o x .
I xRiy}l. T h e r e f o r e we
o b t a i n t h e s o - c a l l e d Condorcet r e l a t i o n o f t h e p r o f i l e : xRy i f f y i s m a j o r i t a r y n r e f e r r e d t o x. F o r m a l l y we can w r i t e : R =
U C' ~5 n+l
P
R
iE
i'
I G ' I= [T1 t h e i n t e r s e c t i o n and t h e u n i o n o f b i n a r y r e l a t i o n s b e i n g d e f i n e d as u s u a l . Otherwise t h a t Condorcet r e l a t i o n l o o k s l i k e a median ( c . f . Barthelemy and llonjardet, (1981)). I n o u r case, we t r y t o summarize t h e s i m i l a r i t y o r d e r s , q i v e n h y each i n d i v i d u a l i n one group, i n t o a l i n e a r o r d e r . each Ri
A t f i r s t , we n o t i c e t h a t when
i s complete, t h e n R i s comolete t o o .
t o be a t r a n s i t i v e r e l a t i o n even i f each Ri
Nevertheless R w i l l n o t need i s so.
T h a t nhenomenon i s
known as " E f f e t Condorcet" ( c . f . G u i l b a u d (1952)) o r ' b a r a d o x o f v o t i n g " . I n o r d e r t o r e f o r m it, i t i s c l a s s i c a l t o proceed as f o l l o w s (when each Ri i s complete): b y i n v e r t i n g t h e d i r e c t i o n o f som a r c s ( i . e . b y t r a n s f o r m i n g
s o w xRy i n t o yRx), we a r e a b l e t o g e t , f r o m R, a t r a n s i t i v e r e l a t i o n
R.
184
Among a l l p o s s i b l e
R,
we choose the r e l a t i o n m i n i m i z i n g the sum
-
z (.(X,.Y) .(Y,X)), ( X ,yFC C denotinq the s e t o f i n v e r t e d arcs. I t i s then s u f f i c i e n t t o break the t i e s ( w i t h o u t creatinc? any c y c l e ) i n order t o o b t a i n a l i n e a r order, the s o - c a l l e d median o r d e r o f the D r o f i l e .
P r o p e r t i e s and j u s t i f i c a t i o n s o f
the use o f median orders may be found i n G a r t h @ l $ y and r'onjardet (1961).
4.2.
CONDORCEAN AldbLY2IS OF TI10 GROUP5
Consider now two orouns G I and G2 o f r e s o e c t i v e l y nl and n 2 oersons.
bt
i s associated the b i n a r y r e l a t i o n on X: the 'bpinion" o f i on the elements o f X . Thus we o b t a i n two D r o f i l e s TI-1 and T 2 each i n d i v i d u a l i i n Gk (k=1,2)
.
The condorwar a n a l y s i s cornnares a median order associated t o c r d e r associated t o
IT
2.
TIt' o
a median
Since a o r o f i l e may admit s e v e r a l median orders,
these two are chosen w i t h a maximal c o r r e l a t i o n ( i n K e n d a l l ' s sense).
This
comparison may i n v o l v e the f o l l o w i n g Dractices: (a)
q u a l i t a t i v e l y : i n o r d e r t o l o c a t e the ( o r d i n a l ) asymmetries, we prono-
se t o :
-
d e t e c t the sequences ( o f consecutive o b j e c t s ) common t o the two median orders and maximal;
-
determine the subsets n1,Q2,
...,QD
of X , c a l l e d ordered blocks, so t h a t
f o r eacli o f the two median orders'Qi < Qi+l ded by each element o f Qi+l), t h e
-
(each element i n Qi
i s boun-
ni beino minimal;
determine the i v t e r s e c t i o n D a r t i a l o r d e r o f the two median orders and examine t h e i n c o m a r a h l e elements i n t h a t i n t e r s e c t i o n .
( h ) q u a n t i t a t i v e l y : beyond the computation o f the K e n d a l l ' s 1, we propose three c o e f f i c i e n t s (more o r l e s s i n s o i r a t e d by Gordon (1979a) and (1979b)).
A l l those c o e f f i c i e n t s w i l l p r o v i d e an e v a l u a t i o n o f the maanitude o f the ( o r d i n a l ) asymmetry.
They take values between 0 and 1 and are d e f i n e d such
as : a/n-I.
a being the sum o f the lengths o f the maximal comnon sequences;
p-lh-1,
p b e i n g t h e number o f minimal ordered blocks;
R/n-1 = n-y/n-1,
6 b e i n g t h e maximal l e n g t h of a maximal chain i n the i n t e r -
s e c t i o n o r d e r and y denoting the minimal number o f elements t o remove i n o r d e r t o make the two median orders c o i n c i d e .
185
.4symmetries ofsimilarity judgments
4.3. P REIlARK ABOUT THE EXPERIVENTAL DESIGN I t i s worth n o t i c i n g t h a t the e s s e n t i a l nature o f the m a j o r i t y r u l e f o r b i d s any merging
4.4.
-
-
both e x n l i c i t o r i m n l i c i t
o f t h e two y o u n s i n t o one.
CnNDORCEAN ANPLYSIS OF THF SIYILASITY ORDERS: QUALITATIVE STUDY
L e t 1,2,.
. .,14, denote
t h e f o u r t e e n Dai r s o f microcomnuters.
The Condorcet
r e l a t i o n a s s o c i a t e d t o t h e b l u e b o o k l e t ( g r o w G1) may be n i c t u r e d as f o l 1ows :
3
+
8
9
+
t
-
5
+
+
2
+
7
+
14
+
10
+
1
13
+
4
+
12
+
1
4.c
11 ft
6 (The arrows a r e comnosed by t r a n s i t i v i t y excented any mention o f t h e c o n t r a ry, f.i. 8
1
5).
5 p r o h i b i t the existence o f 8
T h a t r e l a t i o n admits , j u s t one c y c l e
.548
and t h e median o r d e r s a r e e x t r a c t e d , w i t h a t i e b r e a k i n g , f r o m t h e f o l l o w i n g comolete p r e o r d e r ,
3
+
23
-+
9
+
5
+
2
+
7 + 14 + 10
+
13
+
4 + 12 + 1
f+
I1 4s
6 From t h e green b o o k l e t s we g e t t h e f o l l o w i n g Condorcet r e l a t i o n :
5 +3+8+9+2+7+
1 4 + 1 0 + 1 3 + 6 + 1 + 4 + 12 Sf
11 which admits n o c y c l e . The two f o l l o w i n g median o r d e r s a r e i n maximal c o r r e l a t i o n . Blue -
-
3
<
8
<
9
<
5
<
2
<
7
<
14
<
10
<
13
<
11
<
6
<
4
<
12
Green: 5 < 3 < 8 < 9 < 2 < 7 < 14 < 10 < 13 < 11 < 6 < 1 < 4
<
1 12
/.A? Barthdl;my
186
The maximal common seauences a r e : 3
<
8
c
9 , 2'
<
7 < 14 < 10
c
13 < 1 1 < 6 , 4
<
12
They r e a r e s e n t p i e c e s o f symmetrical iudgments between t h e two rlrouns.
The
m i d d l e sequence n o i n t s o u t a complete agreement (symmetry) f o r seven n a i r s l o c a t e d a t t h e m i d d l e of t h e s c a l e s . o a r c e l !4,12,1)
T h i s sequence i s d o v i n a t e d by a t o n
and dominates a t a i l n a r c e l { 3 , 8 , 9 , 5 } .
I n those two p a r -
cels asymmtries occur. T+e decomoosition o f t h e two o r d e r s i n t o o r d e r e d b l o c k s CcrrobGrates those remarks.
I t p o i n t s t h e t o o o a r c e l O9 = {12,4,1),
t h e t a i l o a r c e l {3,8,9,51,
t h e m i d d l e sequence b e i n o b r o k e n ur) i n t o seven s i n g l e t o n s . The Hasse diaaram o f t h e i n t e r s e c t i o n o f t h e two median o r d e r s i s p i c t u r e d ,
!t c o r r o b o r a t e s t h e n r e v i o u s remarks and p o i n t s o u t asymmetries i n v o l v i n g t h e f o l l o w i n g p a i r s : 12 and 1, 4 and 1 f o r t h e s t i m u l i n e r c e i v e d as t h e most s i m i l a r ; 5 and I), 5 and 8 , 5 and 3 f o r t h e s t i m u l i o e r c e i v e d as t h e less similar.
The asymmetry a t t h e t o o i s d i v e r t i n g enouah.
I t concerns
t h e t h r e e o a i r s i n v o l v i n a t h e same t h r e e microcomputers: 1 = TRS 80 mod 2 /
CENIE 3 , 4 = Z 89 / G E N I E 3 , 12 = Z 89 / TPS 80 mod 2 . 12
I
I I 13 I 10 I
11
14
i 1 5
9
I I
8
3 4 . 5 . CONDORCEAN ANALYSIS: qUANTITPTIVE STUDY The values o f t h e c o e f f i c i e n t s we have oroposed are: T
= .89,
a / n - 1 = .69,
o- 1 = .62,
B / n - l = .85.
187
Asymmetries of similoriy judgments
They i n d i c a t e
8
l a r @ e enough symmetry, each o f them i n v o l v i n g i t s p r o p e r
information. 4.6.
OF THE AVERAGE ORDERS
ANALYSIS
The m a n values o f t h e r a t e s g i v e n t o each D a i r o f microcomputers, induce on each b o o k l e t t h e f o l l o w i n g comalete p r e o r d e r s .
- 3 Green: 5 Blue:
;(
<
8
<
3 < 8 < 9 < 7
<
5 < 9 < 2 < 7 < 14 < 10 < 11 < <
2 < 14 < 10 < 13 < 11
;4
1
<
12
c
6 < 4 < 1 < 12
<
means a 14: d i f f e r e n c e ) .
Comnaring t o t h e median o r d e r s we f i n d a c o i n c i d e n c e excepted f o r : two i n v e r s i o n s 2, 7 and 1, 4 f o r t h e green b o o k l e t s ; two i n v e r s i o n s
9, 5
and 4 , 12 f o r t h e b l u e b o o k l e t s . The asymmetries a r e b e t t e r marked than i n t h e case o f median o r d e r s . The maximal common sequences are: 3 < 8 , 14 < 10, 11 < 6 and 4 < 1. !eads t o e i g h t
Q1 = {3, 5, C),
o6
= (13,
111,
The decomposition i n t o o r d e r e d b l o c k s
ni. O2 = {91,
Q, = (61,
Q3 = {7, 21,
Q, = {14},
O5 = {lo},
Q8 = { 4 , 1, 12).
The Hasse diagram of t h e i n t e r s e c t i o n i s p i c t u r e d .
That n a r t i a l order i s
c o n t a i n e d i n t o t h e i n t e r s e c t i o n o f t h e two median o r d e r s .
I 7 /I4\
\
/ 2
7\
8 I I
3 Kendall
T
remains h i g h :
T
= .87
The o t h e r c o e f f i c i e n t s decrease: a/n-1 = .31, p - l / n - 1
= .54,
~ / n - l = .69.
4 . 7 . AS A (PROVISION~L)
CmcLusIw
From t h e p r e v i o u s s t u d y the f o l l o w i n g n o i n t s emerqe. (i)
I n a c o n t e x t n o t so n r o n i t i o u s as those s u r r o u n d i n q T v e r s k y ' s e x p e r i -
ment:
-
t h e n o t i o n o f s a l i e n c e i s n o t so c l e a r i n t h e f i e l d o f microcomputers, t h e n o t i o n s o f s u b j e c t and r e f e r e n t a r e ambi@uous; asvmmetries keep on t o be observed b u t i n a weaker sense, e i t h e r n u m e r i c a l w i t h an o q t i m a l grou? i n ? ( w h i c h does n o t c o i n c i d e w i t h t h e a o r i o r i s a l i e n c e s ) o r o r d i n a l ( i n d e e d , t h a t weakqess c o n f i r m s , i n s o r e sense, T v e r s k y ' s i n t e r o r e t a t i o n ) .
( i i ) The o r d i n a l a n a l y s i s emphasizes t h e f a c t t h a t t h e asymmetries a r e r e l a k d t c the D o s i t i o n o f t h e n a i r s o f s t i m u l i i n t h e s i m i l a r i t y s c a l e . They a r e observed e s s e n t i a l l y ( e n t i r e l y i n t h e case o f t h e median o r d e r s ) a t t h e head and t h e t a i l o f t h a t s c a l e . ( i i i ) O f course, t h e observed a s y m t r i e s deoend uoon t h e o b s e r v a t i o n methods. They a r e l o w e r f o r median o r d e r s than f o r average o r d e r s . Now, t h e f o r m e r a r e much m r e n u m e r i c a l than t h e l a t t e r .
Is t h i s any c o n n e c t i o n
between the degree o f " n u m e r i c a l i t y " o f t h e o r d e r s and t h e n o s s i b i l i t y o f o b s e r v i n g asymmetries?
PE FE RENCES
[U
Barthelemy, J.P.,
and V o n j a r d e t , B. The median Drocedure i n c l u s t e r
a n a l y s i s and s o c i a l c h o i c e t h e o r v , Math. SOC. S c i
Ifl
Duval , S, and Duval , V .H.,
131
1 ( 1 9 8 1 ) , 235-267.
Consistency and coani t i o n , Lawrence E r l b a u n
ass., H i 1 l s d a l e , London (1983) Gordon, A . G . ,
.,
.
A measure o f t h e aareement between r a n k i n o s , B i o m e t r i k a ,
66, 1 (1979a), 7-15.
I41
Gordon. A.G.,
Another measure o f t h e agreement between r a n k i n g s , R i o -
m e t r i k a , 66, 2 (1979b). 327-332. I51
Guilbaud, G.Th.,
Les t h e o r i e s de l ' i n t e r e t g e n e r a l e t l e Drobleme l o a i -
que de l ' a g r e g a t i o n , Economie ADpliqu@e, 5(4), ( 1 9 5 2 ) . k e p r i n t e d i n Elements de l a t h e o r i e des j e u x , Dunod, P a r i s (1968). E n g l i s h t r a n s l a t i o n i n , Readings i n Y a t h e n a t i c a l s o c i a l s c i e n c e s , Science Research A s s o c i a t e s , Chi caqo ( 1966) [61
K e n d a l l , M.G.,
I71
Tversky, A.,
, 262-307.
Rank c o r r e l a t i o n methods, Hafner, New York, 3 r d e d i t i o n
(1962). 3 2 7-352.
F e a t u r e s o f s i m i l a r i t y , P s y c h o l o g i c a l Review, 84, 4 (1977)
Asymmetries ofsimilarity judgments
[8]
189
Tversky, A, and C a t i , I,S t u d i e s of s i m i l a r i t y , i n E. Rosh and B . L l o y d , eds.,
gories
, E r l baun,
On t h e n a t u r e and t h e p r i n c i p l e o f f o r m a t i o n o f c a t e H i 1 l s d a l e and London. (1979).
APPENDIX The f o u r t e e n D a i r s
1
TRS 80 Mod. 2
2
SIL'Z 3
3
HP85A
4
z
5
T I 99/4A
6
ABC 24
39
7
XEROX 820
8
TX 25
9
V I C 20
10 11
f4 20 XEROX 820
12
Z 89
13
APPLE
14
DAI
rrI
/ / / / / / / / / / / / / /
GENIE 3
APPLE I 1 Plus OSBORNE 0 1
GENIE 3
zx
51
TRS 80 Mod. 2 APPLE 111 GOUPIL 2
DAI ITT 3030 MICRAL 8021 TRS 80 Mod. 2 YICRAL 8021 V I C T O R LAMBDA
The w r i t i n g o r d e r o f each p a i r and t h e o r d e r between t h e ? a i r s a r e those o f the blue booklets. The a n r i o r i s a l i e n c e I n t h e b l u e b o o k l e t s t h e a D r i o r i s a l i e n c e s were s o c o n s t i t u t e d .
(+,-): (-,+):
1, 3, 4 , 9, 13 2, 6, 7, 8, 14.
S t a t i s t i c a l d e s c r i p t i o n of the hooklets
Pair a/b
-r 2
mean s i m i l a r i t y 51 ue
Green
standard deviation Blue
--lT-!Tm-37- - n z T 9.37
Green
s t a v d a r d i zed s i m i 1a r i ty 91 ue
Green
-7s-¶- 1 . 3 5 7-72-
10.80
3.440
4.080
-0.54
-0.17
3
6.45
7.96
4.211
4.750
-1.93
-1.35
4
12.52
13.73
3.202
3.548
0.96
1.04
5
8.20
7.07
3.987
3.703
-1.10
-1.71
6
12.45
13.39
3.416
3.341
0.93
0.90
7
9.71
9.79
3.786
3.918
-0.38
-0.59
8
7.97
8.16
4.701
3.606
-1.21
-1.26
9
8.32
9.22
3.237
3.524
-1.04
-P.83
10
11.06
11.51
4.165
3.762
0.27
0.12
11
12.21
13.38
4.288
3.391
0.82
0.89
12
12.48
14.95
3.048
3.107
0.94
1.54
13
12.45
11.71
3.582
3.983
0.93
0.20
14
10.45
11.22
3.867
3.745
-0.02
-
10.497
11.220
9.097
2.420
0
-
0.00 0
191
Asymmetries of similarity judgments
Ontimal g r o u p i n g Ordered n a i r s (a,b) so t h a t s(a,b)
s ( b y a ) : 1, 5, 6, 7, 8, 10, 13 ( f o r t h e
>
c o n v e n t i o n a l w r i t i n g o r d e r o f each n a i r ) .
A p r i o r i salience
Namely
Agreement w i t h a D r i o r i salience
(TRS 80 Hod 2, GENIE 3)
('y-1
N@
(APPLE II', S I L ' 2 3 )
('9-1
NO
( - ,+I
YES
(-,+I
YES
(OSSOIW
(!:,
hF F5P)
(GENIE 3, Z 89) ( T I 99/4A, Z X 81)
neutral
(ABC 24, TRS 80 Mod 2 )
( -9')
YES
(XEROX 820, APPLE 111)
(
-
3')
YES
( T X 25, GnuPIL 2)
(
-9')
YES
(DAI, V I C 20) (1120, I T T 3030)
neutral
(MICRAL 8021, XERDX 820)
neutral
(TRS 80 Mod 2, Z 89)
neutral
(-9')
(APPLE 111, MICRAL 8021) (VICTOR LAMBDA, DAI)
(+,-I (+*-I
YES
NO
NO
The a u t h o r wishes t o thank HervP Abdi, C a r o l i n e k v s r e n n e , Yves Le Foll and Jawad Mourtada f o r t h e i r h e l p .
This Page Intentionally Left Blank
TRENDS IN MATHEMA1'ICAL PSYCHOLOGY E . Degreef a n d / . Van Bu genhaut (editors) 0 Elsevier Science Dublisfers B.V. (North-Holland), 1984
193
THE AXIOMATIZATION OF ADDITIVE DIFFERENCE MODELS FOR PREFERENCE JUDGMENTS M.A.
Croon
Vakgroeo S t a t i s t i e k en Methoden E Technieken van Onderzoek Subfacul t e i t P s y c h o l o c j e K a t h o l i e k e Hogeschool T i l b u r g F o l l o w i n g t h e approach t a k e n by F i s h b u r n (1980), t h e a l g e b r a i c a x i o m a t i z a t i o n o f an a d d i t i v e d i f f e r e n c e model f o r n a i w i s e choices between mu1 t i - a t t r i b u t e s t i m u l i i s discussed.
1. INTRODUCTION Tversky (1969) showed t h a t , under s p e c i f i e d exDerimental c o n d i t i o n s , c o n s i s t e n t and p r e d i c t a b l e i n t r a n s i t i v i t i e s i n t h e p a i r w i s e choices between m u l t i a t t r i b u t e s t i m u l i can be demonstrated.
Furthermore, he discussed two mo-
d e l s t h a t can account f o r these r e s u l t s .
The f i r s t model which imposes a
j u s t n o t i c e a b l e d i f f e r e n c e s t r u c t u r e o r semiorder on a l e x i c o g r a p h i c o r d e r i n g , was used by h i m i n t h e c o n s t r u c t i o n o f t h e a l t e r n a t i v e s f o r h i s expe-
A s i m i l a r model was s t u d i e d by Ng (1977). ,The
rimental investigations.
second model, c a l l e d t h e a d d i t i v e d i f f e r e n c e model, assumes t h a t t h e c h o i c e between p a i r s o f m u l t i - a t t r i b u t e s t i m u l i i s based on comparisons o f component-wise d i f f e r e n c e s between t h e a l t e r n a t i v e s .
L e t A=Alx.
o f m u l t i - a t t r i b u t e s t i m u l i w i t h elements o f t h e f o r m x=(xl,. y=(yl, Let
>o
...,yn)
where xi
..,xn)
and
r e p r e s e n t s t h e v a l u e o f s t i m u l u s x on a t t r i b u t e i.
be t h e e m p i r i c a l p r e f e r e n c e r e l a t i o n on A such t h a t x t o y i f and
o n l y i f x i s a t l e a s t as much p r e f e r r e d as y.
(A,>,J
..xAn be a s e t
The p r e f e r e n c e s t r u c t u r e
s a t i s f i e s t h e a d d i t i v e d i f f e r e n c e model i f t h e r e e x i s t r e a l - v a l u e d
f u n c t i o n s ul,.
. .,un
and i n c r e a s i n g f u n c t i o n s
. . ,$,
i n t e r v a l s such t h a t x z 0 y i f and o n l y i f : n
where @i(-6)=-$i(6)
E $i IUi(Xi)-Ui(yi)I 5 i= 1 f o r a l l i.
O
d e f i n e d on some r e a l
194
M . A . Croon
A l t h o u g h Tversky (1969) discussed some p r o p e r t i e s o f t h i s a d d i t i v e d i f f e r ence model, he d i d n o t p r o v i d e a complete a x i o m a t i z a t i o n o f i t a l t h o u g h he mentioned t h a t a s e t o f o r d i n a l axioms f o r a symmetric a d d i t i v e d i f f e r e n c e model o f s i m i l a r i t y ( r a t h e r than p r e f e r e n c e ) judgments was g i v e n by Beals, K r a n t z and Tversky (1968).
As a model o f p r e f e r e n c e judgments, t h e a d d i t i v e d i f f e r e n c e model r e c e i v e d l i t t l e attention.
I n h i s d i s s e r t a t i o n , van Acker (1977) reviews t h e l i t e r -
a t u r e on i n t r a n s i t i v e c h o i c e and mentions t h e a d d i t i v e d i f f e r e n c e model and some g e n e r a l i z a t i o n s o f i t , w i t h o u t g i v i n g a thorough d i s c u s s i o n o f t h e i r axiomatization.
R e c e n t l y F i s h b u r n (1980) p r o v i d e d an a x i o m a t i z a t i o n
o f t h e s o - c a l l e d l e x i cograDhi c addi t i ve d i f f e r e n c e model i n which p r e f e r ence judgments a r e g l o b a l l y governed b y a l e x i c o g r a p h i c r u l e which may loc a l l y ( e . g . when t h e two s t i m u l i a r e a u i t e s i m i l a r ) be superseded by an a d d i ti ve d i f f e r e n c e r u l e .
Moreover, t h e p u r e a d d i t i v e d i f f e r e n c e model can
be c o n s i d e r e d as a l i m i t case o f t h i s more g e n e r a l model.
I n h i s paper,
F i s h b u r n r e s t r i c t s h i m s e l f t o t h e case i n which t h e s t i m u l i a r e composed o f two a t t r i b u t e s .
Furthermore, h i s a x i o m a t i z a t i o n i s based on r a t h e r
s t r o n g t o p o l o g i c a l assumptions about t h e component s e t s .
I n t h e p r e s e n t paper, F i s h b u r n ' s approach t o t h e a x i o m a t i z a t i o n o f t h e p u r e a d d i t i v e d i f f e r e n c e model i s pursued.
I n t h e f i r s t p l a c e , an a t t e m p t was
made t o r e p l a c e t h e t o D o l o g i c a l assumptions about t h e component s e t s by some more a l g e b r a i c assumptions s o as t o p r o v i d e an a x i o m a t i z a t i o n o f t h e p u r e a d d i t i v e d i f f e r e n c e model which i s more i n l i n e w i t h t h e approach o f K r a n t z e t a l . (1971).
I n t h e second n l a c e , t h i s paper shows how F i s h b u r n ' s
approach can be g e n e r a l i z e d t o t h e case when t h e s t i m u l i a r e composed o f more t h a n two components.
Due t o space l i m i t a t i o n s , we o n l y g i v e f u l l
p r o o f s o f lemmas and theorems f o r t h e case o f two components. f o r t h e case o f t h r e e component s e t s a r e b r i e f l y sketched.
The p r o o f s
The g e n e r a l
m u l t i - a t t r i b u t e a d d i t i v e d i f f e r e n c e model and some g e n e r a l i z a t i o n s o f i t w i l l be discussed i n a subsequent paper. 2. ADDITIVE DIFFERENCE STRUCTURES ON TWO COMPONENTS SETS
I n t h e sequel we assume t h a t A and X a r e two non-empty s e t s and t h a t 2 o i s a b i n a r y r e l a t i o n on AxX. Elements o f A w i l l be denoted by s m a l l l e t t e r s , e i t h e r p r i m e d o r unprimed, f r o m t h e f i r s t h a l f o f t h e alphabet; s i m i l a r l y , s m a l l l e t t e r s from t h e second p a r t o f t h e a l p h a b e t w i l l be used t o denote
195
Axiomatuation ofadditive differewe models
F i n a l l y , elements o f AxX w i l l be denoted by o r d e r e d p a i r s
elements o f X .
of s m a l l l e t t e r s such as ax, by, The s t r u c t u r e
(
AxX,
>o)
CU,
etc.
i s s a i d t o he an a d d i t i v e d i f f e r e n c e s t r u c t u r e on
two component s e t s i f t h e b i n a r y r e l a t i o n
>o
s a t i s f i e s t h e f o l l o w i n g axioms:
1. Axiom 1 The b i n a r y r e l a t i o n
(1) ax
>o
i s r e f l e x i v e and connected:
>o
ax f o r a l l ax E AxX
( 2 ) ax z o by o r b y
>o
ax f o r a l l ax, by
Note t h a t we do n o t assume t h a t
>o
E
AxX.
i s transitive.
Hence,
>o
i s n o t a weak
order relation. 2. fi.xiom 2 The r e l a t i o n z o s a t i s f i e s t h e f o l l o w i n g t r i p l e c a n c e l l a t i o n (TC) c o n d i t i o n : aw
>,, bx
b y z o a z 1 * cw
zo
dx
cz z o dy 3. Axiom 3 The r e l a t i o n
>o
s a t i s f i e s the f o l l o w i n g quintuple cancellation conditions:
(1) 4 C A if
au
>o
b v and
' b x b o cy
{
b'v
>o
a'u
c'y
>o
b'x
t h e n c z z o aw i m p l i e s c ' z
zo
a'w
(2) QCX if
au z o b v and b v ' CV 2 o
dw
{*,I
t h e n f w lo gu i m p l i e s f w '
zo
au'
>o
cv'
z0
gu'.
I n o r d e r t o f o r m u l a t e t h e n e x t axioms we d e f i n e two new r e l a t i o n s
=o
and
'0.
Definition 1 ax
=o
by i f f a x a0 by and by z o ax
One e a s i l y proves t h a t
=o
i s r e f l e x i v e and symmetric.
However,
n e c e s s a r i l y t r a n s i t i v e : hence i t i s n o t an e q u i v a l e n c e r e l a t i o n ,
=o
i s not
M...i. Croon
196
Definition 2 ax >,
by i f f ax
The r e l a t i o n
>o
by and n o t by
?c
ax
i s irreflexive.
We now f o r m u l a t e t h e s o l v a b i l i t y c o n d i t i o n s .
". S 0 1 V ? h i l i t v
Cpndi t i n p q
( 1 ) U n r e s t r i c t e d s o l v a b i l i t y f o r two unknowns: a. USA: g i v e n x,y
E
X, t h e r e e x i s t a,b E A such t h a t ax
= D
by.
b . USX: g i v e n a,b
E
A , t h e r e e x i s t x,y E X such t h a t ax
=o
by.
( 2 ) R e s t r i c t e d s o l v a b i l i t y f o r one unknown: a. RSA: i f ax dx
=o
>o
by and by 2,
cx, t h e r e e x i s t s d E A such t h a t
by.
6. RSX: i f ax >,, b y and by
5,
a z , t h e r e e x i s t s u E X such t h a t
au =, by.
5 . Archimedean axiom I n o r d e r t o f o r m u l a t e t h i s axiom, we need some f u r t h e r d e f i n i t i o n s .
A sequence (al,a2
,..., an ,...)
o f elements of A i s c a l l e d a s t a n d a r d sequen-
ce f o r component s e t A i f t h e r e e x i s t y, z E X such t h a t f o r a l l n: any =, an-lz. A s t a n d a r d sequence i s s t r i c t l y bounded i f t h e r e e x i s t hl,b2 f o r a l l n t h e r e e x i s t un,vn,wn,xn
(1) blun ( 2 ) anwn
>o
b2vn and alvn
>,
alxn
and blxn
E A such t h a t
E X such t h a t :
anun b2wn.
>,
S i m i l a r d e f i n i t i o n s h o l d f o r t h e second component s e t X .
Then t h e a r c h i -
medean axiom can be s t a t e d i n t h e f o l l o w i n g way: e v e r y s t r i c t l y bounded s t a n d a r d sequence i s f i n i t e . F o r f u r t h e r use we p r o v e t h e f o l l o w i n g LEMMA 1 >o
s a t i s f i e s independence:
( 1 ) aw z o bw i m p l i e s ax 2,
hx
( 2 ) aw a , ax i m p l i e s bw 2o bx
197
Axiomatitation of additive difference models
Proof: (1) Suppose aw z 0 b r .
A l s o ax
=o
ax and aw =, aw.
Then by TC
(2)
ax =,
ax
aw
=o
aw 1 imnly ax
aw
>o
bw
>o
bx
SuDoose aw 5 , ax. Then b y TC bw =, bw aw
=o
aw 1 i m p l y bw 2 o b x
aw 2 , ax We now p r o v e t h a t t h e axioms 1 t o 5 a r e s u f f i c i e n t f o r an a d d i t i v e d i f f e r ence r e p r e s e n t a t i o n of
30.
Given t h a t t h e b i n a r y r e l a t i o n b o s a t i s f i e s
these c o n d i t i o n s , t h e r e e x i s t a f u n c t i o n f on A, a f u n c t i o n g on X and a f u n c t i o n $ such t h a t
The f i r s t s t e p i n t h i s D r o o f c o n s i s t s o f d e f i n i n g r e l a t i o n s RA on AxA and
RX on XxX i n t h e f o l l o w i n g way: Definition 3 F o r a r b i t r a r y a,b,c,d
E A,
(a,b)
RA (c,d)
h o l d s i f t h e r e e x i s t x,y E X such
(w,x)
RX (y,z)
h o l d s i f t h e r e e x i s t a,b E A such
and (XxX,R?
b o t h s a t i s f y t h e axioms o f an
t h a t ax a , by and dy 3 , cx. F o r a r b i t r a r y w,x,y,z
E
X,
t h a t aw 3, bx and b z 2 , ay. We can now p r o v e t h a t (AxA,RA)
a l g e b r a i c - d i f f e r e n c e s t r u c t u r e as s t a t e d by K r a n t z e t a l . (1971, 150-154). We g i v e t h i s p r o o f o n l y f o r (AxA,RA)
1.
.
RA i s a weak o r d e r .
a. RA i s r e f l e x i v e . Given a,b E A, by USX t h e r e e x i s t x,y such t h a t ax b y =,
ax.
),
by.
Then a l s o
=o
by.
Then e i t h e r
So (a,b)RA(a,b).
b. RA i s connected. Given a,b E A, by USX t h e r e e x i s t x,y such t h a t ax (1) cx
=o
dy o r ( 2 ) dy a. cx.
M A . Croon
198
I n case (1): cx t o dy and by I n case ( 2 ) :
ax
ax i m p l y (c,d)RA(a,b).
=o
by ar?d dy t o c x i m p l y (a,b)RA(C9d).
=D
c. RA i s t r a n s i t i v e . Suppose (a,b)RA(C,d)
and (C,d)RA(e,f).
NOW: (a,b)RA(c,d)
i m p l i e s av t o bw and dw t o C V f o r Some
(c,d)R
i m p l i e s cx k 0 dy and fy t o ex f o r SOW X,y
A
(elf)
V,W
E
x and x.
By TC: dw t o c v
cx t o Q 1 i m p l y fw t o ev fy
Then: av
ex
?*
>,, bw and fw
Hence ( a,b)RA( e ,f)
t c ev.
.
Definition 4
i f f(a,b)RA(C,d)
(a,b)IA(c,d)
and (C.d)RA(a,b).
Note: Since RA i s a weak o r d e r , i t f o l l o w s t h a t IA i s an equivalence r e l a tion.
LEMMA 2 (a,b)IA(c,d)
i f f au
=o
bv and cu
=o
dv f o r some u,v
E
X.
Proof:
(1)
(a,b)IA(c,d)
i m p l i e s (a,b)RA(c,d)
and (c,d)RA(a,b).
Now U,V
E
X and
dv and by 2 o ax f o r some x,y
E
X.
(a,b)RA(c,d)
i m p l i e s au t o bv and dv
(c,d)RA(a,b)
i m p l i e s cx
>o
>o
cu f o r some
By TC: au t o bv
by l o ax } i m o l y cu
>o
dv
z0
cx
>o
dy
Hence: cu
=o
dv s i n c e a l s o dv
cu.
Furthermore, a l s o by TC: dv t o cu
1 i m p l y bv t o au
cx t o dy by >co ax Hence: au
=o
b v s i n c e a l s o au
(2)
From au
2.
I f (a,b)RA(c,d),
=n
bv and cu
=o
?o
bv.
dv, i t f o l l o w s t h a t (a,b)IA(c,d).
then (d,c)RA(b,a).
199
Awiomatization ojadditive difference modeL
Proof: (a,b)RA(c,d)
i m p l i e s ax 3 o by and dy
-
o r (d,c)RA(b,a) 3.
cx f o r scme x,y E X .
>o
a0 by
Then a l s o dy 3 o cx and ax
I f (a,b)RA(a' , b ' ) and ( b , c ) R A ( b ' , c ' ) ,
then ( a , c ) R A ( a ' , c ' ) .
Proof: By h y p o t h e s i s : au l o bv and b ' v 2 o a ' u f o r some u,v E X and: bx By USX: aw
z0
By QCA: I f au {
bx
bv and
>o
E
b ' x f o r some x,y E X.
X.
b ' v B~ a ' u c'y ?o b'x
aw i m p l i e s c ' z 3 o a'w.
=o
cz and c ' z
=o
I
a o CY
t h e n cz Hence: aw
cy and c ' y
>o
c z f o r some w,z
=o
>o
a'w,
o r : (a,c)RA(a' , c ' ) .
4.
I f (a,b)RA(cyd)RA(a,a)
, then:
(1) t h e r e e x i s t s d ' E A such t h a t ( a , d ' ) I A ( c , d )
(2) there e x i s t s d"
E
A such t h a t (d",b)IA(c,d)
Proof: By USX: c u
dv f o r some u,v
=o
By h y p o t h e s i s : ax
>o
Then by TC: cu
dv
=o
X.
€
by and dy b o cx f o r some x,y E X.
dy
)o
cx 1 i m p l y au b o bv
ax
>o
by
Also: (c,d)RA(a,a)
i m p l i e s cp
?o
dq and aq
3 o ap f o r
some p,q
(1) by TC: aq 2 ap cp 2 dq l i m p l y av dv = o cu Hence we have av
>o
?o
au
au and au
>o
bv.
By RSA t h e r e e x i s t s d ' E A such t h a t au By Lemma 2: au
(2)
=o
By h y p o t h e s i s : cp
d ' v and c u >o
dq and aq
By Lemma 1 (independence): bq By TC: bq
>o
=o
d'v.
dv i m p l i e s ( a , d ' ) I A ( c , d ) . ?o >o
ap. bp.
bp
cp 3 o dq 1 i m p l y bv du =o cu
?o
bu
Hence we have au z 0 b v and bv
z0
bu.
By RSA t h e r e e x i s t s d " E A such t h a t d " u
=o
bv.
E X.
200
hi A. Croon
Hence d"u
=o
bv and cu
o r ( d" ,b ) I A ( c ,d) 5.
=o
dv
.
Archimedean p r o p e r t y .
,...,ai
L e t (al,a2
,...) be a sequence o f elements such t h a t :
( 1 ) (ai+l,ai)IA(a2,al)
f o r a l l ai ,ai+l
('1
not ( a 2 S a l ) 1 A ( a l , a l ) ( 3 ) t h e r e e x i s t d',d" such t h a t ( d ' ,d")PA(ai
,al)PA(d",d'),
P A b e i n g t h e asymmetric p a r t o f RA.
Then we have t o p r o v e t h a t t h e sequence i s f i n i t e . He f i r s t Drove t h a t t h e r e e x i s t u,v
6
X such t h a t ai+lu
=o
a.v f o r a l l 1
a.1 ,a i+l' Now: (ai+l,ai)IA(a2,al)
i m p l i e s ai+lu
=o
and a2u = o a v f o r some u,v E X. 1 Also: (ai+2,ai+l)IA(a2,al) i m a l i e s ai+2x and a x
2
By TC: ai+2x
aly
=o
=o
ai+1Y
aly
=o
a2x
a2u
=o
alv
a.v 1 a.,+1Y
=o
f o r some x,y E X.
1 f m l v ai+p
=o
ai+lv.
Hence t h e r e e x i s t u,v E X such t h a t ai+lu
=o
a 1. v f o r a l l a.1 ,ai+l
i n the
sequence.
,...,an ,...)
Hence, (al,a2
i s a s t a n d a r d sequence i n t h e s e n s e o f Axiom 5.
The p r o o f t h a t t h i s sequence i s s t r i c t l y bounded, f o l l o w s from t h e e x i s t e n c e o f t h e elements d ' and d" w i t h t h e s t a t e d p r o p e r t i e s .
Hence, by Axiom 5
t h i s sequence i s f i n i t e , w h i c h p r o v e s t h a t t h e archimedean axiom i s s a t i s f i e d i n (AxA,RA). We have p r o v e d t h a t (AxA,R$ structure.
s a t i s f i e s t h e axioms o f an a l g e b r a i c d i f f e r e n c e
Hence, t h e r e e x i s t s a f u n c t i o n f: A
i f f f(a)-f(b)
I n a s i m i l a r way we can p r o v e t h a t (XxX,RX,
algebraic difference structure. such t h a t (u,v)RX(w,x)
Now l e t
QA
Re such t h a t (a,b)RA(c,d)
a l s o s a t i s f i e s t h e axioms o f an
Hence, t h e r e e x i s t s a f u n c t i o n g: X
i f f a(u)-g(v)
= ir:r=f(a)-ffb)
5 g(w)-g(x).
f o r some a,b E A1
and
(s:s=g(u)-g(v) f o r some u,v lrle d e f i n e 6: nA + nX as f o l l o w s : fix =
+
3 f(c)-f(d).
E
XI.
-t
Re
20 1
Axiomatbation ofadditive difference models
g(r) = s i f r = f(a)-f(b) = g(u)-?l(v)
5
and av
bu.
=o
Then we can p r o v e t h e f o l l o w i n g lemmas. LEMMA 3
g i s well-defined.
Let r = f(a)-f(b)
= f ( c ) - f ( d ) and s u m o s e av
=o
b u and cx
=o
dw.
We have
t o prove: g ( u ) - g ( v ) = g(w)-g(x). Now f ( a ) - f ( b ) = f ( c ) - f ( d ) i m p l i e s ( a , h ) I A ( c s d ) . such t h a t ay
=o
b z and cy
=o
dz.
Now: ay
=o
b z and b u
=o
av i m p l y ( z , y ) I x ( u , v )
and: c y
=o
dz and dw
=o
cx imply (z,y)Ix(w,x).
Hence ( u s v ) I x ( w , x ) o r g ( u ) - g ( v ) = g ( w ) - g ( x ) . g(-r) = -g(r).
LEMMA 4
Let r = f(a)-f(b) g(u)-g(v) and av = o bu. 5 =
Hence g ( r ) = s . Then a l s o b u
=o
av
-r = f ( b ) - f ( a ) -5 = g ( v ) - s ( u ) . So -s = g ( - r ) o r g ( - r ) = - g ( r ) .
LEMMA 5
g i s increasing.
Let r = f(a)-f(b), r' = f(c)-f(d)
s = g(u)-g(v), s ' = g(w)-s(x) and av =o b u s c x = o dw. Hence g ( r ) = s and $ ( r ' ) = s ' . Suppose r 5 r ' . Then f ( a ) - f ( b ) 2 f ( c ) - f ( d )
or (a,b)RA(Csd). Hence ay 2o b z and dz By TC: ay
?o
bz
dz cx
>o
cy dw
3o
b x and b v
Now aw
=o
a0 cy f o r some y,z
E X.
I i m p l y ax z0 bw =o
au imply (w,x)KX(usv)
o r g(w)-g(x) ? g(u)-g(v)
So t h e r e e x i s t s y,z E X
30 1
M,4. Croon
or s 5 s' or a(r) 5 a(r'). We can now p r o v e t h e f o l l o w i n g R e o r e s e n t a t i o n Theorem:
Proof: Part I Supoose ax
?o
by.
By USA: cx
=o
dy f o r some, c.d E A.
Then
a If(c)-f(d)l
Furthermore: ax
= g(y)-g(x).
b y and dy
>o
c x i m p l y (a,b)RA(c,d)
=o
or: f ( a ) - f ( b ) 5 f ( c ) - f ( d ) . Since 0 i s increasing: $ [ f ( a ) - f ( b ) l Hence t~f f ( a ) - f ( b ) l
>$lf(c)-f(d)l
.
3 P(Y)-~(x)
o r $lf(a)-f(b)l
2 0.
+ g(x)-g(y)
Part I I . Sunnose $ [ f ( a ) - f ( b ) l By USX: au
=o
+ ? ( x ) - g ( y ) 5 0.
bv f o r some u,v
E
X.
Then @ [ f ( a ) - f ( b ) I = g ( v ) - g ( u ) . By USA: cx
=o
dy f o r some c,d E A.
Then a l f ( c ) - f ( d ) l
= g(Y)-g(x).
By assumption: 4 [ f ( a ) - f ( b ) l Hence g ( v ) - g l u )
g(y)-g(x).
t s(Y)-~x)
o r (V,4RX(YlX). Hence t h e r e e x i s t a ' , b ' : By TC:
2
au
=o
bv
a'v
)o
b'u
a'v
>o
} i m o l y ax
b ' u and b ' x >o
>.,
a'y
by
b'x 5o a'y 3. ADDITIVE DIFFERENCE MODELS ON THREE COMPONENT SETS
I n t h i s s e c t i o n we i n v e s t i g a t e how t h e a d d i t i v e d i f f e r e n c e model can be a x i o m a t i z e d i n t h e case t h e s t i m u l i a r e composed o f t h r e e component s e t s . S i n c e t h e f o r m a l p r o o f s o f most lemmas and theorems a r e v e r y s i m i l a r t o those o f t h e t w o component case, we s k e t c h them o n l y b r i e f l y . L e t A = Ia,b,c,d
,...1 , P
= {p,q,r,s
,...1
203
A x i o m t i z a t i o n of additive difference mod&
and X = {u,v,w,x,
...1
denote t h e t h r e e component s e t s .
Then t h e s e t S o f
...
I. s t i m u l i i s g i v e n by S = AxPxX = { a n u s bqv, CWX, L e t lo b e t h e b i n a r y p r e f e r e n c e r e l a t i o n on S . lrle assume t h a t
>o
satisfies
t h e f o l 1owing axioms.
1. Axiom 1. toi
s a r e f l e x i v e and connected r e l a t i o n .
We l e t
>o
and =.,
denote t h e a-
symmetric and t h e symmetric p a r t of ? o .
2. Axiom 2.
(1)
Trfple cancellation conditions.
TCA-PX: apu
>o
bqv
brw 2o asx
(2)
csx
>o
drw
TCP-AX: aljh
>o
bqV
cqx b o dpy d r y > o csx (3)
} i m o l y cpu
>o
dqv
I imply a r u l o bsv
TCX-AP: apu b o b q v crv
)o
dsu
1 i m p l y apx
3o bqy
dsx 2o c r y 3 . Axiom 3. (1)
Quintuple cancellation condition.
QCA:
apx
?o
if{ b I q v then c r y
(2)
>o
bo'x' and { c l q l v i
asz i m p l i e s c ' r y
QCP: if
'
apx d o bqy a'qx' b q ' y bo a p ' x and { b ' r ' y '
then c r z (3)
2.3
bqv
>o
dpw i m p l i e s c r ' z
>o >o
cq'v' b'p'x'
>,, a ' s z . >, b ' r y ' )o
>o
a'q'x' do'w.
QCX:
apu b o bqv a ' p ' v a. b ' q ' w if I bqv, b o apu' and { b ' q ' w ' > o a ' p ' v ' then
CIW
z0
dsu i m p l i e s c r w '
>o
dsu'.
4. S o l v a b i l i t y c o n d i t i o n s .
For each component s e t we need u n r e s t r i c t e d s o l v a b i l i t y f o r two unknowns and r e s t r i c t e d s o l v a b i l i t y f o r one unknown.
E.g. f o r t h e f i r s t component
s e t A these s o l v a b i l i t y c o n d i t i o n s can be f o r m u l a t e d i n t h e f o l l o w i n g way:
204
M..4. Croon
(1)
USA: g i v e n p , q
(2)
RSA: i f apx
apx
=o
E
P and x,y E X, t h e r e e x i s t a,b E A such t h a t
bqy. bqy and b q y
z0
CDX,
there e x i s t d
E
A such t h a t
dpx =,, bqy. S i m i l a r d e f i n i t i o n s can be s t a t e d f o r t h e component s e t s P and X .
5 . Archimedean c o n d i t i o n s . The d e f i n i t i o n o f a s t r i c t l y bounded s t a n d a r d sequence f o r each component
s e t can be adapted f r o m t h e two component case d e s c r i b e d i n t h e p r e v i o u s section.
Subsequently, t h e archimedean c o n d i t i o n s s t a t e t h a t any s t r i c t l y
bounded s t a n d a r d sequence i s f i n i t e .
zo
'ale now i n v e s t i g a t e what k i n d o f r e D r e s e n t a t i o n f o r
i s i m o l i e d by t h e s e
axioms. I n a way s i m i l a r t o what was done i n t h e two component case, we d e f i n e r e l a t i o n s RA on AxA,Rp on PxP and RX on XxX:
(1) f o r a r b i t r a r y a,b,c,d (2)
A:
E
x,y E X such t h a t apx
;o
f o r a r b i t r a r y p,q,r,s
E P:
i f f t h e r e e x i s t p , q E P and
(a,b)RA(c,d)
bqy and dqy
>o
(p,q)Ro(r,s)
cox, i f f t h e r e e x i s t a ,b
A and
x,y E X such t h a t apx z o bqy and b s y z o a r x ,
( 3 ) f o r a r b i t r a r y v,w,x,y p,q E P such t h a t apv
X: (v,w)RX(x,y)
E >o
bqw and bqy
The r e l a t i o n a l s t r u c t u r e s (AxA,R,.,)
i f f t h e r e e x i s t a,b E A and
zo
apx.
, (PxP,R,J
and
(
XxX,RX,
a l l s a t i s f y the
axioms o f an a l g e b r a i c d i f f e r e n c e s t r u c t u r e as f o r m u l a t e d by K r a n t z e t a l .
(1971).
S i n c e these p r o o f s a r e v e r y s i m i l a r t o t h e p r o o f s o f t h e c o r r e s -
p o n d i n g t h e o r e m i n t h e two component case, we o m i t them h e r e . I t f o l l o w s t h a t t h e r e e x i s t f u n c t i o n s f: A
-+
Re, g: P
( t h e s e f u n c t i o n s b e i n g i n t o t h e r e a l s ) such t h a t :
(1)
(a,b)RA(c,d)
(2)
(p,q)Rp(r,s)
z f(c)-f(d) i f fd p ) - g ( q ) z g ( r ) - g ( s )
(3)
(v,w)RX(x,y)
iff h(v)-h(w) z h(x)-h(y).
i f f f(a)-f(b)
We d e f i n e : f o r some a,b E A }
Q~ = {a; a = f ( a ) - f ( b ) Q~
=
Q~
= 15;
{T;
w = g(p)-g(q) F, =
f o r some p , q
E
P)
h ( x ) - h ( y ) f o r some x,y E X I
Next we d e f i n e a f u n c t i o n
F: RAxQp
+
fix
as f o l l o w s .
-+
Re and h: X
+
Re
205
Axiomtkation of oddifiue difference models
Suppose apx
bqy w i t h a
=o
f(a)-f(b)
n = S(P)-9(4) and 5 = h ( y ) - h ( x ) .
Then 5 = F j a , i ) . We can prove t h a t F has the f o l l o w i n g p r o p e r t i e s (again, p r o o f s are omitted):
1.
F i s w e l l defined. Suppose apx
=o
bqy and a ' p ' x '
=o
b'q'y' with
f ( a ) - f ( b ) = f ( a ' ) - f ( b ' ) and o(p)-g(q) = g ( p ' ) - ! J ( q ' ) . Then we a l s o have: h ( y ) - h ( x ) = h ( y ' ) - h ( x ' ) .
2.
F(-a,-n)
3.
F i s m o n o t o n i c a l l y i n c r e a s i n g i n b o t h arguments:
= -F(a,n) ,
(1)
a 2
a' i f f T(a,n)
(2)
II 5 T T '
2 F(a',n)
i f f F(a,n) 5 F ( a , n ' ) .
F i n a l l y , we have the f o l l o w i n g Representation theorem:
This r e p r e s e n t a t i o n theorem shows t h a t t h e axioms, which were formulated above, are n o t s u f f i c i e n t t o ensure t h a t t h e f u n c t i o n F has an a d d i t i v e structure.
Hence, a t t h i s moment we have n o t y e t a r r i v e d a t a complete
a x i o m a t i z a t i o n o f the a d d i t i v e d i f f e r e n c e model on t h r e e component s e t s . I n o r d e r t o achieve t h i s goal we need some supplementary c o n d i t i o n s , which should i m p l y t h a t t h e r e e x i s t f u n c t i o n s $, F(a,r)
and -$*such t h a t
= $[$I(a)+$2(n)l.
The most obvious way t o d e r i v e these supplementary c o n d i t i o n s c o n s i s t s o f d e f i n i n g a r e l a t i o n g f on oAxnP as follows: (a,rr)
:.)t
I f t h e s t r u c t u r e t n A s np,
( a ' , n ' ) i f f F(a,n)
2 F(a',s').
s a t i s f i e s the axioms o f an a d d i t i v e c o n j o i n t
measurement s t r u c t u r e , the r e p r e s e n t a t i o n we have i n view can be derived.
N o w the f a c t t h a t ?* i s a weak o r d e r and t h a t i t s a t i s f i e s independence and u n r e s t r i c t e d s o l v a b i l i t y merely f o l l o w s from t h e P r o p e r t i e s o f F.
However,
i n o r d e r t o ensure t h a t =t s a t i s f i e s the Thompsen c o n d i t i o n we need d i t i onal cancel 1a t i on c o n d i t i o n :
an-
206
M . 4 . Croon
apu if !alp,u
=o =o
bqv b,,,lv
a'o"w and
then ao"y =" b q " z i f f a"oy
=o
-- o
b'o"x b"qx
b"qz.
=o
The p r o o f t h a t t h i s a d d i t i o n a l c a n c e l l a t i o n axiom i m p l i e s t h e Thompsen cond i t i o n f o r =* i s s t r a i g h t f o m a r d . From anu =,
b q v and a ' n ' u
b ' q ' v i t follows t h a t
=o
o r say F ( a , n ) = F ( a ' , n ' ) . From a'n"w =,
b ' q " x and a"nw
=o
b"clx, i t f o l l o w s :
F [ f ( a ' ) - f ( b ' ) ,g ( D " ) - g ( q " ) I o r say F ( a ' , n " ) From a p ' y
=o
=o
b"qz,
= F(a",n).
S t a t e d i n terms o f if
-P( 0 1
= F(a",n).
b q " z and a"oy
o r say F(a,n")
= F If( a " ) - f ( b " 1 ,?! ( D
=4
t h i s becomes:
=*(a',n')
(a,.)
and ( a ' , n " ) =*(a",n) then
( s , ~ " )
=*(a",v').
=*.
T h i s i s e x a c t l y t h e Thompsen c o n d i t i o n f o r F i n a l l y , the s t r u c t u r e
( nA,QP,3*)
s h o u l d a l s o s a t i s f ! / t ' - e erchimedean con-
dition f o r additive conjoint structures,
By t r a n s l a t i n g t h i s c o n d i t i o n i n
terms o f t h e p r e f e r e n c e r e l a t i o n b o , we a r r i v e a t a s t r o n g e r v e r s i o n of t h e archimedean axiom f o r t h e s t r u c t u r e i n Axiom 5 .
I f (AxPxX.2,)
(
AxPxX,
?>
than o r i g i n a l l y formulated
We w i l l n o t d w e l l on t h i s r a t h e r t e c h n i c a l p o i n t h e r e . s a t i s f i e s t h e f i v e axioms s t a t e d e a r l i e r t o g e t h e r w i t h t h e
a d d i t i o n a l c a n c e l l a t i o n c o n d i t i o n and t h e s t r o n g archimedean c o n d i t i o n , then t h e r e e x i s t f u n c t i o n s J,, *[4+f(a)-f(bN i n which
J,
+
and + 2 such t h a t apx b 0 bqy i f f +$?(o)-g(q))l
+
h(X)-h(Y)
5 0
i s a m o n o t o n i c a l l y i n c r e a s i n g f u n c t i o n o f i t s argument.
a l s o have: apx b o bwy i f f
Then we
207
Axiomaruation of additive difference modelr
ml[f(a)-f(b)l P u t t i n o $ ( 5 ) = -$-'(-c), 3
+ t ~ ~ [ g ( o ) - g ( q ) 2I
m3
-1
[h(y)-h(x)l
we have t h e f i n a l r e s u l t : apx
t ~ ~ [ f ( a ) - f ( b ) l+ m,[g(n)-e(q)l i n which
J,
>o
+ $3[h(x)-h(v)l
.
bqy i f f 0
i s a l s o an i n c r e a s i n g f u n c t i o n o f i t s argument.
REFERENCES Beals, R.,
K r a n t z , C.H. & Tversky, A.,
The Foundations o f Y u l t i d i m e n -
s i o n a l S c a l i n g , P s y c h o l o g i c a l Review, 75 (1968) F i s h b u r n , P.C.
, Lexicogranhic Additive , 191-218.
, 127-142.
D i f f e r e n c e s , J o u r n a l o f Vathe-
m a t i c a l Psychology 2 1 (1980) Krantz, D.H.,
Luce, R.D.,
Suwes. P. & Tversky, A . , Foundations o f
Measurement. V o l . I, New York (1971). Ng, Y.-K.,
Sub-semiorder:
a Model o f M u l t i d i m e n s i o n a l Choice w i t h
P r e f e r e n c e I n t r a n s i t i v i t y , J o u r n a l o f Mathematical Psychology 16
( 1977) , 51-59. Tversky, A . , I n t r a n s i t i v i t y o f Preferences
, Psychological
Review 76
(1969), 31-48. Van Acker, P.,
Models f o r I n t r a n s i t i v e Choice, D o c t o r a l D i s s e r t a t i o n ,
U n i v e r s i t y o f Ni jmegen (1977).
This Page Intentionally Left Blank
TRENDS IN M'4 THEMA TICAL PSYCHOLOGY E . Degreefond J. Van Bu genhaut (editors) 0 Elsevier Science Rcbfisfers B.V. (Nortlr-Holland), 1984
209
CENERA,LIZATI@NSOF INTERVAL ORDERS
Jean-Paul Doi gnon U n i v e r s i t d L i b r e de B r u x e l l e s
survey some r e c e n t , y e t u n p u b l i s h e d r e s u l t s t h a t T e n e r a l i z e i n t e r v a l and semi o r d e r s i n t h r e e d i r e c tions: ( i ) m u l t i a l e thresholds, ( i i ) o a r t i a l s t r u c tures, ( i i i ) i n f i n i t e underlyinp set.
1. INTRODUCTION I n o r d e r t o modelize an i n d i v i d u a l ' s Dreferences, we c o n s i d e r a f i n i t e s e t X o f o b j e c t s on which i s g i v e n an asymmetric r e l a t i o n P.
T h i s r e l a t i o n con-
t a i n s t h e o r d e r e d D a i r s ab such t h a t b i s s t r i c t l y p r e f e r r e d t o a.
We a l s o
have on X t h e i n d i f f e r e n c e r e l a t i o n I t h a t we assume r e f l e x i v e , symmetric and d i s j o i n t f r o m P. The c l a s s i c a l " t h r e s h o l d model" o f Luce (1956) w i l l be g e n e r a l i z e d h e r e . R e c a l l t h e fundamental i d e a : one a t t r i b u t e s a u t i l i t y u ( x ) and a t h r e s h o l d t ( x ) t o each o b j e c t x from X i n such a way t h a t y i s D r e f e r r e d t o x when t h e u t i l i t y o f y s u f f i c i e n t l y exceeds t h a t of x, more o r e c i s e l y when
u(x) t t ( x )
<
u(y).
Two w e l l known r e s u l t s s t a t e d below g i v e necessary and
s u f f i c i e n t c o n d i t i c n s f o r t h e e x i s t e n c e o f such a r e o r e s e n t a t i o n o f P ; t h e t h r e s h o l d t ( x ) i s e i t h e r a l l o w e d t o be dependent on x, o r i t has a c o n s t a n t value.
I n these r e s u l t s , t h e r e o r e s e n t a t i o n of t h e i n d i f f e r e n c e comes from
t h e assumption t h a t among two ob,iects t h e i n d i v i d u a l e i t h e r p r e f e r s one t o t h e o t h e r , o r i s i n d i f f e r e n t between them. l a t e r droD amounts t o I = XxX We w i l l denote by R-'
-
(P u
T h i s a s s u m t i o n t h a t we w i l l
P-').
t h e converse, by
5
t h e complementary and by R
0
the
"converse o f t h e complementary" r e l a t i o n o f a a i v e n r e l a t i o n R, and by RS t h e r e l a t i v e p r o d u c t o f t h e r e l a t i o n s R and S.
z 10
j.-P. Doifnun
THEOREM (Fishburn ( 1 9 7 0 ) ) . There e x i s t manoings u: X+Re a n d t: X
with t ( x )
>
Re
0 such t h a t
-
u(x) + t(x) x I y * u(x) + t(x) x P v
+
'4Y)
< 5
u(v). u(v)
5 u(x) i f and only i f P I J I i s connected 0 a n d D I P C P ( o r enuivalently PP P
C
THEOSEFl ( S c o t t and Sunnes (1958)).
I!OreOVer, t can he taken w i t h a con-
+
t(Y)
P).
swnt valLe i f and only i f IP
2
( o r equivalently
P
pop2
CI
PI.
Pairs ( P , I ) s a t i s f y i n g the f i r s t theorem a r e c a l l e d i n t e r v a l o r e ; those s a t i s f y i n g the second theorem are c a l l e d semi o r d e r s . The conditions involving the u t i l i t i e s and thresholds can he formulated more geometrically. F i r s t , n u t g = u and f = u + t . The i r r e f l e x i v i t y of P imo l i e s a L f , a n d we associate t o each o b j e c t x from X the i n t e r v a l
lcj(x). f ( x ) l on the real l i n e . The preference P i s i n t e r p r e t e d as the precedence of such i n t e r v a l s , nhi le the indifference corresponds t o overlapping. Also the threshold becomes the i n t e r v a l length. Noti cing t h a t
x
Y
* f ( x ) < O(Y)
we come close t o another well known r e s u l t concernino t h i s time the so-called C u t t m a n s c a l e s ( a f t e r e . ? . G u t t m a n (1944)).
.
Given a r e l a t i o n R on THEOREM (Riguet (1948) ; Ducamn and Falmagne (1969) ) a f i n i t e s e t X, there e x i s t two manninqs f and q from X t o Re s a t i s f y i n g
x R y
0
f(x)
?(y)
i f and only i f RR'R c R . Relations s a t i s f y i n g the above conditions w i l l be c a l l e d b i o r d e r s , a f t e r Doiqnon, Ducamp and Falmagne (!983+); Monjardet (1978), i n h i s nice survey paner, c a l l s them Ferrers r e l a t i o n s . Another useful c h a r a c t e r i z a t i o n i s in terms o f the 0 - 1 matrices of the
211
Generalizations of intervalorders
r e l a t i o n : R i s a b i o r d e r i f f f o r some s i m p l e o r d e r i n g o f t h e row elements and some s i m p l e o r d e r i n g o f t h e column elements, t h e r e s u l t i n g m a t r i x has a s t e o t y o e f o r m (see f i q . 1).
x1
x2
'3
...
'n
F i s h b u r n ' s theorem on i n t e r v a l o r d e r s can be t r i v i a l l y deduced f r o m t h e l a s t theorem, because t h e i r r e f l e x i v e b i o r d e r s a r e e x a c t l y t h e asymmetric p a r t s P o f i n t e r v a l orders. cations!).
(Comnare w i t h t h e y e a r s and j o u r n a l s o f p u b l i -
F o r semi o r d e r s one can show t h a t i t i s D o s s i b l e t o use t h e
same s i m n l e o r d e r on row and column elements when l o o k i n g f o r a s t e p t y p e 0-1 matrix.
We r e f e r t h e r e a d e r t o M o n j a r d e t (1978) and Roberts (1979) f o r
t h i s and many o t h e r r e l a t e d r e s u l t s , and now come t o t h e promised s u r v e y o f recent generalizations.
2. HULTIPLE THRESHOLDS Suppose t h a t WE d i s t i n g u i s h among t h e i n d i v i d u a l ' s p r e f e r e n c e s s e v e r a l l e vels.
ItJe then f o r m f o r each o f t h e l e v e l s a r e l a t i o n Pi on t h e s e t X o f
o b j e c t s , and want t o o b t a i n general c o n d i t i o n s f o r t h e e x i s t e n c e o f a r e p r e s e n t a t i o n f o r t h e r e l a t i o n s I , PI,
P2,
..., Pk
usin? a u t i l i t y function
t o g e t h e r w i t h one t h r e s h o l d f u n c t i o n f o r each l e v e l o f o r e f e r e n c e .
(Thus
i n D a r t i c u l a r each Pi w i l l be t h e asymmetric p a r t o f some i n t e r v a l o r d e r ) .
P.s i n the c l a s s i c a l c a s e , the notion of biorder appears as a useful t o o l .
'lithout e n t e r i n g i n t o d e t a i l s , l e t us say t h a t a careful study of the cons t r u c t i o n of the simnle orders leadinq t o a s t e p type matrix e a s i l y gives the qeneralization t c multinle simultaneous biorders. Pere i s an example o f the r e s u l t s one can o b t a i n . pROt'o5ITIUN
riven r e l a t i o n s R1, nz,
...,
Rk on the f i n i t e s e t X, the three following
conditions are equivalent: (i)sow simole order on X f o r the row elements toct-ther u i t h scw simnle order on X f o r the column elements oive simultaneous step tyoe matrices f o r
'1% P ~ S
. a .
%
Rk;
( i i ) there e x i s t manninos f l , f p , x R. y o f i ( x ) 1
and
fi(x)
<
fi(y)
( i i i ) t h e relations U _R2""J and
f k , a from X t o Re s a t i s f y i n g :
~ ( y l
c
-
...,
cj(x) s fj(y);
FIFil
. ..
RT 1F~ u R ; ~ F ~ u~
...
Ilk$ 1
'J
(J
R;'_R~
a r e asymmetric. The r e l a t i o n s involved i n t h i s oronosition t h u s admit a n i c t o r i a l Dresentation examplified i n Figure 2 . I t i s i n t e r e s t i n g to notice t h a t one can use the same simnle order f o r row a n d column elements i f an d only i f the union of the t1.10 r e l a t i o n s aDpearina in ( i i i ) above i s i t s e l f asymmetric. A s a c o r o l l a r y , we obtain a renresentation theorem f o r mu1 t i threshold pre-
ference r e l a t i o n s . COFTLLPW.
and asyrnnetric r e l a t i o n s PI, P p , ..., Pk on Civen a symmetric r e l a t i o n the f i n i t e set X , there e x i s t mannings u , t l , t 2 , ..., tk from X t o Re with t i ( x ) > 0 such t h a t x I y
i
u(x)
+
u(Y)
+
tk(x) tk(Y)
3
u(y)
3
'(')9
213
Ceneralizntions of'i n t e n d orders
x
P.7
y
0
t! 2 t2 5
u(x)
t
ti(X)
... 5
tk'
<
u(y),
and
* u ( x ) + t .J( x ) s U ( Y ) + t j ( Y )
U ( X ) + t i ( X ) < U(Y)+ti(.V) i f and o n l y i f
P 1 (1 P 2 c:
... c
Pk'
I r pk = B,
I
u Pk i s connected,
and t h e r e l a t i o n s FJIP;l C' _P2p;1 u and
p;1p1
u
u
... u
,pkPk-1
... u P i 1rk
a r e symmetric.
x1
x2
...
x3 -.
I
__
X
n -.
I
I
I
0
I-
/- '\
I \
I I
-'
Yn Figure 2
Many o t h e r r e s u l t s can be e s t a b l i s h e d as e a s i l y as t h i s one ( f o r i n s t a n c e , s i m i l a r statements o b t a i n e d b y droDping some o f t h e c o n d i t i o n s ) .
Neverthe-
l e s s t h e r e remain open problems, and we p r e s e n t one v e r y general q u e s t i o n .
214
FRPRLEI'.
F i n d c o n d i t i o n s on
I,
s e n t a t i o n u s i n 9 u, tl,
PI,
P?,
t2,
..., Pk
...,
t h a t ensure t h e e x i s t e n c e o f a r e p r e -
tk w i t h f o r sow o r e s c r i b e d i n d i c e s i, t h e
t h r e s h o l d t . b e i n p c o n s t a n t ( i .e. indegendent o f t h e ob.iect x ) . 1
2. n A Q T I 4 L STRUCTtIPES c o n s i d e r a g a i n a o a i r ( F , I ) o f qreference and i n d i f f e r e n c e r e l a t i o n s on
1.k
X, w i t h P asymmetric and I symmetric and r e f l e x i v e , b u t we do n o t assume
t h a t any p a i r o f two o b j e c t s i s i n cne o f these two r e l a t i o n s . man:!
m o t i v a t i o n s fr.r s t u d y i n o t h i s s i t u a t i o n :
There a r e
f o r instance, the individual
may c o n s i d e r t h a t two o b j e c t s a r e incomparable, o r e l s e he c o n s i d e r s t h a t the:!
a r e comoarable b u t we do n o t have t h e comnlete i n f o r m a t i o n on h i s judge-
ments.
These m o t i v a t i o n s l e a d rloy (1980) t o c o n s i d e r n u m e r i c r e p r e s e n t a -
t i o n s w i t h the double i m o l i c a t i o n r e n l a c e d by a s i n o l e one.
More p r e c i s e l v ,
one l o o k s a a a i n f o r mannings u and t , s a t i s f y i n ? t h i s t i m e : x
aud
p
v
-
L!(X) + t ( x )
x I .y *
u(x) t t ( x ) 5 u(y)
i u(y) + t ( y ) z u ( x ) .
tfistorically, threshold.
u(y)
t h e f i r s t r e s u l t i s due t o Roy (1960) and i n v o l v e s a c o n s t a n t
Then Roubens and Vincke (1983+) s e t t l e d t h e v a r i a b l e t h r e s h o l d
case. THEWEV (Roy (1980)).
There e x i s t manninrs u and t as above, w i t h t h e t h r e s h o l d t ( x ) b e i n g cons t a n t , i f and o n l y i f each c y c l e o f t h e r e l a t i o n
I
P U
I has more p a i r s from
than from P .
TtiFnREr' (qoubens and Vincke (19&3+)). There e x i s t m a m i n o s u and t as above i f and o n l y i f t h e r e l a t i o n P I i s ac y c l ic . LJe have o b t a i n e d ( r e n e r a l i z a t i o n s o f these two theorems i n v o l v i n g t h e n o t i o n o f hiorder.
More n r e c i s e l y , a b i o r d e r i s used t o s e n a r a t e two r e l a t i o n s R
and S; t h e n u m e r i c a l f o r m u l a t i o n i s t h e n e a s i l y d e r i v e d . t h e o r e m a r e D a r t i c u l a r cases w i t h P = R and I = S - I .
Notice t h a t the
Gneralizotiom of interval orders
215
PROPOqITION. q i v e n r e l a t i o n s R and 7 on t h e f i n i t e s e t X, t h e f o l l o c r i n o c o n d i t i o n s a r e eq u i va 1en t : (i)
t h e r e e x i s t maoninns f and
-
* f(x)
x 9 Y
s
x and
v
< O(Y),
f ( x ) 5 a(y),
s q ( y ) and f(y)
g(x)
f r o m X t o Re s a t i s f y i n ?
6 f(x)
=.
x = y;
( i i ) t h e r e e x i s t s a h i o r d e r B such t h a t RCRandSC!, and
0 2
B B c B;
( i i i ) i n each c y c l e o f P, u q-’ t h e r e are s t r i c t l y more a a i r s f r o m S - l than f r o m R. Remark t h a t t h e t h i r d n a r t o f c o n d i t i o n ( i ) i n f a c t avoids i n c l u s i o n between
some i n t e r v a l s ; u s i n o t h e f i n i t e n e s s o f X and assuming R i r r e f l e x i v e i t can be shown t h a t ( i ) i s e q u i v a l e n t t o t h e n o s s i b i l i t y o f choosing f and
g i n such a way t h a t those i n t e r v a l s have c o n s t a n t l e n g t h . PROPOSITIQN
.
Given r e l a t i o n s R and S on t h e f i n i t e s e t X, t h e f o l l o w i n g c o n d i t i o n s a r e enuivalent: (i)
t h e r e e x i s t mannings f and a f r o m X t o Re s a t i s f y i n g
and
x p. Y x S y
-
f ( x ) < !7(Y), f ( x ) 5 q(v);
( i i ) t h e r e e x i s t a h i o r d e r B such t h a t RCBandSC!; (iii)RS-i
i s acyclic.
These n r o p o s i t i o n s a l s o l e a d t o i n t e r e s t i n g s h a m e n i n a s o f t h e theorems due t o Roy and Roubens and Vincke ( a m a n u s c r i p t w i t h A . Ducamp and J.-C. Falmagne i s i n p r e o a r a t i o n ) .
F o r i n s t a n c e , one asks f o r a d e s c r i n t i o n o f t h e p a i r s
xy such t h a t , f o r any manpinos u and t, h o l d s t h e r e l a t i o n u ( x ) + t ( x ) < u(y) ( t h e s e a r e t h e p a i r s t h a t must be added t o t h e observed p r e f e r e n c e s i n o r d e r t o g e t a t h r e s h o l d model). t i o n P U (PI)*P,
whew
Here i s t h e answer: those D e i r s form t h e r e l a -
* symbolizes
t h e t r a n s i t i v e closure onerator.
].A? Doignon
3. INFINITE CASE IIe have assumed f o r convenience t h a t t h e s e t X o f o b j e c t s i s f i n i t e .
But
i n f a c t most o f o u r r e s u l t s a l s o h o l d f o r i n f i n i t e s e t s X , under t h e c o n d i t i o n t h a t t h e o r d e r e d s e t Re be r m l a c e d by
SOIF
sim1.v ordered s e t E ( i . e .
one o n l y s t a t e s t h e e x i s t e n c e o f F: and of t h e m a m i n o s from X t o E ) . More d i f f i c u l t i s t h e oroblem i n v o l v i n g an i n f i n i t e s e t X and manpings f r o m X t o Re; d e n s i t y c o n d i t i o n s w i l l be needed.
For instance, Fishburn (1973)
p i v e s somewhat i n t r i c a t e d c o n d i t i o n s i n t h e case o f an i n t e r v a l o r d e r . S i m n l e r c o n d i t i o n s were r e c e n t l y foroed,
t h a t ann1.v as w e l l f o r b i o r d e r s .
(Eecall t h a t the i n t e r v a l orders are the i r r e f l e x i v e b i o r d e r s ) . P R O P O S I T I O N (Doionon, Ducam and Falmaqne ( 1 9 8 3 ) ) .
F o r a o i v e n r e l a t i o n R on a s e t X, t h e r e e x i s t manninqs f and
from X t o
Re such t h a t x R y
*
f(x) < "y)
i f and o r l l y i f t h e r e e x i s t a c o u n t a b l e s u b s e t I" o f X w h i c h i s s t r i c t l y dense,
that is: f o r any x and y i n X,
there e x i s t s m i n
such t h a t :
x R m and n o t r 'c-lB v , or
m P, ,v and n o t x _RR-l m .
4 . CONCLUSIONS The r e s u l t s i n t h i s b r i e f s u r v e y d e s c r i b e c o n d i t i o n s f o r t h e e x i s t e n c e o f numerical reoresentations o f a c e r t a i n type. taken f r o m j o i n t work w i t h A . Pucamn and J.-C. notion o f biorder.
@ur o e n e r a l i z a t i o n s ( o f t e n Falmaone) a r e based on t h e
From o u r n o i n t o f view, t h i s n o t i o n c l a r i f i e s t h e sub-
j e c t because i t s y n t h e t i z e s d i f f e r e n t asoects o f i t , and p r o v i d e s t h e way
As a c o n c l u s i o n , l e t us Dronose a new maxim: i f y o u a r e l o o k i n g f o r a t h r e s h o l d r e D r e s e n t a t i o n theorem, f i r s t f i n d where t o s i m p l e and p e n e r a l n r o o f s .
the b i order i s ! REFERENCES
[I]
Doiqnon, J . - P . ,
Ducamo, A . , and Falmagne, J.-C.,
On r e a l i z a b l e b i o r d e r s
and t h e b i o r d e r dimension o f a r e l a t i o n , J o u r n a l o f Math. P s y c h o l . , t o amear.
Generalizations of interval orders
[ 21
[ 31
217
Ducamp, A . , and Falmaone, J.-C., Composite measurement, Journal o f EIath, Psychol . , 6 ( 1969), 359-390. Fishburn, P . C . , I n t r a n s i t i v e indifference with unequal indifference i n terval s , Journal o f !lath. psycho1 ., 7 (1970) , 144-149. Fishburn, P . C . , Interval reoresentations f o r i n t e r v a l orders and s e n i o r d e r s , Journal of ‘lath. Psychol ., 10 (1973), 91-105.
cuttman, L . , P basis f o r s c a l i n o q u a t i t a t i v e data, Amr. SOC. Rev., 9 ( 1944), 139-150. Luce, R . D . , Semi-orders and a theory of uti1it.y discrimination, Econometrica, 24 (1956), 178-191. Plonjardet, B . , flxiomatiques e t proprietes des quasi-ordres, !lath. Sci . Humaines, 63 (1978), 51-82. Riquet, J . , Les r e l a t i o n s de F e r r e r s , C . R . P c a d . S c i . P a r i s , 232 (1951), 1729-1730. Roberts, F.S., “easurement theory, Encyclooedia of Mathematics and i t s Agolications, Vol . 7, Rddison-’Slesley, Reading, tjass. (1979). [ 101 Roubens, W . , and Vincke, P h . , On o a r t i a l i n t e r v a l orders, t o aopear. [111 Roy, B . , Preference, i n d i f f e r e n c e , incomoarabi l i te. Documents du LAMSAOE, 9 , Universite de Paris-Dauohine (1980). 1121 S c o t t , D., and Sunnes, P . , Foundational asoects o f t h e o r i e s of measurement, Journal Symbolic Loaic, 23 (1958). 113-128.
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TRENDS I A M ~ A THEMATICALPS y c m r . o c ~ E . Degreefand]. Van flu enhaut (editors) 0 Elsevier Science Publisers fl. V. (North-Holland), 1984
219
PESE P IIX 5 0C Ib.UX GENE RA L I S ES : EN COIIRINANT GRAPHES ET HYPERGRAPHES C1 aude F! amen t D e n a r t e m n t c'e P s y c h o l o g i e Uni v e r s i t S de Provence Centre de Mathematique S o c i a l e E .H .E.S .S, , t l a r s e i l l e Nous evoquons i c i quelques n o t i o n s en r e l a t i o n avec l e s reseaux sociaux, q u i combinent graphes e t hypergraphes s u r un m5me ensemble d ' i n d i v i d u s , e t en a j o u t a n t un ensemble de t r a i t s c a r a c t e r i s a n t ces individus.
Nous t r a i t e r o n s p r i n c i p a l e m e n t l e systeme
de c a r a c t e r i s a t i on.
1. INTRODUCTION Traditionnellement,
un reseau s o c i a l e s t r e p r e s e n t @ p a r un graphe, ou
p l u s i e u r s , d o n t l e s sommets correspondent aux i n d i v i d u s .
Plus recement,
d i v e r s t r a v a u x p r o p o s e n t de c o n s i d e r e r q u ' u n reseau s o c i a l e s t en f a i t un systeme r e p r e s e n t a b l e p a r un hypergraphe (Berge (1970)), c ' e s t - & - d i r e , une f a m i l l e de sous-ensembles de l ' e n s e m b l e des i n d i v i d u s : groupes d ' a f f i n i t e , a s s o c i a t i o n s , membres d ' u n e m5me p r o f e s s i o n , c l a s s e s d ' i n d i v i d u s a y a n t des c a r a c t e r i s a t i o n s e q u i v a l e n t e s i d P e de Bougle (1897), Seidman (1981) (1981)).
...
( k s e n n e (1981), r e o r e n a n t une
, Doreian
(1981)
, Bonacich
e t Domhoff
L ' i n t e r e t de c e t t e n o u v e l l e approche ne d o i t pas f a i r e o u b l i e r l e s a p p o r t s de l ' a n c i e n n e .
Aussi nous evoquons i c i quelques n o t i o n s q u i combinent
graphes e t hypergraphes s u r un mOme ensemble d ' i n d i v i d u s . un ensemble de t r a i t s c a r a c t e r i s a n t ces i n d i v i d u s .
On y a j o u t e
Un t e l systkme p e u t
5 t r e c o n s i d e r e comme un reseau s o c i a l g e n e r a l i s e . Pour l a s i m p l i c i t e , nous c o n s i d k r e r o n s d ' a b o r d separement : graphe e t hypergraphe s u r l ' e n s e m b l e X des i n d i v i d u s , d ' u n e p a r t ; e t d ' a u t r e p a r t , l ' h y p e r g r a p h e s u r X , un hypergraphe s u r un ensemble Y de t r a i t s e t une r e l a t i o n de c a r a c t e r i s a t i o n e n t r e X e t Y.
Mais, mathematiquement, l e
p r e m i e r sous-systeme e s t un cas p a r t i c u l i e r du second : on r e t r o u v e l e p r e m i e r en f a i s a n t X = Y , e t en i d e n t i f i a n t l e s deux hypergraphes.
Nous
t r a i t e r o n s donc p r i n c i p a l e m e n t l e systeme de c a r a c t e r i s a t i o n .
2 . NOTATIONS Un hypergraphe p e u t e t r e c o n s i d @ r e c o m e un graphe b i p a r t i
: de 1 'ensemble
des sommets v e r s l ' e n s e m b l e des a r 6 t e s .
Nous designerons p a r l a meme l e t t r e un graphe, l a r e l a t i o n q u ' i l r e p r e s e n t e , e t l a matrice qui l u i e s t associee.
A i n s i : ( x y ) Z , xGy, ( x y ) = l dans G,
s o n t des f o r m u l a t i o n s e q u i v a l e n t e s . La c o m p o s i t i o n des graphes, ou des r e l a t i o n s , s ' e c r i t comme l e p r o d u i t booleen des m a t r i c e s .
On a : xGHy s i e t seulement s i e x i s t e un z t e l que :
xGz e t zHy.
Nous designons p a r G ' l a m a t r i c e transposGe de l a m a t r i c e G, a i n s i que graphe e t r e l a t i o n c o r r e s p o n d a n t s . S o i t deux ensembles X e t Y ; s u r X , un hypergraphe H; s u r Y , un hypergraphe
K; e n t r e X e t
Y , un graphe b i p a r t i
R.
U n t e l systeme s e r a n o t e [ H , R , K ]
.
S i xiHH'x., c ' e s t que l e s i n d i v i d u s xi e t x . s o n t dans au moins un m6me J J sous-groupe du reseau. De meme, s i yiKK'y., c ' e s t que l e s t r a i t s y . e t J 1 y . s o n t dans au moins une m6me c l a s s e de c a r a c t e r i s a t i o n . J Un graphe de s i m i l i t u d e e s t un graphe r e f l e x i f e t s y m e t r i q u e : s o n t de t e l s graphes.
S i un graphe S de s i m i l i t u d e e s t de p l u s t r a n s i t i f
( i . e . : S S = S ) , c ' e s t a l o r s l e graphe d ' u n e e q u i v a l e n c e .
3 . PLENITUDE D ' U N SYSTEISE D e f i n i t i o n : Un systeme [H,R,K]
-
HH' e t K K '
est :
p l e i n h gauche s i : R = HH'R; p l e i n h d r o i t e s i : R = RKK'; p l e i n s ' i l e s t p l e i n a gauche e t a d r o i t e .
221
Rdseuux socioux gdndrulisis
La t e r m i n o l o g i e e s t j u s t i f i e e p a r l e s r e s u l t a t s s u i v a n t s , q u i g e n e r a l i s e n t ceux de R i b l i e r (1978). Theoreme :
S i un systeme [H,R,,Kl
e s t p l e i n ii gauche, ( x H H ' x ' ) e n t r a i n e
que x e t x ' o n t exactement l e s mtmes successeurs dans R; s ' i l e s t p l e i n 1 d r o i t e , ( y K K ' y ' ) e n t r a i n e que y e t y ' o n t exactement l e s mCmes p r e -
decesseurs dans R . Demonstration : S i R=HH'R : (xHH'x' e t x'Ry)
4
(xHH'Ry) * (xRy).
Comnentaires : On v o i t donc que s i une f l e c h e R p a r t d ' u n e a r t t e de H, il en p a r t , dans l a mtme d i r e c t i o n , de chacun des sonnnets de c e t t e a r t t e . Symetriquement, s i l e systeme e s t p l e i n ii d r o i t e e t s i une f l e c h e R a r r i v e dans une a r t t e de K, t o u s l e s sommets de c e t t e a r t t e r e C o i v e n t une f l e c h e de l a mtme o r i g i n e . S i l e systeme e s t p l e i n , e t q u ' o n e x t r a i t de l a m a t r i c e R l a sous-matrice
c r o i s a n t une a r e t e de H e s t une a r t t e de K, on v o i t que c e b l o c e s t t o t a l e ment compose s o i t de 1, s o i t de zeros.
En remarquant que deux a r t t e s de H (ou de K) peuvent a v o i r une i n t e r s e c t i o n non v i d e , on v o i t que l e s p r o p r i e t e s "mtmes successeurs" e t "m@mes prGdGcesseurs" s ' k t e n d e n t aux c l a s s e s des e q u i v a l e n c e s correspondant aux composantes
connexes de H e t de K.
S i nous sommes dans l e cas d ' u n systeme p l e i n , 00 X=Y e t H=K, on v o i t que
l ' e q u i v a l e n c e correspondant aux composantes connexes de H n ' e s t a u t r e que 1 ' e q u i v a l e n c e s t r u c t u r a l e de L o r r a i n e t White (1971).
Or, b i e n des a u t e u r s ( p a r exemple, S a i l e r (1978)) pensent que c e t t e n o t i o n e s t t r o p f o r t e p o u r C t r e u t i l e dans l ' e t u d e des reseaux. nous proposons une n o t i o n p l u s f a i b l e mais p l u s u t i l e . sans demonstration, un r e s u l t a t f a c i l e P r o p o s i t i o n : Un systeme [ H,R,K]
a
etablir
C ' e s t pourquoi Mais a v a n t , donnons,
.
e s t p l e i n s i e t seulement s i : R=HH'RKK'.
4. REGULARITE D ' U N SYSTEME
D e f i n i t i o n : Un systeme [H,R,Kl
- regulier
-
regulier
a a
est :
gauche s i : RKK' = H H ' R K K ' ; d r o i t e s i : HH'R = HH'RKK';
- regulier s ' i l est regulier a droite e t Theoreme : S o i t un systeme I H,R,K] c r o i s a n t une a r 6 t e de
H
a
gauche.
; c o n s i d e r o n s l a s o u s - m a t r i c e de R
e t une a r 6 t e de K; s i l e systeme e s t r e g u l i e r a
gauche, c e t t e s o u s - m a t r i c e comporte, s o i t seulement cies z e r o s , s o i t au moins un 1 dans chaque l i g n e ; s i l e systeme e s t r e g u l i e r
a
droite, cette
sous-matrice comporte, s o i t seulement des zeros, s o i t au moins un 1 dans chaque colonne.
-
-
Demonstration : S i R K K ' = HH'RKK' : (xHH'x' e t x'RKK'y)
(xHH'RKK'y)
(xRKK'y)
-
(3y':xRy').
P r o p o s i t i o n : S i H H ' ou K K ' e s t une e q u i v a l e n c e , l e systeme [ H,R,K]
est
r e g u l i e r s i e t seulement s i : R K K ' = H H ' R . Demonstration : I 1 s u f f i t de demontrer que R K K ' = H H ' R e n t r a i n e que R K K ' ou HH'R @ g a l e H H ' R K K ' ; supposons que K K ' e s t une e q u i v a l e n c e , d o n t l a t r a n s i t i v i t e s ' e c r i t : K K ' K K ' = K K ' ; a l o r s : HH'RKK' = R K K ' K K ' = R K K ' . Commentaires : Supposons que nous soyons dans l e cas d ' u n systeme de c a r a c t e r i s a t i o n : l e s t r a i t s c o n s t i t u a n t une a r C t e de K s o n t e q u i v a l e n t s ( r e l a t i v e m e n t au probleme t r a i t e ) , e t un i n d i v i d u , p o u r a v o i r l a c a r a c t e r i s a t i o n globale correspondant
a
une a r 6 t e de K, n ' a pas b e s o i n d ' e n
a v o i r tous l e s t r a i t s : il s u f f i t q u ' i l a i t t e l t r a i t OU t e l a u t r e t r a i t de l a c l a s s e .
On v o i t donc que l a r e g u l a r i t P t r a d u i t un modele d i s j o n c t i f
de c l a s s i f i c a t i o n . Supposons m a i n t e n a n t que X = Y e t ti = K; p a r exeniple, X e s t un ensemble de s c i e n t i f i q u e s d ' u n mCme s e c t e u r , e t R e s t l e graphe des c i t a t i o n s .
S i une
a r 6 t e r e p r e s e n t e une e c o l e p a r t i c u l i e r e , on p o u r r a j u g e r q u ' u n i n d i v i d u s ' i n t e r e s s e a c e t t e e c o l e s ' i l c i t e au moins l ' u n de ses membres; il n ' e s t pas n e c e s s a i r e q u ' i l l e s c i t e t o u s .
Riseaux sociaux giniralisds
223
On pourrait developper dans l e sens de l a theorie structurale des rBles, chere a Harrison White e t son equipe (White Gal., (19761, Booman e t %ite (1976)). Enfin ( e t ces problemes sont tous l i e s , chez White comme chez nous), nous considerons que l a notion de bloc regulier (sous-matrice de R avec au moins u n 1 dans chaque ligne e t chaque colonne) peut Gtre mise a l a base d'un nouveau "modPle de bloc" ("Blockmodel"; cf. : Arabie, Boorman e t Levitt (1978), Arabie e t Fcoman (1982)). Flous avions expose cette ioee (dans l e cas 00 H H ' e t KK' sont des equivalences) a l a 9eme reunion du Groupe Europeen de Psychologie Mathematique (Ni jmegen ( 1 9 8 ) ) . 5. PERSPECTIVES
Nous pouvons revenir aux reseaux sociaux generalises : u n ensemble X d'individus, e t sur cet ensemble, une famille de graphes Ri, une famille d'hypergraphes H u n ou plusieurs ensembles de c a r a c t h i s a t i o n Yk, avec j' des hypergraphes K k , e t des relations Q,de X vers Y k . Cette multiplicite d'ensembles, de graphes e t d'hypergraphes offre d'interessantes possibilites, dans l a mesure 00 l a regularit6 se conserve travers plusieurs des operations possibles sur ces graphes e t hypergraphes, come l e montrent l e s proprietes suivantes, f a c i l e s i demontrer s i l e s similitudes associees aux hypergraphes sont des equivalences : - s o i t une famille de systemes reguliers [ H , R i , K] sur X e t Y ; l e systerne [ H,u Ri, K] e s t regul i e r ; - s o i t une famille de systemes reguliers [ H i , R, Ki] sur X e t Y ; l e systeme [ U H i , R , u Ki] e s t regulier. (Note : s i on remplace 1 'union par 1 'intersection, les deux proprietes ne sont plus vraies.) - s o i t deux systemes reguliers, [ H , R , Kl sur X e t Y , e t [ K, Q , L] sur Y e t Z ( X , Y e t Z e t a n t des ensembles d i s t i n c t s ou non); l e systeme, sur X e t Z , [ H , R Q , L] e s t regulier. Cette propriete assure l e maintien de l a regularite travers les compos i t i o n s que propose Lorrain (1975). Mais on peut aussi considerer certains rapports entre systemes, q u i engendrent de la regularite; par exemple :
-
K1 avec i = 1 ou 2 , t e l s que: a l o r s , l e s y s t e m [ti, R1, K1 e s t r e o u l i e r d d r o i t e , e t l e
s o i t , s u r X e t Y, deux svstcmes [H, R.;, tlt!'Rl
= R2KK';
K1 e s t r P ? u l i e r
s y s t e m [H, R2,
a
oauche.
6. PROBLEflES SunPosons une r e l a t i o n R e n t r e X e t Y ; on v e u t t r o u v e r un hyaerqraphe tl st:r X,
un h v n e r y a o h e K s u r Y ,
orobleme n ' e s t pas r e s o l u .
t e l s que l e s y s t e m IF, R, Kl s o i t r e g u l i e r .
Ce -
T o u t montre que l a r e s o l u t i o n du orobleme dans
l e cas siaiole evoque, s ' e t e n d r a i t sans d i f f i c u l t e de o r i n c i n e au cas d ' u n reseau g e n e r a l i s e (nolnhreux ensembles, nombreux o r a p h e s ) . Line s o l u t i o n c o m b i n a t n i r e e x i s t e s i or! acceote que l e s h y w r o r a o h e s r e c h e r -
ches c o r r e s o o n d e n t
a
des eouivalences,.
Mais 1 ' a o p l i c a t i o n
a
des donnees
r e e l l e s montre que nresnue t o u j o u r s , l e s s o l u t i o n s e x a c t e s s o n t t r i v i a l e s , a l o r s q u ' o n t r o u v e ( a a r tatonnemen'ts) :antes.
des s o l u t i o n s a n p r o c h k s t r e s i n t e r c s -
I 1 f a u d r a i t donc t r o u v e r une methode, d l a n t s t a t i s t i q u e e t combi-
n a t o i r e , D e r m e t t a n t de c e r n e r l a n u a s i - r @ g u l a r i t e ( b l o c avec au m i n s un 1 dans nresque chaoue l i m e e t dans Dresque chaque c o l o n n e ) . Remerciements: B i e n des i d e e s de ce t r a v a i l s o n t dues
a
A l a i n Degenne; t o u -
t e s se s o n t developnees en i n t e r a c t i o n c o n s t a n t e avec son p r o o r e t r a v a i l .
[ 11
A r a b i e , P.,
Roorman, S . A . ,
L e v i t t , P.R.,
C o n s t r u c t i n g hlockmodels: how
and why, J o u r n a l o f plathematical Psycholoay, 17 ( 1 9 7 8 ) , 21-63. 121
Arabie, P., H.C.
Boorman, S . A . , Blockmodels: develonments and o r o s p e c t s , i n
Hudson ( E d . ) , C l a s s i f y i n a s o c i a l d a t a , San F r a n c i s c o , Jossey-Bass,
( 1 9 0 2 ) , 177-259. Granhes e t Hynergraphes, P a r i s , Dunod ( 1 9 7 0 ) .
131
Berge, C.,
141
Bonacich, P . , Domhoff,
151
Boorman, S . A . ,
GI.,
L a t e n t c l a s s e s and o r o m membership, S o c i a l
Networks , 3 ( 1 9 8 1 ) , 175-196. IOJhite, H.C.,
S o c i a l s t r u c t u r e s f r o m m u l t i n l e networks,
11, Role s t r u c t u r e s , American J o u r n a l o f S o c i o l o q v , 81 ( 1 9 7 6 ) , 1384-
1446,
PI
Bouqle, C.,
171
Degenne, A.,
Ou'est-ce que l a s o c i o l o g i e , Revue de P a r i s ( 1 8 9 7 ) .
Lcs reseaux de c o o p e r a t i o n e t d'echanqe, Centre d ' A n a l v s e
e t de Nathematioue S o c i a l e , E.H.E.S.S.,
Marseille.
[81
Doreian, P.,
P o l v h e d r a l dynamics and c o n f l i c t m o b i l i z a t i o n i n s o c i a l
n e t w o r k s , Soci a1 Networks, 3 ( 1 9 8 1 ) , 107-116.
191
L o r r a i n , F . , I J h i t e , H.C.,
' j t r u c t u r a l e q u i v a l e n c e o f i n d i v i d u a l s i n so-
c i a l networks, ,lournal o f r4athematical S o c i o l o o y , l ( 1 9 7 1 ) , 49-80.
[lo]
L o r r a i n , F.,
RQseaux s o c i a u x e t c l a s s i f i c a t i o n s s o c i a l e s , P a r i s , Her-
mann ( 1 9 7 5 ) .
[ll]
R i b l i e r . !,I.,
f l u e l q w s dPveloooements de l a t h e o r i e des graohes e t aD-
o l i c a t i o n s ,?lia p r o s n e c t i v e p a r analyse de systeme, These de mathemat i q u e s economiques, U n i v e r s i t e P i e r r e e t M a r i e C u r i e , P a r i s ( 1 9 7 8 ) .
1121
S a i l e r , L.D.,
[131
Seidman, S.B.,
S t r u c t u r a l e o u i v a l e n c e : meaning and d e f i n i t i o n , Compu-
t a t i o n and A D p l i c a t i o n , S o c i a l Networks, 1 ( 1 9 7 8 ) , 73-90. S t r u c t u r e s i n d u c e d by c o l l e c t i o n s o f subsets: a hyper-
oraoh approach, Mathemati c a l Soci a1 Sciences, 1 ( 1981) , 381-396.
[I41 "bite, H.C.,
Roorman, S.A.,
B r e i q e r , R.L.,
S o c i a l s t r u c t u r e f r o m mul-
t i p l e networks, I , Blockmodel o f r o l e s and Dosi t i o n s , American J o u r n a l o f S o c i o l o g y , 81 ( 1 9 7 6 ) . 730-780.
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TRENDS LV MATHEMATICAL PSYCIIOLOG Y E . Degreef and J. Van Bu e n h t (editors) 0 Ekrvirr Science Pirbljsgrs 8.1'. (Abrth-HolIutidj, i984
227
C'h CCKCEPTS OF THE D I f l E N S I n N OF A RELATION I\NP I! GENERALIZED RECOGNITInN EXPERIblENT
Klaus H e r b s t Uni v e r s i t a t Regensbur?
Confusion amonct s t i m u l i i s g e n e r a l i z e d t o a b i n a r y r e l a t i o n R f r o m t h e s e t o f s t i m u l i t o i t s nower s e t . I f R induces a senimodular l a t t i c e o f f i n i t e l e n ? t h , t h e conceot o f l a t t i c e dimension i s m e a n i n g f u l .
In
any case 9 i s t h e i n t e r s e c t i o n of a t l e a s t one s e t o f biorders.
The minimum number o f b i o r d e r s needed
d e f i n e s t h e b i o r d e r dimension o f R.
Data from an
e x p e r i m e n t w i t h Vorse codes e r e analvsed.
1. INTRODI!CTION Consider t h e f o l l o w i n g r e c o o n i t i o n e x o e r i m e n t . s t i m u l i a r e Dresented i n secluence. mulus has been r e p e a t e d o r n o t .
Two elements f r o m a s e t P o f
A s u b j e c t decides whether t h e f i r s t s t i -
The exDerimenta1 desion i s A x A .
d a t a i n d u c e a b i n a r y r e l a t i o n S on A.
L e t the
This empirical r e l a t i o n S i s o f t e n
found t o be non-symmetric l i k e t h e c o n f u s i o n m a t r i c e s which a r i s e from r e n e t i t i o n s of the exneriment. Mu1 t i d i m e n s i o n a l s c a l i n f nrocedures use c o n f u s i o n m a t r i c e s i n o r d e r t o man t h e s e t o f s t i n w l i i n t o a m e t r i c snace such t h a t t h e d i s s i m i l a r i t y between Q a i r s o f s t i m u l i i s reDresented b y a d i s t a n c e f u n c t i o n .
The f o u n d a t i o n o f
these s c a l i n g s i s u n s a t i s f a c t o r y w i t h r e s p e c t t o t h e c h o i c e o f t h e m e t r i c snace and i t s dimension. Beals, Krantz, and Tversky (1966) f i r s t gave a s o l u t i o n o f t h e r e p r e s e n t a t i o n problem f o r a s n e c i f i c c l a s s o f m e t r i c spaces.
B u t t h e y nresuppose
t h a t t h e dimension of t h e s t r u c t u r e i s known i n advance. I n t h i s paper two concents o f t h e dimension o f a r e l a t i o n are discussed, t h e l a t t i c e dimension and t h e b i o r d e r dimension.
228
K. Herbst
2 . /? GENERALIZED R E C O G N I T I O N EXPFR1M:tT The a n a l v s i s r e l i e s on t h e f o l l o w i n o o e n e r a l i z e d r e c o o n i t i o n e x n e r i m e n t : l i k e i n t h e orecedinc! d e s i n n a s i n o l e s t i m u l u s a
C
A i s nresented a t f i r s t . o f elements from A .
R u t now i t i s f o l l o w e d by a randop seouence (b,d,c)
The s u b i e c t decides t r h e t h e r t h e f i r s t s t i m u l u s a has heen r e o e a t e d i n t h e senuence (b,d,c).
The t o t a l nower s e t P(A) i s used such t h a t t h e design o f
t h i s exrleriment i s A
x
L e t t h e d a t a induce a r e l a t i o n R
"(A).
-C A
P(b).
x
3 . LATTICF r u w s I n r ~ F o r s h o r t l e t r.ow P denote t h e nower s e t o f RD = i a
6
A
I
P.
F o r any D F P we s e t
aQo? and d e f i n e a r e l a t i o n on P, Pp C P nR o
F
iff
x
RD C - Rq.
T h i s a l l o w s us t o i n t r o d u c e an e q u i v a l e n c e r e l a t i o n on
i f f Pn = R q , and t o reduce classes.
D
P, bv
D,
such t h a t
D
%
tl
t o a s e t o f renresentatives o f equivalence
Pp i s c a l l e d i r r e d u c i b l e i f f t h e r e e x i s t no two e n u i v a l e n t e l e -
w n t s n and q . Any i r r e d u c i b l e r e l a t i o n R p f u l f i l l s t h e d e f i n i t i o n o f a D a r t i a l o r d e r . Moreover, i t i s a l a t t i c e r l r o v i d e d t h a t e v e r y o a i r n,rl E P has a supremum and an i n f i m u m i n R
P'
DEFI PI1T I nN Rp
i s a lattice iff
P l : Rp i s a p a r t i a l n r d e r A2: For e v e r y
O,(I
F
P e x i s t s a l e a s t unper bound r
P 3 : F o r e v e r y In,q
F
P e x i s t s a o r e a t e s t l o w e r bound s
f
P, r = surt(o,q).
(suoremum)
P, s = inf(o,n).
(infimw! The concent o f l a t t i c e dimension i s based on t h e n o t i o n o f a c h a i n f r o m t h e l e a s t t o t h e g r e a t e s t e l e m e n t i n Rp which has maximal l e n g t h . mal chains i n a l a t t i c e R
I f a l l maxi-
have t h e same l e n g t h m, we c a l l m t h e l a t t i c e
P dimension o f P p , L d i m R p = m.
P s u f f i c i e n t c o n d i t i o n f o r t h e e x i s t e n c e o f Ldim Rp can be f o r m u l a t e d u s i n q the following d e f i n i t i o n .
229
Concepts oftlie dimension of. relatiou
DE FI PiITI ON L e t o,q,r
he elements o f
Fp
and
P
denote t h e n e i a t i c r - o f Cn.
Then D covers
o i f f qRpn and f o r a l l r h o l d s qRpr v r $ p .
F o r . a n y l a t t i c e R e x i s t s a dimension L d i m ? if a t l e a s t one o f t h e f o l l o P P w i n g axioms h o l d s :
A4: I f
n and q c o v e r i n f ( n , n ) , t h e n suo(p,q) covers n and ? .
( u n n e r semimodul a r i t v )
1.5: I f suD(p,q)
covers D and
( l o w e r semimodul a r i t y )
t h e n n and q cover i n f ( n , q ) .
fl,
.
I f a s u b j e c t i n a g e n e r a l i z e d r e c o n n i t i o n e x o e r i m e n t commits n o e r r o r s , which w a n s t h a t no confusion hannens, then R
i s i s o m o r n h i c t o a Boolean
P
l a t t i c e o f s e t s ( ? ’ ( P , ) ,C) - c o n s i s t i n n of t h e Dower s e t o f s t i m u l i and t h e r e The l a t t i c e dimension i s i n t h i s case e q u a l t o
l a t i o n of s e t i n c l u s i o n .
\!.I.
t h e c a r d i n a 1 i t . y o f A, Ldim RP = an examnle w i t h
IAl
The f o l l o w i n g Hasse diagram r e o r e s e n t s
= 3.
ab
ac
IX
However, i f c o n f u s i o n o c c u r s , R s m a l l e r than / A / .
P
2‘1
may s t i l l be a l a t t i c e o f some dimension
Here a r e two examples w i t h
IAl
= 3.
Bv ab ... we denote
j u s t t h a t o E P = P(A) which c o n s i s t s o f t h e elements a,b,
... o f
A.
Examole 1: aRbc, bRac and cRab induces ab
%
ac
Q
bc
Rp reduces t o
%
abc.
P’p\b
a and Ldim Rp
2.
\b/
c
E x a m l e 2: I f aRc, bRc and cRab,
Rp reduces t o
/c\
N o t i c e , t h a t i n examnle 2 t h e c o n f u s i o n i s non-symmetric so t h a t t h e s t i m u l u s c dominates t h e s t i m u l i a aod b.
230
K . Her6st
P . i s asvclnetry i s n r e s e r v e d i n a r e n r e s e n t a t i o n o f ( P , R $
i n a vectorspace.
If A1-.45 a r e v a l i d and two sunnlewentary axioms, lr6 and A 7 l i s t e d below, a r e s a t i s f i e d then
(
D,an) i s hommornhic t o a vectorsnace V over a f i e l d F
(!S67),
(Rirkhoff
Rrosler (1980)).
The elements o f P correspond t o suhsoa-
ces o f V s o t h a t t h e s i n f l l e s t i m u l u s c i n examnle tw rrould he r e n r e s e n t e d n o t Sy a v e c t o r b u t h v a q l a n e i n V . A6: Every c h a i n i n Rp c o n s i s t s o f a f i n i t e number nf elements.
A7: Let
B
he t h e l e a s t and e t h e p r e a t e s t e l e m e n t i n R,,.
e x i s t s a comnlement
fl F
D
such t h a t i n f ( n , n )
=
0
F o r each o
and sun(o,q)
E P
= e.
4. EIPIRICAL E X A W L E
P n e n e r a l i z e d r e c o o n i t i o n e x o e r i w n t was r u n with f i v e s u h l e c t s and t h e f o l l o w i n o s i x fJorse Codes ( H e r h s t ( 1931) ) . a
. .
.
- -
h C
d
e f
1 4 th jP
. . - - - . .
-
=
6 and I P ( A )
1
A
= 64 t h e d e s i m
x
P ( P ) has t h e o r d e r 6
x
64
Tyni c a l v t h e s u b j e c t h e a r d a t f i r s t a s i n o l e code and a f t e r w a r d s a sequence o f codes b o t h chosen a t random.
Then i t d e c i d e d whether t h e s i n g l e code
had been r e n e a t e d i n t h e sequence. The r e l a t i o n R
-P C
x
P was o n e r a t i n n a l l y d e f i n e d bv a% i f f t h e frequencv
o f yes-answers exceeds t h e f r e q u e n c v o f no-answers.
The r e s u l t s o f t h i s
e x n e r i m e n t mav be summarized as f o l l o w s .
I n no case t h e concent o f l a t t i c e dimension can be a n p l i e d .
E i t h e r axiom
P 2 (supremum) i s n o t f i l l f i l l e d o r n o t a l l maximal chains have t h e same lenqth.
The axioms A4 and 115 ( s e m i m o d u l a r i t y ) do n o t h o l d .
The c o n c e o t o f
a l a t t i c e s t r u c t u r e i s t o o r e s t r i c t i v e f o r these data. Ire observe t h a t s u b j e c t s seldom miss a s i g n a l , t h a t means i f a holds.
B u t f a l s e alarms a r e numerous, we f i n d aPn a l t h o u a h a
F
9
p, aRD
There-
p.
f o r e Rp can he reparded as an e x t e n s i o n o f a Boolean r e l a t i o n o f s e t s .
Con-
s e q u e n t l y , R i s r e d u c i b l e i n s o f a r as e q u i v a l e n c e c l a s s e s can be formed i n but not i n A.
The C a r t e s i a n n r o d u c t A
x
P can be reduced f r o m 6
x
64
P
231
Concepts oftlie dimension o f a relation
to 6
x
35 a t most and 6
x
12 a t l e a s t .
Subsets o f A w h i c h a r e n o t o r d e r e d
i n a Roolean r e l a t i o n o f s e t s are o r d e r e d i n t h e e n o i r i c a l r e l a t i o n s generat e d hv t h e f i v e suh.jects.
This leads t o the % l l w i n ? idea.
5 . BInRDER D I M E N S I O N I n an extreme case o f o r d e r
R would reduce t o a t r i a n r r l e P
t h a t means Rp and a c o r r e s n o n d i n q l v d e f i n e d RA a r e connected, i . e . t h e y a r e weak o r d e r s . I n t h i s case R f u l f i l l s t h e d e f i n i t i o n o f a b i o r d e r . DEFINITInN
-
Let R C A
x
R i s a b i o r d e r P i f f f o r a l l a,b
P.
E
A and n,?
P, aRo and
E
091 i m n l i e s aRq o r bRn.
A r e o r e s e n t a t i o n o f A and P i n
E?, t h e s e t o f r e a l numbers, assures t h e f o l -
1owi no theorem. THEOREM I f R i s a b i o r d e r , t h e r e e x i s t two maonings f: A
for all a
E
A and D
E
+lR and
g:
P + R , such t h a t
P h o l d s aRn i f f f ( a ) s g ( o ) .
The b a s i c s u b r e l a t i o n o f a b i o r d e r R i s
o
q
alR :I
w i t h t h e diaoram
b
p a1
/!*
Doignon, nucamp and Falmanne ( t o aooear) show t h a t ifR i s a b i o r d e r , a rel a t i o n () c (A
U
P)
x
(A
P) can be d e f i n e d which has t h e n r o n e r t y o f a
IJ
-
o r d e r . I t s r e p r e s e n t a t i o n i n lR is t h e well-known Guttman s c a l e . The c o r r e s p o n d i n g diagram o f t h e b a s i c examnle j u s t shown i s o I
b a.
&
237-
K Herbs!
ar] n
The h a s i c f o r b i d d e n s u h r e l a t i o n i s
n
h
t h e connlement of which.
! i s i l l u s t r a t e d by: o
n
X h. a
As c o u l d he e x n e c t e d , no s u h i e c t r e a l i z e d a h i o r d e r .
Unidiwnsional renre-
The m a t r i x below shows a t v n i c a l r e l a t i o n R.
sentations are imoossihle.
N o t i c e t h e number o f e n u i v a l e n c e c l a s s e s and t h e f r e r l w n c y o f t h e f o r h i d d e n
s u b r e l a t i on. E x a m l e : r e l a t i o n P o f s u h j p c t n o fl. I\ a
h
c
d
a ;_ _ _ - ~ _ _ _ a b
P R
..
--
I I
I
9
Q
d f
~-
ac ad af
R R R R R R Q
bd be hf cd ce cf df
f
R
C
SC
e
R R
abdf acde acdf acef hcef
P
R
R Q
R R P R
R R R
F!, de
?r
%
ae, ade abf abc
R R R P
R R R
R P R
R R
9
%
abd, ahe, bde, ahde
%
cde
‘L
ef, def
%
bcd, bce, ahcd, ahce, bcde
R
R R R
R
R R R R
R R
R R
R R
R R R
R R R R R
R
R
R R R
P
R
1,
R
L.
R R R
?r
R
P
D R
ah
R
R 9 R
acd ace acr adf ae f hcf hdf cdf ce f
R
%
R R P
?r
I!
%
9 9 ? R
adef b e f , ahef, b d e f , abdef abcf
%
,-” cdef
R R R
Q R R
bcdf
%
abcde, ahcdf, a b c e f , acdef, b c d e f , abcdef
%
233
Concepts o f t h e dimension ofa relation
The concept o f b i o r d e r dimension o r h i d i m e n s i o n o f a r e l a t i o n r e l i e s on t h e f o l 1 o v i n q theorem. THEnREPi c A x P e x i s t two f a m i l i e s o f b i o r d e r s { E i l F o r any r e l a t i o n R Ri = U B!.
such t h a t R =
and {B!} J
J
Ve r e s t r i c t t h e n o t i o n o f h i d i m e n s i o n t o t h e i n t e r s e c t i o n case.
DE F I FJIT I flN The h i d i m e n s i o n o f a r e l a t i o n R
5A
x
P, Pidim
P, i s t h e s m a l l e s t c a r d i n a l
n, such t h a t t h e r e e x i s t s a c o l l e c t i o n of h i o r d e r s i R i
I
i
F
I } w i t h 111 = n
and R = n B i i' The m u l t i d i m e n s i o n a l r e n r e s e n t a t i o n t h e o r e v has been nroven hy Foignon, n u c a m and Falmagne ( t o anoear). THEOREM L e t R he a c o u n t a b l e r e l a t i o n and B i d i m R = n, then t h e r e e x i s t two mappinps
f : A +lRn and g: P +IRn, i=l,.
. .,n
such t h a t aRo i f f f i ( a )
o f the vectors f ( a ) ,
O(D) E
s gi(o),
for
fl
components
Rn.
I n a g e n e r a l i z e d r e c o g n i t i o n e x o e r i m e n t we m i n h t e x n e c t t h e s u b j e c t t o o i v e
-
a n o s i t i v e answer e x a c t l y i f a l l s t o r e d and h i a h l y e v a l u a t e d f e a t u r e s o f t h e f o r e g o i n ? s i n q l e s t i m u l u s a r e j u d o e d t o he i n c l u d e d i n t h e f o l l o w i n g seouence o f s t i m u l i . B i d i m R may be i n t e r n r e t e d as t h e c o n j o i n t dimension o f t h e c o n f u s a b i l i t y o f t h e s i n g l e s t i m u l i on t h e one hand and o f t h e Dower o f t h e subsets o f s t i m u l i t o l e a d t o c o n f u s i o n on t h e o t h e r hand. The e v a l u a t i o n o f t h e b i d i m e n s i o n o f a r e l a t i o n i s a Droblem which o f t e n t u r n s o u t t o be h a r d . I f R i s i r r e d u c i b l e , t h a t means Rp an RA a r e i r r e d u c i b l e , an u m e r bound i s g i v e n b y B i d i m R 6 min
(\Al,IPl).
F o r P = P ( A ) we have B i d i m R s I A I ,
the c a r d i n a l i t y o f the s e t o f s t i m u l i .
I n o r d e r t o g i v e a l o w e r bound, t h e
following d e f i n i t i o n i s useful. DEFI NIT1 ON
-
Let A ' c - A and P ' C P . merit o f K, denoted b y
-
Then K C A '
R,
x
-
P ' i s c a l l e d a crown i f f t h e comole-
i s a one t o one corresnondence between A ' and
P'.
K . tkrbst
'34
I n a crown K e x i s t s f o r each o a i r a,b t h a t (a,b)
E
A ' e x a c t l y one n a i r D,q
i s a forhidden s u b r e l a t i o n .
(D,?)
P ' such
E
This leads t o a standard
resul t. LEt4Yn
The h i d i m e n s i o n o f a crown K r- P.'
x
P ' i s enual t o the c a r d i n a l i t y o f i t s
domain and range, R i d i m K = I I \ ' I = I P ' I . ilow, l e t Kmax
5R
be t h e p r e a t e s t crown i n R such t h a t f o r a l l crowns
9 holds lKmaxl
t I K . 1 , and l e t R he i r r e d u c i b l e . J case c o n s i d e r e d h e r e . we have t h e i n e q u a l i t y
K.
Then, i n t h e s p e c i a l
1 -
s Ridim R 5
R i d i m Ymax
in\.
I f B i d i m R > 2 t h e e v a l u a t i o n o f t h e h i d i m e n s i o n i s c a l l e d a NP-comolete problem.
T h a t w a n s , t h e c o p n u t i n n t i m e t o check w h e t h e r R i d i m R > n > 2
i s an e x a o n e n t i a l f u n c t i o n o f t h e i n h e r e n t c o n d i t i o n s o i v e n by R (Yannakakis ( t o anpear)).
DoiGnon, Ducamo and Falmaqne ( t o aanear) f o r m u l a t e these con-
d i t i o n s i n t h e lanouaoe o f granh t h e o r y .
I n n r i n c i n l e , t h e elements o f
( t h e comnlement o f P ) a r e r e g a r d e d as v e r t i c e s and s p e c i a l sequences i n
Now, t h e v e r t i c e s o f H ( R ) a r e c o l o u r e d
o f the basic forbidden s u b r e l a t i o n . 1
!?
These sequences a r i s e from g e n e r a l i z a t i o n s
as edges o f a hypernraph H ( R ) . by a f a m i l y o f c o l o u r s {C.
P
I i6 Ii
such t h a t no edge i s monochromatic, i . e .
i s c o l o u r e d hy one c o l o u r o n l y . The minimum number o f c o l o u r s
I
which a l l o w such a c o l o r a t i o n , i s c a l l e d
the c h r o m a t i c number Chrom H ( R )
.
I t can he shown t h a t R i d i m R = Chrom t i ( R ) .
The main i d e a o f t h e p r o o f way he s k e t c h e d as f o l l o w s .
IfR i d i m R = Chrom H ( R ) = n, each c o l o r a t i o n Ci extended t o a maximal h i o r d e r
5R,
with i=l,
...,n,
can be
Ri.
That means, f o r a l l o t h e r e x t e n s i o n s
R{ o f Ci we have
R 1! c- ii.cR. 1The cornolemefit o f Ri
and R = n Bi. i T h i s c o l o r i n g p r o c e d u r e was s u c c e s s f u l l y a m f i e d i n two cases of Morse code i s t h e n a m i n i m a l b i o r d e r Ri
data. F o r the remaing t h r e e d a t a s e t s t h e q u e s t i o n whether R i d i m Kmax
Bidim R
I
was l e f t open a f t e r 10 hours CPU-time. An a l t e r n a t i v e p r o c e d u r e t u r n e d o u t t o be e f f e c t i v e i n a l l cases.
Let
235
Cancepts of the dimension of a relntion
{@i
I iE.I}
R.
be a p a r t i t i o n o f
R =
R
Then
u Di and
=
i
mi
I r-ln
i
i
I f now {DiI i s a f a r r i l v o f b i o r d e r s , each
i
pi
= R. i s a b i o r d e r and R = 1
t!owever, these b i o r d e r s Di a r e n o t n e c e s s a r i 1v maximal comnlements Bi need n o t
.
i Therefore, t h e i r
be minimal and 111 i s o n l y an unner bound of
tc!
I f v e denote t h e minimal number o f c o l o u r s Di
R i d i m R = Chrom H(R).
Ri.
needed
i n t h i s orocedure by ndiln I' we have B i d i m Kmax ,< R i d i m P ,< k?r7 F . T o r a l l f i v e d a t a m a t r i c e s o f llorse codes h o l d s R i d i m Kmax = Ddim R so t h a t Ddim R = R i d i m p . B i d i m R v a r i e s between 3 and 5 f o r d i f f e r e n t s u b i e c t s .
A c o l o r a t i o n o f the
e m o i r i c a l example shown above w i t h B i d i m R = 4 i s g i v e n below. The c o l o r s Di
are ahhreviated t o the i n d i c e s i=1,
...,4 .
Example: R e l a t i o n R o f s u b j e c t nP 4 ( c c l o r e d ) :
P a
h
c
d
e
f
l
l
l
1
B a
l
b
R
R
1
1
1
1 1 sab
c
3
3
R
2
2
1
d
3
3
1
R
R
f
2
2
2
2
2
a
P
R
c
R
ad
R
af
R
bc
R
b
d
2
R
2
1
2
1
3
1
R
R
1
'L
2
2
2
2
R
sebf
1
sabc
R R
2
2
R
R
l
R
l
be
R
R
1
R
R
1
9
R
4
2
2
R
d
ce c
df
3 3
f
R
R R
R
3
4
3 3
3
3
3
R
ae, ade
1
bf c
se,de
R
2
1
2
R 2
R
R
R
R
1
s
abd, abe, bde, abde
scde sef,
def
236
K. Herbst
A a
b
c
d
e
f
acd a ce
%
p
hcd, hce, ahcd, abce, bcde
acf adf
%
adef
ae f
1,
b e f , a b e f , h d e f , ahdef
t. cf
%
abcf
%
cdef
h6f cdf CP f
ahdf acde bcdf
acdf acef
abcde, a b c d f , abcef, acdef,
hcef
b c d e f , ahcdef The c o l l e c t i o n o f h i o r d e r s {Ri$,
which i s i n d u c e d by t h e c o l o r a t i o n process,
i s c a l l e d a h i o r d e r r e n r e s e n t a t i o n o f 9.
Unfortunately,
i n most cases t h e
h i o r d e r r e n w s e n t a t i o n o f a h i n a r y r e l a t i o n i s i n c o n t r a s t t o the bidimens i o n n o t unique.
There mav even e x i s t a l a r o e f a m i l y o f r e Q r e s e n t a t i o n s .
I n o r d e r t o s e l e c t a s n e c i a l one which a l l o w s a c o n v i n c i n g i n t e r n r e t a t i o n , we need a more e l a h o r a t e d theor!!
about the r e c o n n i t i o n nrocess.
6. 4ESU~F A l a t t i c e s t r u c t u r e a s s o c i a t e d w i t h t h e conceot of l a t t i c e dimension i s an
a o o e a l i n q t h e o r e t i c a l d e v i c e t o r e n r e s e n t t h e e m n i r i c a l r e s u l t o f a general i z e d recoqni t i o n e x n e r i m n t . vectorsoace.
T h i s i s because o f i t s corresnondence t o a
A semimodular l a t t i c e , however, imnoses severe c o n s t r a i n t s on
t h e d a t a which a r e h a r d t o he s a t i s f i e d . A mu1 t i d i m e n s i o n a l b i o r d e r r e o r e s e n t a t i o n w i t h t h e c o n c e n t o f b i o r d e r dimen-
-
s i o n e x i s t s f o r each h i n a r v r e l a t i o n .
The e v a l u a t i o n t e c h n i q u e s remain t o
be a m e l i o r a t e d s o t h a t r e l a t i o n s o f l a r g e r s i z e can he a n a l y z e d i n a s u i t a h l e time.
I n order t o f i n d a s o e c i f i c multidimensional b i o r d e r reoresentation
which has a r i c h e m n i r i c a l meaninn, t h e t h e o r y o f t h e r e c o o n i t i o n p r o c e s s must be developed f u r t h e r .
Concepts of rlie dinmrsion o j u relnrion
237
REFERE tlCES
Beals, R., Krantz, D.H., and Tversky, A . , Foundations of multidimensional s c a l i n o , Psychol. Review, 75 (1968), 127-142. Bi rkhol'f, G . , Lattice theory, Providence, American blathematical Societ.v (1.567). Doignon, J-!'. , Ducamn, A . , a n d Falmagne, k c . , On r e a l i z a b l e biorders and the biorder dimension o f a r e l a t i o n ( t o annear). Driisler, J . , Grundlaaen des Fiour-Erkennungsexneriments, Zei t s c h r i f t Fir Exn. und Angew. Psycho1 . , 27 (1980), 1-25. Herbst, K . , Empirische Untersuchung eines verbandstheoretisch begrundeten mehrdirensionalen Skalierungsverfahrens, Zei t s c h r i f t f i r Ex?. und Angew. Psychol ., 28 ( 1 9 8 l ) , 232-254. Yannakakis, W . , The comolexitv of t h e Dartial order dimension oroblem ( t o apoear) .
This Page Intentionally Left Blank
TRENDS lh' MATHEMATICAL PSYCHOLOGY E . Depeef and I . Van flu genhaui (editors) 0 Elsevier Science Publisfers B. I.' (North-Holland), 1984
239
ISOTONIC REGRESSION ANALYSIS AND ADDITIVITY
Ranald R . Flacdonald Deoartment o f Psychology I!ni v e r s i t,y o f S t i r l i n ?
T h i s naner reviews tl7e 1iterat.ut-e on i s o t o n i c r e g r e s s i o n w i t h a view t o f i n d i n g t e s t s f o r a d d i t i v i t y i n d a t a whose p o n u l a t i o n values a r e unioue o n l y up t o a monotonic t r a n s f o r m a t i o n .
Several
anproaches a r e c o n s i d e r e d and a s e q u e n t i a l one i s recommended
-
f i r s t t e s t i n g m o n o t o n i c i t y , and then
t e s t i n g t h e double c a n c e l l a t i o n axiom.
Two t e s t s
a r e p r e s e n t e d f o r t e s t i n g m o n o t o n i c i t y and an a l g o r i thm f o r d e t e c t i n g s i g n i f i c a n t denartures from double c a n c e l l a t i o n i s o u t l i n e d .
1. INTRODUCTION For a number o f y e a r s mathematical p s y c h o l o g i s t s have s t u d i e d a x i o m a t i c approaches t o a d d i t i v i t y .
Under t h e heading o f c o n j o i n t measurement t h e y
have been concerned w i t h whether t h e combined e f f e c t s o f a p a i r o f independ e n t l y m a n i p u l a t e d v a r i a b l e s a r e c o m p a t i b l e w i t h an u n d e r l y i n g a d d i t i v e r e presentation.
To p u t i t a n o t h e r way, what a r e t h e necessary and s u f f i c i e n t
c o n d i t i o n s under which monotonic f u n c t i o n s o f each o f two v a r i a b l e s e x i s t such t h a t when added t o g e t h e r t h e i r sum i s i t s e l f a monotonic f u n c t i o n o f t h e combined e f f e c t s ?
A much q u o t e d example i s t h a t o f a s i m p l e b e t w i t h Can t h e s u b j e c t i v e v a l u e o f
some o r o b a b i l i t y o f w i n n i n g a s p e c i f e d amount.
such b e t s be expressed i n terms o f a monotonic f u n c t i o n of t h e p r o b a b i l i t y p l u s some monotonic f u n c t i o n o f t h e D o t e n t i a l w i n n i n g s ? One r e p o r t of a s o l u t i o n t o t h e p r o b l e m i s g i v e n i n t h e f i r s t paper t o be p u b l i s h e d i n t h e J o u r n a l o f Mathematical Psychology, Luce and Tukey (1964). and o t h e r accounts a r e w i d e l y a v a i l a b l e o f which t h e most comorehensive i s K r a n t z , Luce, Suppes and T v e r s k y (1971).
The s o l u t i o n t o t h e Droblem was f o r m u l a t e d i n
terms o f t h e o r d e r r e l a t i o n s which must h o l d between a l l p o s s i b l e combined
243
R.R. MdcDonuld
e f f e c t s o f the variables. hold o r not.
I d i t h r e a l d a t a these c o u l d be f o u n d e i t h e r t o
A p a r t f r o m a numher o f u n t e s t a b l e t e c h n i c a l axioms t h e d a t a
must conform t o t y o o r d e r r e s t r i c t i o n s . t o t h e independent v a r i a b l e s .
They must be o r d e r e d w i t h r e s p e c t
T h a t i s , i f one v a r i a b l e i s h e l d c o n s t a n t ,
an i n c r e a s e i n t h e o t h e r v a r i a b l e s h o u l d l e a d t o an i n c r e a s e i n t h e v a l u e
of t h e data.
This i m n l i e s t h a t a d d i t i v e data are monotonically r e l a t e d t o
t h e indenendent v a r i a b l e s and t h ? r e s t r i c t i o n w i l l be r e f e r r e d t o as t h e monotonicity r e s t r i c t i o n .
I n a d d i t i o n t o t h e m o n o t o n i c i t y r e s t r i c t i o n Luce
and Tukey (1964) r e o u i r e t h a t t h e double c a n c e l l a t i o n axiom be met. double c a n c e l l a t i o n axiom s t a t e s t h a t if a ' , a " , a"' dependent v a r i a b l e A and b ' , b " , b " ' variable
The
a r e l e v e l s of t h e i n -
are l e v e l s o f the o t h e r independent
then i f i n e n u a l i t i e s (1) and ( 2 ) :?olrl, so s \ ~ u ! d ( 3 )
where uab i s t h e v a l u e o f t h e d a t a c o r r e s p o n d i n g t o l e v e l a o f A and b of R.
The double c a n c e l l a t i o n axiom a l s o i s r e q u i r e d t o h o l d w i t h > r e p l a c e d
by < .
T h i s i s o f l i m i t e d use when d e a l i n g w i t h t h e f a l l i b l e d a t a o f expe-
rimentalists.
I n such cases i t would b e u s e f u l t o know i f any v i o l a t i o n s
i n t h e o r d e r r e s t r i c t i o n s c o u l d be e x n l a i n e d by t h e i n h e r e n t v a r i a b i l i t y o f the data o r whether they represent r e a l effects. There i s a n o t h e r p r o b l e m i n t e s t i n g i f t h e above o r d e r r e s t r i c t i o n s h o l d i n r e a l data.
As t h e number o f l e v e l s o f t h e independent v a r i a b l e s s t u d i e d i s
f i n i t e , one cannot check t h e double c a n c e l l a t i o n axiom f o r a l l p o s s i b l e l e v e l s of A and B .
I n f i n i t e cases double c a n c e l l a t i o n may be s a t i s f i e d
w h i l e more complex c a n c e l l a t i o n s a r e n o t .
Any v i o l a t i o n o f a c a n c e l l a t i o n
no m a t t e r how complex r e n d e r s an a d d i t i v e r e p r e s e n t a t i o n i m p o s s i b l e .
Ar-
b u c k l e and L a r i m e r (1976) and M c C l e l l a n d (1977) p r o v i d e o r o b a b i l i t i e s o f complex c a n c e l l a t i o n s b e i n g v i o l a t e d g i v e n t h a t l r o n o t o n i c i t v and double c a n c e l l a t i o n h o l d i n randomly o r d e r e d m a t r i c e s . crease as t h e m a t r i c e s i n c r e a s e i n s i z e . c e l l a t i o n s would be a hideous t a s k .
a D a r t i a l way o u t .
However, Luce and Tukey (1964) p r o v i d e
B a r e s e p a r a t e d by equal aB f o r m a d u a l s t a n d a r d sequence) m o n o t o n i c i t y and
Where t h e l e v e l s o f A and
mounts ( t e c h n i c a l l y A,
These a r o b a b i l i t i e s i n -
T e s t i n g a l l a o s s i b l e complex can-
24 1
kotoiiic repenion arlalysis and additivity
double c a n c e l l a t i o n a r e necessary and s u f f i c i e n t conc'itions t o ensure an a d d i t i v e r e r J r e s e n t a t i o n o v e r t h e l e v e l s of A and B s t u d i e d .
Thus
i t i s d e s i r a b l e t o design experiments so t h a t t h i s c o n d i t i o n i s approxima-
t e l y met.
I f t h e l e v e l s of A and B do form a dual s t a n d a r d sequence, t h e n
i t i s o n l y necessary t o t e s t m o n o t o n i c i t y and double c a n c e l l a t i o n ;
i n 3:s
case complex c a n c e l l a t i o n s may be i g n o r e d .
I t i s w o r t h d i s t i n g u i s h i n g between t h e p r e s e n t c o n c e p t i o n o f a d d i t i v i t y w h i c h concerns t h e e x i s t e n c e o f a d d i t i v i t y h o l d i n g i n some monotonic t r a n s f o r m a t i o n o f t h e d a t a and a d d i t i v i t y which h o l d s i n u n t r a n s f o m d data. S t a n d a r d t e s t s o f i n t e r a c t i o n s i n a n a l y s i s o f v a r i a n c e , T u k e y ' s (1964) add i t i v i t y t e s t an6 "anc'el's (1971) work on n o n a d d i t i v i t y a r e a l l concerned w i t h t h e second sense o f a d d i t i v i t y .
The c o n j o i n t measurement Droblem i s
concerned w i t h t h e f i r s t sense and i f a d d i t i v i t y e x i s t s i n t h i s sense i t
w i l l a l s o do so a f t e r any monotonic t r a n s f o r m a t i o n has been a o p l i e d .
For
t h e r e s t o f t h e d i s c u s s i o n a d d i t i v i t y w i l l be used o n l y i n t h e more g e n e r a l sense.
A l s o i n t h e a n p l i c a t i o n s t o be c o n s i d e r e d i t w i l l be assumed t h a t
t h e o r d e r i n g o f t h e l e v e l s o f t h e indeoendent v a r i a b l e s i s known i n advance. Bartholomew (1961) p r e s e n t e d a method f o r s t a t i s t i c a l i n f e r e n c e under o r d e r r e s t r i c t i o n s which he c a l l e d i s o t o n i c r e g r e s s i o n a n a l y s i s .
More r e c e n t r e -
views o f t h e area a r e g i v e n i n Barlow, B a r t h o l o w w , E r e m e r and Brunk (1972) and Smith and Macdonald (1983).
A major a p o l i c a t i o n o f i s o t o n i c regression
a n a l y s i s i s t o p r o v i d e t e s t s o f d i f f e r e n c e s between means s u b j e c t t o c e r t a i n order restrictions.
I t would seem l i k e l y t h a t t h e s t a t i s t i c a l l i t e -
r a t u r e on i s o t o n i c r e g r e s s i o n w o u l d be r e l e v a n t t o t h e oroblems in a p p l y i n g c o n j o i n t measurement t o f a l l i b l e d a t a .
T h i s oaper i s designed t o examine
t h e e x t e n t t o which t h i s i s t r u e and t o suggest some s t a t i s t i c a l t e s t s f o r f i t t i n o an a d d i t i v e c o n j o i n t model. The aims o f t h e s t a t i s t i c i a n s w o r k i n g on i s o t o n i c r e g r e s s i o n a r e somewhat d i f f e r e n t f r o m those o f t h e c o n j o i n t neasurers.
Tests developed i n i s o t o n i c
r e g r e s s i o n a r e designed t o i n c r e a s e t h e power o f t h e t e s t when t h e o r d e r i n g o f t h e means can be p r e d i c t e d i n advance.
C o n j o i n t measurement i s concer-
ned w i t h whether t h e r e a r e any d i f f e r e n c e s between t h e means i n c o m p a t i b l e w i t h a d d i t i v i t y r a t h e r than showing whether an a d d i t i v e model g i v e s a b e t t e r f i t t h a n no model a t a l l .
242
R.R.MacDonald
2. NOTATION
The f o l l o w i n g n o t a t i o n a n o l i e s t o a two way a n a l y s i s o f variance, and t o a one way a n a l y s i s when the s u h s c r i p t k i s omitted:
J
i s the number o f l e v e l s o f v a r i a b l e A
i s the number o f l e v e l s o f v a r i a b l e B
I(
Yijk
i s the i t h o b s e r v a t i o n i n the j k t h treatment
i s the number o f observations i n each jk t h treatment (assumed
n
equal)
N
i s t h e t o t a l number o f observations
u
i s the t h e o r e t i c a l o v e r a l l mean
ir
i s the observed o v e r a l l mean
ujk
ir .jk ir j
irk
i s the t h e o r e t i c a l mean i n the j k t h treatment i s the observed mean i n the j k t h treatment
i s the observed mean i n the j t h l e v e l of A i s the observed mean i n the k t h l e v e l o f R
i-u
i s the i s o t o n i c regression estimate o f u . jk Jk' i*ujk i s derived w i t h respect t o some order r e s t r i c t i o n s ; u j , k , < u ~ , , ~f,o,r
some s p e c i f i e d j ' , j " , k ' , k " .
Barlow e t a1 ( 1 9 7 2 ) show t h a t the i s o t o n i c
regression e s t i m a t e i s the maximum l i k e l i h o o d e s t i m a t e o f u
w i t h these r e s t r i c t i o n s
jk
.
consistent
L i s the number o f d i f f e r e n t values taken by the i s o t o n i c regression e s t i mates i-u. Jk. Considering f i r s t the u n i v a r i a t e case, the usual s t a t i s t i c used t o t e s t f o r e q u a l i t y o f treatment means i s : CI
1 n ( D j-O)
F(J-1,N-J)
" / ( J-1)
= j
z ij
(4) ( Y -G.)*/(PI-J)
"
I n analysis o f variance n o t a t i o n : F =
SSbet/dfbet - MSbet SSw/dfw -
ww
Eartholomew (1961) proposes the s t a t i s t i c E: z n(i-u.-D) 2 J E J2 = j 1 (Yij-D)' ij
(5)
243
Isotonic regression analysis m d additivity
I n o u r n o t a t i o n t h i s i s expressed:
2 SSmon EJ = E
(7)
2 J d i f f e r s f r o m F i n two i m p o r t a n t r e s p e c t s .
F i r s t l y the i s o t o n i c regres-
s i o n e s t i m a t e s a r e used t o compute t h e between t r e a t m e n t v a r i a n c e through SSmon.
Secondly, t h e between t r e a t m e n t v a r i a t i o n u n s o e c i f i e d by t h e o r d e r
r e s t r i c t i o n s ( w h i c h i s r e f e r r e d t o as SSres = SSbet t o t h e e r r o r term.
-
SSmon) c o n t r i b u t e s
The second d i f f e r e n c e has t h e a f f e c t o f i n c r e a s i n g t h e
) when t h e d i f f e r e n c e s between t h e j" when j ' < j " ) . Idhen means a r e i n t h e o r d e r i n g s p e c i f i e d i n H1(uj, < u j" power o f t h e t e s t a g a i n s t Ho(uj,=u
t h e r e a r e d i f f e r e n c e s between t h e means which a r e n e i t h e r s p e c i f i e d n o r p r e c l u d e d i n H1 a t e s t u s i n g MSw as t h e e r r o r t e r m may be more p o w e r f u l 2 than EJ. The e x t e n s i o n t o t h e b i v a r i a t e case i n n o m a l analyses o f v a r i a n c e i s t o p a r t i t i o n SSbet i n t o t h a t due t o v a r i a b l e A ( S S A ) , t h a t due t o v a r i a b l e B (SSB) and a r e s i d u a l c a l l e d t h e i n t e r a c t i o n ( S S A B ) .
The degrees o f f r e e -
dom a s s o c i a t e d w i t h each a r e known and s e p a r a t e F values a r e used t o t e s t
each o f t h e components o f t h e between v a r i a t i o n a g a i n s t t h e e r r o r term, see Winer (1971). The a p p l i c a t i o n o f i s o t o n i c r e g r e s s i o n t o t h e two way a n a l y s i s i s consider e d b y Barlow e t a1 (1972) f o l l o w i n g Shorack (1967). 2
They recommend u s i n g
EJ as d e f i n e d i n e q u a t i o n (8) r a t h e r than ( 7 ) .
T h i s i s t o t e s t Ho
uj,
= u
jl'
a g a i n s t HI
ujl
u
j"
f o r some s p e c i f i e d jl and
j " . The ( A ) a f t e r SSmn denotes t h a t t h e o r d e r r e s t r i c t i o n s i n v o l v e o n l y
l e v e l s o f A.
S i m i l a r t e s t s a r e developed i n v o l v i n g t h e B v a r i a b l e b u t no
t e s t s o f d i f f e r e n c e s between means i n t h e i n t e r a c t i o n a r e d i s c u s s e d i n t h i s context, 3. UNIVARIATE TESTS OF MONOTONICITY
L e t us f i r s t c o n s i d e r t h e u n i v a r i a t e case.
T h i s i s n o t e n t i r e l y degenerate
s i n c e i f m o n o t o n i c i t y were v i o l a t e d t h e d a t a c o u l d n o t be r e p r e s e n t e d by an a d d i t i v e model.
A demonstration o f nonmonotonicity orecludes the possibi-
lity o f a d d i t i v i t y
.
R . R . MacDonald
144
I n u n i v a r i a t e i s o t o n i c r e g r e s s i o n t h e i s o t o n i c r e g r e s s i o n e s t i m a t e s Tu. J . . J are found u s i n g t h e 0001 a d j a c e n t
v i t h r e s p e c t t o t h e o r d e r 1, 2,.
v i o l a t o r s a l g o r i t h m d e s c r i b e d i n B a r l o w e t a? (1972). M o n o t o n i c v a r i s t i o n 2 i n t h e means i s t h e n demonstrated u s i n g EJ as d e f i n e d i n e q u a t i o n ( 6 ) . I n t h e a p p l i c a t i o n o f c o n j o i n t measurement we a r e concerned i n d e m o n s t r a t i n g I t seems n a t u r a l t o t a k e SSres = z n(ir i-u.)' as 2 j- J an e s t i m a t e o f t h e p o D u l a t i o n v a r i a n c e u Nonmonotonic j v a r i a t i o n w i l l nonmonotonic v a r i a t i o n .
.
i n c r e a s e t h e v a l u e o f SSres and i f i t i s s i g n i f i c a n t l y g r e a t e r t h a n w o u l d be e x p e c t e d f r o m t h e w i t h i n c o n d i t i o n v a r i a n c e , t h e nonmonotonic v a r i a t i o n has been demonstrated.
I f Ho, uj,=uj,, 2 2
SSres i s d i s t r i b u t e d as no
f o r a l l j ' and j " , i s t r u e then
A p r o b l e m a r i s e s when t h e n u l l hypo-
t h e s i s t o be t e s t e d i s H i : e i t h e r u j , = uj,, o r u j , < uj,,,
f o r .i'< j " .
Suppose u < u i t i s s t i l l possible t h a t b., > and as such t h e J j'" j ' j'" and c o u l d c o n t r i b u t e t o SSres. The e x p e c t e d v a r i a t i o n between iI j' J v a l u e o f such v a r i a n c e w o u l d be l e s s t h a n t h a t i f u = u Thus SSres i s 2 2 j' j". d i s t r i b u t e d as l e s s t h a n n u x ( ~ - ~ ) T. h i s argument i s developed more f o r m a l l y by H a r t i g a n (1967) who e s t a b l i s h e s f o r SSres t h e u p p e r bound 2 2 nc x ( ~ - ~ ) .Thus an a p p r o p r i a t e t e s t s t a t i s t i c f o r cigClt be F I L as de::,I,
f i n e d i n e q u a t i o n ( 9 ) . though i t w i l l be c o n s e r v a t i v e t o an unknown e x t e n t f o r reasons g i v e n above.
F I L (J-L,N-JK)
=
SSres / (J-L)
(9)
MSw
Bartholomew (1961) c o n s i d e r e d u s i n g a F I L s t a t i s t i c as a t e s t f o r a d d i t i v i He r e j e c t e d t h i s approach as L i s a random v a r i a b l e w h i c h i n c r e a s e s as 2 t h e p r o b a b i l i t y o f Ho decreases and H1 i n c r e a s e s . EJ t h e recommended t e s t ,
ty.
incorporates L i n t o the t e s t s t a t i s t i c .
I n t h e monotonic case n o n - l i m i t i n g
values o f L w i l l be d e t e r m i n e d m a i n l y by t h e monotonic v a r i a t i o n .
The non-
monotonic t e s t i s concerned w i t h t e s t i n g t h e s i g n i f i c a n c e o f whatever nonmonotonic
v a r i a t i o n e x i s t s i n d e p e n d e n t l y o f t h e s i z e o f t h e monotonic
L
variation.
I t i s n o t t h e r e f o r e a p p r o p r i a t e t o use
statistic.
S i m i l a r l y i t i s a p p r o p r i a t e t o use SSw as t h e e r r o r t e r m as i t
as p a r t o f t h e t e s t
i s d e s i r e d t o show t h a t t h e nonmonotonic v a r i a t i o n i s g r e a t e r t h a n t h e e s t i mate o f t h e w i t h i n c o n d i t i o n v a r i a t i o n .
T h i s p o i n t has been d i s c u s s e d above.
The c o n s e r v a t i s m o f t h e p r o p c s e d t e s t ( e q u a t i o n ( 9 ) ) i s due t o t h e presence o f s m a l l amounts o f v a r i a t i o n between means w h i c h a r e t r u l y m o n o t o n i c .
It
245
Isototiic regression unulysis und additivity
w o u l d seem p r e f e r a b l e t o use a t e s t which gave g r e a t e r w e i g h t t o t h e l a r ger nonmnotonic v a r i a t i o n .
Such a t e s t i s t h e s t u d e n t i z e d range t e s t -see
b l i n e r (1971). To use t h i s t e s t we f i n d t h e r e s i d u a l e s t i m a t e s o f t h e t r e a t ment means ( r i i ~ . ) as d e f i n e d i n e q u a t i o n (10).
J
n'bi. =. 0j
-
Ti. J t G
(10)
The l a r g e s t and s m a l l e s t o f these a r e s u b s t i t u t e d i n t o t h e t e s t s t a t i s t i c q as i n e q u a t i o n (11). n-ularFest q =
"smallest MSwIn
(11)
The s t u d e n t i z e d range t e s t i s designed t o d e t e c t d i f f e r e n c e s b e b e e n k t r e a t m e n t means f r o m t h e d i f f e r e n c e between t h e l a r g e s t and s m a l l e s t mean. Since i n t h e p r e s e n t case t h e r e a r e J t r e a t m e n t s and
L different isotonic
e s t i m a t e s , t h e a p p r o p r i a t e v a l u e t o use f o r k when l o o k i n g up t h e t a b l e s i s J-Ltl. F o r reasons analogous t o those a p p l y i n g t o FIL t h e range t e s t w i l l a l s o be c o n s e r v a t i v e b u t s i n c e i t g i v e s most w e i g h t i n g t o t h e l a r g e s t d i f f e r e n c e i t s h o u l d be m r e p o w e r f u l t h a n F ( L .
The r e l a t i v e powers o f t h e two t e s t s and t h e e x t e n t o f t h e i r c o n s e r v a t i s m cannot he a s c e r t a i n e d
i n general, s i n c e i t w i l l depend on t h e amount o f t r u e monotonic v a r i a t i o n p r e s e n t i n t h e p o p u l a t i o n means. I n d e e d t h e c o n s e r v a t i s m o f these t e s t s w i l l depend on t h e t r u e values o f t h e p o p u l a t i o n means and t h e p r o b a b i l i t y t h a t p o p u l a t i o n means i n t h e p r e d i c t e d o r d e r w i l l g i v e r i s e t o sample means o u t o f o r d e r .
I f t h e values o f
t h e i s o t o n i c e s t i m a t e s o f t h e p o p u l a t i o n means a r e t r e a t e d as p o p u l a t i o n values, t h e e x t e n t o f t h e c o n s e r v a t i s m c o u l d be e s t i m a t e d by s i m u l a t i o n . From r e p e a t e d s i m u l a t i o n s t h e e x p e c t e d values o f SSres and L c o u l d be found where t h e r e i s no nonmonotonic v a r i a t i o n . equation (13) t o a l l o w f o r t h i s .
C o r r e c t i o n s c o u l d be made t o
However, t h e r e q u i r e m e n t t o r u n a simu-
l a t i o n on each t e s t r u l e s t h i s o u t as a r o u t i n e procedure. 4. BIVARIATE TESTS OF MONOTONICITY T h i s s e c t i o n a t t e m p t s t o g e n e r a l i s e these r e s u l t s t o b i v a r i a t e m o n o t o n i c i t y . The s i m p l e s t approach i s t o r e l y on o n l y t h e o v e r a l l means f o r each of t h e l e v e l s o f A and 5. T h i s i s e s s e n t i a l l y t h e aoproach suggested by Shorack
(1967) mentioned above, who t r e a t e d t h e two way a n a l y s i s as i f i t were two
246
R . R . AfacUuiiald
indenendent one way analyses. found w j t h r e s p e c t t o 1, 2,
2
The i s o t o n i c r e g r e s s i o n e s t i m a t e s i-u. a r e
..., J
J
and monotonic v a r i a t i o n w i t h r e s p e c t
t o A t e s t e d u s i n g EJ as d e f i n e d i n e q u a t i o n ( 8 ) .
To t e s t t h e presence o f
nonmonotonic v a r i a t i o n w i t h r e s p e c t t o A one c o u l d t h e r e f o r e use fined i n equation ( 9 ) . t o variable B.
FIL as de-
The t e s t i n g procedure can be r e n e a t e d w i t h r e s p e c t
The two t e s t s a r e m u t u a l l y independent when t h e number o f
o b s e r v a t i o n s i n each o f t h e c o n d i t i o n s i s e q u a l .
Barlow e t a1 (1972) d i s -
cuss t h e case where t h e numbers o f o b s e r v a t i o n s a r e unequal. James (1961). comnenting on Partholomew's work, suggested ways t h a t i s o t o n i c r e g r e s s i o n m i g h t be extended t o i n c l u d e i n t e r a c t i o n s .
Rartholomew
(1961) r e p l i e d t h a t he had c o n s i d e r e d t h e p r o b l e m b u t f o u n d i t f o r b i d d i n g . H i r o t s u (1978) f o l l o w e d
UD
these i d e a s by c o n s i d e r i n g v a r i o u s comDarisons
o f t r e a t m e n t means which c o n t r i b u t e d t o t h e i n t e r a c t i o n , e . p .
(l.,')
I f a l l such comparisons a r e n o s i t i v e ( o r n e g a t i v e ) t h e d a t a a r e monotonic. This i s a stronger r e s t r i c t i o n r e q u i r i n g n o t o n l y t h a t the data monotonicall y i n c r e a s e s w i t h t h e indenendent v a r i a b l e s b u t t h a t t h e i n c r e a s e w i t h each
successive l e v e l o f an independent v a r i a b l e i s i t s e l f m o n o t o n i c a l l y i n c r e a s i n g ( o r decreasina).
Another way o f p u t t i n g i t i s t h a t t h e change due t o
+he independent v a r i a b l e s i s a c c e l e r a t i n g e i t h e r n o s i t i v e l y o r n e g a t i v e l y . Consequently any t e s t which showed a l l such comparisons t o be p o s i t i v e o r n e g a t i v e would i n d e e d e s t a b l i s h m o n o t o n i c i ty ( t h o u g h n o t double c a n c e l l a t i o n ) , b u t a n e g a t i v e r e s u l t would l e a v e open t h e p o s s i b i l i t y o f monotonicity.
Unless t h e r e i s reason t o e x p e c t an a c c e l e r a t i n g e f f e c t o f t h e i n -
dependent v a r i a b l e s , t h i s approach i s p r o b a b l y n o t r e l e v a n t . Shorack's procedures o u t l i n e d above a r e r e l a t i v e l y s t r a i g h f o t w a r d b u t t h e y ignore p o t e n t i a l l y r e l e v a n t v a r i a t i o n c o n t r i b u t i n g t o the i n t e r a c t i o n .
In
o t h e r words, u n l e s s t h e n o n m n o t o n i c v a r i a t i o n i n one v a r i a b l e i s p r e s e n t averaged o v e r a l l l e v e l s o f t h e o t h e r , t h i s anproach w i l l f a i l t o d e t e c t the nonmonotonicity.
I s o t o n i c r e g r e s s i o n e s t i m a t e s i u can be found w i t h
respect t o the p a r t i a l o r d e r j , k
< j',k'
i f j < j ' and k < k ' .
Several
algorithms t o f i n d i s o t o n i c regression estimates w i t h respect t o p a r t i a l o r d e r s a r e r e p o r t e d i n B a r l o w e t a1 (1972).
The p o o l a d j a c e n t v i o l a t o r s
a l g o r i t h m which worked w i t h simpl'e o r d e r s f a i l s i n t h e more g e n e r a l case, As i s o t o n i c r e g r e s s i o n e s t i m a t e s can be found t h e SSbet can be s p l i t i n t o
247
Isotonic regression malysis and additiuity
t h e sums o f squares between t h e i s o t o n i c r e g r e s s i o n e s t i m a t e s SSmon and SSres, a r e s i d u a l due t o nonmonotonic v a r i a t i o n .
Again u s i n g t h e upper
bound f o r SSres ( H a r t i g a n ( 1 9 6 7 ) ) an o v e r a l l t e s t based on F I L i s p o s s i b l e This i s presented i n equation (13). FIL(JK-L,N-JK)
=
S i m i l a r l y t h e m u l t i p l e range t e s t can be a p p l i e d t o t h e b i v a r i a t e case. Again t h e r e s i d u a l ticnn!onotcni c e s t i v a t e s cC ttse t r e a t m e n t means can he found and t h e l a r g e s t and s m a l l e s t s u b s t i t u t e d i n t o e q u a t i o n ( 1 1 ) . v a l u e f o r k i n t h i s case i s JK-L+1.
The
5 . EXAMP1.F The t e s t s o u t l i n e d above w i l l be i l l u s t r a t e d u s i n g a r t i f i c i a l d a t a produced by Cunningham (1982) and g i v e n i n t a b l e 1 below.
I t w i l l be assumed t h a t
t h e raw d a t a a r e t h e means o f t e n independent o b s e r v a t i o n s ( n = 10). Isotonic
Data
Estimates B 3 7
6
9
6.5
6.5
9.0
B 2 3
1
8
2.57
2.=7
Q.?
BIZ
4
5
2.0
2.67
5.0
A1 A2
A1
A3
A2
A3
Table 1 F i c t i t i o u s example f r o m Cunningham (1982) A p p l y i n g S h o r a c k ' s approach t h e i s o t o n i c r e g r e s s i o n e s t i m a t e s w i t h r e s p e c t 3.83,
N o n a d d i t i v i t y i n A w o u l d be t e s t e d by F(1.81) =
0.66/MSw.
7.33.
T h i s r e s u l t s i n an SSmon (A) o f 140
t o A ( t h e i u ' s ) a r e 3.83, and a SSres (A) o f 0.66.
I t w o u l d r e q u i r e tlsw t o be l e s s t h a n .166 t o i n d i c a t e s i g n i f i c a n t
n o n a d d i t i v i t y a t t h e 0.05 l e v e l ,
T h i s approach w o u l d produce no evidence
o f n o n a d d i t i v i t y i n B, since a l l t h e v a r i a t i o n i s monotonic and SSres (B) e q u a l s zero. Usin? t t e i s o t o n i c r e g r e s s i o n e s t i m a t e s w i t h r e s p e c t t o t h e p a r t i a l o r d e r for j ' , j " and k ' , k " g i v e n i n t a b l e 1, SSmon and SSres j ",k"
UjlkI <
248
R.R. MucDonald
come o u t as 549 and 51. /MSw.
N o n a d d i t i v i t y c c u l d be t e s t e d b y F(3,81)
= (51/3)
A MSw o f l e s s t h a n 6.83 w o u l d be r e q u i r e d f o r t h e r e s u l t t o be s i p I f t h e s t u d e n t i z e d range t e s t were a p p l i e d
n i f i c a n t a t t h e 0.05 l e v e l .
c r l e s s , w o u l d be r e q u i r e d f o r s i g n i f i c a n c e
o = 9.48/MSw and a Ffsw of 7.€8
a t t h e 0.05 l e v e l . 6 . DOUBLE CANCELLATION A X I O M
! l o n o t o n i c d a t a a r e n o t n e c e s s a r i l y a d d i t i v e as t h e example due t o K r u s k a l i n S c h e f f e (1959) shows.
Here m a t r i x A g i v e s t h e r a n k o r d e r o f t h e d a t a
from smallest t o l a r g e s t .
A i s m n o t o n i c and R and C a r e m a t r i c e s formed
by a s u b s e t o f t h e elements i n A.
B
A
C
[: :] I f t h e d a t a a r e assumed t o be a d d i t i v e , f r o m
from C l s t + 8 t h = 4th+2nd.
m a t r i x B l s t + 7 t h = 3 r d + d t h and
The l e f t hand s i d e o f t h e f i r s t e q u a t i o n i s a l -
ways l e s s t h a n t h e l e f t hand s i d e o f t h e second under any monotonic t r a n s formation o f the data.
The r e v e r s e i s t r u e o f t h e r i g h t hand s i d e .
l y t h e a d d i t i v i t y assumption i s u n t e n a b l e .
Clear-
Data must be shown n o t t o de-
D a r t from m o n o t o n i c i t y and n o t t o v i o l a t e double c a n c e l l a t i o n b e f o r e we can conclude t h e y a r e a d d i t i v e . Having c o n s i d e r e d t e s t s o f m o n o t o n i c i t y we now l o o k a t ways o f t e s t i n g double c a n c e l l a t i o n assuming t h a t t h e d a t a a r e monotonic.
I n i t s most ge-
n e r a l form we must t e s t a l l t h e 3 x 3 m a t r i c e s f o r i n e q u a l i t i e s (1) t o ( 3 ) . I f we assume m n o t o n i c i t y many fewer m a t r i c e s need t o be examined.
3 x 3 m a t r i c e s where a ' < a " < a"'
and b ' < b " < b"'
Only i n
i s m o n o t o n i c i t y compa-
t i b l e w i t h i n e q u a l i t i e s ( 1 ) and ( 2 ) b e i n q t r u e and ( 3 ) b e i n q f a l s e o r v i c e vprsa.
S t i l l f u r t h e r r e d u c t i o n s on t h e number o f 3 x 3 m a t r i c e s which need
t o be examined a r e n o s s i b l e .
An a l a o r i t h m f o r f i n d i n n a l l t h e v i o l a t i o n s o f
double c a n c e l l a t i o n i n a s e t o f d a t a which a r e c o m p a t i b l e w i t h m o n o t o n i c i t y i s g i v e n i n t h e appendix. Procedures f o r f i n d i n g i s o t o n i c r e g r e s s i o n e s t i m a t e s r e q u i r e a p r i o r i o r d e r i n g o f t h e c o n d i t i o n s even i f i t i s o n l y p a r t i a l .
The double c a n c e l l a -
t i o n axiom imposes a r e s t r i c t i o n c o n d i t i o n a l on o t h e r v a l u e s o f t h e d a t a .
249
Isotonic regression amlysis and additivity
As t h i s i s n o t an a p r i o r i o r d e r i n g t h e e s t i m a t i o n a l g o r i t h m s g i v e n i n
Barlow q t a1 (1972) would have t o be extended i n o r d e r f o r t h e i s o t o n i c r e g r e s s i o n e s t i m a t e s t o be found,
Ift h e a l g o r i t h m s were so extended t h e
r e s u l t i n g procedure f o r t e s t i n g f o r n o n - i s o t o n i c v a r i a t i o n would be t h e t e s t s g i v e n i n e q u a t i o n s ( 9 ) and (11).
I n t h e absence o f such e x t e n s i o n s one
must l o o k f o r s t a t i s t i c a l t e s t s o f double c a n c e l l a t i o n u s i n g a more conv e n t i o n a l approach. Having t e s t e d m o n o t o n i c i t y u s i n g one of t h e techniques g i v e n above and f a i l e d t o f i n d a s i g n i f i c a n t v i o l a t i o n , i t i s now assumed t h a t t h e d a t a a r e monotonic.
The i s o t o n i c r e g r e s s i o n e s t i m a t e s w i t h r e s p e c t t o b i v a r i a t e
m o n o t o n i c i t y a r e f e d i n t o t h e a l g o r i t h m i n t h e appendix.
This gives a l l
t h e 3 x 3 submatrices f r o m t h e complete d a t a m a t r i x which v i o l a t e t h e double c a n c e l l a t i o n axiom.
F o r t h e v i o l a t i o n t o be s t a t i s t i c a l l y s i g n i f i -
c a n t each c f t h e t h r e e i n e q u a l i t i e s c o n t r i b u t i n g t o t h e v i o l a t i o n s h o u l d be s i g n i f i c a n t i n i t s own r i g h t .
Various methods e x i s t f o r e s t a b l i s h i n g
t h e c o n f i d e n c e i n t e r v a l s f o r t h e d i f f e r e n c e s between means a f t e r an a n a l y s i s o f v a r i a n c e (see Winer ( 1 9 7 1 ) ) .
Statistically significant violations
o f t h e double c a n c e l l a t i o n axiom would have been demonstrated i f t h e confidence i n t e r v a l s o f t h e d i f f e r e n c e i n a l l three i n e q u a l i t i e s c o n t r i b u t i n g t o any v i o l a t i o n d i d n o t i n c l u d e z e r o , Indeed, t h e a l g o r i t h m f o r f i n d i n g v i o l a t i o n s i n double c a n c e l l a t i o n c o u l d be r e w r i t t e n f o r l o o k i n g f o r s t a t i s t i c a l l y s i g n i f i c a n t v i o l a t i o n . w o u l d r e q u i r e t h a t t h e i n e q u a l i t y s i g n s < a n d > be r e p l a c e d by <
A l l this
s and
> s
i n d i c a t i n g s t a t i s t i c a l l y s i g n i f i c a n t d i f f e r e n c e s and t h e i n e q u a l i t i e s b e i n g t r e a t e d as equal when t h e d i f f e r e n c e s were n o t s i g n i f i c a n t . 7. CONCLUSIONS The o v e r a l l t e s t i n g procedure f o r a d d i t i v i t y which seems t o make most use o f t h e data i s a s e q u e n t i a l one.
The e x p e r i m e n t s h o u l d b e designed so t h a t
t h e e f f e c t s o f s u c c e s s i v e l e v e l s o f t h e independent v a r i a b l e s a r e a p p r o x i mately equal.
A t e s t f o r monotonicity using e i t h e r FIL (equation (13)) o r
t h e s t u d e n t i z e d range t e s t ( e q u a t i o n (11)) s h o u l d n e x t be performed.
If
t h i s r e s u l t s i n a s i g n i f i c a n t e f f e c t m o n o t o n i c i t y and hence a d d i t i v i t y have been v i o l a t e d .
I f no s i g n i f i c a n t e f f e c t s a r e found,
t h e double c a n c e l l a -
t i o n axiom i s t e s t e d assuming m o n o t o n i c i t y as o u t l i n e d above.
Ifn e i t h e r
o f t h e above t e s t s produces s i g n i f i c a n t e f f e c t s , t h e d a t a have n o t been shown t o s i g n i f i c a n t l y d e n a r t f r o r a d d i t i v i t y .
3 0
?F FF PF Ft CF
?r\uc'.!e,
:.,
ar?8
!.ariwr, ?.,
l'-e
rl.i+?r
o c t v o wav t a h l e s s a t i s f y i n g
c e r t a i n addi t i v i t'/ axioms, J o u r n a l o f r!athematical Psycholooy, 13 (?.-:-,),
P9-100. Rartholomew, D.J., Bremner, J.M. and Brunk, H.D.,
Barlow, R.E.,
Sta-
t i s t i c a l I n f e r e n c e under Order R e s t r i c t i o n s , W i l e y , New York ( 1 9 7 2 ) . Partholomew, D.J., A t e s t o f homogeneity o f means under r e s t r i c t e d a l t e r n a t i v e s , J o u r n a l o f t h e Royal S t a t i s t i c a l S o c i e t y , S e r i e s B, 23
( 1 9 6 1 ) , 239-281. Cunningham, J . P . ,
M u l t i o l e monotone r e g r e s s i o n , Psych. B u l l . , 92 (1982)
7917800. H a r t i g a n , J.A.,
D i s t r i b u t i o n o f t h e r e s i d u a l sums o f squares i n f i t t i n g
i n e q u a l i t i e s , R i o m e t r i k a , 54 ( 1 9 6 7 ) , 69-84. Hirotsu, C.,
Ordered a l t e r n a t i v e s f o r i n t e r a c t i o n e f f e c t s , B i o m e t r i k a ,
65 ( 1 9 7 8 ) , 561-570. C o n t r i b u t i o n t o d i s c u s s i o n o f t h e naper by D.J. B a r t h o l o -
James, G.S.,
mew, J o u r n a l o f t h e Royal S t a t i s t i c a l S o c i e t y , S e r i e s B, 23 ( 1 9 6 1 ) ,
278-279. Krantz, D . H . ,
Luce,
R.D., SuDpes, P. and Tversky, ',., Fcundations o f
wasurewiit. ka+tvic
P r c c s , I'el-1 Yor!< (1971.).
Luce, R.D. and Tukey, J.I.I., Simultaneous c o n j o i n t measurement: a new t y p e o f fundamental measurement, J o u r n a l o f Math. P s y c h o l . ,1 ( 1 9 6 4 ) ,
1-27. Mandel, J . , A new a n a l y s i s o f v a r i a n c e model f o r n o n - a d d i t i v e d a t a , Technometrics,
13 ( 1 9 7 1 ) , 1-18.
McClelland, G., A Note on A r b u c k l e and L a r i w r , " T h e number o f two-way t a b l e s s a t i s f y i n g c e r t a i n a d d i t i v i t y axiom;:
J o u r n a l o f Math. Psychol .,
1 5 , ( 1 9 7 7 ) , 292-295. The A n a l y s i s o f Variance, W i l e y , New York ( 1 9 5 9 ) .
Scheffe, H., Shorack, G . R . ,
T e s t i n g a g a i n s t o r d e r e d a l t e r n a t i v e s i n model 1 a n a l y -
s i s o f v a r i a n c e : normal t h e o r y and n o n p a r a m e t r i c , Ann. Math. S t a t i s t . ,
38 ( 1 9 6 7 ) , 1740-1753. Smith, P.T.
and Macdonald, R.R.,
Methods f o r i n c o r p o r a t i n g o r d i n a l i n -
formation i n t o analysis o f variance: g e n e r a l i z a t i o n s o f o n e - t a i l tests, B r i t i s h J o u r n a l o f Mathematical and S t a t i s t i c a l PsycholoSy, 36 ( 1 ? 8 3 ) ,
1-21. Tukey, J.W..
One degree o f freedom f o r n o n a d d i t i v i t y , B i o m e t r i c s , 5
( 1949) , 232-242.
Isotonic
251
regression analysis and additivity
[I51 V i n e r y R.J., S t a t i s t i c a l P r i n c i p l e s i n Experimental Design, PlcGrawH i l l (1971). APPEND1 X A l q o r i t h m f o r f i n d i n g v i o l a t i o n s i n double c a n c e l l a t i o n c o n s i s t e n t w i t h monotoni c i t y
In what f o l l o w s i n e q u a l i t i e s a r e taken t o be t r u e ifand only i f t h e y h o l d I n o t h e r words i f b o t h s i d e s o f an i n e q u a l i t y a r e
i n t h e s t r i c t sense.
e v a l u a t e d and found t o be equal t h e i n e q u a l i t y w i l l be d e s c r i b e d as b o t h " n o t t r u e " and " n o t f a l s e " . L e t A be t h e complete J x K m a t r i x
c o n t a i n i n g a l l t h e d a t a and l e t A j k be
t h e s u b m a t r i x formed f r o m A w i t h columns j , j t l and j + 2 and rcws k, k - 1 L e t us s t a r t by c o n s i d e r i n g t h e s u b m a t r i x A j k and t e s t i n e q u a l i -
and k-2.
t i e s (I\,(?) and ( 3 ) i n t h i s m a t r i x .
(I:) a r e t r u e or n o t ( l ) , n o t
I f and o n l y i f e i t h e r !1),(2)
( 2 ) and ( 3 ) a r e t r u e i s douhle c a n c e l l a t i o n v i o -
''e n e x t c o n s i d e r a l l t h e ' x 3 s u h r a t r i c e s f r o m
lated.
and ?, and rows Y,
pl
and n o t
fib w i t h
and n (k-rw). Tte t r u t h o f ( 1 ) , ( 2 )
c o l u m s l , 2,
and ( 3 )
;7
has
i m p l i c a t i o n s f o r t h e i r t r u t h i n t h e submatrices we a r e c o n s i d e r i n g as spec i f i e d i n (14) t o ( 1 6 ) below: '2,K
'1,K-1 '1,K
> '3,K-1
=. d
> '2,K-2
> '3,K-2
'2,K
=9
' '3,m
'1,K-1 u
l,K
> '2,n
> u
3,n
m
<
K-1
n <. K-2
(15
n <. K-2
W i t h these r e s t r i c t i o n s i n mind we s h a l l c o n s i d e r a l l p o s s i b l e s t a t e s f o r i n e q u a l i t i e s ( l ) a n d ( 3 ) i n A. State
1 : ( l ) a n d ( 3) t r u e .
I n e q u a l i t i e s ( l q a n d (16) guarantee t h a t i n e q u a l i t i e s (1) and (3) w i l l b e t r u e f o r a l l t h e submatrices under c o n s i d e r a t i o n .
Thus t h e double c a n c e l -
l a t i o n axiom i s n e v e r v i o l a t e d . State
2 :(1) f a l s e and ( 3 ) t r u e .
From (16) i n e q u a l i t y (3) w i l l be t r u e i n a l l t h e submatrices under d i s c u s sion.
Because o f m o n o t o n i c i t y and i n e q u a l i t y ( 3 ) we have:
252
R . R . MdcDorurld
'2,K
> '1,K
> '3.K-2
This means t h a t i n e o u a l i t y ( 1 ) w i l l be t r u e f o r m = K-2 and hence I I ~ C R U o fS ~ p o n o t o n i c i t y f o r a l l v a l u e s o f m n o t equal t o K - 1 .
V i o l a t i o n s o f double
c a n c e l l a t i o n can t h e r e f o r e o n l y o c c u r when i n e q u a l i t y ( 2 ) i s f a l s e a t
m = K-I.
IJe t h e r e f o r e t e s t i n e q u a l i t y ( 2 ) a t
1;1
=
K-1 f o r successively
d e c r e a s i n g values o f n s t a r t i n g a t n = K-2 u n t i l i n e o u a l i t y ( 2 ) i s found t o be n o t f a l s e .
M o n o t o n i c i t y quarantees t h a t i n e q u a l i t y w i l l c o n t i n u e n o t t o
he f a l s e f o r a l l s m a l l e r values o f n.
Where i n e q u a l i t y ( 2 ) has been f o u n d
t o be f a l s e , v i o l a t i o n s o f double c a n c e l l a t i o n have o c c u r r e d . State
3:
(1) t r u e and ( 3 ) f a l s e .
By ( 1 4 ) i n e q u a l i t y ( 1 ) w i l l be t r u e f o r a l l t h e s u b m a t r i c e s and v i o l a t i o n s
w i l l o n l y o c c u r where i n e q u a l i t y ( 3 ) i s f a l s e and ( 2 ) i s t r u e .
The values
o f n f o r w h i c h i n e q u a l i t y ( 3 ) i s f a l s e a r e found by t e s t i n g t h i s i n e q u a l i t y
for s u c c e s s i v e l y d e c r e a s i n g v a l u e s o f n s t a r t i n g a t K-2 u n t i l i t i s n o l o n I n e q u a l i t y (2) s h o u l d now be t e s t e d f o r each v a l u e o f n f o r
oer false.
which i n e q u a l i t y ( 3 ) i s f a l s e .
T h i s i s done f o r a l l p o s s i b l e values o f m
d e c r e a s i n g from K - 1 u n t i l i n e q u a l i t y ( 2 ) i s no l o n g e r t r u e .
Each t i m e i n -
e q u a l i t y (2) i s found t o be t r u e , a v i o l a t i o n o f double c a n c e l l a t i o n has occurred. State
4:
b o t h (1) and ( 3 ) f a l s e .
To b e g i n w i t h , i n e q u a l i t y
(1) i s t e s t e d f o r d e c r e a s i n g values o f m u n t i l a t
m = c ' i t i s f o u n d t o be n o t f a l s e and a t m = c " i t i s f o u n d t o be t r u e :
(c',~").
A t t h i s p o i n t we know we know t h a t i n e q u a l i t y ( 3 ) must be f a l s e
a t n = c ' + l (18) and s i n c e (1) i s a l s o f a l s e double c a n c e l l a t i o n w i l l h o l d f o r n u r e a t e r than c '
.
We n e x t t e s t i n e q u a l i t y ( 3 ) a t n = c ' .
Ift h i s i s t r u e i n e q u a l i t y (1) i s
f a l s e f o r a l l values o f m from K - 1 t o c ' + l and t h e procedures adopted i n state
2
f o r m = K - 1 s h o u l d be r e a p p l i e d h e r e f o r a l l v a l u e s o f m f r o m
K-1 t o c ' + l .
No v i o l a t i o n o f t h e t y p e 1: 2 and n o t 3 i s o o s s i b l e s i n c e 1
and 3 a r e f a l s e f o r a l l values o f n g r e a t e r t h a n c ' by m o n o t o n i c i t y and 3 i s t r u e f o r a l l values o f n g r e a t e r t h a n c ' .
I f c ' does n o t equal c " i n e q u a l i t y (1) w i l l be equal f o r values o f n f r o m
253
Isotonic regression analysis and additivity
Double c a n c e l l a t i o n w i l l n e v e r be v i o l a t e c ' i n these cases.
c"t1 to c'.
I f however, i n e q u a l i t y ( 3 ) was found t o be f a l s e a t n = c ' no v i o l a t i o n
Occurs as i n e q u a l i t y (1) i s a l s o f a l s e f o r a l l values o f m g r e a t e r t h a n c ' .
A v i o l a t i o n i s P o s s i b l e a t m = c " (where i n e q u a l i t y (1) i s t r u e ) i f i n e q u a l i t y ( 3 ) i s f a l s e f o r n = c"-1. ? should
!3e
Here t h e Procedures adonted i n s t a t e
amlied.
There s t i l l remains t h e p o s s i b i l i t y of a v i o l a t i o n o f double c a n c e l l a t i o n i f i n e q u a l i t y (1) i s f a l s e f o r a l l values o f m.
Here by r e a s o n i n g s i m i l a r -
l y t o ( 1 7 ) i n e q u a l i t y ( 3 ) w i l l be f a l s e f o r n g r e a t e r than 1.
I f however,
i n e q u a l i t y ( 3 ) were found t o be t r u e a t n = 1, i n e q u a l i t y ( 2 ) s h o u l d be checked f r o m m = 2 u n t i l K - 1 o r i n e q u a l i t y ( 2 ) i s found t o be t r u e .
Double
c a n c e l l a t i o n w i l l have been v i o l a t e d f o r e v e r y f a l s i f i c a t i o n of i n e q u a l i t y
(2)
*
S t a t e 5 : one o r more o f i n e q u a l i t i e s (1) and ( 3 ) a r e e y a l . I f e i t h e r i n e q u a l i t y ( 1 ) o r ( 3 ) i s t r u e o r b o t h a r e equal t h e s i t u a t i o n i s like state
I n e q u a l i t i e s (14) and (16) a r e t r u e i n t h e weak f o r m (where
1.
Thus
t h e g r e a t e r t h a n s i a n s a r e reDlaced by g r e a t e r t h a n o r equal s i g n s ) .
i n t h i s s i t u a t i o n double c a n c e l l a t i o n w i l l n e v e r be v i o l a t e d as n e i t h e r i n e q u a l i t y (1) n o r (3) can be f a l s e . I f i n e q u a l i t y (1) i s f a l s e and i n e q u a l i t y ( 3 ) i s equal t h e n o s i t i o n i s s i milar t o state
2
.
The weak v e r s i o n o f i n e q u a l i t y ( 1 7 ) guarantees inequa-
l i t y ( 1 ) i s n o t f a l s e f o r m = K-2. V i o l a t i o n s o f double c a n c e l l a t i o n can t h e r e f o r e o n l y o c c u r when m = K-1, i n e q u a l i t y ( 3 ) i s t r u e and i n e q u a l i t y
( 2 ) f a l s e f o r some v a l u e o f n. I n e q u a l i t y ( 3 ) s h o u l d be t e s t e d f o r m = K-1 f o r s u c c e s s i v e l y d e c r e a s i n g values o f n u n t i l i n e q u a l i t y (3) i s f o u n d t o be true a t n
I
c ' a f t e r w h i c h i t w i l l be t r u e f o r a l l s m a l l e r values.
Inequa-
l i t y ( 2 ) s h o u l d then be e v a l u a t e d a t m = K - 1 f o r values o f n d e c r e a s i n a from n = c ' u n t i l i n e q u a l i t y ( 2 ) i s no l o n g e r f a l s e .
Each f a l s i f i c a t i o n
o f i n e q u a l i t y ( 2 ) corresponds t o a v i o l a t i o n o f t h e double c a n c e l l a t i o n axiom. The f i n a l p o s s i b i l i t y i s where i n e q u a l i t y (1) i s e o u a l and ( 3 ) i s f a l s e . Here t h e procedure i s as i n s t a t e
4 where c ' and c " a r e d i f f e r e n t .
Ine-
q u a l i t y (1) i s t e s t e d f o r s u c c e s s i v e l y d e c r e a s i n g values o f m u n t i l a t rn = c " i t i s t r u e .
I n e q u a l i t y ( 3 ) i s e v a l u a t e d a t n = c"-1 and i f f a l s e
t h e Drocedure f o r s t a t e
3 i s adonted.
224
R.R. MdcDonald
Ile have now checked f o r double c a n c e l l a t i o n i n a l l t h e 3 x 3 r r a t r i c e s w i t h columns 1, 2 and 3 and rows K, F, and n. p and rows K , K-1 and
lues o f o
c
o < J.
Each m a t r i x w i t h columns 1, o and
K-2 can be t r e a t e d i n t h e same way as A, f o r a l l va-
T h i s w i l l e n a b l e t h e t e s t i n a o f double c a n c e l l a t i o n i n
a l l m a t r i c e s w i t h rows 1, o and p and columns K,
pl
and n.
T h i s comoletes
uIK.
I n o r d e r t o c o n s i d e r e v e r y poss i b l e v i o l a t i o n t h e nrocedures s h o u l d be r e o e a t e d f o r a l l s u b m a t r i c e s A j k , t e s t i n q a l l 3 x 3 matrices i n c l u d i n n
j
,I-1 and k
2.
>
ACKN@'*lLF:DGEIIE NTS The a u L ' @ rwishes t o exnress h i s thanks t o D r . P.T. the m r u c c r i n t .
rmit':
f o r comments on
TREh’DS I N MATHEM.-ITICAL PSYCHOLOCY E . Degreef and J. Van Bu gerthaut (editors) 0 Iflreuier Science Publisfers B. V. (North-Holland), 1984
255
TESTING FECHNERIAN SCALARILITY BY f4AAXIHUf1 LIKELIHOOD ESTIMATION flF ORDERED RINOflIAL PARAFETTERS Rai ne r %us f e l d I J n i v e r s i ty o f Bonn
A q e n e r a l i z a t i o n o f t h e customary m d e l s f o r n a i r comoarison d a t a i s o r e s e n t e d i n a Darameter f r e e way.
F o r t h i s g e n e r a l model, c a l l e d Fechnerian
s t r u c t u r e , a maximum l i k e l i h o o d e s t i m a t i o n n r o c e dure i s nronosed based on a two nhase i s o t o n i c r e gression t e c h n i q w
.
1. INTRODUCTION There i s a s p e c i f i c c l a s s o f p s y c h o l o g i c a l experiments h a v i n g t h e p r o p e r t y t h a t t h e generated d a t a can be r e p r e s e n t e d i n a n a t u r a l way as r e l a t i v e f r e q u e n c i e s sampled f r o m a s u i t a b l y chosen D r o b a b i l i ty d i s t r i b u t i o n .
Dis-
c r i m i n a t i o n experiments i n nsychonhysics and s i a n a l d e t e c t a b i l i ty t h e o r y , c h o i c e s between gambles s e r v e as exanples.
, nreference
s t u d i e s and l e a r n i n g e x p e r i m e r t s nay
I n these cases t h e e m p i r i c a l r e l a t i o n a l system i s a l -
ready qiven numerically.
I n i t s most s i m p l e f o r m ( b i n a r y resDonse d a t a )
t h i s system c o n s i s t s o f a s e t o f s t i m u l i (ai)i
and a b i n a r y dominance r e l a -
t i o n which by independent r e p l i c a t i o n s leads d i r e c t l y t o a n u m e r i c a l f u n c t i o n o t h a t s a t i s f i e s : p(ai,aj)
+ P(aj,ai)
= 1 f o r a l l i , j i n the correspow
d i n g index s e t . I f w i t h r e s p e c t t o t h i s dominance r e l a t i o n each s t i m u l u s i s assumed t o be r e p r e s e n t a b l e by a r e a l number, and i f t h e dominance p r o b a b i l i t i e s a r e a monotone f u n c t i o n o f t h e d i f f e r e n c e o f t h e r e s p e c t i v e s t i m u l u s narameters, t h e model d e s c r i b e d i n t h e f o l l o w i n a may be a D p r o p r i a t e (
D i s~ s h~o r t h a n d
f o r p(ai , a j ) ) .
2. FECHNERIAN STRUCTURES DEFINITION L e t A,R be nonerrpty f i n i t e s e t s and D be a r e a l v a l u e d f u n c t i o n p:AxB+(O,l). The s t r u c t u r e ( AxR.d
i s a Fechnerian s t r u c t u r e , i f t h e r e p x i s t s a d i s t r i -
R. Mausfeld
256
b u t i o n f u n c t i o n 0, which i s strong1.v monotone, whenever i t i s d i f f e r e n t f r o m 0 and 1, and c o i i t i n u o u s , such t h a t D(0) = 1 / 2 and f o r a l l A and b k , ,h. J
",ai
F
D the f o l l o w i n o c o n d i t i o n holds:
,'^K?
i )
The d e f i n i t i o n above i s i n d e n e n d e n t o f t h e n u m e r a t i o n o f elements o f
A or 9 . i i ) !' e c h e r i a n s t r u c t u r e i s d e f i n e d h e r e hy a system of c o n s t r a i n t s on t h e resnonse p r o b a h i l i t i e s w i t h o u t any e x o l i c i t r e f e r e n c e t o u n d e r l y i n a scales.
These can be i n t r o d u c e d h v
f o r any k , k '
iii From t h e i n j e c t i v i t y o f D on ( 0 , l ) i t f o l l o w s t h a t f o r a f i x e d rea
[1
the
v a l u e d f u n c t i o n s u and v a r e unique e x c e n t f o r a common a d d i t i v e con-
s t a n t , i. e . t h e y a r e d i f f e r e n c e s c a l e s . Thus,a s t r u c t u r e ( A x B,n) i s F e c h n e r i a n - s c a l a b l e i f t h e dominance o r o b a b i l i t i e s a r e a monotone f u n c t i o n o f t h e s c a l e v a l u e d i f f e r e n c e .
For a d e t a i -
l e d a n a l v s i s o f t h i s model, see Luce-Suopes (1965) and, i n a somewhat d i f f e r e n t c o n t e x t , Falmaqne (1971).
Fechnerian s c a l a b i l i t y i s obviously a
s n e c i a l case o f s i m l e s c a l a b i l i t v , a conceot f i r s t i n t r o d u c e d by K r a n t z (1964) and f u r t h e r i n v e s t i a a t e d by Tversky-Russo (1969) and T v e r s k y ( 1 9 7 2 ) . The o b s e r v a b l e o r o p e r t i e s o f s t o c h a s t i c c o n s i s t e r c v and o t h e r necessary c o n d i t i o n s on t h e observed r e l a t i v e f r e q u e n c i e s can e a s i l y be f o r m u l a t e d i n t h i s more g e n e r a l case.
F o r i n s t a n c e , s t r o n o s t o c h a s t i c t r a n s i t i v i t y may
be d e f i n e d as f o l l o w s . DEFINITI'IW The system ( A x B,n) s a t i s f i e s t h e c o n d i t i o n o f s t r o n g s t o c h a s t i c c o n j o i n t t r a n s i t i v i t y (SSCT) i f f o r any a,b n(b,x)
6 min [ o ( a , x ) , ~ ( b , y ) I
E
A, x,y
* o(a,y)
Any Fechnerian s t r u c t u r e s a t i s f i e s SSCT. lit y model, (Luce-SuVnes
P:
E
3
max [ ~ ( a , x ) , ~ ( b , . v ) l
I n t h e case o f a Fechner an u t i -
( 1965)), t h i s c o n d i t i o n reduces t o t h e usua
s tochas t i c t r a n s i t i v i t y c o n d i t i o n .
strong
25 7
Testmf Fechnerian scalability
B y i n t r o d u c i n g s c a l e s i n t o a Fechnerian s t r u c t u r e i t can be seen t h a t i t i s an a d d i t i v e s t r u c t u r e i n the sense of c o n j o i n t measurement t h e o r y , The system o f c o n s t r a i n t s on t h e resnonse n r o b a b i l i t i e s i s i n f a c t a system o f c a n c e l l a t i o n c o n d i t i o n s t h a t guarantees t h e e x i s t e n c e o f s c a l e s i n case o f an a d d i t i v e s t r u c t u r e .
To show t h i s c o n s i d e r t h e veak o r d e r r e l a t i o n
A x R t h a t i s induced by t h e o r d e r r e l a t i o n
on
5 on t h e dominance a r o b a h i l i -
ties:
( T h i s d e f i n i t i o n i s n o t q u i t e r e a l i s t i c because o f t h e a r t i f i c i a l p r e c i s i o n o f t h e measurement o f dominance n r o h a b i l i t i e s .
I n order t o e s t a b l i s h ine-
q u a l i t y a minimal t h r e s h o l d d i f f e r e n c e t h a t has t o be exceeded i s t o be defined.) I t can e a s i l y b e shown (Mausfeld-Stumpf
(1983)) t h a t t h e system ( A x By;)
s a t i s f i e s an axiom scheme g i v e n by S c o t t (1964) f o r a f i n i t e a d d i t i v e s t r u c ture, comprising a l l nossible c a n c e l l a t i o n conditions.
Thus, w i t h r e s p e c t
t o t h e q u a l i t a t i v e laws i n v o l v e d , t h e uniqueness of s c a l e i n a Fechnerian s t r u c t u r e i s t h a t o f an o r d e r e d m e t r i c s c a l e , whose t i g h t n e s s i n c r e a s e s w i t h t h e number o f elements i n A and B .
2 . VODELS OF FECHNERIAN STRUCTURES A Fechnerian s t r u c t u r e i s a g e n e r a l i z a t i o n o f a v a r i e t y o f models, i n c l u d i n q T h u r s t o n e ' s case V, t h e RTL-model, t h e Rasch-model, monotone p a i r comp a r i s o n systems and s t r i c t u t i l i t y models.
By p r o o e r c h o i c e o f t h e f o r m o f
t h e response f u n c t i o n o r by D a r t i c u l a r i n t e r p r e t a t i o n o f t h e s e t s A and B o r t h e d i s c r i m i n a t i o n i n d e x n these cases may be r e c o v e r e d . The s e t A may be i n t e r o r e t e d as a s e t o f s t i m u l i , and R c o u l d he a s e t o f Dersons o r (as i n d e t e c t i o n nroblems) a s e t o f background v a l u e s .
I n an
o t h e r c o n t e x t A and B may be s e t s o f s t i m u l i o f d i f f e r e n t m o d a l i t i e s obt a i n e d b y a cross m o d a l i t y comparison, o r even i d e n t i c a l s t i m u l i under d i f f e r e n t experimental conditions.
I n q e n e r a l , a p a r t i c u l a r mathematical f o r m
o f t h e response f u n c t i o n i s assumed, t h e c h o i c e o f which denends Dredomin a n t l y on mathematical convenience and i t s s u i t a b i l i t y f o r aarameter e s t i m a t i o n ( c f . Thurstone (1927), p . 373; Rasch (1966),
D.
3).
Once a p a r t i c u -
l a r f u n c t i o n i s chosen, Drocedures f o r t h e e s t i m a t i o n o f parameters and f o r e v a l u a t i n g goodness o f f i t can be c o n s t r u c t e d (see e.g. Bock-Jones (1968), Andersen (1980)).
258
R . Mousfeld
I f we s t a r t w i t h an assumption a h o u t t h e f u n c t i o n a l form o f t F c response
f u n c t i o n t h a t i s more o r l e s s a r b i t r a r y as f a r as n s y c h o l o ? i c a l t h e o r y i s concerned, some nroblems may a r i s e .
I t i s well nossible t h a t a nostulated
s t a t i s t i c a l model i s a n o w e r f u l and narsimonious t e c h n i n u e t o summarize comvlex d a t a on a d e s c t - i n t i v e l e v e l n o t w i t h s t a n d i n g t h a t i t mav n o t have any meanin? i n terms o f s u b s t a n t i v e t h e o r y .
Havinn o b t a i n e d an a c c e o t a h l e
goodness o f f i t we a r e f a c e d w i t h t h e q u e s t i o n : does t h e i n v a r i a n c e o f s c a l e values ( t h a t i s a f o r m a l i m p l i c a t i o n o f t h e assunintion o f a o a r t i c u l a r mathematical f u n c t i o n ) corresnond ti. m y i n v a r i a n c e s o f t h e h y p o t h e z i sed p s y c h o l o g i c a l s t r u c t u r e ?
I n o t h e r words: which i s t h e ? r , v ? r i a n c e o f
the e m n i r i c a l system r e s u l t i n ? o n l y from p u r e l y q u a l i t a t i v e n s y c h o l o o i c a l laws t h a t a r e assumed t o be c o r r e c t ? L e t us c o n s i d e r t h e f o l l o w i n n examnle: < o r t h e s e t o f p a i r e d comparison d a t a o f Table 1, G u i l f o r d (1954) o b t a i n e d a s o l u t i o n f o r T h u r s t o n e ' s case V .
S t i mu1 us
1
2
3
4
5
6
7
1 2
.4a
.65
.7a
.92
.93
.95
.99
-
.50
.69
.80
.85
.97
.94
. 4 ~ .63
-
-
.52
.72 .67
.a9 .78
.9i .86
-
.54
.64
.74
-
.46
.62
-
.5F
-
3 4 5 6
7
-
-
Table 1 P a i r comoarisons m a t r i x f o r seven 1if t e d wei o h t s The c o r r e s o o n d i n a l i k e l i h o o d r a t i o t e s t has a v a l u e o f 10.18 ( d f = 1 5 ) , so T h u r s t o n e ' s model i s n o t r e j e c t e d .
N e v e r t h e l e s s , as f a r as t h e u n d e r l y i n g
o s y c h o l o q i c a l t h e o r y i s concerned, t h e c h o i c e o f a normal response f u n c t i o n
i s arbitrary.
The p s y c h o l o n i c a l laws o n l y r e f e r t o a F e c h n e r i a n s t r u c t u r e .
A s o l u t i o n o b t a i n e d i n t h e g e n e r a l case by t h e method t o be d e s c r i b e d below i s shown i n Table 2.
The c o r r e l a t i o n between t h e two s o l u t i o n s i s .998.
The t h e o r e t i c a l p r o n o r t i o n s i n Table 2, however, a r e c l o s e r t o t h e d a t a i n t h e sense o f mean a b s o l u t e d i s c r e n a n c y . I n t e s t i n g Fechnerian s c a l a b i l i t y t h e response f u n c t i o n i s o b v i o u s l y n o t unique.
I f an a c c e n t a b l e f i t i s o b t a i n e d under T h u r s t o n e ' s case V , t h e r e
a r e i n f i n i t e l y many o t h e r response f u n c t i o n s t h a t f i t t h e d a t a .
259
Testing Fechnerian scalubility
-1.13
Scale v a l u e S t i mu1us
1
1 2 3 4 5 G
-
-
-.33 2 .6500
-.24
3 .79@0 .6900
-
-
-
7
4 .3875 .7900 .6300
-
-
..-,7
.P'
-
5 .9300 .PP75 .7200
.6550
-
-
-
-
n-,
.br.
1.fG
6 .9525 .9525 .8875 .7800 .6550
-
7 .9900 .9525 .9100 .8600 .7400 .6 200
-
-
Table 2 Scale values and t h e o r e t i c a l n r o n o r t i o n s f o r a Fechnerian s o l u t i o n f o r t h e d a t a o f Table 1. Consequently, t h e n r o n e r i n v a r i a n c e c l a s s o f s c a l e values cannot be d e t e r mined by a r b i t r a r i l y f i x i n q D , b u t r a t h e r t h e whole c l a s s o f a o p r o o r i a t e resnonse f u n c t i o n s has t o be taken i n t o account.
On t h e o t h e r hand, i f we
have o b t a i n e d a bad f i t , how t o decide i f t h i s m i s f i t i s p r i m a r i l y due t o v i o l a t i o n s o f a r b i t r a r y mathematical assumptions o r
-
much more i m p o r t a n t
-
due t o v i o l a t i o n s o f s u b s t a n t i v e q u a l i t a t i v e laws t h a t a r e sunnosed t o h o l d ?
To assume a res7onse f u n c t i o n t h a t does n o t have an i n t e r n r e t a t i o n i n a subs t a n t i v e t h e o r y "presurmoses measurenent as t h e goal and i s t h e r e f o r e even more s u b j e c t t o t h e danger o f i n a p o r o o r i a t e t h e o r i z i n g " ( K r a n t z (1972)). To show t h a t d i f f e r e n t models a r e e q u a l l y w e l l s u i t e d f o r d e s c r i p t i v e DurDoses and t h a t i t i s d i f f i c u l t t o a i v e reasons on t h e o r e t i c a l grounds t o a d o p t a o a r t i c u l a r model, c o n s i d e r a s e t o f p a i r comparison d a t a r e p o r t e d a l s o by C u i l f o r d (1954) and analvsed b y l i l o s t e l l e r (1958), Hohle (1964) and Iblinsbero-Ramsey (1981) under v a r y i n a model assumptions. V o s t e l l e r (1958) e s t i m a t e d t h e c o r r e s p o n d i n g s c a l e values by each o f t h e f o l l o w i n g response f u n c t i o n s : t h e u n i f o r m on t h e i n t e r v a l 0 t o 1, t h e arcsin
4, the
normal, 1/2 e - I X I , t h e t - d i s t r i b u t i o n w i t h 10 d . f . ,
and he
showed t h a t t h e c o r r e l a t i o n s between t h e s c a l e values f o r any two methods i s always g r e a t e r .99 f o r t h e g i v e n data.
S i m i l a r r e s u l t s aDply t o t h e
parameters e s t i m a t e d under t h e BTL-model and under t h e g e n e r a l method t o be proposed below.
Even though t h e r e s u l t i n g s c a l e s a r e n e a r l y l i n e a r t r a n s -
formations o f each o t h e r ( T a b l e 4 ) , t h e reproduced response o r o b a b i lit i e s under t h e v a r i o u s models a r e q u i t e d i f f e r e n t ( T a b l e 5 ) , and so i s any goodness o f f i t t e s t e x p e c t e d t o be.
Vege t a b 1e
1
2
3
4
5
6
7
8
9
1. 2. 3. 4. 5.
-
.818
.770 .631
.311 .723
.892 .736
.;378
,V99
.E92
.743
-311
.845
.926 .858
-
.551
.676 .58S
.736 .561
,345 .676
.797 .601
.818 .730
-
.507
.628
.682
.628
-
,574
.709
.764
-
-
.527
.642
-
.628
Turnips Cahbaae Ecetc AsDaragus
-
-
Soinach
-
6 . Carrots 7. 5 t r i n a beans
a.
Peas
-
-
-
-
-
9. Corn Tahle 3
Preference judgements from 148 o b s e r v e r s f o r each q a i r f r o m n i n e vegetahles. ~~
uniform
arcsin6
Case
v
1/2e-IXJ
t10
BTL
__ Fechneri an scalphility
1. T u r n i a s
0
0
0
0
0
0
0
2 . Cabbaqe
.284 .374
.306 .390
.321
.402
.362 .433
.327 .4OG
.329 .409
.403 .462
.578 .683 .669
.593 .695 ,679
.601 .702
.629 .726
.605 .705
.607
,706
.856
-688 .860
.690
Feas
.849 .882
.685 .859
.6ia .713 .720
.G85
.886
.890
Corn
1
1
1
1
3. 4. 5. 6. 7. 8. 9.
Peets Asparaqus Soinach Carrots Beans
.871
.887 1
.707 .856 .886
.865
1
1
.880
Table 4 Scale v a l u e s f o r Table 3 under d i f f e r e n t model assumntions.
F o r t h e g r o p o r t i o n s i n Table 5 t h e sum of a b s o l u t e d e v i a t i o n s f r o m t h e obs e r v e d values i s t h e l o w e s t f o r t h e t h e o r e t i c a l p r o p o r t i o n s o b t a i n e d by t h e parameter e s t i m a t i o n under t h e F e c h n e r i a n s c a l a b i l i t y model t o be d i s c u s s e d below.
The average a b s o l u t e d e v i a t i o n s i n Table 5 f o r t h i s model a r e ,026.
Refore p r e s e n t i n g a method o f e s t i m a t i n g parameters i n t h e g e n e r a l case o f a Fechnerian s t r u c t u r e , a s h o r t o u t l i n e of t h e n a r c m e t e r e s t i m a t i o n i n t h e normal o r l o g i s t i c case m i g h t be u s e f u l .
26 1
Testing Feclinerian scalability
Observed
U n i f o r m a r c s i n 6 Case V
1/2e-IX1
t10
ETL Fechneri an s c a l a b i l ity ~
(1,6)
.878
.863
.866
.868
.870
.869
.867
.885
(2,6)
.743
.709
.718
,724
.741
.707
.726
.737
(3,6)
.736
.660
.671
.673
.702
.681
.682
.686
(4,6)
.561
.549
.552
.554
.568
.556
.556
.583
(5,6)
.507
.492
.491
.489
.481
.488
.489
.507
(6,7)
.574
.598
.6 06
.611
.636
.614
.6 10
.607
(6,8)
.709
.6 16
.623
,628
.649
.630
.630
.686
(6,9)
.764
.6 80
.689
.695
.716
.699
.699
.696
Table 5 Comparison o f observed and t h e o r e t i c a l n r o n o r t i o n s f o r t h e " C a r r o t " o b s e r v a t i o n s (column 6 and row 6 o f Table 3 ) . 4. PARAVETER ESTIMATION I N THE CASE OF NORMAL OR LOGISTIC RESPONSE
FUNCTIONS L e t N be t h e number o f indeoendent r e n l i c a t i o n s f o r each comearison and nij
t h e ( a b s o l u t e ) frequency by which a s t i m u l u s ai
-
dominates a s t i m u l u s
a L e t pij = n. ./N. The random v a r i a b l e nij f o l l o w s a b i n o m i a l d i s t r i j' 1J t u t i o n w i t h parameters N and D ~ t ~h e , unknown dominance p r o b a b i l i t i e s t o be estimated.
I n a complete D a i r c o m a r i s o n exDeriment a l l p a i r s o f m s t i m u l i
a r e presented, l e a d i n p t o );( o f binomial p r o b a b i l i t i e s L e t p be t h e v e c t o r n = (nlly
...,nijY...,n
D =
indenendent e s t i m a t e s
Eij
o f an e q u a l number
D~ j.
( ~ ~ ~ , . . . , p ~ ~ , . . . , p ~ - ~and , ~ )n t h e v e c t o r
m-1 ,m ) .
Then t h e l i k e l i h o o d o f t h e e n t i r e s e t o f i n -
dependent o b s e r v a t i o n s i s
Omitting a d d i t i v e constants t h e l o g - l i k e l i h o o d i s :
m L i s a f u n c t i o n on t h e parameter sDace 0 , 0 c R n y n = ( * ) . l i h o o d estimate
0
€
e i s t o b e found such t h a t
A maximum l i k e -
262
R . Muusfeld
I f t h e r e a r e no c o n s t r a i n t s on t h e response p r o b a b i l i t i e s t h e I1L-estimates
p . . o b v i o u s l y a r e t h e r e l a t i v e frequencies 6 . .. IJnder t h e assumytion o f a 1J
1J
n a r t i c u l a r response f u n c t i o n the unknown dominance n r o b a b i l i t i e s i n ( 3 ) a r e s u b s t i t u t e d by t h e resoonse f u n c t i o n , whose parameters, i .e. t h e s c a l e val u e s , a r e t o b e e s t i m a t e d hy m a x i m i z i n g ( 3 ) . F o l l o w i n a t h e e s t i m a t i o n o f parameters a r a t i o o f l i k e l i h o o d s can be c a l c u l a t e d : the l i k e l i h o o d o f t h e d a t a under t h e s o e c i f i c resaonse f u n c t i o n , t h a t i s i n t h e r e s t r i c t e d a a r a l l e t e r snace l y qenera
Then the
e,
t o t h e l i k e l i h o o d under a complete-
model w i t h parameter space ?. ikelihood ratio s t a t i s t i c is:
A =
max L ( n J n ) cea
m-1
N-n..
m
i=l j=i+l
1J
I'nder c o n d i t i o n s t h a t a r e f u l f i l l e d i n t h e normal and l o n i s t i c case t h e d i s t r i b u t i o n o f - 2 I n ;r convernes i n p r o b a b i l i t y t o t h e c h i - s n u a r e d i s t r i b u t i o n as N tends t o i n f i n i t y .
The degrees o f freedom o f t h e l i m i t i n g c h i - s q u a r e
d i s t r i b u t i o n a r e enual t o t h e number o f independent e s t i m a t e d n a r a n a t e r s i n '
minus t h e c o r r e s n o n d i n g nun-ber i n 0 .
I n t h e case d i s c u s s e d h e r e , t h e numi n 'I equal m - 1 .
h e r o f indenendent e s t i m a t e d narameters i n 0 i s m(m-1)/2,
F o r some models t h e v a l u e s o f -2 I n a a r e shown i n T a b l e 6 based on t h e d a t a o f Table 3 .
A c c o r d i n g t o t h i s goodness of f i t s t a t i s t i c t h e two c l a s s i c a l
models and the s p l i n e model f a i l t o a c c o u n t f o r t h e e m n i r i c a l d a t a .
To r e -
j e c t Fechnerian s c a l a b i l i t y , however, i t i s n o t s u f f i c i e n t t o r e j e c t t h e BTL-model o r T h u r s t o w ' s case V o r any o t h e r model w i t h a n a r t i c u l a r respon-
se f u n c t i o n . -2 I n A
df
T h u r s t o n e ' s Case V
52.91
28
BTL-mode 1
46.76
28
n n
model (Winsbera-Ramsey)
42.95
25
p < .05
Fechnerian s c a l a b i l i t y
23.04
-
-
Model
p-value <
.01
<
.05
I n t e g r a t e d B-sDline-
263
Testing Peclinerian scalability
5 . PC.EPr1ETER ESTIP'ATION I N FECHNERIAN STRUCTURES I n a Fechnerian s t r u c t u r e t h e o n l y r e q u i r e m e n t on t h e response f u n c t i o n D i s t o be continuous and strcnql!r monotonic.
Any such f u n c t i o n w a r e n t e e s
t h e e x i s t e n c e o f an a d d i t i v e r e p r e s e n t a t i o n .
The advantaoe o f i n c l u d i n g
a l l f u n c t i o n s o f t h e o r e t i c a l i n t e r e s t i n t h i s d e f i n i t i o n has t h e drawback t h a t t h e resnonse f u n c t i o n cannot h e s p e c i f i e d a n a l y t i c a l l y i n t h e g e n e r a l case.
So t h e e s t i m a t i o n o f parameters b y maximum l i k e l i h o o d methods i s n o t
straightforward. I n t h e f o l l o w i n g we w i l l nronose a method t o achieve such an e s t i m a t i o n o f parameters.
F o r sake o f s i m o l i c i t y l e t now A = B.
Assume t h a t t h e rank
o r d e r o f t h e dominance n r o b a b i l i t i e s w i t h r e s n e c t t o t h e u n d e r l y i n g s c a l e s i s a l r e a d y known, o o s t n o n i n q t h e d e t e r m i n a t i o n o f t h i s o r d e r f o r a moment. Then each o f these b i n o m i a l narameters can be e s t i m a t e d by maximum l i k e l i hood e s t i m a t i o n i n t h e f o l l o w i n g way. Any resnonse f u n c t i o n 0 o f a Fechnerian s t r u c t u r e g i v e s r i s e t o a system o f i n e n u a l i t y c o n s t r a i n t s on t h e resnonse o r o b a b i l i t i e s t h a t r e s t r i c t s t b ~ r c c e l oarameter space, D e f i n e a v e c t o r - v a l u e d f u n c t i o n manning f r o m R n i n t o Rnm1 by:
with hr(n) = h r ( n l , . = "r+l
-
.- ,Pn) -
1< r ~ n - 1
Pr
...
Let 0 2 n 1 z p 2 -< n n ( a t l e a s t one i n e q u a l i t y s t r i c t ) be an o r d e r of t h e response p r o b a b i l i t i e s t h a t i s c o n s i s t e n t w i t h t h e model assumntion o f a Fechnerian s t r u c t u r e , t h e n t h e model narameter snace e
e = { D ~ F O ,h ( p )
>
c
B
S:
01
Under these cons t r a i n t s exDression ( 3 ) c o u l d n u m e r i c a l l y be maxi m i zed by any n o n l i n e a r o p t i m i z a t i o n a l g o r i t h m (see e.g. G i l l - M u r r a y
1974)).
Because t h e l i k e l i h o o d f u n c t i o n i s continuous i n i t s aroument i t has i t s maximum on t h e c l o s e d bounded s e t d e f i n e d b y t h e system of i n e q u a l i t y constraints.
To show t h a t t h e maximum i s unique and t h a t t h e ML-estimates a r e
c o n s i s t e n t , we r e l y on a correspondence between maximum l i k e l i h o o d e s t i m a t i o n and i s o t o n e r e g r e s s i o n i n t h e case considered.
1-64
R. hfausfeld
THEOREPI
L e t ( A x A , p ) he a Fechnerian s t r u c t u r e and l e t 0
2 D~ 2 n 2 5
.. . 2 pn
51
be an o r d e r on the resnonse n r o b a h i l i t i e s t h a t s a t i s f i e s t h e model con-
Let
straints.
6,
be t h e mean o f a sample o f
from a f i e r n o u l l i d i s t r i b u t i o n v i t ' i mean p t .
N
indenendent o b s e r v a t i o n s t Then t h e maximum l i k e l i h o o d
e s t i m a t e o f p t under t h e n r e s c r i b e d o r d e r i s f u r n i s h e d u n i q u e l y b y t h e i s o tone r e g r e s s i o n o f
bt
w i t h weights N
t'
T h i s tlreorem i s a s D e c i a l case o f theorem 2.12.
by Rarlow e t a l . (1972)
,
see a l s o Ayer e t a l . (1955), v.Eeden ( 1 9 5 6 ) .
As t h e
6..
1J
are consistent estimators o f the
D..
1J
i n t h e u n r e s t r i c t e d case,
t h e y a r e c o n s i s t e n t e s t i m a t o r s i n t h e r e s t r i c t e d case.
F o r a n r o o f , see
theorem 2 . 2 i n B a r l o w e t a l . ( 1 9 7 2 ) . T h i s r e s u l t reduces a l q o r i t h m i c nroblems c o n s i d e r a b l y , because t h e paramet e r s can be c a l c u l a t e d e a s i l y by means o f t h e " p o o l - a d j a c e n t - v i o l a t o r s ' ' a1 o o r i thm f o r i s o t o n e r e o r e s s i o n
.
ilow t o determine t h e l i n e a r o r d e r t h a t i s c o n s i s t e n t w i t h a known p a r t i a l
o r d e r and c o m p a t i b l e w i t h a Fechneri an s t r u c t u r e . Sunoose t h e rank o r d e r o f t h e s t i m u l i i s known i n advance, as i t g e n e r a l l y i s t h e case i n p s y c h o o h y s i c a l s t u d i e s .
I f t h e s t i m u l i a r e elements o f a
Fechnerian s t r u c t u r e , t h e o r d e r on t h e s t i m u l i i n d u c e s a p a r t i a l o r d e r on t h e dominance p r o b a b i l i t i e s . bility.
T h i s o r d e r has t h e o r o p e r t y o f s i m p l e s c a l a -
Thus, i f t h e rank o r d e r o f t h e s t i m u l i i s known, t h e i n d u c e d p a r -
t i a l o r d e r on t h e dominance o r o b a b i l i t i e s i s g i v e n a D r i o r i .
I f a l l the
o t h e r o r d e r r e l a t i o n s , besides t h i s o a r t i a l o r d e r , c o u l d be chosen a r b i t r a -
r i l y , a l i n e a r o r d e r c o u l d be c o n s t r u c t e d t h a t i s c o n s i s t e n t w i t h t h e p a r t i a l o r d e r and whose i s o t o n e r e g r e s s i o n i s i d e n t i c a l t o t h a t one o f t h e p a r t i a l o r d e r ( f o r t h e c a l c u l a t i o n o f t h i s l i n e a r o r d e r , see Barlow e t a l . (1972, p. 8 2 f f ) . Any f u r t h e r r e s t r i c t i o n on t h e c l a s s o f c o n s i s t e n t l i n e a r o r d e r s w i l l l e a d t o a c o n s i s t e n t l i n e a r o r d e r such t h a t t h e c u m u l a t i v e sum diagram (CSD) i s " l e s s convex" and t h e s l o p e s o f t h e CSD a r e i n a l e a s t squares sense l e s s c l o s e t o t h e s l o p e s o f t h e g r e a t e s t convex m i n o r a n t (GCM) ( f o r d e t a i l s , see Barlow e t a l . (1972))
Thus t h i s s o l u t i o n w i l l be d i f f e r e n t f r o m t h e i s o -
tone r e g r e s s i o n w i t h r e s p e c t t o t h e p a r t i a l o r d e r .
If a goodness o f f i t t e s t f o r t h e i s o t o n e r e g r e s s i o n w i t h r e s p e c t t o t h i s
265
Testing Fechnerian scalability
r e s t r i c t e d c o n s i s t e n t l i n e a r o r d e r has proved t o he s a t i s f a c t o r y ,
a test
f o r t h e s o l u t i o n w i t h r e s o e c t t o t h e o a r t i a l o r d e r w i l l Drove t o be s a t i s factory either.
On t h e o t h e r hand, r e j e c t i n g t h e r e s t r i c t e d s o l u t i o n leads
t o r e j e c t i o n o f a l l s o l u t i o n s t h a t a r e r e s t r i c t e d e i t h e r e q u a l l y o r more severe 1y
.
I n t h e t h e o r y considered, t h e c o n s t r a i n t t o be imposed on t h e l i n e a r o r d e r i s t h a t i t s h o u l d he c o m p a t i h l e w i t h t h e assumption o f a Fechnerian s t r u c ture.
L e t us c a l l such an o r d e r a Fechnerian o r d e r .
Supoose t h e i s o t o n e
r e g r e s s i o n has been c a l c u l a t e d f o r a l l Fechnerian o r d e r s t h a t a r e c o n s i s t e n t w i t h t h e g i v e n p a r t i a l o r d e r ; then choose t h a t Fechnerian o r d e r , f o r which t h e s l o p e o f t h e CSD i s c l o s e s t t o t h e s l o p e of t h e 6CY.
The r e s u l t i n g so-
l u t i o n i s the isotone regression w i t h respect t o the p a r t i a l order i n the c l a s s o f a l l Fechnerian o r d e r s , t h a t i s , i t minimizes t h e sum
i n the class o f isotone functions
D
t h a t s a t i s f y c o n d i t i o n (1).
F o r p r a c t i c a l purposes, a s u i t a b l e s o l u t i o n can be achieved b y u s i n g Krusk a l s s t r e s s minimizing algorithm.
I t does n o t , o f course, n e c e s s a r i l y l e a d
t o t h e " b e s t " Fechnerian o r d e r as d e f i n e d above.
But, as a c o n s e r v a t i v e
procedure, it can be e x p e c t e d t o p r o v i d e a g a i n s t wrong c o n c l u s i o n s
.
I d i t h r e g a r d t o t h e goodness o f f i t s t a t i s t i c t h e r e remain some
objections,
whether t h e t o p o l o g i c a l o r o o e r t i e s o f t h e model oarameter soace
- which
oend on t h e number o f s t i m u l i
-
cie-
might i n f l a t e the l e v e l o f s i g n i f i c a n c e .
I f n o a o r i o r i o r d e r i n g o f t h e s t i m u l i i s known i n advance, as i t i s t h e case w i t h t h e vegetable data, t h e b e s t l i n e a r o r d e r has t o be e s t i m a t e d e n t i r e l y from the data.
The use o f t h e t e s t s t a t i s t i c t h a t w i l l be g i v e n
i n t h e n e x t s e c t i o n has no r i g o r o u s j u s t i f i c a t i o n i n t h i s case. Results f o r the Guilford-data F o r t h e d a t a o f Table 3, t h e l i n e a r o r d e r t h a t i s n e a r e s t t o t h e d a t a i n t h e sense of m i n i m i z i n g s t r e s s i s determined b y t h e UNICDN-program (Roskam ( 1 9 7 4 ) ) , w i t h a r e s u l t i n g s t r e s s o f .11.
The ML-estimates under t h i s o r d e r
r e s t r i c t i o n s a r e shown i n Table 7. The mean absol Ute d i s c r e o a n c y between observed and t h e o r e t i c a l o r o p o r t i o n s i s .021, compared t o .034 and .032 i n t h e normal and l o g i s t i c case respecti v e l y
.
Veae tab1 e
2
3
4
5
6
~
.8075
1. T u r n i p s
.80 75
.5833
2 . Cabbaae 3. Reets
-
4 . Asoaragus
5. c n i n a c h 6. Carrots
3345 6861 6066
.8955
.7360 .656 1
.a110
-
9
.8955 .a315
.9260
.GO75
.8315
.6a6i
.?365
~~
.8850
-5833
-
a
7
.go75 .6a6 1
~~
~
.a345
,6066 ,6861 .6960 .6066 .cia61 .6960
7. Beans
.5270
8. Peas
.6066 .6066
9 . Corn Table 7 HL-es t i m a t e s f o r a a e n e r a l Fechnerian s t r u c t u r e . I t i s i n t e r e s t i n c t c n o t e t h a t t h e r e n k o r d e r o f t h e s t i m u l i , which i s t h e
same f o r a l l Yesnorse f u n c t i o n s g i v e n i n Table 2, i s r e v e r s e d f o r s t i m u l i 5 and 6 i n t h e Genera; case o f a Fechnerian s t r u c t u r e , as can be seen f r o m t h e s c a l e values i n Table 2, and t h e comparison ( 5 , 6 ) i n Table 3. T h i s r e s u l t i n d i c a t e s a v i o l a t i o n o f t h e model assumptions, b e c a m e
ir,
a
Fechnerian s t r u c t u r e a l l s o l u t i o n s have t h e i n v a r i a n c e o f an o r d e r e d m e t r i c scale. 6 . EVPLUATIW CWDNESS OF F I T
FOP FECHNERIAK STRUCTURE<
Under t h e model h y p o t h e s i s o f a o a r t i c u l a r l i n e a r o r d e r on t h e response p r o b a h i l i t i e s t h e parameter sDace u n f o r t u n a t e l y does n o t s a t i s f y t h e c o n d i t i o n s t h a t a r e necessary f o r t h e usual a s y m p t o t i c r e s u l t s o f t h e l i k e l i h o o d r a t i o The s t a t i s t i c - 2 I n
test statistic.
t h a t i s c a l c u l a t e d f o r t e s t i n g the
Fechnerian model a o a i n s t a comr.letely qeneral model i n 53 ( T a b l e 4 ) i s n o t asymototically
x 2- d .i s t r i b u t e d
In the case o f
a n o r m a l l y d i s t r i b u t e d random v a r i a b l e w i t h known v a r i a n c e
and c o n s e q u e n t l y i s o f d e s c r i p t i v e v a l u e o n l y .
Rarlow e t a l . (1972) develooed a t e s t s t a t i s t i c f o r t e s t i n g o r d e r hypotheses. Under t h e n u l l - h y p o t h e s i s o f t h e e q u a l i t y o f m a n s o f d i f f e r e n t e x p e r i m e n t a l x 2- d i s t r i -
c o n d i t i o n s t h e d i s t r i b u t i o n o f t h i s s t a t i s t i c i s a weicrhted sum of butions given by:
261
Tesring Fechnerian scalability
-*
where k i s t h e number o f m a n s b e i n ? compared, ui i s t h e i s o t o n e r e g r e s s i o n o f t h e ith sample mean w i t h r e s n e c t t o t h e l i n e a r o r d e r t h a t i s t e s t e d , and i s t h e maximum l i k e l i h o o d e s t i m a t o r o f t h e o o n u l a t i o n means under t h e n u l l h y p o t h e s i s , t h a t i s t h e grand samole mean. I n o r d e r t o u t i l i z e t h i s t e s t s t a t i s t i c t h a t i s a p n r o n r i a t e f o r t h e model considered, t h e c o n d i t i o n s o f a normal d i s t r i b u t e d random v a r i a b l e w i t h known v a r i a n c e a r e t o be f u l f i l l e d .
I n t h e o r e s e n t case the number o f i n -
dependent r e p l i c a t i o n s i s l a r o e enough t o j u s t i f y t h e a o o r o x i m a t i o n t o t h e norlnal d i s t r i b u t i o n as o i v e n by t h e c e n t r a l l i m i t theorem.
As t h e variances o f t h e p r n n c r t i o n s under Ho deDend on t h e unknown t r u e dominance p r o b a b i l i t i e s , t h e monotone t r a n s f o r m a t i o n 2 a r c s i n f i i s performed on t h e d a t a .
I t i s w e l l known t h a t t h i s t r a n s f o r m a t i o n has a v a r i a n c e w h i c h
i s indeDendent o f p , namely 1 / N and t h e t r a n s f o r m e d values a r e approximate1j d i s t r i b u t e d as a normal d i s t r i b u t i o n w i t h mean 2 a r c s i n & and v a r i a n c e 1/N. I f t h e d a t a a r e i n t h e o r d e r d e s c r i b e d by H1,
t h e er;p;rical
values a r e a l -
ready t h e i s o t o n e r e g r e s s i o n w i t h r e s p e c t t o t h i s o r d e r and t h e t e s t s t a t i s t i c w i l l have i t s maximum.
An e m p i r i c a l o r d e r which d o e s n ' t d i f f e r mar-
k e d l y f r o m t h e t e s t e d o r d e r , d o e s n ' t s e r i o u s l y a f f e c t t h e power o f t h i s t e s t (see Barlow e t a l . (1972, D. 1 6 5 f f ) ) .
I f , on t h e o t h e r hand, t h e em-
p i r i c a l d a t a a r e i n such an o r d e r t h a t t h e number o f d i s t i n c t values t a k e n by t h e i s o t o n e r e g r e s s i o n tends t o one, t h e Dower o f t h e t e s t tends t o a p o s i t i v e v a l u e n o t exceeding i t s s i g n i f i c a n c e l e v e l , o r t o zero.
Conse-
o u e n t l y , Ho i s a u s e f u l dummy h y p o t h e s i s as i t cannot be r e j e c t e d i f t h e r e are strong departures from the r e s t r i c t i o n s o f t h e a l t e r n a t i v e hypothesis. Results The d a t a o f Table 3 were t r a n s f o r m e d a c c o r d i n g l y , and t h e v a l u e o f
$,
which reduces t o
can be computed. F o r t h e G u i l f o r d - d a t a o f Table 3 t h i s t e s t s t a t i s t i c has a v a l u e of -2 x~~ = 373.57. Under Ho t h e o r o b a b i l i t y t h a t t h e t e s t s t a t i s t i c takes on a value n o t l e s s than t h i s one i s :
R . Lfirusfeld
268
where P ( j , 3 6 )
36
!X236
Pr
373,571
=
c
P(j,36)
2
Pr
::73,571
j=2
i s the n r o b a b i l i t y
tt-pt
the i s o t o n i c regression f u n c t i o n
ct
takes e x a c t l y j d i s t i n c t values ( s e e Barlow e t a l . (1972, p . 1 2 6 f f ) ) . By c a l c u l a t i o n o f r ( j , 3 6 )
and b y a n n r o x i m a t i o n s o f t h e necessary x2-values
i t can be seen t h a t t h e o r o b a h i l i t y o f o h t a i n i n q a r e s u l t as l a r q e as t h i s
one i s beyond any r e a s o n a b l e s i ani f i cance l e v e l
.
I t can s a f e l y be concluded t h a t t h e s t i m u l i can be r e n r e s e n t e d as elements o f a Fechnerian s t r u c t u r e , t h a t i s , t h e p r e f e r e n c e s c a l e used i n these comp a r i s o n s can be r e o a r d e d as one-dimensional and t h e n r e f e r e n c e v r o b a b i l i t i e s a r e monotone f u n c t i o n s o f t h e r e s a e c t i v e s c a l e v a l u e d i f f e r e n c e . I f a F e c h n e r i a n s t r u c t u r e c a n n o t he acceoted, t h e
des t e s t s f o r s i m p l e s c a l a b i l i t y , t o o . i s t o t e s t t h e necessary c o n d i t i o n
ii
test s t a t i s t i c provi-
A weak t e s t o f s i m n l e s c a l a b i l i t y
t h a t t h e means o f t h e c e l l e n t r i e s which
are p a r a l l e l t o the m a t r i x diaaonal are i n increasino o r d e r .
A test that
i s e q u i v a l e n t t o s i m p l e s c a l a b i l i t y i s t o t e s t t h e comolete i n d u c e d p a r t i a l order v i a isotone regression. REFERENCES Andersen, E.R.,
D i s c r e t e s t a t i s t i c a l models w i t h s o c i a l s c i e n c e a r t u l i -
c a t i o n s , Amsterdam, N o r t h H o l l a n d ( 1 9 8 0 ) . Ayer, PI.,
Brunk, H.D.,
Ewina, G.M.,
Reid, V.T.,
and Silverman, E.,
An
e m p i r i c a l d i s t r i b u t i o n f u n c t i o n f o r samDlinq w i t h i n c o m p l e t e i n f o r m a t i o n . The Annals o f Flathematical S t ? t i s t i c s , 26 ( 1 9 5 5 ) , 541-647. Barlow, R.E.,
Rartholomew,
D.J., Sremner, J.M.,and
Rrunk, H.D.,
Statis-
t i c a l i n f e r e n c e under o r d e r r e s t r i c t i o n s , London, L l i l e y (1972). Bock,
R.D., and Jones, L.Y.,
The measurement and D r e d i c t i o n o f j u d g -
w n t and c h o i c e , San F r a n c i s c o , Pol den-Day ( 1968)
.
Eeden, C . van., rdaximum l i k e l i h o o d e s t i m a t i o n o f o r d e r e d D r o b a b i l i t i e s , I n d a q a t i o n e s mathematicae, 18 (1956) Falmagne, J.C.,
, 444-455.
The g e n e r a l i z e d Fechner a r o b l e m and d i s c r i m i n a t i o n ,
J o u r n a l o f f l a t h e m a t i c a l Psycholoqy, 8 (1971), 22-43.
G i l l , P.E.,
and Murray,
El,
(Eds.),
Numerical methods f o r c o n s t r a i n t
o o t i m i z a t i on, fiw York, Academic Press (1974). Guilford, (1954).
J.P.,
Psychometric methods (2 e d . ) ,
New York, McGraw-Hill
269
I'esting Fechnerian rcalabifity
l : o k l e , R.H.,
An e m n i r i c a l e v a l u a t i o n and comparison o f two models f o r
d i s c r i m i n a b i l i ty scales, J o u r n a l o f Vathematical Psycholoay, 3 (1966) ,
174-183. K r a n t z , D.H.,
The s c a l i n g o f small and l a r g e c o l o r d i f f e r e n c e s , Unpub-
l i s h e d doctGra1 d i s s e r t a t i o n . U n i v e r s i t y o f Pennsylvania (1964). K r a n t z , D.H.,
Measurement s t r u c t u r e s and n s y c h o l o a i c a l laws, Science,
175 ( 1972), 1427-1435. Luce, R.D.,and l i t y , In R.D.
Suopes. P., Luce, R.R.
Preference, u t i l i t y and s u b j e c t i v e p r o b a b i -
Bush and E . G a l a n t e r (Eds.),
Handbook of Ma-
t h e m a t i c a l Psycholoqy, Vol. 111, New York, h!iley (1965), 249-410. Plausfeld, R.,
and Stumnf, H.,
Latent trait-rlodelle m i t additiver Struk-
t u r , Z e i t s c h r i f t Fir D i f f e r e n t i e l l e und D i a p n o s t i s c h e Psychologie, 4
(1983), 87-111. F., The m y s t e r y o f t h e m i s s i n g corpus, Psychometrika, 23
?Iosteller,
(1958) , 279-269. Rasch, G.,
An i n f o r m a l r e p o r t on t h e p r e s e n t s t a t e o f a t h e o r y o f ob-
j e c t i v i t y i n comparisons, I n Proceedings of t h e NUFFIC I n t e r n a t i o n a l Summer s e s s i o n i n Science a t ' H e t oude H o f ' , The Hague (1966). Roskam, E., Unidimensional c o n j o i n t measurement (UNICON) f o r m u l t i f a c e t e d designs, Unpublished r e p o r t , P s y c h o l o g i c a l Laboratorium, U n i v e r s i ty o f Nijmegen (1974). S c o t t , D.,
Fleasurement s t r u c t u r e s and l i n e a r i n e q u a l i t i e s , J o u r n a l o f
H a t h e m a t i c a l Psychology (1964), Thurstone, L.L.,
233-247.
Psychophysical a n a l y s i s , American J o u r n a l o f Psycho-
l o ~ y ,38 (1927), 368-389. Tversky, A.,
Choice by e l i m i n a t i o n , J o u r n a l o f Mathematical Psycholoay,
9 (1972), 341-367. Tversky, A.,
and Russo, J.E.,
S i m i l a r i t y and s u b s t i t u t a b i l i t y i n b i n a r y
choices, J o u r n a l o f Mathematical Psychology, 6 (1969), 1-12.
, A n a l y s i s of p a i r w i s e D r e f e r e n c e , Psychometrika, 46 (1981), 171-186.
I.linsberg, S . , and Rar:cey, J.O. using i n t e g r a t e d B-snlines
data
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TRENDS lN MATHEMATICAL PSYCHOLOGY E. Degreef and J. Van Bu mhaut (editors) @Elsevier Science h b l i c g r s B. V. (NorthHolland), 1984
271
PROBABILIST1 C CONSISTENCY, HOMOGENEOUS FAVILIES OF RELflTIONS AND LINEAP A-RELATIONS
B . ''on j a r d e t U n i v e r s i t e de P a r i s V Centre de tlathematique S o c i a l e
The n o t i o n s o f l i n e a r d i s c r i m i n a t i o n s t r u c t u r e and o f homogeneous f a m i l y o f semi-orders have been i n troduced i n the theory o f p r o b a b i l i s t i c consistency b y Luce and Roberts, r e s p e c t i v e l y .
Here, we
s t u d y t h e more g e n e r a l n o t i o n s o f l i n e a r A - r e l a t i o n s and homoaeneous f a m i l i e s o f S.T.F.
r e l a t i o n s , such
n o t i o n s b e i n n a b l e t o be u s e f u l i n t h e t h e o r y o f p r o b a b i l i s t i c c o n s i s t e n c y (non f o r c e d o r i n d i f f e rence c h o i c e s ) and i n s e v e r a l o t h e r s i t u a t i o n s ( c o l l e c t i v e c h o i c e , non-symmetrical s i m i l a r i t y ,
...).
1. INTRODUCTION The Roberts book, l k a s u r e m e n t Theory (1979), c o n t a i n s an e x c e l l e n t s t a t e m e n t o f t h e t h e o r y o f p r o b a b i l i s t i c c o n s i s t e n c y , t o which we r e f e r t h e r e a d e r f o r
background and r e f e r e n c e s .
Here, we o n l y r e c a l l some i d e a s o f t h i s t h e o r y .
I t d e a l s w i t h " p a i r comnarison system" a r i s i n g of r e p e a t e d b i n a r y c h o i c e o f
an i n d i v i d u a l .
C e n e r a l l y , these b i n a r y choices a r e " f o r c e d " , so t h a t i f we
denote b y p(x,y) t h e r e l a t i v e frequency w i t h which x i s chosen a g a i n s t y, we have D(x,Y)+P(Y,x) = 1 (and, f o r convenience, i t i s o f t e n assumed P ( X , X ) =1 ~ ) . I t can apnear a c e r t a i n " c o n s i s t e n c y " i n t h e observed b e h a v i o r o f an i n d i v i -
1
1
1
), and t h e t h e o r y wants account o f i t . The t h e o r y has t h r e e main i n g r e d i e n t s : e x p l i c a t i v e models, observable p r o p e r t i e s , and theorems c o n n e c t i n g models dual
( f o r example, D(X,Y) 5
and p r o p e r t i e s .
and o(y,z)
3
always i m D l y D(X,Z) 5
The e x p l i c a t i v e models a r e based on a n u m e r i c a l r e p r e s e n t a -
t i o n o f compared o b j e c t s , t h e " u t i l i t y " f u n c t i o n , t h e g e n e r a l i d e a b e i n g t h a t t h e more d i f f e r e n t a r e t h e u t i l i t i e s u ( x ) and u ( y ) , t h e m r e e s s i l y x and y can be " d i s c r i m i n a t e d " , quencies p(x,y)
and p ( y , x ) .
and t h e more d i f f e r e n t a r e t h e observed f r e F o r example, i n t h e monotone s c a l a b i l i t y model
( F i s h b u r n ( 1 9 7 3 ) ) , t h e r e e x i s t twfi mans u : X
+IR, a :
u(X)
x
u(X)
[0,11,
+
w i t h t non d e c r e a s i n ? ( n o n i n c r e a s i n g ) i n i t s f i r s t (second) arpument, and
n ( x , y ) = ~ [ u ( x ) , u ( y ) ] . One can d i s t i n g u i s h two k i n d s o f o b s e r v a b l e p r o p e r ties.
F i r s t l y , D r o n e r t i e s o f t h e ~ ( x , y ) ' s such
t i c t r a n s i t i v i t y nroqerty;
as
t h e above weak stochas-
a n o t h e r examole i s t h e weak indenendence o r o p e r -
t y (Fishburn (1973)): n(x,z)
n(v,z) i m p l i e s a ( x , t )
>
5
v(y,t),
Secondlv, D r o n e r t i e s o f t h e " t h r e s h o l d r e l a t i o n s " R X = ! ( x , y ) :
for a l l t . n(x,y)>X!,
a s s o c i a t e d w i t h t h e n ( x , y ) ' s ; f o r examole, Roberts (1971b) d e f i n e s t h e p r o 1 *:RX, El7, 11 ? i s an "homogeneous f a m i l y " o f s e m i o r d e r s . F i n a l l y ,
pert!{:
t h e theorems a s s e r t t h a t sow o b s e r v a b l e p r o p e r t i e s a r e necessary and suff i c i e n t ( " t e s t a b l e " ) c o n d i t i o n s f o r t h e e x i s t e n c e o f some e x n l i c a t i v e models. F o r i n s t a n c e , t h e weak indenendence, homogeneous f a m i l y o f semiorders and monotone s c a l a b i lit v q r o n e r t i e s a r e e q u i v a l e n t s t a t e m e n t s ( R o b e r t s and Fishburn (1971-73)). P!OIJ,
these l a s t r e s u l t s a r e p r o v e d f o r mathematical o b j e c t s c a l l e d " d i s c r i -
p i n a t i o n s t r u c t u r e " (Luce ( 1 9 6 8 ) ) , " n r o b a b i l i s t i c r e l a t i o n " , orohability function", etc.
..
"binary choice
b u t , i n f a c t , these oh.jects a r e s n e c i a l cases
o f the general natkematical n o t i o n o f A-relation,
i . e . a ( b i n a r y ) r e l a t i o n of
vrhic5 t h e o r d e r e d n a i r s a r e v a l u e d ( " w e i q h t e d " ) by the elements o f a s e t
' (some soeak o f " f u z z y r e l a t i o n " ) .
So, we can ask two q u e s t i o n s : can we
q e n e r a l i z e t h e above r e s u l t s t o a r b i t r a r y A - r e l a t i o n s ? !s i t u s e f u l ?
The
f i r s t answer i s o o s i t i v e ( w i t h h a l i n e a r l y o r d e r e d s e t ) and c o n t a i n e d i n theorem 13 o f t h i s n a o e r , c h a r a c t e r i z i n g t h e " l i n e a r " 7 - r e l a t i o n s , s u c h
a
n o t i o n h a v i n g been i n t r o d u c e d b y Luce ( 1 9 6 8 ) , see a l s o Roubens and Vincke
(1983).
F i r s t l y , i n the context o f oro-
Ve now answer t h e second q u e s t i o n .
h a b i l i s t i c c o n s i s t e n c y , t h i s g e n e r a l i z a t i o n a l l o w s t o t a c k l e non f o r c e d choices ( n ( x , y ) o(x,y)
t
+ n(y,x)
n(y,x)
<
1) and i n d i f f e r e n c e c h o i c e s ( t h a t can l e a d t o
> 1)situations.
Secondly.there
a r e lrany o t h e r c o n t e x t s i n -
d u c i n g ; . - r e l a t i o n s where t h e n o t i o n o f l i n e a r n - r e l a t i o n can be i n t e r e s t i n g ( f o r e x a m l e , c o l l e c t i v e c h o i c e , o r nonsymmetrical d i s s i m i l a r i t y ) . i t hapaens a f r e q u e n t f a c t i n mathematics:
Finally,
t h e ?enera1 t h e o r y i s s i m p l e r
and e n l i g h t e n s t h e n r e v i o u s r e s u l t s . I n s e c t i o n 2, we g i v e b a s i c n o t i o n s and r e s u l t s on b i n a r y r e l a t i o n s : R-board, s t e p - t y o e R-board, t r a c e T R , " c o m o a t i b i l i t y " w i t h a weak o r d e r , F e r r e r s r e l a t i o n s ( P r o p o s i t i o n 1 and k f i n i t i o n 2 ) , s e m i t r a n s i t i v e F e r r e r s r e l a t i o n s ( P r o p o s i t i o n s 3, 6 and D e f i n i t i o n s 4 and 5 ) .
Section 3 studies the general
n o t i o n o f homogeneous f a m i l y o f r e l a t i o n s ( D e f i n i t i o n 7 and Theorem 8), and
273
Probabilistic consistency
t h e s n e c i a l case o f a n e s t e d f a m i l y
-
o r chain
-
o f relations (Proposition
9 ) . F i n a l l y , i n s e c t i o n 4, we d e f i n e A - r e l a t i o n s , A-chains o f r e l a t i o n s ( t h e y a r e one t o one), and we use t h e r e s u l t s o f s e c t i o n 3 t o o i v e e l e v e n c h a r a c t e r i z a t i o n s o f l i n e a r A - r e l a t i o n s (theorem 13); i n c o n c l u s i o n , we q l j i c k l y D o i n t o u t f o r m e r r e s u l t s t h a t a r e s o e c i a l cases o f t h i s theorem.
N.B.
We s h a l l use t h e f o l l o w i n q terms f o r c l a s s i c a l b i n a r y r e l a t i o n s :
qumiohdeh
= r e f l e x i v e and t r a n s i t i v e r e l a t i o n ; iutnk
ohdeh = complete q u a s i -
The l i n e a r o r d e r a>b>c>d,
o r d e r ; f i n e m ofideh = a n t i s y m m e t r i c weak o r d e r . i s denoted abcd.
The dual o f a r e l a t i o n R i s t h e r e l a t i o n Rd = { ( x , y ) : ( y , x )
PQ o f r e l a t i o n s P and 0 i s d e f i n e d by x P Q
9 RI.
The p r o d u c t
y i f f ( i f and o n l y i f ) t h e r e
e x i s t s z w i t h x P z and z r) y. 2. SE!IITRANSITIVE FERRERS RELATIONS
I n t h e f o l l o w i n g s e c t i o n s , X always denotes a f i n i t e s e t , and R a b i n a r y r e l a t i o n on X . damental.
The n o t i o n o f "board r e o r e s e n t a t i o n " o f such a r e l a t i o n i s f u n -
So, we use t h e f o l l o w i n g d e f i n i t i o n s .
A R-bomd i s an o r d e r e d 3 - u o l e (R,L1,L2)
where
E i s a h i n a r y r e l a t i o n on X,
and L1 and L2 a r e t w o l i n e a r o r d e r s on X.
F R-hoard i s o t e p - Q p e i f f f o r a l l x,y,z,t
E
X,
[x L1 y, y R z and z L2 t ] * [ x
R tl.
I n t u i t i v e l y , a R-board corresponds t o a r e o r e s e n t a t i o n o f t h e r e l a t i o n R by a 0-1 t a b l e a u ( F i g u r e I ) , which l i n e s (columns) a r e l a b e l l e d b y t h e e l e ments o f X, ranked a c c o r d i n g t o t h e o r d e r L1(L2). A R-board i s s t e p - t y p e i f t h e r e e x i s t s a b o r d e r l i n e s e n a r a t i n q t h e 0 ' s f r o m t h e 1 ' s on t h e 0 - 1 t a b l e a u (Figure 2).
F i g u r e 1: (R,cdab ,dcba) I f , i n t h e R-board (R,L1,L2),
b
d
a
0
0
1
0
0
0
c
01
F i g r e 2: (R,bdca,hdac)
L1 = L 2 = L, we s h a l l s i m p l y w r i t e i t (R,L).
I n t h i s case, t h e a s s o c i a t e d r e p r e s e n t a t i o n i s an usual 0 - 1 i n c i d e n c e m a t r i x .
214
8. Monjardet
Ve now d e f i n e t h e n o t i o n o f F e r r e r s r e l a t i o n . PROPnSITIOPl I L e t R be a b i n a r y r e l a t i o n on X; t h e f o l l o w i n g statements a r e e q u i v a l e n t ( 1 ) For a l l x,y.z,t
X,
[ x b and z R t l
( 7 ) R has a s t e n - t y n e R-hoard,
( 3 ) There e x i s t two naps f.9: xRv e* q ( x ) > f ( v ) .
[xRt o r z R y l
=)
.
(R,L1,L2). X +lR such t h a t f o r a l l x,y,
F X,
"FFI?!ITINI 2 A F e x c e t ~! ~ d a t i o ni s a r e l a t i o n s a t i s f y i n g t h e c o n d i t i o n s o f o r o p o s i t i o n 1. F e r r e r s r e l a t i o n s have t e e n d e f i n e d by R i a u e t (1951) w i t h s e v e r a l e q u i v a l e n t They have been s t u d i e d by s e v e r a l
c o n d i t i o n s and e s n e c i a l l y ( 1 ) and (2).
a u t h o r s : 9ouchet (1971) who o i v e s c o n d i t i o n s ( 2 ) and ( 3 ) , r ' o n i a r d e t (1978), Co?is (1980,1982), biorders.
floionon, flucamn and Falrnaane (19O4), who use the term
I n the case o f a b i n a r v r e l a t i o n between two d i f f e r e n t s e t s , t h e
c o r r e s n o n d i n g n o t i o n has been used b y 5uttman ( i t i s t h e well-known Guttman s c a l e ( 1 9 4 4 ) ) and s t u d i e d by f l u c a m and Falmaqne ( 1 9 6 9 ) . It i s worthwhile t o n o t i c e t h a t the " a n t i r e f l e x i v e " ( o r " i r r e f l e x i v e " ) Fer r e r s r e l a t i o n s a r e t h e well-known ( a n t i r e f l e x i v e ) i n t e r v a l o r d e r s , whereas t h e r e f l e x i v e F e r m r s r e l a t i o n s a r e t h e " r e f l e x i v e i n t e r v a l o r d e r s " ( i .e. the dual o f i n t e r v a l o r d e r s ) . I f R i s a r e l a t i o n on X, we s e t
R f ( x ) = i\/ F X : xPy1
R-(x) =
{V
X : yRx1
and we d e f i n e the tllnce TR o f R by xTRy * R+(x)
2
R+(y) and R - ( x ) C_: P - ( y ) .
N o t i c e t h a t T R i s a q u a s i o r d e r b u t n o t n e c e s s a r i l y a weak o r d e r . PR0POSITInM 3 ( C o c i s ( 1 9 8 0 ) ) L e t R be a r e l a t i o n ; t h e f o l l o w i n o statements a r e e q u i v a l e n t .
-
( 1 ) R i s a Ferrers r e l a t i o n s a t i s f y i n g the f o l l o w i n g Dronerty: f o r a l l x,y.z,t
C
X,
[xRv and yRzl
[ x R t o r tRz1.
( 2 ) R has a s t e p - t y n e R-hoard ( R , L ) . ( 3 ) There e x i s t two matx f,n: X +IR, such t h a t f o r a l l x,y E X xRy and
0
a(x) > f(y)
.
f ( x ) 6 f ( Y ) * ?(X) < 4. Y ) . ( 4 ) The t r a c e TR i s a weak o r d e r .
275
Probabilistic consistency
DEFINITION 4 A nernLOLnnnLLive F e m m kelntion (S.T.F)
i s a r e l a t i o n s a t i s f y i n g t h e con-
d i t i o n s o f p r o p o s i t i o n 3. The ( n o t v e r y good) term " s e m i t r a n s i t i v e " i s due t o Chinman ( 1 9 6 0 ) . s u l t s o f t h i s d e f i n i t i o n t h a t the a n t i r e f l e x i v e S.T.F. we1 1-known ( a n t i r e f l e x i ve) semiorders (A.S .O ,)
, whereas
r e l a t i o n s a r e t h e r e f l e x i v e semiorders (R.S.O.), The r e l a t i o n R of F i g u r e s 1 and 2 is a S.T.F. (R,bdca)
It re-
r e l a t i o n s are the t h e r e f l e x i v e S .T.F
i.e. t h e dual o f semiorders.
r e l a t i o n , because t h e R-board
i s s t e o - t y p e , b u t i t i s n o t a semiorder.
ble s h a l l need another c h a r a c t e r i z a t i o n o f a STF r e l a t i o n .
L e t R be a r e l a -
t i o n : we s e t xPy * xRy and yRCx; x I y * xRy and yRx xJy * xRCy and yRCx DEFINITION 5
A weak o t d e h I4 i s compn.tLble i d t h n m,fkLLon R i f f t h e f o l l o w i n g c o n d i t i o n s are satisfied:
-
-
- R
1.1 C
(J J
xWylJz and x I z
-
x I y and y I z
xbJyldz and xJz * xJy and y J z .
PRPPOSITION 6
A r e l a t i o n R i s a s e m i t r a n s i t i v e F e r r e r s r e l a t i o n i f f t h e r e e x i s t s a weak order comnatible w i t h R.
Moreover, i n t h i s case, t h e s e t o f weak o r d e r s
c o n p a t i b l e w i t h R i s t h e s e t o f weak o r d e r s i n c l u d e d i n t h e t r a c e TR, and t h e s e t o f l i n e a r o r d e r s L such t h a t t h e R-board (R,L)
i s sten-type i s the
s e t o f l i n e a r o r d e r s i n c l u d e d i n TR. N.B.
The above d e f i n i t i o n o f c o m n a t i b i l i t y and p r o n o s i t i o n 6 g e n e r a l i z e t h e
c l a s s i c a l Roberts r e s u l t s f o r semiorders (1971a).
3. HOMOGENEOUS FAMILIES OF RELATIONS We c a l l &zndy 06 &elations on X, a n - u p l e
F
= (R1,...,Ri,...,Rn)
r y ) r e l a t i o n s on X. The & m e TF o f such a f a m i l y o f r e l a t i o n s i s d e f i n e d by n
o f (bina-
216
B. Monjardef
where Ti
i s t h e t r a c e o f t h e r e l a t i o n Ri.
= T
Ri The f o l l o w i n g d e f i n i t i o n g e n e r a l i z e s t h e Roberts d e f i n i t i o n n f an homogeneous f a m i l y o f semiorders (1971b).
DErINITIn!i 7 A ~ a m i P ~F/0 4 t e . P n Z h n n i s I ~ f - l z o r n o ~ e ~ wiofuf~ t h e r e e x i s t s a weak o r d e r ( i =1,.
comnati b l e w i t h each Ri
IJ
. .,n) .
THEnREt1 3
,..., Rn)
L e t F = ( R 1 ,..., Ri
be a f a m i l y o f r e l a t i o n s ; t h e f o l l o w i n g s t a t e -
ments a r e e q u i v a l e n t . (1)
F i s +homoqeneous.
(2)
There e x i s t s a weak o r d e r
(3)
For a l l i,j, Ti
'31
5
TF.
T . i s a weak o r d e r . 1
(5) (6)
i s a S.T.F. r e l a t i o n and f o r a l l i # j Pi(Ri+Ji)P. U P.(Ri+Ili)Pi U P . Pi(R.tJi) IJ(RitJi)Pi P. C P J J .1 1 J j' The t r a c e TF i s a rroak o r d e r ( c o m n a t i h 1 e w i t h each Ri). F i s a W-homoaeneous f a m i l y o f S.T.F. r e l a t i o n s .
(7)
There e x i s t s n + l marx fl
(6) F o r e v e r y i, Ri
i, and f o r a l l x,y xRiy
-
g(x)
>
,...,fi ,... ,fn, q:
X - R such t h a t f o r e v e r v
X
E
fib), and
fi(x)
* P(X) d "(Y).
i fi(y)
Foreover, i f these c o n d i t i o n s a r e s a t i s f i e d ,
t h e weak o r d e r s
'I
such t h a t F
i s 14-hompeneous a r e t h e weak o r d e r s i n c l u d e d i n T F . tiow, we c o n s i d e r t h e s a e c i a l case where t h e f a m i l y F i s n e s t e d ( f o r a l l Ve c a l l such a f a m i l y a chain 0 4
i,j, Ri c R . o r P C Ri). J 'j
m h t i o n n and
we w r i t e
F = (R1 3 R2
... 3
R.
1
... 3
Rn).
I n t h i s case, we d e f i n e a man nF: X 2 nF (x,y)
=
I!i
E:[l,,..,nl:
+
10
< 1
xRiyl[
...
< nl
, by,
f o r a l l x,y,
(a)
.
= Flax ( 0 , max !i E [ l , . .,nl : xRiyll
(b)
:hut, nF i s a A - r e l a t i o n (see s e c t i c n 4 ) , w i t h A = 1le d e f i n e t h e nF-scc)re o f an e l e m e n t x, by:
[o
<
1<
... <
nl.
277
Probobilktic consistency
Then, we have t h e f o l l o w i n q r e s u l t (where we use t h e d e f i n i t i o n s o f l i n e a r i -
t y and m o n o t o n i c i t y o f a A - r e l a t i o n g i v e n i n s e c t i o n 4 ) . PRnPOSITIDN 9 I f F = (R1,
...,Pi, ... ,Rn)
i s a c h a i n o f r e l a t i o n s , t h e f o l l o w i n g statements
a r e e o u i v a l e n t t o t h e statements o f theorem 1.
(8)
TF = SF.
(9)
nF i s l i n e a r .
( 1 0 ) nF has a monotone nF-board. floreover, i f i n t h i s case, Rn i s r e f l e x i v e , s t a t e m e n t ( 5 ) becomes: ( 5 ' ) F o r e v e r y i. Ri i s a r e f l e x i v e semiorder, and i f we s e t Oij = P -P j i' for all i < j
nirinij
u Q ~ ~ Iu ~0.P.P.I.~
u I ~ P c ~ PQ ~ ~ j' N.B. I f F i s an a r b i t r a r y f a m i l y o f r e l a t i o n s , we can always d e f i n e t h e r e 1J 1 1
I n t h i s case, t h e statements of prono-
l a t i o n nF b y t h e above f o r m u l a ( a ) .
s i t i o n 9 a r e necessary b u t n o t s u f f i c i e n t i n o r d e r
4. APPLICATIONS
TO
F
i s ls'-homgeneous.
A-RELATIONS: LINEAR A RELATIONS
L e t ( A , s ) be a l i n e a r l y o r d e r e d s e t , w i t h a l e a s t element O A and a g r e a t e s t e l e m e n t 1.,
A A - h e h f i o n b on t h e s e t X i s a trap 1: X 2
t h e image by l o f t h e o r d e r e d p a i r ( x , y ) . XY F o l l o w i n g Luce (1968), we d e f i n e t h e t r a c e T
-+ A ;
we denote by
l
xTp
* f o r every z
o f t h e A - r e l a t i o n 1 , by:
E X, lxz 3 b and bzx s b YZ ZY *
C l e a r l y , TL i s a q u a s i o r d e r , n o t n e c e s s a r i l y c o m l e t e .
If
A CIR. we d e f i n e t h e
l - s c o t e o f an element o f ( X , b ) by:
and t h e jcom xelulion Sb b y XSLY * Sb(X) 5 S,(Y). C l e a r l y , Sb i s a weak o r d e r , and Tb
5 sb,
DEFINITION 10
A A-relation
e
i s Pinem i f f , f o r a l l x,y,z,t
[bxz > by, o r L z x
<
l,yl
=,
[Lxt
+ byt
E X
and btx
s btty'
.
278
B. Monjardet
Ile c o u l d d e f i n e a n o t i o n of P-board,
w n e r a l i z i n o t h e n o t i o n o f R-board.
Put, here. r e o n l y use t h e f o l l o w i n g d e f i n i t i o n . DEFIFlITInN 11
P A - r e l a t i o n l has a monotone !-hoard X, such t h a t f o r a l l x,y,z,
6
i f f , t h e r e e x i s t s a l i n e a r o r d e r L on
X,
xLy * P x z :: Pyz and Pzx I P
ZY
*
I n t u i t i v e l y , when l has a r o n o t o n e (-board,
i t can be r e o r e s e n t e d by a
t a b l e a u whose t h e l i n e s and t h e columns a r e l a b e l l e d a c c o r d i n g t o
L, and
w i t h nondecreasina m o n o t o n i c i t y o f t h e elements i n a l i n e o r i n a column ( s e e P x m D l e , F i q t r e 3).
Fiyre 3 Ire use t h e f o l l o w i n g v a r i a n t o f t h e F i s h b u r n d e f i n i t i o n o f monotone s c a l a -
b i l i t y (1.372).
DEFINITIOU 12 A '-relation 7 : u(X)
P i s nionctonic&r/
u(X)
6 c d n b l e i f f t h e r e e x i s t two mans u: X +iR,
+
h , such t h a t f o r a l l x,y.
X, B
=@[u(x),u(y)l w i t h @ XY non i n c r e a s i n g (non d e c r e a s i n o ) i n i t s f i r s t (second) araument. x
E
Now, we show t h a t t h e n o t i o n o f \ - r e l a t i o n i s e q u i v a l e n t t o t h e n o t i o n o f A-chain o f r e l a t i o n s . A f-chniri c < hctktio~t,! on X i s a c h a i n C o f r e l a t i o n s o f X, i n d e x e d by an i n c r e a s i n q sequence o f elements of A : 0, ,< a1 <
x2 ,...,
<
xi
<
... x n < h 1 '
'Je w r i t e :
z
C = X > R X ' R
1
x2
... > R
'i
...
>@.
R 'n
There e x i s t s a c a n o n i c a l b i j e c t i o n between t h e s e t o f . ( - r e l a t i o n s on X and t h e s e t of !:-chains
-
o f r e l a t i o n s on X . 2 ) =
I f E i s a h - r e l a t i o n , and i f !(X
t i n c t values taken by relations o f
P:
x1
<
x2
P., we d e f i n e t h e h - c h a i n
... <
Xn i s t h e s e t o f d i s -
Ce b y means o f t h e " t h r e s h o l d "
279
Probabilistic cotuisteiicy
2
= X
C,
2
R, 1
... 2
= ~ ( x , y ) E x': 'i N o t i c e t h a t i f we s e t T
R, i
... 3 -
3
fl
'n
t x y5 a.1. 1
where F
, we
= Trace o f R
'i
-
R
U
have T,
=
'i
I f C i s a A-chain o f r e l a t i o n s , i n d e x e d by
x1
<
... <
Ti i=l i Q
.
... <
Xi
A n y we de-
f i n e a A - r e l a t i o n l , , by: Lc(xyy) = N.B.
OAV(Vlx
E
ikl,...,
(XJ)
An):
E
RX)).
The above b i j e c t i o n i s an isomorphism between t h e two s e t s o f A - r e l a -
t i o n s and A-chains c f r e l a t i o n s , endowed w i t h n a t u r a l o r d e r s . Now, we can use t h e s e c t i o n 3 r e s u l t s t o c h a r a c t e r i z e l i n e a r . ( - r e l a t i o n s and answer a q u e s t i o n i n i t i a t e d by Luce (1958).
THEnREP1 13 L e t C be a A - r e l a t i o n on X, and Ce t h e a s s o c i a t e d A-chain o f r e l a t i o n s ; t h e f o l l o w i n g statements z r e e q u i v a l e n t .
(1)
i s W-homogeneous.
C,
( 2 ) There e x i s t s a weak o r d e r bI 5 T., 2 n Txj i s a weak o r d e r . ( 3 ) F o r a l l xi, x j E t ( X ) , T 2 'i (4) F o r e v e r y xi € [ ( X ) , R i s a S.T.F. r e l a t i o n , and i f we s e t 'i
i s a weak o r d e r ( c o m p a t i b l e w i t h each R,
(5)
The t r a c e T,
(6)
C,
(7)
There e x i s t s n + l mam fl,.
i s a bl-homoaeneous f a m i l y o f S.T.F.
a l l x,y XR+Y
(8)
T,
.. ,fi ,. . .,f n,
relations.
). i
q such t h a t f o r e v e r y i and f o r
X
* gi(x)
> fb),
and fi(x)
d fi(y)
* dx) s dy).
= .S,
i s linear. has a monotone ,-board. i s monotonically scalable. t h a t c o n d i t i o n ( 8 ) g i v e s a v e r y e f f i c i e n t way t o r e c o g n i z e i f a A-re-
t
i s l i n e a r : f o r a l l x,y w i t h xS,y,
t o t e s t i f xT,y.
zso
B. Monjardct
No\,, we c o n s i d e r p a r t i c u l a r cases o f l i n e a r A - r e l a t i o n s . = a. t h e n f o r e v e r y 1 2 a ( , ( 1 ) If f o r e v e r y x , t X x ( a n t i r e f l e x i ve) s e w i o r d e r .
< a),
R,
i s a reflexive
+ Pyx = 5 , we say ( 2 ) I f A i s a s e t o f numbers, and i f f o r a l l x,y, lxy B t h a t f? i s i p s o d u a l : then, t h e t h r e s h o l d r e l a t i o n s R,, for < 7 , are the
dual o f t h e R,,
for
2
c h a i n Cq o f r e l p t i o n s .
0
.
Thus, we can o n l y c o n s i d e r t h e " h a l f " o f t h e
The r e s u l t s o b t a i n e d i n t h i s case by Jacquet-Laqreze
(197s) and Roubens and Vinkce ( 1 9 8 3 ) , corresaond r e s n e c t i v e l y t o e q u i v a l e n ces o f statements ( 6 ) , ( 8 ) , ( 1 0 ) and ( 7 ) , ( 9 ) . ( l o ) , o f theorem 13. B ( 3 ) If f o r a l l x,y. .! + .! = B, and moreover P = i m p l i e s xTly and YX XY XY yT x, one o b t a i n s r e s u l t s o f Roberts on homgeneous f a m i l i e s o f semiorders
. e
(1971b). (4)
F i n a l l y , i n t h e case o f an homoqeneous f a m i l y o f two semiorders
(P u n,P),
c o n d i t i o n ( 4 ) becomes P I (2 u 0 I P
CJ
O P I
LJ
I P r)
CI
P.
This
case has been s t u d i e d b y Cozzens and Roberts ( 1 9 8 2 ) , and Roy and Vincke (1904) ; f o r example, t h e y b o t h g i v e necessary and s u f f i c i e n t r e n r e s e n t a t i o n s i n o r d e r t h a t n u m e r i c a l r e n r e s e n t a t i o n s w i t h two c o n s t a n t t h r e s h o l d s e x i s t ( i . e . t h e r e e x i s t fiyg
with
xPy i f f g ( x ) > f 2 ( y ) , x ( e x ) ) y i f f g ( x ) > f l ( y ) ,
and f 2 - g
E
b f 1- g :p>O).
REFERENCES Bouchet, A . , Etude c o m b i n a t o i r e , w p l i c a t i o n s , These, l J n i v e r s i t 6 Ycient i f i q u e e t M e d i c a l e de Grenoble ( 1 9 7 1 ) . Chioman, J.S.,
Consumption t h e o r v w i t h o u t t r a n s i t i v e i n d i f f e r e n c e , i n
P r e f e r e r ~ c e s , U t i l i t y and Demand, New York, t l a r c o u r t Brace (1971). Coqis, O.,
@ t e r m i n a t i o n d ' u n o r e o r d r e t o t a l c o n t e n a n t un n r e o r d r e e t
contenu dans une r e l a t i o n de F e r r e r s l o r s o u e l e u r domaine commun e s t f i n i , C.R.A.S., Cogis, O . ,
283 S 6 r i e A (1976), 1007-1009.
A nronos des q u a s i - o r d r e s
-
Note, t l a t h . S c i
. Hum.,
72 (1980)
107-111. Cogis, O.,
F e r r e r s d i g r a o h s and t h r e s h o l d graDhs, D i s c r . Math. 38
( 1982) , 33-46. Cozzens, M.B.,
and Roberts, F.S.,
Double s e p i c r d e r s and double i n d i f -
f e r e n c e GraDhs, SIAW, J . A l g . D i s c r . Math., Doignon, J.-P.,
Ducamp, A . and Falmagne, J.C.,
3-4 (1982). 566-583. On r e a l i z a b l e b i o r d e r s
and t h e b i o r d e r dimension o f a r e l a t i o n , J. Math. Psych., (1984).
t o appear
281
Probabilistic consistency
Ducamp, A.,
and Falmagne, J.C.,
Composite measurement, J . Math. Psych..
6 (1969) , 359-390. F i s h b u r n , P.C.,
B i n a r y c h o i c e o r o b a b i l i t i e s : on t h e v a r i e t y o f stochas-
t i c t r a n s i t i v i t y , J. Path. Psych. Jacquet-Laareze,
E.
,
10-4 (1973), 327-352.
, Reoresentations
de q u a s i - o r d r e s e t de r e l a t i o n s
D r o b a b i l i s t e s sous forme s t a n d a r d e t e t h o d e s d ' a p p r o x i m a t i o n , Math. S c i . Hum., 63 (1978), 5-24. Luce, R.C.,
A P r o b a b i l i s t i c t h e o r y o f u t i l i t y , Econometrica, 26 (1958),
193-224. M i r k i n , B.G.,
D e s c r i p t i o n o f some r e l a t i o n s on t h e s e t o f r e a l - l i n e
i n t e r v a l s , J . 'lath. Psych., 9 (1972), 243-252. Monjardet, B.,
and J a c q w t - L a g r e z e ,
e t q u a s i - o r d r e s , Vath. S c i Ilonjardet, B.
. Hum.,
, Axiomatiques
E.
, Modelisation
des p r @ f & r e n c e s
62 (1978) , 5-10.
e t D r o p r i e t e s des q u a s i - o r d r e s , Math. S c i .
Hum., 63 (1978), 51-82. R i g u e t , J., Les r e l a t i o n s de F e r r e r s , C.R.A.S.,
232, P a r i s (1951) ,
1729-1730. Roberts, F.S., J. Comb. Th., Roberts, F.S.
On t h e c o m n a t i b i l i t y between a granh and a s i m p l e order,
11 (1971a), 248-263.
, Homogeneous
fami l i e s o f semiorders and t h e t h e o r y o f
p r o b a b i l i s t i c c o n s i s t e n c y , J. Math. Psych., Roberts, F.S.
, tkasurement
8 (1971b), 248-263.
t h e o r y , Encyclopaedia o f mathematics and
i t s a p p l i c a t i o n s , Reading, Mass., Addison-Wesley (1979). Roubens, M., and Vincke
, Ph. , L i n e a r
f u z z y graphs, Fuzzy s e t s and s y s -
tems, 10 (1983), 79-86. Roy, B.,
and Vincke, Ph.,
Pseudo-orders:
d e f i n i t i o n , n r o p e r t i e s and
n u m e r i c a l r e p r e s e n t a t i o n (1983), s u b m i t t e d .
This Page Intentionally Left Blank
TRENDS IN MATHEMATICAL PSYCHOLOGY
E. DegreefandJ. Van Bu g e n h u t (editors) 0 Ekevier Science hblisfers B.V. (North-Holland), 1984
283
APPLICATIONS OF THE THEORY OF MEANINGFULNESS TO ORDER AND MATCHING EXPERIMENTS Fred S . Rnberts Department o f Ya thema t i cs Rutqers U n i v e r s i t y New Brunswick, New Jersey, USA
The t h e o r y of meaningfulness of statements u s i n g numerical scales i s applied t o the analysis o f s t a t e ments i n v o l v i n q s c a l e s r e p r e s e n t i n g t h e semiorders and i n d i f f e r e n c e graphs which r e s u l t from o r d e r and matching e x n e r i m e n t s ,
1. INTRODUCTION The conceDt o f meaninqfulness i s among t h e most u s e f u l conceDts o f measurement t h e o r y .
T h i s concent has found a p p l i c a t i o n s t o such areas as measure-
ment o f average performance, imoortance r a t i n g s , consumer p r i c e i n d i c e s , n o l l u t i o n measures, i n d i c e s o f consumer confidence, nsychophysical s c a l i n g , a n a l y s i s o f s o c i a l networks, s t r u c t u r a l modeling i n decisionmaking, and t h e dimensional i n v a r i a n c e o f s c i e n t i f i c laws. veyed i n Roberts (1979, 1983a).
Such a p p l i c a t i o n s a r e s u r -
I n t h i s paper, we a o n l y t h e t h e o r y o f
meaningfulness t o t h e a n a l y s i s o f o r d e r and matching experiments when matching i s not transitive. The paper i s o r g a n i z e d as f o l l o w s . o r d e r and matching experiments.
S e c t i o n 2 d e s c r i b e s what we mean b y
S e c t i o n 3 summarizes t h e r e l e v a n t concepts
f r o m t h e t h e o r y of meaningfulness i n measurement.
F i n a l l y , S e c t i o n 4 con-
t a i n s t h e main r e s u l t s and S e c t i o n 5 g i v e s t h e D r o o f s .
2. ORDER AND MATCHING EXPERIMENTS A v a r i e t y o f experiments i l i v o l v e judgements such as "Iam i n d i f f e r e n t between x and y", " x and y a r e s i m i l a r " , " x and y a r e t h e same", " I cannot t e l l x and y apart", periments.
o r "x and y match".
We s h a l l c a l l these m a t c h i n q ex-
R e l a t e d exoeriments i n v o l v e judgements o f o r d e r , such as " I
p r e f e r x t o y " , " x i s b i g g e r than y " , " x i s b e t t e r than y " , and so on. s h a l l c a l l these o r d e r e x D e r i n e n t s .
We
Coombs (1964) d i s t i n g u i s h e s these two
284
F.S. Roberts
k i n d s o f experiments bv t a l k i n q a b o u t D r o x i m i t y r e l a t i o n s and o r d e r r e l a tions.
F o r d i s c u s s i o n o f o r d e r and matchirlc e x p e r i m e n t s , see, f o r i n s t a n c e ,
G a l a n t e r (1956). Foodman (1951), o r R e s t l e (1959). I t was observed by Armstrona (1939) t h a t observed m a t c h i n g i s o f t e n n o t
transitive. century.
S i m i l a r o b s e r v a t i o n s oo back t o Poincari. i n t h e n i n e t e e n t h
I n resDonse t o such o b s e r v a t i o n s , Luce (1956) i n t r o d u c e d t h e no-
t i o n of s e m i o r d e r t o a c c o u n t f o r o r d e r ( p r e f e r e n c e } judoements when matchina (indifference) i s not transitive.
I n particular, i f A i s a f i n i t e set
and P i s a b i n a r y r e l a t i o n on A , (A,P) f u n c t i o n f: A + Re s a t i s f y i n g
i s a s e m i o r d e r i f we can f i n d a
aPb * f ( a ) > f ( b ) + 6 ,
(1)
where 6 i s a f i x e d n o s i t i v e number measuring t h r e s h o l d o r . j u s t n o t i c e a b l e difference.
Semiorders were c h a r a c t e r i z e d b y Luce (1956) and by S c o t t
and Surtnes ( 1 9 5 8 ) .
Analogous t o s e m i o r d e r s a r e i n d i f f e r e n c e flraphs.
A i s a f i n i t e s e t and I i s a b i n a r y r e l a t i o n on A, ( A , I )
-
graoh i f we can f i n d a f u n c t i o n f: A aIb
-
-+
If
i s an i n d i f f e r e n c e
Re s a t i s f y i n a
I f ( a ) - f ( b ) l s 6,
(2)
where a g a i n 6 i s a f i x e d D o s i t i v e number measurinq t h r e s h o l d o r j u s t noti ceable d i f f e r e n c e .
I n d i f f e r e n c e qraDhs were c h a r a c t e r i z e d b y R o b e r t s
(1969). I n t h i s Daper, we s h a l l s t u d y t h e meaningfulness o f v a r i o u s a s s e r t i o n s i n v o l v i n g t h e f u n c t i o n s f w h i c h s a t i s f y Equations ( 1 ) and ( 2 ) . 3. MEANINGFULNESS The t h e o r y o f meaningfulness o f s t a t e m e n t s i n v o l v i n g s c a l e s o f measurement s t e m s f r o m the work o f Suones (1959) and Supoes and Zinnes (1963), though i t has i t s r o o t s i n t h e n o t i o n o f i n v a r i a n c e w h i c h goes back much e a r l i e r .
The t h e o r y i s sumnarized i n Roberts (1979).
We g i v e o n l y t h e most b a s i c
d e f i n i t i o n s h e r e , r e f e r r i n g t h e r e a d e r t o Roberts (1979) f o r a l l u n d e f i n e d terms. Sunnose 4 and 8 a r e two r e l a t i o n a l systems, w i t h A t h e u n d e r l y i n g s e t o f
A and B t h e u n d e r l y i n q s e t o f . 8 .
I f f: A
-+
B i s a homomornhism f r o m A
285
Theory of meaningfulness
into
8, we c a l l t h e t r i o l e (A,E,f),
and sometimes j u s t t h e f u n c t i o n f, a
scale. -
I t i s a numerical s c a l e i f R i s a s e t o f r e a l numbers.
4: f ( A )
+
a scale.
A function
B i s c a l l e d an a d m i s s i b l e t r a n s f o r m a t i o n o f t h e s c a l e f i f
$of
is
Suppes (1959) and Suppes and Zinnes (1963) say t h a t a s t a t e m e n t
i n v o l v i n g n u m e r i c a l s c a l e s i s meaningful i f i t s t r u t h v a l u e i s unchanged i f admissible transformations are a n o l i e d t o every scale i n the statement.
U n f o r t u n a t e l y , as Roberts and Franke (1976) o o i n t o u t , t h e r e a r e some d i f ficulties with this definition.
To see t h i s , i t s u f f i c e s t o c o n s i d e r t h e
semiorder d e f i n e d by
A = {a,h,cl,
P = {(c,a),(c,h)l.
(3)
Then two homomorphisms f and q s a t i s f y i n g E q u a t i o n (1) w i t h 6 = 1 a r e g i v e n b.Y f ( a ) = 0, f ( b ) = 0, f ( c ) = 2
(4)
and a ( a ) = .l,g ( b ) = 0, g ( c ) = 2. I t follows t h a t
and moreover t h a t
Hence, by t h e Suppes-Zinnes t h e s t a t e m e n t ( 6 ) is m e a n i n g f u l . However, we a l s o have
f o r any a d m i s s i b l e t r a n s f o r m a t i o n 4 o f f. definition,
The a s s e r t i o n t h a t a and b r e c e i v e t h e same s c a l e values i s s a t i s f i e d f o r one homomorphism and v i o l a t e d f o r a n o t h e r .
I t seems meaningless. T h i s We say a s t a t e m e n t i n v o l v i n a
l e a d s us t o m o d i f y t h e d e f i n i t i o n as f o l l o w s ,
286
F.S. Roberts
n u m e r i c a l s c a l e s i s meaningful ifi t s t r u t h v a l u e i s unchanoed if e v e r y s c a l e i n t h e s t a t e m e n t i s r e p l a c e d b v a n o t h e r a c c e n t a b l e one. more q e n e r a l d e f i n i t i o n , s t a t e m e n t ( 6 ) i s meaninqless.
i s w i d e l y acceoted, a t l e a s t i n s n i r i t .
Under t h i s
This d e f i n i t i o n
F o r a l t e r n a t i v e f o r m u l a t i o n s , see
Adams, e t a1 ( 1 9 6 5 ) , Falmaone and Narens ( 1 9 8 3 ) . Luce ( 1 9 7 8 ) , Narens (1981), Pfanzaql (1968). Roberts (1980), Roberts and Franke (1976), and Robinson
f 1963). A s c a l e (A,Y,f)
i s c a l l e d r e a u l a r i f f o r any o t h e r s c a l e (b,F,q),
a f u n c t i o n 6 on f ( A ) s o t h a t q =
$of.
I t i s e x a c t l y the i r r e q u l a r i t y o f
t h e homomorphism f o f E q u a t i o n ( 4 ) w h i c h causes d i f f i c u l t i e s . no
6
on f ( A ) so t h a t o =
$of,
there i s
f o r q as d e f i n e d i n ( 5 ) .
There i s
Roberts and Franke
(1976) observe t h a t i f e v e r y s c a l e i n q u e s t i o n i s r e g u l a r , t h e n t h e o r i g i n a l Supoes-Zinnes d e f i n i t i o n o f meaningfulness can be used, b u t o t h e r w i s e t h e more g e n e r a l d e f i n i t i o n i s necessary. 4 . THE ANALYSIS OF ORDER AND WATCHING EXPERIMENTS I n t h i s s e c t i o n , we a p p l y t h e conceot o f meaningfulness t o statements i n v o l v i n a s c a l e s s a t i s f y i n g E q u a t i o n s (1) and ( 2 ) .
Perhaos t h e most common
k i n d o f a s s e r t i o n u s i n g s c a l e s o f measurement i s t h e a s s e r t i o n
See Roberts (1983b) f o r a p e n e r a l t r e a t m e n t o f t h e meaningfulness o f t h e assertion (8).
We s h a l l i n v e s t i g a t e t h i s and r e l a t e d a s s e r t i o n s h e r e .
F o r t h e i n d i f f e r e n c e graoh r e n r e s e n t a t i o n ( 2 ) , t h e c o n c l u s i o n ( 8 ) i s i n g e n e r a l meaningless.
For i f f s a t i s f i e s (2).
t h e n so does - f .
The s c a l e s
f and g i n Equations ( 4 ) and ( 5 ) show t h a t a c o n c l u s i o n ( 8 ) can a l s o be
meaningless f o r t h e s e m i o r d e r r e o r e s e n t a t i o n ( 1 ) .
For f ( a ) > f ( b ) f a i l s ,
b u t g(a) > g(b) holds. Supnose n e x t t h a t we d e f i n e a s e m i o r d e r b y t a k i n g
A = {a,b,c),
P = {(a,b),(b,c),(a,c)l.
Then f o r e v e r y f s a t i s f . y i n g ( l ) , we have f ( a ) > f ( b ) s i o n f ( a ) > f ( b ) i s meaningful. m e a n i n q f u l f o r e v e r y (x,y)
E AxA.
+
6,
so t h e c o n c l u -
I n f a c t , i n t h i s examole, f ( x ) > f ( y ) i s We s h a l l see under w h a t c o n d i t i o n s a
287
Theory of meaningfulness
s i m i ’ l a r c o n c l u s i o n can be drawn.
lie say t h a t xEy h o l d s iff o r a l l z, xPz * yPz and ZPX * zPy. I t i s c l e e r t h a t (A,E) i s an e q u i v a l e n c e r e l a t i o n . I n t h e example (j), aEb, we nave t h e f o l 1 owing r e s u l r: Theorem 1: SliprrOSe A i s f i n i t e and f i s a h o r o m r p h i s m on ( A , P ) s a t i s f y i n g
(1).
Then f ( x )
> f(y)
is meaningful f o r a l l (x,y) F
AxA
i f and o n l y i f
whencv.er x # y, ~ x E y . F o r a D r o o f o f Theorem 1, see t h e n e x t s e c t i o n . O f course, we c o u l d n o t e x p e c t t h e c o n c l u s i o n f ( a ) > f ( b ) t o be m e a n i n g f u l
f o r a matching exoeriment, i . e . , o r d e r judgements.
one where we j u s t have matching b u t no
On t h e o t h e r hand, t h e n o t i o n o f o r d e r i s o f t e n cap-
t u r e d by judgments o f betweenness. and we m i g h t e x p e c t judgments o f betweenness t o be m e a n i n g f u l . We say t h a t a r e a l number 6 i s between two r e a l numbers a and y i f a < 6 < y o r y < 6 < a.
If f satisfies (2), i s i t
m e a n i n g f u l t o conclude t h a t f ( b ) i s between f ( a ) and f ( c ) ? To answer t h i s q u e s t i o n , l e t us say xEy h o l d s i f f o r a l l z, x I z * y I z . Again, i t f o l l o w s t h a t (A,E)
i s an e q u i v a l e n c e r e l a t i o n .
F o r i n s t a n c e , suppose
A = {a,b,u,v,a,5)
Then we have aE6. t h e r e a r e x1,x2
We say t h a t ( A , I ) i s connected i f f o r a l l x and y i n A ,
,..., x P
so t h a t xIxl,xlIx2
,...,X ~ - ~ I X(A.1) ~ , oX f ( 9~ ) I ~ .
i s n o t connected, as t h e above f a i l s w i t h x = a, y = u.
we have t h e f o l -
lowing r e s u l t : Theorem 2: Suppose A i s f i n i t e and f i s a homomorphism on (A,I)
satisfying
(2), and suppose t h a t I f ( A ) I 2 3. Then t h e c o n c l u s i o n t h a t f ( y ) i s between f ( x ) and f ( z ) i s meaningful f o r a l l x,y,z conditions hold: ( a ) F o r a l l x # y, sxEy. ( b ) (A,I)
i s connected.
i n A i f and o n l y i f t h e f o l l o w i n g
288
F.S.
Roberts
F o r a n r o o f o f Theorem 2 , see t h e n e x t s e c t i o n .
To i l l u s t r a t e t h i s theo-
rem, l e t us t a k e & = 1 and d e f i n e f and g f o r ( A . 1 )
o f ( 9 ) as f o l l o w s :
f ( a ) = 0, f ( b ) = .6, f ( u ) = 4 , f ( v ) = 5 . 1 , f ( a ) = 4 . 2 , f ( 8 ) = 4 . 3 , and o ( a ) = 10, g ( b ) = 9.4,
o ( u ) = 3 , q ( v ) = 4 . 1 , q ( a ) = 3.3, g ( 6 ) = 3 . 2 .
Then we have f ( a ) between f ( a ) and f ( v ) , b u t g ( u ) n o t between g ( a ) and a ( v ) ; we a l s o have f ( a ) hetween f ( u ) and f ( B ) ,
b u t g(a) n o t between g ( u ) and g ( 8 ) .
Both c o n d i t i o n s o f Theorem 2 a r e v i o l a t e d : we have aE8 and we have ( A , I ) disconnected. To summarize:
Betweenness c o n c l u s i o n s can be drawn f r o m m a t c h i n g e x p e r i -
ments, b u t o n l y under c e r t a i n c o n d i t i o n s .
5.
PROOFS OF THEOREK 1 AND 2
To Drove Theorem 1, we use t h e n o t i o n o f s i m o l e o r d e r (A,R)
compatible
w i t h t h e b i n a r y r e l a t i o n (A,P) w h i c h i s i n t r o d u c e d i n Roberts (1971). Suonose (A,R)
i s a s i m n l e o r d e r and (A,P)
i s an asymmetric b i r a r y r e l a t i o n .
D e f i n e I on A by x I y * zxPy & %YPX. We say (A,R) i s c o m p a t i b l e w i t h (A,P)
if
xPy * xRy and xRyRz & XIZ xIy & yIz.
(11)
=)
kssuming A i s f i n i t e , Roberts (1971) p r w e s t h e f o l l o w i n g :
111
I f (A,P)
i s an asymmetric b i n a r y r e l a t i o n , t h e n t h e r e i s a s i m p l e o r -
d e r R on A c o m p a t i b l e w i t h (A,P)
I21
i f and o n l y i f (A,P)
i s a semiorder.
Every c o m p a t i b l e s i m p l e o r d e r R on A i s o b t a i n e d as f o l l o w s :
there is
289
Theory ofmeoningfufness
lr
a unique s i m o l e o r d e r i n g R
of e q u i v a l e n c e c l a s s e s under E, and we
then o b t a i n R f r o m R* by o r d e r i n g elements w i t h i n an e q u i v a l e n c e c l a s s arbitrarily
(31
.
I f R i s a c o m a t i b l e s i m p l e o r d e r on A, then t h e r e i s a homomorphism f ( f u n c t i o n s a t i s f y i n g (1)) so t h a t
homomorphism and R i s d e f i n e d by ( 1 2 ) . and
Conversely, i f f i s a
sxEy f o r a l l x # y , then R i s a c o m p a t i b l e s i m o l e o r d e r on A. SuDpose xEy, x # y .
P r o o f o f Theorem 1.
By
[a
above, t h e r e i s a s i m p l e
o r d e r R on A, compatible w i t h P, i n which x f o l l o w s y. and a n o t h e r R ' i n which y f o l l o w s x.
By [31 above, t h e r e a r e homomornhisms f and g so t h a t
f ( x ) > f ( y ) and g ( x ) < g ( y ) .
+y
Conversely, suppose x
c o m p a t i b l e s i m p l e o r d e r R. l a t e d t o R by ( 1 2 ) .
We conclude t h a t f ( x ) > f ( y ) i s meaningless. Then by [21, t h e r e i s a unique
i m l i e s cxEy.
Moreover, by [3], ever,y homomorphism f i s r e -
Thus, f o r ever.v (x,y)
E AxA,
and f o r a l l homomorphisms
f and g, f ( x ) > f ( y ) i f fg ( x ) > g ( v ) .
0.E.D. To p r o v e Theorem 2, supoose ( A , I ) i s r e f l e x i v e and symmetric.
L e t us say
t h a t a s i m p l e o r d e r R on A i s c o m n a t i b l e w i t h I i f Equation (11) h o l d s . Then assuming A i s f i n i t e , Roberts (1571) proves t h e f o l l o w i n g :
I f (A,I)
i s a r e f l e x i v e , symmetric b i n a r y r e l a t i o n , then t h e r e i s a
s i m n l e o r d e r on (A,I)
c o m a t i b l e w i t h (A,I)
i f and o n l y i f ( A , I ) i s
an i n d i f f e r e n c e graDh. Supoose t h a t (A,I)
i s connected,
Then e v e r y c o m p a t i b l e s i m p l e o r d e r
R on A i s o b t a i n e d as i n [2] above, e x c e p t t h a t f o r each such s i m p l e
order, reversing the e n t i r e order gives another comoatible simple o r der.
I f (A,I)
i s n o t connected, i t s connected comoonents a r e i t s maximal
i n d u c e d connected s u b r e l a t i o n s .
R on A as f o l l o w s .
We f i n d a l l c o m p a t i b l e s i m p l e o r d e r s
F i n d one c o m o a t i b l e s i m o l e o r d e r f o r each connect-
ed component o f ( A , I ) ,
and t h e n o r d e r these o r d e r s a r b i t r a r i l y .
190
171
F.S. Roberts
Statement [3] ahove h o l d s ( w i t h homomornhisms h e i n s f u n c t i o n s s a t i s fyinq ( 2 ) ) . Sunnose x # v and xEy.
F r o o f o f Theorem 2.
z # x,y.
S i n c e l f ( l \ ) I z 3, we can f i n d
Now by 151 and 161, t h e r e a r e c o m o a t i b l e s i m n l e o r d e r s R. i n
which x f o l l o w s y f o l l o w s z, and R ' , i n which y f o l l o w s x f o l l o w s z . by 171, t h e r e a r e homomornhisms f and q w i t h f ( x ) q(vl > s(x) > q(z).
>
f(y) > f(z)
Thus,
and
The c o n c l u s i o n t h a t f ( y ) i s hetween f ( x ) and f ( z ) i s
waninsless, Sunnose C = ( A , I ) comments.
i s n o t connected.
Thus, t h e r e a r e a t l e a s t two connected
I f t h e r e a r e t h r e e connected comoonents, K1,
Kz, K3,
these
comnonents csn t e o r d e r e d a r h i t r a r i l v i n a c o m p a t i h l e s i m n l e o r d e r ( b y 161 1. F i x x i n K1,
y i n Kz,
z i n Kj.
Then v i s between x and z i n some coninati-
h l e s i m o l e o r d e r and n o t i n a n o t h e r .
Then b v [ 7 1 , f ( v ) i s between f ( x )
and
f ( z \ i n some homomornhism, h u t q ( y ) i s n o t between q ( x ) and q(z) i n a n o t h e r homomorohism.
Next, sunnose t h e r e a r e two connected ccmnonents, K1 and K 2 .
Since l f ( 4 ) ' 2 3. a t l e a s t one o f these comnonents, say K1, L e t z be i n K2.
ments x and y .
has two e l e -
Then by 151, t h e r e a r e c o m p a t i b l e s i m p l e
o r d e r s o f K1 i n which x f o l l o w s y and i n w h i c h y f o l l o w s x.
I f we o r d e r
K2 b e f o r e K1, we o b t a i n c o m n a t i h l e s i m o l e o r d e r s f o r ( A , I ) i n w h i c h x f o l l o w s v f o l l o w s z and i n which y f o l l o w s x f o l l o w s z .
As above, we can
f i n d homomornhisms f and q so t h a t f ( y ) i s between f ( x ) and f ( z ) and g ( y ) i s n o t between a ( x ) and o ( z ) . F i n a l l y , suppose ( a ) and ( b ) h o l d .
Then 151 savs t h a t t h e r e a r e e x a c t l y
two c o m o a t i b l e s i m n l e o r d e r s on ( A , I ) ,
one t h e r e v e r s e o f t h e o t h e r .
Hence, y i s between x and z i n b o t h o f t h e s e , or i n n e i t h e r .
Thus by 171 ,
f ( v ) i s between f ( x ) and f ( z ) i n a l l homomornhisrns o r i n none, O.E.D.
REFERENCES [I]
Adams, E.W.,
Faqot. R.F.,and
Robinson, R.E.,
A Theory o f A p p r o p r i a t e
S t a t i s t i c s , Psychometrika, 30 (1965), 99-127.
I21
Armstrong, W.E., Jrnl
. , 49
The Determinateness o f t h e U t i 1 i t . y F u n c t i o n , Econ.
(1939), 456-467.
I31
Coombs, C.H.,
141
Falmagne, J.C.,
A Theory o f Data, !.liley, New York,
(1964).
and Narens, L., Scales and Meaninafulness o f Q u a n t i -
t a t i ve Laws, Synthese , 55 ( 1983)
, 287-325.
nleory of meanirigfulness
291
Galanter, E . H . , An Axiomatic and Exoerimental Study of Sensory Order a n d Yeasure, Psychol. Rev., 63 (1956), 16-28. Goodman, N . , Structure of Aonearance, Harvard Uni versi ty Press, Cambridge, (1951). Luce, R . D . , Semiorders and a Theory of U t i l i t y Discrimination, Econometrica, 24 (1956), 178-191. Luce, R . D . ,
Dimensionally Invariant Numerical Laws Correspond t o Mean-
ingful Q u a l i t a t i v e Relations, Philos. S c i . , 45 (1978), 1-16. Narens, L . , A General Theory of Ratio S c a l a b i l i t y W i t h Remarks About the Measurement-theoretic Concept of Meaningfulness , Theory and Decis i o n , 13 (1981) , 1-70. Pfanzagl, J . , Theory of Measurement, bliley, New York, (19681 Restle, F . , A Metric and an Ordering on S e t s , Psychometrika, 24 (1959), 207-220. Roberts, F . S . ,
Indifference Graohs, in F. Harary ( e d . ) , Proof Tech-
niques i n Graph Theory, Academic Press, New Yorh (1969), 139-146. Roberts, F.S., On the Comnatibility between a Graoh and a Simple Order, J . Comb. Theory, 11 (1971), 28-38. Roberts, F.S. , Hkasurement Theor,y, w i t h Aoolications t o kcisionmaking, U t i l i t y , and the Social Sciences, Addison-Wesley, Reading, MA (1979). Roberts, F.S., On Luce's Theory of Meaningfulness, Philos. S c i . , 47 (1980), 424-433. Roberts, F. S. , Aoplications of the Theory of Pleaningfulness t o Psychology , mimeographed, Department of Mathematics, Rutgers University, New Brunswi ck, NJ, (1983a). Roberts, F.S., On the Theory of Meaningfulness of Ordinal Comparisons, Measurement, (1983b) , t o apoear. Roberts, F.S. and Franke, C . H . , On the Theory of Uniqueness i n Measurement, J . Hlath. Psychol , 14 (1976), 211-218.
.
Robinson, R . E . ,
A Set-theoretical Approach to Emnirical Meaningfulness o f Empirical Statements, Tech. R e p t . No. 55, I n s t i t u t e f o r Mathematical Studies i n the Social Sciences, Stanford University, Stanford, C A , (1963). S c o t t , D.,
and Suppes, P . , Foundational Aspects o f Theories of k a s -
u r e m e n t , J . Symbolic Logic, 23 (1958), 113-128. Suppes, P . , Measurement, Emnirical Meaningfulness and Three-Valued Logic, i n C.W. Churchman and P . Ratoosh ( e d s . ) , Measurement: Definitions and Theories, I!iley, New York (1959), 129-143.
291
F.S. Roberts
1221 Sunpes, P . , and Zinnes, J., B a s i c Measurement Theorv, i n R.D.
Luce,
P.R. Bush, and E . G a l a n t e r ( e d s , ) , Handbook o f ! ' i t h e m a t i c a l " s y c h o l ogy, V o l , 1, ' J i l e y , New York (1953). 1-76.
ACKNOIJLE DSEME NTS The a u t h o r would l i k e t o thank t h e N a t i o n a l Science F o u n d a t i o n f o r i t s SUDp o r t under G r a n t Number IST-83-01496 t o Rutgers U n i v e r s i t y .
He would a l s o
l i k e t o thank A r u n d h a t i Ray-Chaudhuri, L a r r y Harvey, 5aw Rosenbaurn. and J e f f 5 t e i f f o r t h e i r h e l o f u l cowments.
TRJZNDS Zh' MATIfEMATlCAL PSYCHOLOGY E. figreef and J. Van Bu enhaut (editors) 0 Elsevier Science D u b l i s f k €IV.. (North-Holland), 1984
293
A NEl.1 DERIVATION nF THE RASCH HnDEL
Fdw. E . Roskam Department o f Psycholony, U n i v e r s i t,y o f N i jmepen
and Paul G.I.1. Jansen R i j k s Psycholoaische D i e n s t , Den Haacl The Rasch model i s u s u a l l y d e r i v e d from s t a t i s t i c a l requirements o f e s t i m a t i o n o f i t s Darameters.
The
o r i n c i p l e o f ' s o e c i f i c o b j e c t i v i t y ' or, equivalently,
' s amp 1e indeoenden ce ' r e o u i r e s t h a t s ubje c t narame t e r s can be e s t i m a t e d indeoendent from t h e i t e m Darameters and v i c e versa.
This i n t u r n requires t h a t s u f f i c i e n t
s t a t i s t i c s e x i s t f o r t h e Darameters, from which t h e Rasch m d e l f o l l o w s .
Rather than u s i n g a s t a t i s t i c a l
araument, t h e n r e s e n t lraner Dresents a d e r i v a t i o n o f t h e Rasch model based on t h e r e q u i r e m e n t t h a t t h e n r o h a b i l i t y o f i n f e r e n t i a l o r d e r i n g o f s u h j e c t s ( o r items,
reso.) i s samnle indenendent.
The d e r i v a t i o n o a r a l l e l s
Ducamp & Falmaqne's a x i o v a t i z a t i o n o f t h e Guttman s c a l e (comnosite m e a s u r e m n t ) ; i t i s f i r s t g i v e n f o r t h e o r d e r i n a o f two i t e m s o r s u b j e c t s , and subsequently gener a l i z e d t o the o r o h a b i l i t y o f a m u l t i D l e ordering. The core e q u a t i o n i n t h e d e r i v a t i o n i s , o f course, n o t d i f f e r e n t f r o m t h e c o r e e q u a t i o n i n t h e s t a t i s t i c a l der iv a t i o n .
1. INTRODUCTION
As a m o t i v a t i o n f o r u s i n q t h e Rasch model, t h e measurement argument o f snec i f i c o b j e c t i v i t y i s n o t seDarated v e r y w e l l f r o m t h e s t a t i s t i c a l argument o f sufficiency.
The measurement argument s t i o u l a t e s t h a t s u b j e c t measure-
ment s h o u l d n o t be i n f l u e n c e d bv t h e s e l e c t i o n @ f a s n e c i f i c measuring aaent ( i t e m ) from t h e reference u n i v e r s e , whereas t h e s t a t i s t i c a l argument s t r e s ses t h e s t a t i s t i c a l advantapes o f u s i n g c o n d i t i o n a l e s t i m a t i o n orocedures (e.g.
c o n s i s t e n t e s t i m a t o r s and t e s t s t a t i s t i c s w i t h known and d e s i r a b l e
294
E.E. Roskarn and P.C. W.Ionsen
nroperties).
I n p a r t i c u l a r , t h i s requires the existence o f s u f f i c i e n t sta-
t i s t i c s f o r t h e p a r a m t e r s o f t h e Rasch model. There i s a need f o r an e x a c t d e f i n i t i o n o f s n e c i f i c o h j e c t i v i t y i n a Droh a b i l i s t i c c o n t e x t whose f o r v u l a t i o n i s indenendent o f o u r e l v s t a t i s t i c a l considerations.
A r e c e n t l y nronosed d e f i n i t i o n (Roskav (1983)) o f a spe-
c i f i c a l l y o b . j e c t i v e o r o b a b i l i s t i c comparison i s d i s c u s s e d and e l a b o r a t e d i n s e c t i o n 2.
I n s e c t i o n 3 we aroue s u h s e a u e n t l y why we c o n s i d e r t h e o r i n -
c i q l e o f s n e c i f i c o b j e c t i v i t y t o he such n n i m o r t a n t r w u i r e n e n t o f s c i e n t i f i c comnarisons, i n c l u d i n o t h e one5 o f a t t i t u d e t h e o r y . RASCH'? FQWlIILPTInY nF 5pECIFIC PRlECTIVITY Fasch ( 1 9 6 6 ~4~- 5 ) i n t r o d u c e d s n e c i f i c o b j e c t i v i t y i n t e n s o f a t h e o r y about o b j e c t i v i t y i n d e t e r m i n i s t i c c o m a r i s o n s between o b j e c t s by ineans o f an aoent.
An examole o f a s p e c i f i c a l l y o b j e c t i v e cornoarison i s the r e l a t i o n between mass m, f o r c e
F and a c c e l e r a t i o n a, i n Newton's Second Main Law: (!I
= pa
Thus: t h e f o r c e F . t h a t a c t s unon a body v w i t h mass nv causes an a c c e l e r a 1
ti on
a which can be observed.
vi
Fi
-5
-
Plow suonose we want t o compare two o b j e c t s v and w
w i t h r e s o e c t t o t h e i r masses rnv and m
W
respectively.
To t h i s end, we l e t a
f o r c e , Fi say, a c t unon these o b j e c t s , and observe t h e r e a c t i o n s avi
a . \41
*
and
The r a t i o comnarison
i s , due t o ( 2 ) , independent o f t h e s n e c i f i c f o r c e Fi t h a t was used t o b r i n g about t h e r e a c t i o n s avi
and awi:
-a -v i - Fi'mv aw i
rnw
Fi/mw=iq
(4)
Thus t h e comparison o f t h e o b j e c t s i s independent o f the aaent i: i t i s sDecifically objective.
T h i s examole i s due t o Rasch (1960, ch. V I I ) .
h i s t r e a t m e n t he s t r e s s e s t h e m u l t i n l i c a t i v e c o n j u n c t i o n o f Fi
and mv i n
In
(Z),
295
A new derivation of the Rasch model
which i s understandable c o n s i d e r i n g ( 4 ) . Then he s t a t e s (1.c):
I w i s h t o n o i n t o u t t h a t i n t h i s c o n t e x t o n l y t h e o b j e c t s and/or t h e aaents a r e s u b j e c t t o comnarison, w h i l e the d a t a themselves a r e n o t d i r e c t l y compared; t h e y o n l y serve as i n s t r u m e n t s f o r t h e comparison aimed a t .
The consenuences o f these two concents: ( s o e c i f i c ) compari-
sons and s n e c i f i c o b j e c t i v i t y , comnleted b y t h e r e n u i r e n e n t t h a t a cnmp a r i s o n i s alwavs D o s s i b l e and t h e r e s u l t always unambiguous, a r e r e a l l y overwhelminq.
This "una&i?uity"
may be c l a r i f i e d bv means o f t h e f o l l o w i n g q u o t a t i o n
f r o m Qasch (1960, 9 ) : I f a r e l a t i o n s h i n between two o r m r e s t a t i s t i c a l v a r i a h l e s i s t o be c o n s i d e r e d r e a l l y i m o o r t a n t , as more than an ad hoc d e s c r i o t i o n o f a
.
v e r y l i m i t e d s e t of d a t a ( . .) , t h e r e l a t i o n s h i n s h o u l d be found i n s e v e r a l s e t s o f d a t a which d i f f e r m a t e r i a l l y i n some r e l e v a n t r e s o e c t s . I n g e n e r a l , t h e comoarison o f o b j e c t s v and w from a u n i v e r s e o f o b j e c t s i s s a i d t o be s p e c i f i c a l l y o b j e c t i v e when t h e r e s u l t o f t h i s comparison i s t h e Same 'or a l l aoents i ,...,K from u n i v e r s e o f aaents t h a t a r e used t o e l i c i t r e a c t i o n s x ( v , i ) f r o m these o b j e c t s .
So analooous t o t h e r e l a t i o n
between mass, f o r c e and a c c e l e r a t i o n , we have:
xvi
=
ev
)t
E
(5)
i
and a necessary and s u f f i c i e n t c o n d i t i o n f o r s o e c i f i c a l l y o h j e c t i v e measurement o f e i t h e r o b j e c t s o r agents.
Thus, i n a d e t e r w i n i s t i c c o n t e x t ,
x ( v , i ) s h o u l d be m u l t i p l i c a t i v e l y o r a d d i t i v e l y decomoosable t o a l l o w f o r s p e c i f i c a l l y o b j e c t i v e c o m a r i s o n s ( c f . Rasch (1979) , 79). Ramsay (1975), i n h i s r e v i e w o f K r a n t z e t a l . (1971)
, states
t h a t t h i s book
" d e a l s w i t h t h e f a c t t h a t v i r t u a l l y a l l t h e laws o f p h y s i c s can be exDressed n u m e r i c a l l y as m u l t i n l i c a t i o n s o r d i v i s i o n s o f measurements" (o.c., And on
D.
258). 262 he u t t e r s aaain h i s amazement a t
" t h e remarkable f a c t t h a t t h r o u p h o u t t h e g i p a n t i c range o f p h y s i c a l knowledge numerical laws assume a remarkable s i m o l e f o r m o r o v i d e d f u n -
.
damental measurement has taken n l a c e ( . . ) " .
E.E. Rorkum und P.C.W. Jamen
19b
Thus i t seems t h a t s n e c i f i c o b . j e c t i v i t y i s a n r o o e r t y o f ' v i r t u a l y a l l ' comparisons i n n h y s i c s . Above, s n e c i f i c o b . i e c t i v i t y has been d e f i n e d i n a d e t e r m i n i s t i c c o n t e x t . I n a n r o b a b i l i s t i c model, t h e narameters A , 8 and servarle.
E
a r e e s s e n t i a l l y unob-
Therefore. i n the l i t e r a t u r e the reauirement f o r s o e c i f i c ohjec-
t i v i t y i s r e f r a m e d as t h e demand t h a t t h e measurement model s h o u l d a l l o w f o r " D r o b a h i l i t y exDressions which denend on o n l y one s e t o f p a r a m t e r s a t a t i m e " (Douglas (1380), l ) , and c o n s e o w n t l v s o e c i f i c o b j e c t i v i t y i s de-
f i n e d as f o l l o w s f o r a r o b a h i l i s t i c models: " A two parameter s t a t i s t i c a l node1 f o r measurino i s d e f i n e d t o be sne-
c i f i c a l l y o b j e c t i v e i f s t a t i s t i c a l i n f e r e n c e orocedures e i i s t such t h a t i f we i n t e n d t o draw s t a t i s t i c a l i n f e r e n c e a b o u t o n l y one o f t h e narameters i t does n o t denend on o t h e r p a r a n e t e r s "
(Scheiblechner,
(1979), 2 1 ) . and
.
" ( . . ) t h e c o n d i t i o n a l d i s t r i b u t i o n g i v e n s u f f i c i e n t s t a t i s t i c s can be taken as a s p e c i f i c a l l y o b j e c t i v e s t a t i s t i c a l model"
(o.c.,
22).
A s i m i l a r d e f i n i t i o n i s n i v e n b y F i s c h e r 8 Pendl ( ( 1 9 8 0 ) , 173-174).
In a
n r o b a b i l i s t i c c o n t e x t , s n e c i f i c o b j e c t i v i t y i s a l s o c a l l e d 'samnle indeoendence' o r ' s a m o l e - f r e e measurement'.
By these d e f i n i t i o n s s o e c i f i c o b j e c -
t i v i t y i n a p r o b a b i l i s t i c c o n t e x t becomes e o u i v a l e n t t o t h e s t a t i s t i c a l n r o p e r t y o f s u f f i c i e n c y , so t h a t , i n f a c t , s o e c i f i c o b j e c t i v i t y c o i n c i d e s w i t h t h e Rasch model.
Suooes ( i r : Soada 8 Kemnf ( 1 9 7 7 ) , 22) q u e s t i o n e d t h e
p o s s i b i l i t y o f findina a reneral formulation f o r s n e c i f i c o b j e c t i v i t y t h a t c o u l d b e s e o a r a t e d frcm t h e Rasch model i t s e l f .
I n a r e p l y F i s c h e r (o.c.,
36) a d m i t t e d t h a t " i t i s d i f f i c u l t t o g i v e a c o m n l e t e l y g e n e r a l d e f i n i t i o n o f soecific objectivity".
Thus i t seems t h a t t h e measurement arqument o f s o e c i f i c o b j e c t i v i t y and t h e s t a t i s t i c a l argument o f s u f f i c i e n c y a r e n o t w e l l s e o a r a t e d i n t b e l i t e r a t u r e o f t h e Rasch model.
2. A NElJ FORYULATION OF SPECIFIC OBJECTIVITY I N PPOBARILISTIC COMPARISONS Roskam (1961, 57; 1983, 81-83) demonstrated t h a t t h e Rasch model i s e q u i v a l e n t t o the r e q u i r e m e n t t h a t t h e a r o b a b i l i t y o f i n f e r r i n g t h e i t e m o r d e r i n g i R j i s t h e same f o r a l l s u b j e c t s f r o m t h e r e f e r e n c e o o o u l a t i o n .
I n agree-
ment w i t h t h e f i r s t q u o t a t i o n o f Rasch, two i t e m s i and j ( " o b j e c t s " ) a r e
297
A new derivation of the Rasch model
ordered ("comr,ared") on account o f the (dichotowous) responses avi o f a suhset v ( " a a e n t " ) from the reference o o p u l a t i o n .
and a vj
L e t avi, a denote the (random) event t h a t s u b j e c t v responds o o s i t i v e l y vj t o i t e m i o r i t e m j , resn. L e t rv denote the number o f p o s i t i v e ansln'ers given by s u b j e c t v. To meet the requirement t h a t an o r d e r i n g o f i t e m i s "always o o s s i b l e " , i t i s demanded t h a t avi
# avj, e.q. rV
= 1 f o r the two-item t e s t ( i , j ) ;
thus
the o r d e r i R j i s c o n d i t i o n a l on rv = 1, and the p r o b a b i l i t y may be w r i t t e n as o ( i R j
I rv =
1).
Rasch's requirement o f "unamhiguity" i s then formulated
as n ( i R j l r v = 1)
1 v,
(6)
which may be considered as an exact f o r m u l a t i o n o f the second q u o t a t i o n from Rasch. Relow, i n our p r e s e n t a t i o n o f the a c t u a l d e r i v a t i o n , we w i l l demonstrate t h a t the Rasch model can be d e r i v e d as the o r o b a b i l i s t i c analogue o f the Guttman scalogram. I n the Guttman scaloaram, the o r d e r o f the dichotomous items i and j on a dimension o f d i f f i c u l t y (denoted as i R j : i i s harder than j ) can be determined unequivocally i n a d e t e r m i n i s t i c sense when some sub.ject v dominates j (denoted as: vSj) b u t does n o t dominate i ( - v S i ) , assuming the domina-
t i o n r e l a t i o n s c o m l y w i t h "comDosi t e t r a n s i t i v i t y " (Ducamn & Falmagne
( 1969) )
.
COMPOSITE TRANS I T 1VITY The r e l a t i o n ' v dominates i ' ,w r i t t e n as vSi, i s observed when, e.g.
object
v solves Droblem i,o r when s u b j e c t v accepts statement i. W t h o u t l o s s
o f g e n e r a l i t y , the o r d e r i n g can be reversed and the data i n t e r p r e t e d as the complementary r e l a t i o n . Ducamr, & Falmagne (1969) have formulated the f o l l o wing d e t e r m i n i s t i c oroperty, which we c a l l composite t r a n s i t i v i t y .
It i s
the same as Guttman's (1950) p r i n c i p l e o f s c a l a b i l i t y , and i t defines a Guttman scale: i f (vSi , n o t w S i & wSj) then vSj
(7)
I t says t h a t , i f v dominates i,w does n o t dominate i,b u t w dominates j , then v must dominate j .
From t h i s property, i t f o l l o w s , as shown by Ducamo
& Falmapne t h a t a simple composite o r d e r i n g o f v, w
,...
and i,j
,... e x i s t s .
298
E. E . Roskum und P.C. W. Jansen
Thus, by ( 7 ) :
, 1v
i R j =: v S j & - v S i
(8)
i s a s i m o l e o r d e r , and i f i R j f o r some v, then i R j f o r any v .
T a k i n g (8) as
t h e d e f i o i t i o n o f t h e o r d e r i n q amonn two i t e m s , we can determine t h e i r o r d e r whenever some s u b j e c t v dominates j b u t n o t i . quch a d i f f e r e n t i a l response can o n l y he observed w i t h s u b j e c t s who s c o r e
r
V
= 1 i n the i t e m o a i r ( i , j ) .
I n such a case, t h e i t e m o r d e r
R j i s esta-
u on l b l i s h e d i n d e p e n d e n t l v o f t h e s u b j e c t sampled f r o m t h e r e f e r e n c e ~ o ~a ti
( 5 amo le- indcr.Er,den t 1v ) : ( v S j 8 -vSi
I
rv = 1)
1v.
S i n i l a r l y , t h e o r d e r i n g amonrr two s u b i e c t s i s d e f i n e d hv vQw =: v q i & - w S i
(10)
and i s ohserved whenever two s u b j e c t s respond d i f f e r e n t l y t n a s i n g l e i t e m , r e g a r d l e s s o f which i t e m t h a t i s . PROBARILISTIC GENERALIZATION
FOR K=2
L e t K denote t h e number o f i t e m s c o n s i d e r e d . r a l i z e d t o a p r o b a b i l i s t i c c c n t e x t as f o l l o w s .
F o r K.2,
e?. ( 8 ) may be gene-
The p r o b a b i l i t y o f t h e e v e n t
i P i can be d e t e r m i n e d u n 2 q u i v o i a l l y i n a s t o c h a s t i c sense when
p ( i R j ) =: o(vS.j & -vSi
I
r v = 1)
1 v.
(11)
The o r d e r i n g w i l l be c a l l e d a s t o c h a s t i c a l l y c o n s i s t e n t i t e m o r d e r i n g . The r i o h t n a r t o f ( 1 1 ) i s i d e n t i c a l t o F i s c h e r ' s e n u a t i o n (12.4.1)
196) where i t i s d e r i v e d f r o m t h e r e q u i r e m e n t o f s u f f i c i e n c y . i t i s as much as a d e f i n i t i o n o f s u f f i c i e n c y .
(1974,
I n effect,
The p r e s e n t d e r i v a t i o n i s ,
however, n o t based on s u f f i c i e n c y as a s t a t i s t i c a l r e q u i r e m e n t b u t on t h e r e q u i r e m e n t o f a c o n s i s t e n t s t o c h a s t i c o r d e r i n g of t h e iterns, analogous t o t h e o r d e r i n g d e r i v e d f r o m comnosi t e t r a n s i t i v i t y i n t h e scaloqram s t r u c t u r e .
From (11) we can d e r i v e :
299
A new derivation o f t h e Rasch model
-
p ( v S j & -vSi) p(vS5 & - v S i ) + o ( - v S j & v S i )
-
p(vSj)(l-n(vSi)) p(vc:)(l-p(vSi)) + p(vSi)(l-p(vSj))
l
v
(12)
assuming l o c a l s t o c h a s t i c indeDendence.
'Jhen we d e f i n e n ( v S i ) as t h e ICC
fi ( F;) = fi o f i t e m i, (12) becomes
f .( l-fi) J I v - ) + f i(l-f,i) + j ( l fi from which, assuminp f t o be continuous a n d - d i f f e r e n t i a b l e , and nowhere equal t o zero, i t f o l l o w s t h a t fi i s g i v e n by t h e i t e m c h a r a c t e r i s t i c curve
of t h e Rasch model (Roskam (1983), 82): fi(E)
where e = e x p ( c ) , and the apnendix),
E
= exD(-o).
=
1/ I l+exp(o.-S )I o r 1 v
(Details o f the d e r i v a t i o n are given i n
Thus t h e Rasch model can be d e r i v e d f r o m t h e r e q u i r e m e n t o f
s t o c h a s t i c a l l y c o n s i s t e n t i k e m o r d e r i n p s , and i n t h i s sense t h e model can be c o n s i d e r e d as . t h e p r o b a b i l i s t i c c o u n t e r p a r t o f t h e scalogram: t h e (probab i l i t y o f the) ordering i R j i n the t e s t ( i , j ) the s e l e c t i o n o f a subject.
i s e s t a b l i s h e d independent o f
Since t h e Rasch model i s symmetrical i n i and
v, ( 1 1 ) c o u l d be r e o l a c e d b y i t s analogue f o r a s t o c h a s t i c a l l y c o n s i s t e n t s u b j e c t ordering. Note t h a t (12) i s i n t h e Rasch model equal t o D ( i R j ) =: p ( v S j & -vSi = pij(O,llrv
I
rV = 1)
= 1, K=2) =
+
(15) J Eq. ( 1 5 ) i m p l i e s t h a t t h e p r o b a b i l i t y p ( i R j ) conforms t o t h e Luce c h o i c e r u l e ( c f . A n d r i c h , ( 1 9 7 8 ~ ) )and ~
D(iRj)
E ~ / ( E ~ E.)
E.
W T = : i s o b v i o u s l y independent o f what o t h e r i t e m s a r e i n c l u d e d i n t h e t e s t .
(16) This
means t h a t p ( i R j ) computed a c c o r d i n g t o ( 1 5 ) i s b o t h independent o f v and o f the o t h e r items i n the t e s t .
300
E.E. Roskam and P.G.W.Jansen
The d e r i v a t i o n oresented above concentrates or? the o r d e r i n g o f two items. One miaht consider the o o s s i h i l i t y o f g e n e r a l i z i n ? t h i s d e r i v a t i o n t o mult i o l e i t e m o r d e r i n q s iRjRk
... by
a m l y i n g s t o c h a s t i c t r a n s i t i v i t y on the
i t e m n a i r n r o b a h i l i t i e s a ( i R j ) , emoloving a reasonina t h a t would be s i m i l a r t o Ducamp R Falmagne's (1969) t r e a t w n t o f conoosi t e measurement. However, such a g e n e r a l i z a t i o n would n o t be c o r r e c t , because i n t h i s way a m u l t i p l e o r d e r i n g o f p r o b a b i l i t i e s would he o b t a i n e d i n s t e a d o f the r e quired n r o b a b i l i t y o f a multinle ordering.
Therefore we orooose i n the
n e x t s e c t i o n a g e n e r a l i z a t i o n o f p ( i R , j ) t o m u l t i o l e i t e m o r d e r i n g s which i s analogous t o Roskam'.s (1983) d e r i v a t i o n f o r the o r d e r i n g o f a n a i r o f items.
2.1. W L T I P L E ITEll OQDERINGS The case K=3 '%re f i r s t discuss t h e case K=3.
I n the Guttman scalooram, the m u l t i o l e o r d e r
iRjRk can be determined u n e q u i v o c a l l y o n l y when two s u h j e c t s are observed w i t h resoonse v e c t o r s ( 0 , 0, 1) and (0, 1, 1).
Conversely,
a multiole order
o f s u b j e c t s unvOw i s d e f i n e d w i t h the two i t e m resoonse v e c t o r s (0, 0, 1) and (0, 1, 1 ) .
As the argument i s e n t i r e l y symmetrical i n s u b j e c t s and
items, we g i v e the f o l l o w i n o d e r i v a t i o n s i n terms o f t h e i t e m o r d e r i n g i 9 j 9 k . By analogy, a g e n e r a l i z a t i o n o f (11) i s D(iRjRk) =: o(vSk & - v S j & -vSi
PI wSk & w S j & - @ S i I ry = 1 C rw = 2 )
o(vSk & - v S j & -vSi
I
r v = 1)
*
D(wSk & w S j & - w S i
I
=
r,q = 2)
-I v.
(17)
I n the Rasch model o r o b a b i l i t y ( 1 7 ) i s indeed independent o f t h e s u b j e c t s v, w:
I
p(vSk & -vSj R -vSi
O ; ~ ~ ( ( , , OI , Irv = I )
Ek
Ei + " j + "k
*
rv = 1) a (o(wSk & w S j & - w S i
*
1
pijk(O.l,I
rw= 2 ) =
j'k EiEk
+
EiEj
+
EjEk
'
P r o b a b i l i t y (18) can be r e w r i t t e n as ( u s i n g ( 1 5 ) )
I
rw= 2) =:
301
A new derivation ofthe Ratch model
2
o ( i R j ) o ( jRk)D( iRk)
-
(19)
1 t o(iRi)n(jRk)(l-n(iRk)\
Note t h a t n ( i R j ) i s d e f i n e d a c c o r d i n o t o ( 1 5 ) , which i s independent o f K . Since n ( i R j ) comnlies w i t h t h e o r o d u c t r u l e
we can d e r i v e from (19) :
which i m p l i e s
E q u a t i o n ( 2 2 ) f o r m u l a t e s t h e decolnposi t i o n r u l e a c c o r d i n ? t o which t h e b i
-
n a r y o r o b a b i l i t y o ( i R j ) may be o b t a i n e d f r o m t h e t r i n l e D r o h a b i l i t i e s p ( i R j 9 k ) i n t h e Rasch model cornnlying w i t h t h e d e f i n i t i o n s i n (11) and ( 1 7 ) . I n t h e i r overview o f c e n e r a l D r o b a b i l i s t i c r a n k i n q t h e o r i e s , Luce and Suppes ((1965), 351-358) p r e s e n t a few examoles o f decomposition r u l e s .
E.g. an
obvious c h o i c e seems t o be
p ( i r l j ) = n(iRjRk)
+ n(iRkRj)
which does n o t comply w i t h ( 2 2 ) however.
t
o(kRiRj)
(23)
O b v i o u s l y (11) and ( 1 7 ) as w e l l
as t h e i r g e n e r a l i z a t i o n t o r a n k i n g s of K > 3, a c t as a r a n k i n q o o s t u l a t e . G e n e r a l i z a t i o n o f (11) and ( 1 7 ) t o K > 3 Formula (15) can be w r i t t e n as*
(i1
o(iRj) =
Ti -
Y1
*
I n t h e f o l l o w i n g e q u a t i o n s , t h e Y f u n c t i o n s a r e d e f i n e d as f o l l o w s :
Y(isj)
r
i s t h e sum o f a l l o r o d u c t s
EkEIEh...,
each c o n s i s t i n g o f r terms
302
E.E. Roskarn and P G . W. Jamen
and (19) as
Note t h a t ( 2 4 ) and ( 2 5 ) a r e n o t d i r e c t l y comoarable s i n c e
y1
i s cornouted
. i n ( 2 4 ) b u t as E~ + E t E~ i n ( 2 5 ) . J j Generalizing ( 1 7 ) and ( l G ) , t h e n r o b a b i l i t y o f an o r d e r i n g o f K i t e m s i s
as ci
t F
n(lR2R
...RK)
=:
.,(O,O .... ,0,1 j * n 12.. .K ( O , q .....O , l , l * .......... * .......... * o 12.. .K ( O , l , l , . ..,I I o
12..
(1,. - y1
.., K - l )
(1,.
* y2
Y1
K-1
= n
r=l)
I r=2)
r=k)
..,K-2)
(1) 'K- 1 * ... * -
'K- 1
y2
.., K - r }
(1,.
yr
I
r- 1
'r
which i s a o a i n i n d e o e n d e n t o f t h e s u b j e c t samole; a l l y - f u n c t i o n s i n ( 2 6 ) are computed on t h e K i t e m n a r a m t e r s
E
~
... ,
, E ~ .
3 . SOf1E ALTERNATIVES
I n the o r o c e d i n g s e c t i o n , o u r main arclument c o n c e n t r a t e d on a d e f i n i t i o n o f the o r d e r r e l a t i o n a m n g i t e m s ( o r s u b j e c t s , r e s o . ) .
I t can be argued t h a t
o t h e r d e f i n i t i o n s o f t h e o r d e r r e l a t i o n i R j a r e o o s s i b l e , and m i g h t l e a d t o
other models f o r measuring s u b j e c t s and i t e m s .
'ale d i s c u s s two such p o s s i -
bilities. The f i r s t i s : i R j 9 p(vS.j & -vSi The second p o s s i b i l i t y i s iRj
9
I
(27)
r v = l ) > .50 f o r a l l v
(28)
n ( v S j ) > p ( v S i ) f o r a l l v.
The l a t t e r d e f i n i t i o n i s e q u i v a l e n t t o n o n - i n t e r s e c t i n n itern response c u r v e s . t a k i n g a l l (K,2) c o m b i n a t i o n s f r o m yb=l,
by d e f i n i t i o n .
..
E ~ , E ~ . C ~ , E ~ , , E. ~ . c x c l u d i n s E~ ,E
The f o l l o w i n g r e l a t i o n h o l d s :
..
.I
303
A new deriuation of the Rarch model
Both d e f i n i t i c n s 20 n o t atmear t o be f e a s i b l e as a d e f i n i t i o n .
The oroblem
w i t h these d e f i n i t i o n s i s , t h a t t h e v d e f i n e i R j throuah unknown o r o b a b i l i ties.
These d e f i n i t i o n s make t h e e v e n t o r assessment o f i R j dependent c r
uncbservable c o n d i t i o n s ,
I n c o n t r a s t t o these d e f i n i t i o n s , o u r approach i n
t h e Drecedinq s e c t i o n was based on an e m p i r i c a l d e f i n i t i o n o f i R j , f o l l o w e d b y t h e r e q u i r e m e n t t h a t t h e n r o b a b i l i t v of t h i s e v e n t i s independent o f t h e observed s u b j e c t s . content.
O t h e r d e f i n i t i o n s , such as i n (28) have no e m o i r i c a l
One can, o f course, e s t i m a t e t h e D r o h a b i l i t i e s i n (28), and i n f e r
i R j , b u t t h a t would be a -n o s t hoc d e f i n i t i o n , and would he as good o r bad as, e.g.
s a y i n g t b a t i K j i f t h e frequency o f v S j i s l a r g e r then t h e frerruen-
cy o f vSi i n some r e n r e s e n t a t i v e samnle ( w i t h no c r i t e r i u m as t o t h e r e p r e s e n t a t i v e n e s s o f t h e samale). I n t h e c o n t e x t o f p r o b a b i l i s t i c d a t a , any s t a t e m e n t c o n c e r n i n g s u b j e c t s and/ o r i t e m s can o n l y be a p r o b a h i l i s t i c statement.
I n o r d e r t o be meaningful
and c o n s i s t e n t , such p r o b a b i l i s t i c statements have t o s a t j s f y c e r t a i n conditions.
T h i s was t h e anproach adooted i n t h e n r e c e d i n g s e c t i o n .
Any d e r i -
v a t i o n which does n o t t r e a t i R j as a o r o b a b i l i s t i c e v e n t , would n o t be v a l i d . By t h i s r e a s o n i n g we do n o t , o f course, e x c l u d e t h a t models can be f o r m u l a t e d which imoly a s i m n l e o r d e r i n g o f s u b j e c t s and i t e m s . model i m o l i e s such o r d e r i n g .
Fny h o l o m r p h i c
I n f a c t , even a non-homomorphic model i s com-
o a t i h l e w i t h a d e f i n i t i o n o f an o r d e r i n g of t h e i t e m s : e . g . i n t h e Birnbaum model, we can d e f i n e t h e o r d e r i n q o f t h e i t e m by t h e s o - c a l l e d d i f f i c u l t y parameter, b u t we m i g h t as w e l l d e f i n e i t b y t h e d i s c r i m i n a t i o n oarameter. The i s s u e i s n o t whether we can d e f i n e an i t e m o r d e r i n g based on some model, b u t whether we can base a model on a p r o p e r d e f i n i t i o n o f t h e i t e m - o r d e r i n g , t a k i n g i n t o account t h a t we deal w i t h o r o b a b i l i s t i c data, and t h a t we want n r o b a b i l i s t i c statements about i t e m s independent o f t h e s u b j e c t sample.
4. SPECIFIC OBJECTIVITY AS A VETHODOLOGICAL REQUIREtlENT The d e r i v a t i o n i n s e c t i o n 2 shows t h a t a s p e c i f i c a l l y o b j e c t i v e model p r o duces a c o n s i s t e n t s u h j e c t o r d e r w i t h r e s p e c t t o some c o n s t r u c t o r t r a i t . S u b j e c t measurement i s , i n a s t o c h a s t i c sense, indenendent o f t h e s o e c i f i c s e t of i t e m s s e l e c t e d f r o m t h e u n i v e r s e of c o n t e n t d e f i n i n g t h e p e r t i n e n t b e h a v i o r a l domain.
A conseauence o f t h i s i s t h a t a s o e c i f i c a l l y o b j e c t i v e
measurement model i s t h e o n l y meaningful i n s t r u m e n t f o r d e f i n i n g and meas u r i n g a t h e o r e t i c a l c o n s t r u c t suaposed t o u n d e r l y b e h a v i o r i n t h a t domain.
304
E.E. Roskczm and P.G. W.j a w c n
Before we e l a b o r a t e t h i s 'meaningfulness a r g u m e n t ' , r e have t o r e f u t e one v o s s i b l e counter-arnument. I t nay be t h a t i n t h e b e h a v i o r a l s c i e n c e s , s p e c i f i c o b j e c t i v e measurement can
he e s t a b l i s h e d f a r l e s s f r e q u e n t l y than i n p h y s i c a l s c i e n c e s .
I n general,
the Rasch model imposes r a t h e r s t r o n g r e q u i r e m e n t s on t h e d a t a .
This i s ,
however, an e m p i r i c a l argument, and as such cannot match t h e m e t h o d o l o g i c a l s t a t u s o f the p r i n c i n l e o f s o e c i f i c o b j e c t i v i t y .
P. w t h o d o l o o y i s a proaram
i n t h e f i r s t p l a c e , a o r e s c r i n t i o n f o r o b t a i n i n g m e a n i n g f u l r e s u l t s f r o m ema i r i c a l research.
E m o i r i c a l d a t a e i t h e r a r e o b t a i n e d under t h i s program
and t h e y do n o t r e f u t e i t
-
-
o r t h e y a r e n o t o b t a i n e d under i t , i n which case
i n f e r e n c e s drawn -Frori t h e n can s i m n f y n o t b e c o n s i d e r e d s c i e n t i f i c a l l y meaningful.
S p e c i f i c o b i e c t i v i t y can o n l y be r e f u t e d by o t h e r m e t h o d o l o q i c a l
n r i n c i n l e s which, e s s e n t i a l l y , araue wby i t w o u l d n o t make sense.
An exam-
n l e o f such an argument m i g h t be t h a t a p r i n c i p l e t h a t n e v e r works because i t i s n o t adequate i n view o f t h e n a t u r e o f t h e ohenomena under s t u d y ,
s h o u l d he d i s c a r d e d .
P,,C n r e s e n t , i t seems t o o e a r l y f o r such a c o n c l u s i o n .
Statements a b o u t s u b j e c t s a r e g e n e r a l o n l y when t h e y a r e indeoendent o f t h e s o e c i f i c measurement ' n r o b e ' , domain o r u n i v e r s e o f c o n t e n t .
' a n e n t ' , o r ' i t e m ' samnled f r o m t h e p e r t i n e n t
I f a g e n e r a l t r a i t u n d e r l i e s t h e responses
o f s u b j e c t t o i t e m s o r nrobes f r o m t h a t u n i v e r s e o f c o n t e n t , then any sample o f i t e m s f r o m t h a t u n i v e r s e o f c o n t e n t s h o u l d be e q u a l l y v a l i d t o measure t h e t h e o r e t i c a l c o n s t r u c t t h a t i s addressed b y t h e n .
I f , f o r i n s t a n c e , an
a t t i t u d e l i k e ' c o n s e r v a t i s m ' i s t o he m e a n i n g f u l a t a l l , and i s d e f i n e d w i t h r e s o e c t t o a d e f i n e d domain o f s o c i a l i s s u e s , we s h o u l d he a b l e t o measure s u b j e c t s ' c o n s e r v a t i s m by o b s e r v i n g t h e i r resnonses t o i t e m s from t h a t domain, and t h e comparative i n f e r e n c e c o n c e r n i n ? two n e o n l e ' s degree o f c o n s e r v a t i s m s h o u l d n o t depend on t h e i t e m s samnled f r o m t h a t domain. F o r , i f i t were dependent on t h e samole o f i t e m s , one c o u l d as w e l l conclude t h a t one Derson i s nwre c o n s e r v a t i v e t h a n t h e o t h e r ,
I f t h a t were t h e case
-
BS
t h e o t h e r way around.
w h i c h o f course i s a m a t t e r t o d e c i d e e m p i r i c a l l y
-
the c o n s t r u c t o f ' c o n s e r v a t i s m ' w o u l d n o t be a u n i t a r y , u n i d i m e n s i : n a i cons t r u c t , h u t a combination o r amalgation of several t r a i t s o r d i s p o s i t i o n s , making t h e v e r y conceDt o f ' c o n s e r v a t i s m ' evanescent, and n o t m e a n i n g f u l as a t r a i t w i t h r e s p e c t t o which o e o o l e can be o r d e r e d . denend on t h e i t e m s s a m l e d t o assess i t .
The o r d e r i n g would
O f course, s o e c i f i c o b j e c t i v i t y
and s a n l e independence s h o u l d n o t be confused w i t h t h e r e l i a b i l i t y and t h e d i s c r i m i n a t i v e power o f a c e r t a i n s a m l e of i t e m s a d m i n i s t e r e d t o a p a r t i c u -
A new derivation of the Rasch model
l a r o r o w o f subjects.
30.5
E.p. i t e m s which, though h e l o n g i n ? t o a homogeneous
u n i v e r s e o f c o n s e r v a t i s m i t e m s and s a t i s f y i n p t h e Rasch vodel, a r e r e j e c t e d by ( a l m o s t ) everybody, would n o t he much good i n a c o n s e r v a t i s m q u e s t i o n -
n a i r e , e x c e o t w i t h t h e most c o n s e r v a t i v e s u b j e c t s . I n s u m a r y , we f i n d t h a t s n e c i f i c o b j e c t i v i t y , and hence t h e Rasch model,
i s a b a s i c m e t h o d o l o o i c a l c o n d i t i o n f o r t h e d e f i n i t i o n and measurement o f t h e o r e t i c a l c o n s t r u c t s (e.n. v e r b a l a b i l i t y , i n t e r n a l c o n t r o l , machiavelism).
I t i s o f s n e c i a l imnortance f o r t h e t h e o r e t i c a l s t u d y o f a t t i t u d e s
and o e r s o n a l i t y t r a i t s , w h e w i t c o u l d be v e r y h e l n f u l i s o l a t i n q and meas u r i n q such c o n s t r u c t s .
To what e x t e n t t h e r e o u i r e m e n t o f s n e c i f i c o b j e c -
t i v i t v can a c t u a l l y he met, i s a m a t t e r f o r e m o i r i c a l r e s e a r c h t o decide. I n t h a t way, i t i s a l s o a t e s t f o r t h e meaningfulness o f n e r s o n a l i t y con-
structs
.
APP E N D I X
Assuming t h a t fi,
fj,
... a r e
c o n t i n u o u s l y d i f f e r e n t i a b l e and naqhere equal
t o zero, t h e r e c i p r o c a l o f eq. (13) can be w r i t t e n 8s:
Exoanding and s i m p l i f y i n g leads to: f!(f.-fi) J
2
1
-
2
f!(f.-f.) 1
J
fj
J
0;
f;
m =q.(-lTJ J J So, f o r e v e r y i,j, we have: f; a c o n s t a n t , indeDendent o f 5. This d i f f e r e n t i a l equation i s solved by
1 fi( 5 ) = I t e x p ~ - c 5( - a i ~ l where c i s an a r b i t r a r y c o n s t a n t .
306
E.E. Rorkam a i d P.C.IV. Janren
REFE RE I.iCES
R e l a t i o n s h i o s between Thurstone and Rasch a m r o a c h e s t o
Andrich, D.,
i t e m s c a l i n g , P a n l i e d P s y c h o l o v i c a l Measurement, 2 (1978), 449-460.
C o n d i t i c n a l i n f e r e n c e i n a g e n e r i c P?sch model , I n v i -
Douglas, G . A . ,
t a t i o n a l Seminar on t h e I m r o v e m e n ? o f tqeasuremnt i n E d u c a t i o n and Psycholoay, Pelbourne (1980) , 22-23 May. Ducamp, A . & Falmagne, J.-C.
, Comoosite
measurement, ,lournal o f Pathe-
m a t i c a l Psycholoay, 6 (1969), 359-390. F i s c h e r , C.H., E i n f u h r u n g i n d i e T h e o r i e n s y c h o l o a i s c h e r T e s t s , Huher,
!Jim (1974). & Pendl, P.,
F i s c h e r , G.H.,
I n d i v i d u a l i z e d t e s t i n g on t h e b a s i s o f t h e
dichotomous Rasch model, I n L.J.Th.
Ili
van d e r Kamn, 'J.F. Lanqerak &
de Grui j t e r , Psychometrics f o r e d u c a t i o n a l dehates, New York,
D.N.Y.
l e y (1980). 171-188. Chanters 2 , 3 , 6 , 8 and 9 i n 5 . S t o u f f e r e t a l . ( e d s . ) ,
Cuttman, L.,
Measurement and P r e d i c t i o n , P r i n c e t o n , U n i v e r s i t,y o f P r i n c e t o n Press
(1950). Jansen, P.G.W.,
qasch a n a l v s i s o f
a t t i t u d i n a l data, Doctoral Disser-
t a t i o n , U n i v e r s i t y o f Ni.jmeaen, P r i n t : R i j k s P s y c h o l o g i s c h e D i e n s t , The Hague, The N e t h e r l a n d s (1983). K r a n t z , D.H.,
Luce, R.D.,
Suopes, R.,
& Tversky, A .
, Foundations
of
measurement, V o l . 1, A d d i t i v e and p o l y n o m i a l r e o r e s e n t a t i o n , New York, Academic Press ( 1971).
R Sunoes, P.,
Luce, R.D.,
l i t y , I n R.D.
Luce, R.R.
P r e f e r e n c e , u t i l i t y and s u b j e c t i v e D r o b a b i Bush, R
Psycholoay, V o l . I I 1 , New York Ramsay, Z.C., e t a1
E.
G a l a n t e r , Handbook o f ? a t h e m a t i c a l
, I J il e y
(1965).
Peview o f f o u n d a t i o n s o f measurement by D a v i d
., Psychome tri ka , 40
H. K r a n t z
(1975) , 257-262.
Rasch, G . , P r o b a b i l i s t i c models f o r some i n t e l l i g e n c e and a t t a i n m e n t t e s t s , Copenhaauen: Danish I n s t i t u t e f o r E d u c a t i o n a l Rescerch (1?6C), r e p r i n t e d (1960). Rasch, G . , An i n f o r m a l r e o o r t on a t h e o r y o f o b j e c t i v i t y i n comparisons, I n Proceedings o f t h e NUFFIC I n t e r n a t i o n a l summer s e s s i o n i n s c i e n c e a t ' H e t Oude H o f ' , The Hague, 14-28 July,(1966). Rasch, C.,
On s n e c i f i c o b j e c t i v i t y : an a t t e m n t a t f o r m a l i z i n g t h e r e -
q u e s t f o r g e n e r a l i t y and v a l i d i t y o f s c i e n t i f i c s t a t e m e n t s , Danish Yearbook o f P h i l o s o p h y , 14 (1977) , 56-94.
A new derivation of the Ratch model
[141
307
Roskam, Edw. E . , Ilethodenleer, I n t1.C.J. Duijker h P.A. Vroon ( r e d . ) Codex Psycholoqicus, Amsterdam, Elsevier (1981), 31-68.
[15]
Roskam, T&. E . , flllqemeine Datentheorie, In H . Feqer 81 H . Bredenkamp ( E d . ) , Ffessen und Testen; Band 3 der S e r i e Forschungsmethoden der Psycholoqie der Fnzyklopadie der Psycholoaie, Gottinqen, Hogrefe (1983)
[16]
1-135. Scheiblechacr, t'.
[la
Journal of Pathematical Psycholooy, 19 (1979), 15-38. Spada, H . , ?. k m f , I.f.F., (Eds.), S t r u c t u r a l modeis of thinking and
, S n e c i f i c a l l y objective s t o c h a s t i c latency mechanism,
learning , Bern, Huber ( 1377) .
The oaper i s based on Roskam (1983), and on a chapter from the doctoral d i s s e r t a t i o n of the f i r s t author (,!ansen ( 1 9 8 3 ) ) .
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TRENDS I N M A THEMATICAL PSYCHOLOGY E . DegreefandJ. Van Bu enkaut (editors) 0 Elsevier Science Publisgrs E . V. (hbrth-Holland), 1 9 8 4
309
A DFFIrdITInN OF PARTIAL INTERVPL ORDEPS
'larc Rouhens Facul t @polytechninue de F'ons , Re1 ni UCI Philinne Vincke Universite Libre de Bruxe?les, Relqium
Preference r o d e l l i n g often uses concents such as oreorders, semiorders and i n t e r v a l orders and, in most cases, i t i s assumed t h a t the r e l a t i o n s a r e comolete (no inconnarahili t y ) . However, incomarahi l i t i e s n a t u r a l l y anpear i n many decision nroblems, as f o r e x a m l e , in m u l t i c r i t e r i a nrohlems. That i s the reason whv the study o f p a r t i a l r e l a t i o n s becomes more and pore imnor-. tant. In this naoer, we oronose a d e f i n i t i o n of o a r t i a l i n t e r v a l order which extends the concent of i n t e r val order, which i s comnatible w i t h a numerical reoresentation a n d which i s defined hy Dronerties very easy to veri f.y. 1. IMRODUCTION
Preference modellino uses the concents of preorder, semiorder, i n t e r v a l ord e r , Dseudo-order and fuzzv order t o a l a r a e e x t e n t . In most cases, i t i s assumed t h a t the r e l a t i o n s are c o m l e t e (no incomoarability). Uhen one i s confronted with outrankinq r e l a t i o n s i l l u s t r a t i n a in the b e s t wav nossible the oreference o r the olobal judament on the s e t A of potential a c t i o n s , one i s in f a c t nrovided w i t h incomolete r e l a t i o n s . Defining p a r t i a l i n t e r v a l orders i s conseouently worth doing in the r e l a t i v e l y current cases when one meets only with r e l a t i o n s of oreference ( P ) , indifference ( I ) and incomaarabi l i ty (R) . In this naner we oronose a d e f i n i t i o n of p a r t i a l i n t e r v a l order which extends the concept of i n t e r v a l o r d e r , which i s comoatible w i t h a numerical reoresentation and which i s defined by o r o p e r t i e s very easy t o v e r i f y .
310
M. Rovbr1rc and Pli. Vinrke
2 . HLI"F?ICAL
REPPE5ECTPTInW OF CW'LETE PPiD PPQTI.AL PI?CO"DFRS
! t i s w e l l known t h a t , p i v e n an a s v m e t r i c r e l a t i o n P and a r e f l e x i v e and
s m w t r i c r e l a t i o n I on a f i n i t e s e t A , t h e c o u n l e ( P , ! )
i s a comnlete n r e -
order i f f
a ) 4' a , h C A : a P h o r b bl
r,
a o r a I b (exclusive or)
i s transitive
p
i s transitive:
c)
i f f a f u i c t i o n o e x i s t s such t h a t , \. a,h C
a
F:
rr(a) > n ( h \ ,
h
a I h * o(a\ = n(h\.
on the o t 9 e r hand, ( 0 , I ) i s a n a r t i a l o r e o r d e r i f ft h e c o n d i t i o n s h ) an? c \
a r e s a t i s f i e d and if( " , I ) i s a n a r t i a l a r e o r d e r , t'len a = u n c t i o n 9 e x i s t s such t h a t , V a,h i
r, b
a
=,
a I h
C-
A:
q(a) > n(h), o(a) = n(b).
S n , t h e conceot o f n a r t i a l n r e o r d e r i s s u f f i c i e n t b u t n o t necessary t o have t h e l a s t n u n e r i c a l r e a r e s e n t a t i o n , because t h i s r e o r e s e n t a t i o n does n o t i m n l l f t h e t r a n s i t i v i t v o f p.
I n "act,
t h e r,ecessar\t an& s u f f i c i e n t c o n d i t i o n
t o have t h i s n u m e r i c a l r e o r e s e n t a t i o n i s t h a t t h e r e l a t i o n P does n o t cont a i n anv c i r c u i t and t h a t I i s t r a n s i t i v e . exist
a*,
elements
...,
(P
an such t h a t ai
c o n t a i n s a c i r c u i t ift h e r e
P ai+l,
u i
and an P a l ) .
This
i s an immediate conTenuence o f a w e l l -known theorern of qraa'7 t h e o r y ( w e
141
).
? . S W E CHARACTERIZATInNS OF TPE IF!TE?VAL nRDER5 L e t a s a i n P be an a s v m t r i c r e l a t i o n and I a r e f l e x i v e and symmetric r e l a t i o n d e f i n e d on a f i n i t e s e t A .
L e t 9 , 5+ and C - be t h e r e l a t i o n s d e f i n e d
as f o l lows. a R b i f f a 3 b, h 1
a
$+
T
c ,
d: a I d and d
r,
b.
a S- b i f f Let
= (A,U)
a P b i
a
1 b,
a and a
b i f f 3 c: a P c and c
0
I b*
1
be t h e oran4 a s s o c i a t e d t o ( P , I ) (a,b) E LI and (b,a) (a,b)
5
U
and (h,a)
as f o l l o w s :
4
U (P-type a r c ) ,
6
11 ( I - t y n e a r c ) .
A definition oJpurtial interval orders
Let
call a circuit o f
IS
a s e t o f couoles {(aiy
311
ai+l)y
i=l,...,kI
o f ele-
ments o f A such t h a t
al’ ai+l o r ai
{ ak+l =
a.
p
1
I ai+l,
Y i,
o f t h e c i r c u i t k i n a k.
t h e lennt!, THEOWI 1
If (i)
” = 0,
the f o l l o n i n o orooerties are equivalent.
V a,h,c,d
I
p
P : a P b, b
f
” c P);
I
c, c P d =. a P d (which i s denoted
( i i ) I n G , e v e r y c i r c u i t o f l e n o t h 4 c o n t a i n s a t l e a s t two c o n s e c u t i v e I - t v oe arcs; ( i i i ) I n G , e v e r y c i r c u i t c o n t a i n s a t l e a s t two c o n s e c u t i v e I - t v p e arcs; ( i v ) The r e l a t i o n
$-
does n o t c o n t a i n any c i r c u i t ;
( v ) The r e l a t i o n S, does n o t c o n t a i n any c i r c u i t ; ( v i ) There e x i s t t v o f u n c t i o n s 9 and q such t h a t , V a,h E A: a P b
0
g(a) > n(b) + n(b)
a I b * g(a) s o(b) + a ( b ‘ a ( b ) 6 cl(a) + q ( a
, Y
q ( a ) 5 0. These o r o o e r t i e s d e f i n e what i s u s u a l l y c a l l e d an i n t e r v a l o r d e r ( Ill
,[21 ) .
Proof: (i)
=)
(ii):
L e t (a,b),
(b,c),
(c,d),
(d,a) E U; i f t h i s c i r c u i t does n o t c o n t a i n
two c o n s e c u t i v e I - t v n e a r c s , t h e n we have a P h and c P d ( o r e v e r v s i m i l a r s i t u a t i o n w i t h o t h e r c o u n l e s ) which-ir:plies o f ( i ) and o f t h e f a c t t h a t h
w i t h (d,a)
f
a P d (consequence
P c o r b I c); t h i s i s i n c o n t r a d i c t i o n
U.
( i i ) * (iii): I f a c i r c u i t of
does n o t c o n t a i n two c o n s e c u t i v e I-t,yoe a r c s , then
i t c o n t a i n s elements a,b,c,d
such t h a t a P b, h I c , c P d, which i v -
o l i e s , b y ( i i ) , a P d: t h i s d e f i n e s a n o t h e r c i r c u i t o f c, t h e l e n g t h o f which i s s m a l l e r t h a n t h e l e n a t h o f t h e i n i t i a l c i r c u i t .
M. Roubens arid PJi. Vincke
312
Reneatin! v!i tli ( i i)
t h i s o n e r a t i o n a o a i n and again, we o b t a i n a c o n t r a d i c t i o n
.
( i i i ) =. ( i v ) : e v i d e n t . ( i v ) = (v): evident. (v) = (vi): q i n c e S + has no c i r c u i t , a f u n c t i o n g e x i s t s such t h a t ,
Y a,h
-
A:
C
a 7,
h
n ( a ) > o(b)
(cf.
52).
To each e l e m e n t a of A , we a s s o c i a t e t h e numher q ( a ) such t h a t l
a(a)
+
a(a)
+ q(a)
q(a)
<
a ( b ) , V b: h P a,
2 o ( c ) , Y c: c
I a:
t h i s i s always n o s s i b l e because b P a and c I a i m a l v b S+ c, thus moreover q ( a ) ; 0 because a I a .
q(b) > p ( c ) :
'Je o h t a i n so two f u n c t i o n s a and q such t h a t , 4' a,h
t
E
n:
a P h = a(a) > o(b) + q(b), a I b
=,
a(a) 6 o(b) + q ( b ) , {g(b) s o(a) + o(a);
as R =
0,
t h e s i m n l e i m n l i c a t i o n s mcy be r e n l a c e d by double i m p l i c a -
tions. (vi)
(i): a P h , b I c and c P d i m p l y o(a) > ?(b)
+
q(b) 2 o(c)
>
q(d)
+
q ( d ) , so t h a t a P d.
0.E.D.
4 . TPlIARDS A D E F I N I T I W OF PARTI4L INTERVAL nRDFR I n terms o f p r e f e r e n c e s , an i n t e r v a l o r d e r assumes t h a t o i v e n two elements o f A, vou must have n r e f e r e n c e o r i n d i f f e r e n c e b e t w e n them.
t o define a conceot which allows incornoarabilities ( 9
+ a)
Ve want now
and w h i c h ex-
tends t h e conceot o f i n t e r v a l o r d e r i n t h e same wav as a o a r t i a l p r e o r d e r extends a comolete n r e o r d e r .
I n t h i s nurpose, i t i s i n t e r e s t i n q t o see
w h i c h a r e t h e c o n n e c t i o n s hetween t h e s i x n r o n e r t i e s o f theorem 1 when
R # 0.
Note t h a t i n t h i s case, n r o n e r t y ( v i ) must be r m l a c e d by
( v i ) ' There e x i s t two f u n c t i o n s g and q such t h a t , V a,b
F
A:
313
A definition ofpartial interval orders
P b * g(a)
c(b) + n(b), a I b * g(a) 6 g(b) + n(b),
a
{
{
? ( b ) s o ( a ) + n(a!, n ( a ) 2 0.
I f r! # 8 , then 1") ( i ) * ( i i i ) * ( i v ) 2") ( i i i ) (i)
0
+
(v) * ( v i ) '
5
(ii)
3 " ) (ii) # (iii). Proof: -
* ( i v ) and t h a t ( i i i ) 5 ( i v ) =. ( v ) . l o ) I t i s e v i d e n t t h a t (i) ( v i ) ' i s s i m i l a r t o t h e n r o o f o f ( v ) =* ( v i ) b u t The o r o o f o f ( v ) =$
s t o o s b e f o r e t h e use o f t h e f a c t t h a t R = The f a c t t h a t ( v i ) '
8.
( i i i ) i s v e r y easy t o v e r i f y , as w e l l as t h e f a c t
that (vi)' * ( i i ) . Consider a P h, h I c, c P d, a R d, a R c and b R d: t h i s example
2 O )
shows t h a t ( i i i ) j i ( i ) .
3")
Consider a P b , b I c, c bRd,bRe,
P
d, d I e , e P f,
f
I a, a R c, a R d, a P e.
b R f, c P e, c R f, d R f: t h i s examnle shows t h a t
(ii) # (iii).
Q.E.D. Ue see t h a t severa de r
.
\eve r t h e l ess
rravs e x i s t f o r t h e d e f i n i t i o n o f a p a r t i a l i n t e r v a l o r n i v e n t h e imoortance o f t h e n u m e r i c a l r e a r e s e n t a t i o n o f
a D w f e r e n c e s t r u c t u r e , we t h i n k t h a t D r o o e r t y ( i i i ) ( o r ( i v ) o r ( v ) ) must be imnosed.
On t h e o t h e r hand, i t i s known t h a t t h e rJassape o f t h e comolete
n r e o r d e r t o t h e i n t e r v a l o r d e r i s nerformed t o t h e d e t r i m e n t o f t h e t r a n s i t i v i t y o f I w h i l e the t r a n s i t i v i t y o f P i s maintained.
T h i s i s t h e reason
why we Drooose t o m a i n t a i n t h e t r a n s i t i v i t y o f P i n t h e d e f i n i t i o n o f a D a r t i a l i n t e r v a l order. T h i s does n o t mean t h a t a s t r u c t u r e where
P
i s n o t t r a n s i t i v e does n o t o r e -
s e n t any i n t e r e s t , b u t we t h i n k t h a t such a s t r u c t u r e must have a n o t h e r name.
These c o n s i d e r a t i o n s l e a d t o t h e f o l l o w i n a d e f i n i t i o n .
M. Roubetis and Ph. I'inrke
114
rEF I 1; I T !n'' ( P , I ) i s a p a r t i a l i n t e r v a l order i f f
1") the r e l a t i o n S+ ( o r 2") D i s transitive.
) does not contain anv c i r c u i t ;
P E V 9K
The choice of ( v ) i n s t e a d of ( i i i ) (which i s e q u i v a l e n t ) in the d e f i n i t i o n i s due t o algorithmic considerations: i t i s much e a s i e r t o h u i l d 5+ and t o verifv the absence o f c i r c u i t in i t (algorithms e x i s t i n y r a n h t l e o r y ) t h a n to v e r i f y nroperty ( i i i ) . 5 . CnNCLLlYInNS
're have nrooosed in t h i s naaer a d e f i n i t i o n of n a r t i a l i n t e r v a l o r d e r , which has the followino c h a r a c t e r i s t i c s :
- if
R =
(no i n c o m a r a b i l i t y ) , the n a r t i a l i n t e r v a l o r d e r i s an i n t e r v a l
order:
- the conditions definina a p a r t i a l i n t e r v a l order are verv easy t o v e r i f y ; - the numerical renresentation of a a a r t i a l i n t e r v a l order i s s i m i l a r t o
-
the renresentation o f an interval order: the d i f f e r e n c e c o n s i s t s i n the renlacement of double imolications by simole i m l i c a t i o n s , as i n the oassaae from complete t o p a r t i a l nreorders; every n a r t i a l i n t e r v a l order ( P , I I can be comnleted i n an i n t e r v a l order ( p ' , I ' ) such t h a t P c P ' a n d I c 1 ' , by renlacinrr in a judicious way i n comnarahilities bv nreferences o r i n d i f f e r e n c e s ; i n f a c t , we have only to put a P' b 9 ?(a) > n(b) + o(b), ( a I ' b * g ( a ) s o(h) + o ( b ) , i g(b) 6 a ( a ) + cl(a), where g and q aw the functions defined i n ( v i ) ' .
REFERENCES
111 121
Fishburn, P . C . , I n t r a n s i t i v e i n d i f f e r e n c e w i t h unequal indifference i n t e r v a l s , Journal of Pathematical P s y c h o l o y , 7 ( 1 9 7 0 ) , 144-149. Roberts, F.S., nn non t r a n s i t i v e i n d i f f e r e n c e , Journal of Uathematical Psychology, 7 ( 1 9 7 0 ) , 243-258.
.4 definition ofpartial interval orders
[3]
141 [51
[6]
[7]
315
Rouhens, W . , Analyse e t anreqation des nreferences: mod@lisation, ajustement e t resume de donnees r e l a t i o n n e l l e s , Pevue Reloe de S t a t . , d'Tnformatique e t de Recherche n G r a t i o n n e l l e , 20 (1980), 36-67. Roy, B., AlgPbrQ moderne e t theorie des rrraptles, 2 V O ~ . , Dunod, (1969). Roy, 8 . , Preference, i n d i f f e r e n c e , i n c o m a r a b i l i t e : r e l a t i o n s b i n a i r e s e t s t r u c t u r e s fondamentales , document 9 , L 4 p W 7 F , Ilniversite Dauphine, Paris (1980). Vincke, P h . , Vrais, quasi , vseudo e t p r e c r i t P r e s dans un ensemble f i n i : nronrietes e t aloori thmes, Cahiers du LAWFDE, llniversi t C p a r i s IX Daunhine (1980). Vincke, P h . , Preference modellino: a survev and an exneriment, Onerational Sesearch 8l,J-.R Brans ( e d . ) , Porth-Holland (1981).
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TRENDS IN MATHEMATICAL PSYCHOLOCY E. Dexreef and J. Van Bu e n h u t (edilors) 0 Ekevier Science Publisgrs B. V. (North-Holland), 1984
317
CAUSAL LINEAR STOCHASTIC DEPENDENCIES: THE FORMAL THEORY
R o l f Steyer University o f Trier T r i e r , Federal R e n u b l i c o f Germany
The f o r m a l background o f t h e t h e o r y o f causal l i n e a r s t o c h a s t i c deoendence i s p r o v i d e d , which was i n t r o d u c e d by S t e y e r (1984).
The t h e o r y p r e s e n t e d i s
concerned w i t h those k i n d s o f deDendencies which can be d e s c r i b e d by s o e c i f y i n g t h e f u n c t i o n a l form of a c o n d i t i o n a l e x o e c t a t i o n E(Y1X).
This includes also
those s i t u a t i o n s i n which X i s a m u l t i d i m e n s i o n a l random v a r i a b l e .
The main concepts o f t h e t h e o r y
a r e causal and weak causal l i n e a r s t o c h a s t i c dependencies, t h e d e f i n i t i o n o f which i s based on t h e p r e - and equiorderedness r e l a t i o n s o f s i g m a - f i e l d s and s t o c h a s t i c v a r i a b l e s , on t h e n o t i o n o f p o t e n t i a l d i s t u r h i n q s i g m a - f i e l d s and v a r i a b l e s , as w e l l as o n t h e i n v a r i a n c e and on t h e average c o n d i t i o n s
.
These
concepts a r e f o r m a l l y d e f i n e d and t h e i r p r o o e r t i e s a r e s t u d i e d i n some d e t a i l . Causal l i n e a r stochast i c dependence i s d e f i n e d b y t h e preorderedness cond i t i o n t h a t t h e i n f l u e n c i n g v a r i a b l e i s antecedent t o t h e i n f l u e n c e d v a r i a b l e and by t h e i n v a r i a n c e cond i t i o n , whereas weak causal l i n e a r s t o c h a s t i c dependence i s d e f i n e d by t h e oreorderedness and average condi ti ons
.
Both, t h e in v a r i ance and t h e average
c o n d i t i o n s , and t h e r e f o r e b o t h k i n d s o f causal hynotheses, can e m i r i c a l l y be t e s t e d i n e x n e r i m e n t a l as we1 1 as i n nonexperimental o b s e r v a t i o n a l s t u d i e s . 1. INTRODUCTION
“ C o r r e l a t i o n does n o t Drove c a u s a l i t y ” , i s a s t a t e m e n t u n i v o c a l l y found in
318
R . Steyer
t e x t b o o k s on a p p l i e d s t a t i s t i c s .
However, whenever one l o o k s f o r a d e f i -
n i t i o n o f causal dependence, one e i t h e r f i n d s t r e a t m e n t s on n h i l o s o p h i c a l t h e o r i e s o n l y l o o s e l y r e l a t e d t o t h e concepts o f c o r r e l a t i o n , r e q r e s s i o n , and t h e e x o e r i m e n t a l c o n t r o l t e c h n i o u e s such as r a n d o m i z a t i o n (see e . g . Basozzi (1980) o r H e i s e ( 1 9 7 5 ) ) , o r t r e a t m e n t s o f e x n e r i m e n t a l and q u a s i e x o e r i m e n t a l c o n t r o l t e c h n i a u e s ( s e e e.g. Cook R Camobell ( 1 9 7 9 ) ) .
Althouah
these d i s c u s s i o n s a r e v e r y u s e f u l and i n s t r u c t i v e i n many r e s n e c t s , t h e r e
i s no f o r m a l t h e o r e t i c a l c o n n e c t i o n between t h e c o r r e l a t i o n , r e q r e s s i o n , and a n a l y s i s o f v a r i a n c e models a n n l i e d and t h e c o n t r o l t e c h n i o u e s d i s c u s sed.
Mhat i s , i n f o r m a l terms, t h e d i f f e r e n c e between an a n a l y s i s o f v a r i -
ance model in a n u r e l y o b s e r v a t i o n a l s t u d y a t one hand and i n a randomized exoeriment a t the other? S t e y e r (1983, 1984) has t a k e n some f i r s t s t e n s t o h r i d g e t h e pan between i d e a s o f causal denendence, s t o c h a s t i c models , and e x n e r i m e n t a l c o n t r o l technioues.
The b a s i c Q u e s t i o n r a i s e d i s , which a r e t h e f o r m a l o r o p e r t i e s
t h a t make a s t o c h a s t i c model a causal one?
R e s t r i c t i n g the discussion t o
l i n e a r s t o c h a s t i c deoendencies, i.e. t o t h o s e denendencies w h i c h can be d e s c r i b e d by c o n d i t i o n a l e x n e c t a t i o n s , S t e y e r (1984) nronosed such p r o p e r t i e s d e f i n i n a two tyoes o f nonsnurious o r causal l i n e a r s t o c h a s t i c dependencies, a weak and a s t r o n g one, each o f w h i c h a r e d i s t i n g u i s h e d f r o m noncausal l i n e a r s t o c h a s t i c deoendencies by two c o n d i t i o n s .
-
Beside t h e p r e -
orderedness c o n d i t i o n t h a t t h e i n f l u e n c i n g v a r i a b l e X i s a n t e c e d e n t t o t h e i n f l u e n c e d v a r i a b l e Y , t h e c r u c i a l c o n d i t i o n f o r causal l i n e a r s t o c h a s t i c dependence of a random v a r i a b l e Y on a, p o s s i b l y m u l t i d i m e n s i o n a l , v a r i a b l e X i s t h e i n v a r i a n c e c o n d i t i o n p o s t u l a t i n g t h a t E(YIX,IJ)
random
as=
E(YIX) + HOW h o l d s f o r a l l o o t e n t i a l d i s t u r b i n g v a r i a h l e s 14, where HOW i s a !\r-measurable c o m o o s i t i o n o f a f u n c t i o n H w i t h W .
The i n v a r i a n c e c o n d i -
t i o n s t a t e s t h a t , i f any n o t e n t i a l d i s t u r b i n g v a r i a b l e W i s i n c l u d e d as another c o n d i t i o n i n g v a r i a b l e E(Y!X,W),
,it
o n l y adds t o E(Y1X) i n t h e e q u a t i o n f o r
leaving E ( Y I X ) invariant.
The c r u c i a l c o n d i t i o n f o r weak causal l i n e a r s t o c h a s t i c deoendence, on t h e o t h e r hand, i s t h e average c o n d i t i o n D o s t u l a t i n q t h a t , f o r a l l D o t e n t i a l disturbing variables almost a l l E ( Y I X = x ) .
GI,
E(YIX=x) = /E(YIX=x, W=w)
t h e c o n d i t i o n a l e x p e c t a t i o n s E(YIX=x, W=w) across PX-almost a l l E(YIX=x). t i o n can be w r i t t e n :
PW(dw) h o l d s , f o r Px-
T h i s c o n d i t i o n means t h a t E(YIX=x) i s t h e average o f t h e v a l u e s w o f W, f o r
I f W i s a d i s c r e t e s t o c h a s t i c v a r i a b l e , t h i s equa-
Garsat linear stochactic dependencies
E(YIX=x) =
Iv, E ( Y I X = x , W=W)
319
P(I,l=w).
I t w i l l be shown t h a t t h i s average condition holds, f o r examnle, i f X and a l l p o t e n t i a l d i s t u r b i n a variahles GI a r e s t o c h a s t i c a l l y independent, a cond i t i o n which may be assumed t o hold i n randcmized exDeriments, where X i n dicates group membershin and the potential disturbina variahles 14 represent pronerties of the experimental units ( s u b j e c t s ) before o r a t the tine of the treatment. This nrovides the desired link t o the experimental control technioues such as randomization and matching.
In the formulation of the nreorderedness, invariance and average conditions above, formally undefined terms have been used t h a t have t o be eliminated i n order t o construct a formal theory consisting only of mathematically well-defined terms. So f a r , we have no formal c r i t e r i a t o decide whether o r not a s p e c i f i e d s t o c h a s t i c variable X i s antecedent t o a sDecified s t o c h a s t i c variable Y , and whether o r not a s p e c i f i e d s t o c h a s t i c variable W 2 i s a p o t e n t i a l disturbing v a r i a b l e . Can W1 = X be a notential disturbing variable w i t h respect t o the dependence of Y on X? Can 1J2 = X+Z be a pot e n t i a l disturbinq variable? Are variables mediating between X and Y pot e n t i a l disturbing v a r i a b l e s , o r variables which a r e influenced by Y? I n the following s e c t i o n s , we f i r s t summarize some b a s i c concents of probab i l i t y theory and introduce notational conventions. Then, we give a formal d e f i n i t i o n of the pre- and equiorderedness r e l a t i o n s and study t h e i r prooert i e s . Next, we define notential disturbing sigma-fields a n d variables, which w i l l make the averaae and invariance conditions discussed above well defined formal concepts, too. Then, we give formal d e f i n i t i o n s of causal and weak causal l i n e a r s t o c h a s t i c denendencies , and i n v e s t i g a t e the properties o f these concepts. 2 . SOME BASIC CONCEPTS OF PROBABILITY THEORY AND NOTATION
I n this s e c t i o n , a b r i e f summary and notational concepts o f p r o b a b i l i t y theory a r e given, which a proDer understanding of the theory proposed. the reader i s r e f e r r e d t o Bauer (1974) , Breiman (1977) , Halmos (1969) , o r Loeve (1977, 1978).
conventions of some basic seem t o be e s s e n t i a l f o r For d e t a i l e d introductions , (1968) , GBnsler and S t u t e
3 20
W. Steyer
The fundamental assumntion o f e v e r v s t o c h a s t i c s u b s t a n t i v e ( i .e. p s y c h o l o n i c a l , s o c i o l o g i c a l , e t c . ) model i s t h a t t h e e x n e r i m e n t , o r more g e n e r a l l y , t h e n a r t o f r e a l i t y t o be d e s c r i b e d , can be r e p r e s e n t e d by a n r o b a b i l i t y sDace, t h e d e f i n i t i o n o f w h i c h i s based on t h e f o l l o w i n g concents. Let
F:
=
A s i g m a - f i e l d A on n i s
i1,Z. . . . I be t h e s e t o f n a t u r a l numbers.
d e f i n e d t o be a s e t o f subsets o f I! with t h e f o l l o w i n g t h r e e p r o p e r t i e s . ( b ) I f A E A, t h e n AC E A , where AC := R - A denotes t h e cornC A. (a) Dlement o f A . ( c ) I f (Ai, i € N) i s a sequence o f elements Ai o f A , t h e n t h e i r u n i o n uia
Ai
i s an element o f A .
The i n t e r s e c t i o n o f a f a m i l y (Ai,
i c I ) o f s i a m a - f i e l d s Ai
on n i s a l s o a
s i a m a - f i e l d on n.
I f E i s a s e t o f nonempty s u b s e t s o f R , t h e n tRe s i p s E i s d e f i n e d t o be t h e i n t e r s e c t i o n o f a l l those s i g m a - f i e l d s on R , w h i c h c o n t a i n E as a s u b s e t . f i e l d A ( E ) g e n e r a t e d by t h e s e t system
A measurable space i s d e f i n e d t o be a p a i r (Q,A)
o f a s e t Q and a sigma-
f i e l d A on ?. L e t ( : ? , A ) be a measurable soace.
A o r o b a b i l i t y m a s u r e P: A
+
[0,11 i s de-
f i n e d t o be a f u n c t i o n a s s i g n i n g each A E A a r e a l n o n n e g a t i v e number w i t h (a) P(0) = 0.
the following three o r o o e r t i e s . sequence o f elements o f A w i t h Ai P(Uia4
Ai)
P(Ai).
= Eia
n A
j ( c ) P(R) = 1.
=
0
( b ) I f (Ai,
iEN) i s a
f o r i # j , then
A o r o b a b i l i t y space can now be d e f i n e d t o be a t r i p l e (n,A,P)
o f a set
c a l l e d t h e s e t of e l e m e n t a r y e v e n t s , a s i g m a - f i e l d A o f s u b s e t s o f n, where t h e elements A
E
A a r e c a l l e d e v e n t s , and a D r o b a b i l i t y measure P: A
which a s s i g n s t h e n r o b a b i l i t y P(A) t o each e v e n t A L e t (C,A) and ( n ' , A ' ) (A,A')-measurable,
be measurable snaces.
E
+
[0,11
A.
P lraoping X: R
+
n' i s c a l l e d
iff, f o r a l l A' E A',
X-'(A')
:=
{U f
C:
X ( W ) E A ' ) E A.
1 1 The s e t X- ( A ' ) := { X - ( A ' ) : A ' E A ' ) i s c a l l e d t h e s i g m a - f i e l d g e n e r a t e d 1 by X and A ' . X- ( A ' ) may a l s o be denoted by A ( X , A ' ) . I f X i s real-valued 1 and N ( 1 ) - d i m e n s i o n a l , A(X,A') = X- ( A ' ) i s a l s o denoted by A ( X ) and A' i s onlRN('), which i s defined t o N(I) be t h e s i g m a - f i e l d g e n e r a t e d b y t h e s e t system o f a l l open i n t e r v a l l s o f understood t o b e t h e Bore1 s i a m a - f i e l d 6
RN('),
N(I) EN.
321
Causal linear stochastic dependencies
L e t (Ri,Ai),
i E I , be a f i n i t e o r i n f i n i t e sequence o f measurable spaces. i E I ) o f (A,Ai)-measurable
A s i g m a - f i e l d generated b y a sequence (Xi, p i n g s Xi:
+
Ri
i.e.
t h e s i g m a - f i e l d s XY1(Ai),
-1
i E I ] := AIUiEIXi
A[(Xi,Ai),
(A,)]
.
I f I = ~ l , . . , , N ( I ) ~ i s f i n i t e and ni
=IR,
a l t e r n a t i v e n o t a t i o n A(X1,...,XN(I))
i n s t e a d o f A[(Xi,Ai),
L e t @,A)
mao-
i s defir.ed t o be the s i g r a - f i e l d generated by t h e u n i o n c f
and ( n ' , A ' )
f o r a l l i E I,we a l s o use t h e
be measurable spaces.
i E I].
An ( n ' , A ' ) - s t o c h a s t i c
i s d e f i n e d t o be a mapDing X: R
X on t h e p r o b a b i l i t y space (R,A,P)
variable +
fi'
A N(1)-dimensional r e a l - v a l u e d s t o c h a s t i c v a r i -
t h a t i s (Aye')-measurable.
)-stochastic variable. N(I) The m e a s u r a b i l i t y c o n d i t i o n i n t h e d e f i n i t i o n o f a ( 2 ' , A ' ) - s t o c h a s t i c
, ) ' i (s Na I @ a b l e , f o r example,
R
vari-
able i m p l i e s t h a t
1 X- (A')
:=
{U
E R:
X(W) E A ' ) E A , f o r a l l A '
E
A'.
T h i s a l l o w s t o d e f i n e t h e d i s t r i b u t i o n Px: E , ' + t0.11 o f a ( R ' , A ' ) - s t o c h a s t i c v a r i a b l e X by
1 PX(A') = P[X- ( A ' ) ] , f o r a l l
A'EA'.
The d i s t r i b u t i o n P x o f X i s a D r o b a b i l i t y measure on A ' .
3. PRE- AND EQUIORDEREDNESS
A necessary c o n d i t i o n f o r a s t o c h a s t i c dependence o f Y on X t o be causal i s t h a t X i s antecedent o r , synonymously, p r e o r d e r e d t o Y . I n exoeriments, f o r example, t h e t r e a t m e n t v a r i a b l e s a r e m a n i p u l a t e d b e f o r e t h e e f f e c t s on t h e dependent v a r i a b l e s a r e assessed, and i n nonexperimental s t u d i e s , t o o , t h e i n f l u e n c i n g v a r i a b l e s have t o be antecedent t o t h e i n f l u e n c e d v a r i a b l e s , i f a causal s t a t e m e n t s h o u l d make sense a t a l l .
Even r e c i p r o c a l causal de-
pendence can much b e t t e r be t h o u g h t o f as a process o f mutual i n f l u e n c i n g o f t h e v a r i a b l e s i n v o l v e d (see e.g. S t e y e r ( 1 9 8 2 ) ) , where, a t each p o i n t o f t h e process, t h e c a u s i n g v a r i a b l e i s D r e o r d e r e d t o t h e caused one.
Thus,
r e p r e s e n t i n g r e c i p r o c a l causal r e l a t i o n s by dynamic models n o t o n l y a l l o w s t h e p r e s e r v a t i o n o f a s y m e t r y o f i n f l u e n c e (see e.g. Simon ( 1 9 5 2 ) ) f o r r e c i p r o c a l c a u s a l i t y , b u t i s a l s o much more congruent w i t h t h e dynamic n a t u r e
322
R . Steyer
Preorderedness o f a s t o c h a s t i c v a r i a b l e X t o a second one,
Y,
i s t h e f i r s t concent t o b e d e f i n e d w h i c h draws on t h e f a c t t k l a t (n,A,P)
is
of r e a l i t y .
assumed t o r e p r e s e n t a n r o c e s s .
A ( ~ ~ ' , A ' ) - s t o c h a s t i cnrocess on a n r o b a b i l i t v snace be a f a m i l y (Xt,
t C T) o f (n',A')-stochastic
(n,A,p)
i s defined t o Hence,
v a r i a h l e s on (n,A,P).
one m i o h t t h i n k o f d e f i n i n q Xs t o be n r e o r d e r e d t o Xt, However, such a concent w o u l d be t o o r e s t r i c t e d .
i f s < t, s , t E T .
Oftentimes, variables
have t o be c o n s i d e r e d t h a t a r e d e f i n e d , f o r examnle, bv X := X
S
. Xt,
s#t.
O b v i o u s l y , t h e d e f i n i t i o n o f nreorderedness c o n s i d e r e d above c o u l d n o t be a n n l i e d t o X, because n o t o n l y one b u t two D o i n t s o f t i m e a r e i n v o l v e d i n t h e d e f i n i t i on o f X
.
These problems a r e n o t o n l y o f t h e o r e t i c a l , b u t a l s o o f much n r a c t i c a l i n terest.
I f , f o r examnle,
the e f f e c t o f a psycholooical V e s t r e n t (theraoy,
t r a i n i n g , e t c . ) i s t o be i n v e s t i q a t e d , such a t r e a t m e n t u s u a l l y extends over several sessions.
Hence, i f we l e t T be t h e i n d e x s e t o f t h e s e s s i o n s ,
t h e n t h e t r e a t m e n t v a r i a b l e , i n d i c a t i n g whether o r n o t a n e r s o n r e c e i v e s a t r c t a t w n t , cennot he a s s i a n e d t o one s i n g l e t E T . i f we l e t
T
The same n r o b l e m o c c u r s ,
denote t h e c o n t i n u o u s t i m e covered hy one s e s s i o n .
t i o n o f t h e D s y c h o l o o i s t cannot be a s s i g n e d t o a s i n a l e t o u t l i n e d above.
E
An i n t e r v e n T i n t h e way
S i m i l a r l y , reDeated e v e n t s , such as b e i n o cheated repeated-
l v h v one's mother, may he a cause o f someone b e i n g d i s t r u s t f u l .
Again,
t h e c a u s i n g v a r i a b l e cannot be a s s i a n e d t o a s i n q l e P o i n t of t i m e , a p r o blem which Suppes (1970) seems t o n e o l e c t ( c f . S t e y n i i l l e r (1983) p. 6 0 2 ) ) .
3! .J.
-
!
F i g u r e 1.
A m o n o t o n i c a l l y i n c r e a s i n g f a m i l y (At,t
E
T ) o f sigma-fields.
323
Grusal linear stochartic dependencies
I n o r d e r n o t t o r u n i n t o these d i f f i c u l t i e s ,
we b a s i c a l l y f o l l o w t h e ap-
nroach proposed by S t e y e r (1983). which i s , however, c o r r e c t e d i n some p o i n t s and s t u d i e d i n m r e d e t a i l . (Xt,
T h i s apnroach i s n o t based on a process
t E T ) , b u t on a m o n o t o n i c a l l y i n c r e a s i n g f a m i l y (At,
fields,
t E T) o f siama-
(see F i g u r e 1). L e t T be a s u b s e t o f t h e s e t R o f r e a l numbers.
A monotonically increasina f a m i l y ( A t ,
t E T) o f s i p m a - f i e l d s i s d e f i n e d
t o be a f a m i l y o f s i g m a - f i e l d s At ofi a s e t R, t E T C X R , w i t h t h e p r o p e r t y that, i f s
t,
f
s,t E
T, then As
C At.
Such a m o n o t o n i c a l l y i n c r e a s i n g
f a m i l y o f s i g m a - f i e l d s i s o b t a i n e d , f o r examole, i f we d e f i n e At t o be t h e s i g m a - f i e l d generated by a l l Xs, s E T, s 6 t, where (Xt, t o be a r e a l - v a l u e d s t o c h a s t i c p r o c e s s . t i c nrocess (Xt, (At,
t
E
t
E
T ) i s assumed
However, we do n o t need a stochas-
T) t o construct a monotonically increasing f a m i l y
t E T) o f siama-fields,
as i s shown i n t h e f o l l o w i n g examDle.
3.1. EXAMPLE Consider an e x p e r i m e n t i n which two c o i n s , each h a v i n g one metal and one o l a s t i c s i d e , a r e t o s s e d o n t o a p l a t e h a v i n g t h e p r o p e r t i e s of an e l e c t r o maanet which i s on o r o f f w h i l e t h e two c o i n s a r e tossed. ( F o r a more det a i l e d d e s c r i p t i o n o f t h i s example, see S t e y e r , (1984)).
I n t h i s applica-
t i o n , we may choose R t o be t h e s e t o r o d u c t o f t h e s e t s R l = Ial,a21, Ibl,b2}, and n3 = {c1,c21. Hence, each element (ai,b.,c ) o f R denoJ k t e s one o f t h e e i g h t elementary events t h a t Coin 1 shows s i d e i, Coin 2
O2 =
shows s i d e j, and t h e e l e c t r o magnet i s o f f ( k = l ) o r on ( k = O ) .
Further-
t o be t h e s e t o f a l l subsets o f a .
more, we choose t h e s i g m a - f i e l d A on
I f X denotes t h e s t o c h a s t i c v a r i a b l e i n d i c a t i n a t h e s t a t e o f t h e e l e c t r o magnet and t h e v a r i a b l e s Yi,
i=1,2,
i n d i c a t e t h e outcome o f t o s s i n g c o i n i,
a m o n o t o n i c a l l y i n c r e a s i n g f a m i l y (At, t E T ) o f s i c m a - f i e l d s i s e a s i l y obt a i n e d i f we d e f i n e T := {1,21, A1 := A(X), and A 2 := A(X,Y,,Y,), A(X,Y1,Y2)
where
denotes t h e s i g m a - f i e l d generated by t h e r e a l - v a l u e d s t o c h a s t i c
v a r i a b l e s X, Y1,
Y2.
The example above, i n v o l v i n c j o n l y t h r e e v a r i a b l e s X, Y1 and Y 2 and t h e i r generated s i g m a - f i e l d s ,
i s very simple.
However, i t w i l l h e l p t o i l l u s -
t r a t e t h e concept o f oreorderedness t h a t i s d e f i n e d as f o l l o w s . 3.2.
DEFINITION
L e t (n,A,P)
be a p r o b a b i l i t y space, l e t T be a s u b s e t o f t h e s e t l R of r e a l
313
R . Steyer
numbers, l e t (At, f i e l d s w i t h A(uET b l e s on (c?,A,P), (i)
Lfe say t h a t
t F T) be a m o n o t o n i c a l l y i n c r e a s i n g f a m i l y o f sigma-
At) C A , l e t X,
'$1
be ( P ' , A ' ) - ,
(n",A")-stochastic
varia-
r e s n e c t i v e l y , and f i n a l l y , l e t C , 0 C A be two s i g m a - f i e l d s . C i s preordered t o
P w i t h r e s p e c t t o (A t '- t
E T), i f f
( ' i f f ' i s an a b r e v i a t i o n f o r ' i f and o n l y i f ' ) (a) there i s a
s
E
A s , fl
T, w i t h C
( b ) t h e r e i s an element t
E
T, t >
p
AS and
s , w i t h 1) C At,
As, for a l l
0
S E T , s < t.
( i i ) 'Je say t h a t X i s n r e o r d e r e d t o W w i t h r e s p e c t t o (At,
t
T),
iff
C o n d i t i o n s ( a ) and ( b ) h o l d w i t h 1 ( C ) C = X- ( A ' ) , 1 = U - I ' A ' ).
I f no c o n f u s i o n i s p o s s i b l e , t h e e x n l i c i t r e f e r e n c e t o (At, t t T) may be A c c o r d i n g t o C o n d i t i o n s ( a ) and ( b ) of E e f i n i t i o n 3.2,
omitted.
a smallest t E T f o r which
s
F
T, s
<
D is
there i s
a s u b s e t o f A t and t h a t t h e r e i s an element
t, such t h a t C i s a s u b s e t o f A,.
I n t h e examnle d i s c u s s e d
! and above, where a l l v a r i a b l e s i n v o l v e d a r e r e a l - v a l u e d ( i . e . R' = 9'' = R A' = A " = 8 , where B denotes t h e Bore1 s i g m a - f i e l d on R ) , X i s o r e o r d e r e d 1 t o Y 1 and Y2, because A ( X ) : = X- ( R ) A1, whereas A(Y1) : = Y,-l(R) ! .$ A1 -1 I n t h i s example, Y1 and A ( Y 2 ) : = Y 2 ( B ) A1 , b u t A(Y1), A(Y2) C A2. and Y 2 a r e e q u i o r d e r e d i n t h e f o l l o w i n g sense. 3.3. DEFINITION L e t t h e Dresumptions and n o t a t i o n s o f 3.2 be v a l i d . (i)
We say t h a t C and 0 a r e e q u i o r d e r e d w i t h r e s p e c t t o (At, ( a ) t h e r e i s an element t E T w i t h C, ( b ) t h e r e i s no element s E T,
s
D c At
t E T), i f f --
and
< t, w i t h C C
AS o r 1) C A,.
( i i ) We say t h a t X and I d a r e e q u i o r d e r e d w i t h r e s p e c t t o (Att,
t E T), i f f
C o n d i t i o n s ( a ) and ( b ) h o l d w i t h ( c ) C = X- 1( A ' ) and D = k J - ' ( A ' ' ) . A c c o r d i n g t o C o n d i t i o n s ( a ) and ( b ) o f D e f i n i t i o n 3.3, t h e r e i s a s m a l l e s t element t E T w i t h C and P b e i n g b o t h subsets o f At, and t h e r e i s no e l e ment s E T, s < t, such t h a t C o r 0 a r e subsets o f A s . I t i s e a s i l y seen t h a t t h e diagram o f any r e c u r s i v e o a t h a n a l y s i s model can
be t r a n s l a t e d i n t o a m o n o t o n i c a l l y i n c r e a s i n g f a m i l y o f s i g m a - f i e l d s so t h a t t h e concepts o f p r e - and equiorderedness can be a p p l i e d . nram o f F i g u r e 2, f o r examole, we may d e f i n e T := {1,2,31,
For the path diaA 1 := A(Z
1' Z 2 ) ,
Causal linear stochattic dependencies
325
.
A2 := A(Z1,Z2,Z3) , a n d A3 := A(Z,,. . ,Z4). Obviously, Z1 and Z2 a r e equiordered, Z1 and Z2 a r e both preordered t o Z3 and Z4, and Z3 i s nreordered t o Z4 with respect t o ( A t , t E T) ( s e e Definitions 3.2 and 3 . 3 ) .
Fipure 2.
Path diagram f o r a recursive model w i t h four v a r i a b l e s .
We now t r e a t some formal Drooerties of the pre- and equiorderedness relat i o n s . The f i r s t one i s t h a t preorderedness of sigma-fields, as well as o f s t o c h a s t i c v a r i a b l e s , i s a s t r i c t order r e l a t i o n . Note t h a t a l l propos i t i o n s in the followincl theorems a r e made w i t h respect t o a given family ( A t , t E T ) of sigma-fields. 3.4. THEOREM
Let the presumptions and notations of 3.2 be v a l i d , l e t F C A be a sigmaf i e l d a n d l e t X, W, U be ( n ' , A ' ) - , (n",A")-, and (n"' ,A"')-stochastic variables on the probability space ( n , A , P ) , respectively. ( i ) I f C i s preordered t o 77, then D i s not preordered t o C (asymmetry). ( i i ) I f C i s preordered t o t, and Q i s preordered t o F, then C i s a l s o preordered t o F ( t r a n s i t i v i t y ) ( i i i ) Preorderedness of sigma-fields i s a s t r i c t order r e l a t i o n on the s e t of a l l sigma-fields being subsets of A. ( i v ) Propositions ( i ) t o ( i i i ) a l s o hold f o r s t o c h a s t i c variables X, C' and U taking the r o l e of C, Q, and F, resoectively.
.
Proof. -
We only have t o show t h a t the properties ( i ) and ( i i ) hold f o r sigmaf i e l d s , because a s t r i c t order r e l a t i o n i s defined by the assymetry and t r a n s i t i v i t y p r o p e r t i e s , and preorderedness of s t o c h a s t i c variables i s defined through t h e i r generated sigma-fields. I f C i s preordered t o D w i t h respect t o (At, t E T ) , then there (i)
326
R . Stcyer
i s an element s
E
T with C
C
AS, 0
p
A S . A s (At,
c a l l y i n c r e a s i n o , t h e r e i s n o element t C
E
t E T ) i s monotoni-
T, t > s , such t h a t D
p At, which i m n l i e s t h a t P i s n o t p r e o r d e r e d t o If C i s D r e o r d e r e d t o D, t h e n l a ) there i s a s F T w i t h C c A S , D As.
C
At,
C.
(ii)
i s p r e o r d e r e d t o F , then
If
P
T, t > s, w i t h
(b) there i s a t
c At,
st
f
At,
F c
( c ) t h e r e i s an element u F T, u > t, u i t b
and
AL, F p A s , f o r a l l
s F T , s < u .
Hence, C o n d i t i o n s ( a ) t c ( c ) i m p l y t h a t t h e r e i s an element s F p/ith C C A c ,
F
c A U'
F
F
iA
$T A , ,
and t h e r e i s an e l e m e n t u
~ f, o r a l l s
T, s
<
C
T, u
T
> s, with
u.
Theorem 3.4 does n o t i m o l y t h a t a l l s i g m a - f i e l d s b e i n q subsets o f A a r e i n t h i s r e l a t i o n , b u t t h a t t h e asymmetry and t r a n s i t i v i t y c o n d i t i o n s a r e f u l fil led.
3.5.
THEOREM
L e t t h e presumptions and n o t a t i o n s o f 3 . 2 and 3.4 be v a l i d . ordered t o
D, then
C i s a l s o preordered t o F := A ( C
U
I f C i s pre-
U).
Proof. -
I f C i s preordered t o
D,
then
( a ) t h e r e i s an element s E T w i t h C c A s , U $ A s , and ( b ) t h e r e i s an element t E T, t > s , w i t h P C A D ' (At, t E T ) b e i n q m o n o t o n i c a l l y i n c r e a s i n g , i m p l i e s
t
( c ) C c At,
F c At,
because C ,
D c At imDlies F
C
p As, d
S E T . s
At.
( d ) F i s n o t a s u b s e t o f A S , s E T, s i t, because o f ( b ) . Hence, t n e r e i s an e l e m e n t s t T w i t h L' C A S , F $T AS (see ( a ) and ( a ) ) , and t h e r e i s a t
T, t > s , w i t h F C At,
F
AS, V s t
T, s
< t (see
( c ) and ( d ) ) , which proves t n e theorem A c c o r d i n q t o t h e f o l l o w i n g theorem, equiorderedness i s an e q u i v a l e n c e r e l a tion, 3 . 6 . THEOREM
L e t t h e oresumptions and n o t a t i o n s o f 3 . 2 and 3 . 4 be v a l i d . ( i ) I f t h e r e i s a t 6 T w i t h C C A t and no s E T, s < t, w i t h C C i s equiordered t o i t s e l f ( r e f l e x i v i t y ) .
C
A s , then
&sol
327
linear stochatic dependencies
( i i ) I f C and D a r e e q u i o r d e r e d , t h e n D and C a r e e q u i o r d e r e d (symmetry). ( i i i ) I f C and P as w e l l as D and F a r e e q u i o r d e r e d , t h e n C and F a r e e q u i ordered ( t r a n s i t i v i t y ) . ( i v ) Equiorderedness o f s i g m a - f i e l d s i s an e q u i v a l e n c e r e l a t i o n on t h e s e t o f a l l s i g m a - f i e l d s Ci Ci
c At and no s
C
A , f o r which t h e r e i s an element t
E
T with
T, s < t, w i t h Ci c A S .
E
( v ) P r o n o s i t i o n s ( i ) t o ( i v ) a l s o h o l d f o r equiorderedness o f s t o c h a s t i c v a r i a b l e s X, W, 11 t a k i n g t h e r o l e s o f C , D, and F, r e m e c t i v e i y . Proof. (i) (ii)
C o n d i t i o n s 3.3a and 3.3b a r e o b v i o u s l v t r u e f o r C = P . I s obvious.
( i i i ) I f C and U a r e e q u i o r d e r e d , then ( a ) t h e r e i s a s m a l l e s t t E T such t h a t C , P
C
At and no s E T,
C As o r D c: As. F are equiordered, t h i s imDlies t h a t also
s < t, w i t h C
If
1)
anL
( b ) F c At and t h a t ( c ) t h e w i s no s E T, s < t, w i t h P C As o r F
(1
As.
( a ) t o ( c ) i m p l y t h a t t h e r e i s a t E T w i t h C, F C At,
whereas ( a )
F C A,. F a r e e q u i o r d e w d , which i m p l i e s t h e t r a n s i t i v i t y
and ( c ) i m p l y t h a t t h e r e i s no s E T, s < t, w i t h C C AS o r Hence, C and omperty
.
(iv)
An e q u i v a l e n c e r e l a t i o n i s d e f i n e d b y r e f l e x i v i t y , s y m t r y , and
(v)
T h i s o r o n o s i t i o n i s i m o l i e d by 3.6i t o 3 . 6 i v Y i f we d e f i n e -1 C := X ( A ' ) , P := l!J-'(A"), and F := U-'(A''I ) .
transitivity.
I t s h o u l d be n o t e d t h a t n o t e v e r y s i g m a - f i e l d C
c A i s equiordered t o i t -
I t may hapaen t h a t C i s n o t i n t h e equiorderedness r e l a t i o n , because
self.
t h e r e i s no s m a l l e s t t E T w i t h C c At.
I f T = R , C = IB,n}, f o r example,
t h e r e i s no s m a l l e s t t E T w i t h I @ s ~ C} At.
The e q u i v a l e n c e r e l a t i o n o f
equiorderedness i s n o t d e f i n e d on t h e s e t o f a l l s i g m a - f i e l d s
CiC
A, b u t
o n l y on t h e s e t o f a l l Ci C A , f o r which t h e r e i s a s m a l l e s t t E T w i t h Ci c A t .
? .7. THEOREM L e t t h e n w s u m p t i o n s and n o t a t i o n s o f 3.2 and 3.4 be v a l i d . I f C and
D a r e e q u i o r d e r e d , t h e n C and F
: = A(C
u P ) , as w e l l as D and
328
R. Steyer
T-' a r e e n u i o r d e r e d . Proof. I f C and ? a r e e q u i o r d e r e d , then t h e r e i s a s m a l l e s t t
F
T such
tlltlt
( a ) C,
P c A t , and
( b ) t h e r e i s no s F T, s < t, w i t h C C AS o r f i
C o n d i t i o n ( a ) i m p l i e s F C At,
C
because F i s t h e s m a l l e s t s i g m a - f i e l d
c o n t a i n i n q b o t h r and P as s u b s e t s .
Condition (h) i m o l i e s t h a t
t h e r e i s no s F- T, s < t, w i t h C, ?, o r F C A 5 . C aid
A5.
This implies t h a t
F as w e l l as D and F a r e e q u i o r d e r e d .
3.P,. THF09Fb1 L e t t h e n w s u m o t i o n s and n o t a t i o n s o f 3.2 and 3.4 be v a l i d . ( i ) I f C i s preordered t o dered t o F .
D, and
i, and
F a r e e q u i o r d e r e d , then C i s n r e o r -
( i i ) I f C , P a r e e q u i o r d e r e d and P i s n r e o r d e r e d C.c F, t h e n C i s D r e o r d e r e d
t o F. ( i i i ) P r o o o s i t i o n s ( i ) and ( i i ) a l s o h o l d f o r s t o c h a s t i c v a r i a b l e s X,
01
and
I1 t a k i n o t h e r o l e s o f C , D and F , r e s o e c t i v e l y . Proof. (i)
I n t h i s case, ( a ) t h e r e i s an element s F
t E T, t > s, w i t h P
C
T with At,
C
C
0 $ At,
F
As and an element f o r a l l s E T, s < t .
As, I,
Furthermore, (b) F
C
A t and t h e r e i s no s E T, s < t, w i t h ?
for all s
F
C
As o r F C AS,
T, s < t .
( a ) and ( b ) i m p l y t h a t t h e r e i s a s E T w i t h C C A?.
r
an c i e f i e r t t E T, t > s, w i t h F E At,
F p AS, f o r a l l s
which i m p l i e s t h a t C i s o r e o r d e r e d t o
F (see 3 . 2 ) .
(ii)
A s , atiu E
T, s < t,
Is o b v i o u s .
( i i i ) T h i s o r o o o s i t i o n i m m e d i a t e l y f o l l o w s f r o m 3 . 8 i and 3 . 8 i i , define
r
:= Y - l ( A ' ) ,
? := W - ' ( A " ) ,
an(!
F := \I-'(A'''
i f we
).
Another conceot needed i n t h e f o l l o w i n g s e c t i o n s i s t h a t o f e o u i - o r p r e o r deredness, where ' o r ' denotes t h e l o c i c a l d i s j u n c t i o n . edness i s a weak o r d e r r e l a t i o n .
Equi- o r nreorder-
Aoain, t h e o r o o o s i t i o n s a r e always made
w i t h r e s m c t t o an u n d e r l y i n q f a m i l y (At,
t E
T) o f s i g m a - f i e l d s , t h e
Grusal linear stochastic dependencies
329
exnlicit reference t o which i s omitted only for reasons of convenience. 3.9. DEFINITION
Let the nresumptions and notations of 3.2 be valid. ( i ) Isle say t h a t C i s equi- or nreordered t o 0 , i f f ( a ) C and 0 are enuiordered, or ( b ) C i s preordered t o D. ( i i ) Ire say t h a t X i s enui- or oreordered t o 'J, i f f C i s equi- or preordered t o P a n d ( c ) C = X- 1(A') and P = l , f - l ( A " ) . 3.10. THEOREPI
Let the nresumptions and notations of 3.2 and 3.4 be valid. ( i ) I f there i s a t E T with C C: At such t h a t there i s no s E T , s < t. with C C A,, then C i s equi- or prcordered t o i t s e l f (reflerivi t y ) . ( i f ) If C i s equi- or oreordered t o 0 , and D i s equi- or nreordered t o C , then C and U are equiordered (antisymnetrv). ( i i i ) I f C i s equi- or oreordered t o P , and P i s equi- or oreordered t o F, then C i s equi- or preordered to F ( t r a n s i t i v i t y ) . ( i v ) Equi- or preorderedness o f s i y a - f i e l d s i s a weak order relation on the s e t of all sicma-fields Ci C A , for which there i s a t E T with Ci C A
t such t h a t there i s no s T, s < t, with C C A,. ( v ) Pronositions 3.10i t o 3.10iv also hold for equiorderedness of stochast i c variables X, W , U taking the roles of C, P, and F, resnectively. Proof. -
ProDerty 3.10i follows from 3.6i. In order t o prove 3.1Oiiy we consider four cases. First, b o t h C and U are preordered t o one another, w h i c h can be excluded, because of 3.4i. Second, C and D are equiordered. In this case, the pronosition 3.10ii i s imnlied by 3.6ii. Third, C i s preordered t o P and D i s equiordered t o C . This contradicts Condition ( b ) of Definition 3.3. Fourth, U i s oreordered t o C , and C i s equiordered t o U . Again, this case cannot occur, because of 3.3b. The proof of prooerty ( i i i ) follows the same line of arguments as t h a t o f property 3.10ii. For the f i r s t case t h a t C , 0 and 0, F are equiordered, the property i s imolied by3.6iii. For the second case t h a t C i s preordered t o C and U i s nreordered t o F, ProDerty 3.lOiii i s
R. Steyer
330
i m n l i e d by 3 . 4 i i i .
F o r t h e t h i r d case t h a t C i s n r e o r d e r e d t o
U , and
0 and F a r e e q u i o r d e r e d , i t i s i m o l i e d hy 3 . 8 i , and f o r t h e l a s t case t h a t C and I! a r e e q u i o r d e r e d , whereas D i s o r e o r d e r e d t o F , i t i s i m ( i v ) A weak o r d e r r e l a t i o n i s r'efined by r e f l e x i v i t y ,
o l i e d by 3 . 8 i i .
antisymmetry, and t r a n s i t i v i t y . t o ( i v ) , i f we d e f i n e f : = X - ' ( A ' ) , 3.11.
( v ) This n r c n o s i t i o n f o l l o w s from ( i )
D
: = ~ ~ ~ " ( A " and ) , F := U-'(A''').
THEOREM
L e t t h e nresumptions and n o t a t i o n s o f 3.2 and 3.4 be v a l i d .
o r o r e o r d e r e d t o P , then C i s a l s o e q u i - o r o r e o r d e r e d t o F
I f C i s equi:= A ( C
u 27).
?roof. ( a ) I f C i s p r e o r d e r e d t o P , then, accordin!
t o 3.5i, C i s also preor-
dered t o F . ( b ) I f C and
D
a r e e q u i o r d e r e d , then, a c c o r d i n g t o 3 . 7 i , C and F a r e
also equiordered.
( a ) and ( b ) i m l y t h e theorem.
4 . POTENTIAL DISTURRING SIGVA-FIELDS AND VPRIABLES
Ey i n t u i t i o n , i t i s obvious t h a t nreorderedness o f X t o
Y i s a necessary
b u t n o t s u f f i c i e n t c o n d i t i o n f o r t h e s t o c h a s t i c denendence o f Y on X t o he causal.
Two a l t e r n a t i v e second necessary c o n d i t i o n s , t h e i n v a r i a n c e and
a v e r a w c o n d i t i o n s have been d i s c u s s e d by S t e y e r (1984).
To s t a t e e i t h e r
of them o r e c i s e l y , we need a f o r m a l d e f i n i t i o n o f a n o t e n t i a l d i s t u r b i n g v a r i a b l e and a p o t e n t i a l d i s t u r b i n g s i q m a - f i e l d .
The aim i s t o d e f i n e v a r i -
a b l e s which a r e p o s s i b l y confounded w i t h X and thus make t h e denendence o f Y on X a s p u r i o u s one. !n t h e f o l l o w i n g d e f i n i t i o n we need t h e conceDt o f p r o b a b i l i t y measure t h a t i s n o n t r i v i a l w i t h r e s p e c t t o a s i g m a - f i e l d C. w h i c h i s d e f i n e d t o b e a Drob a b i l i t y measure Q on f such t h a t t h e r e i s a t l e a s t one C
0
q(c)
<
E
C with
1.
4.1. DEFINITION L e t (n,A,P)
be a p r o b a b i l i t y sDace, l e t (Aty t E T ) , T C l R , be a monotoni-
c a l l y increasing family o f sicma-fields w i t h A denote s i g m a - f i e l d s , variables
on (n,A,P),
(
~ At)J c ~ A,~ l e t C y D , F c A
and f i n a l l y l e t X, W be (n' , A ' ) - , respectively.
(n",A")-stochastic
331
Cuusal linear stochatic dependencies
D i s c a l l e d p o t e n t i a l d i s t u r b i n g sigma-field w i t h respect t o -(Aty t E T) and C l o r , ifC = X-l(A')y w i t h r e s D e c t t o (Aty t E T) and X I , (i)
i f f t h e f o l l o w i n g two c o n d i t i o n s h o l d : (a)
D
i s e q u i - o r n r e o r d e r e d t o C , and
( b ) t h e r e i s a p r o b a b i l i t y measure r) on A t h a t i s n o n t r i v i a l w i t h r e s c e c t t o b o t h C and D such t h a t C and 0 a r e s t o c h a s t i c a l l y independent. (ii)
-(At,
181
i s c a l l e d a D o t e n t i a l d i s t u r b i n g variable w i t h resDect t o
t
E
T) and C [ o r , i f C = X-l(A')y
w i t h r e s n e c t t o (At,
t
E
T)
and 8 ,
i f f C o n d i t i o n s ( a ) and ( b ) h o l d , as w e l l as ( c ) U = GI- 1( A " ) . C o n d i t i o n 4 . l a i s chosen i n o r d e r t o e x c l u d e m e d i a t o r v a r i a b l e s f r o m t h e s e t o f potential disturbing variables.
M e d i a t o r v a r i a b l e s , say U, a r e
v a r i a b l e s f o r which X i s n r e o r d e r e d t o U and U i s Dreordered t o Y .
These
v a r i a b l e s do n o t t h r e a t e n t h e v a l i d i t y o f a causal p r o p o s i t i o n b u t may e l a b o r a t e i t . They a r e i m p o r t a n t t o d i s t i n g u i s h between d i r e c t and t o t a l e f 2 f e c t s . C o n d i t i o n 4 . l b i s added t o e x c l u d e s t o c h a s t i c v a r i a b l e s l i k e X, X , ax, e t c . f r o m t h e s e t o f p o t e n t i a l d i s t u r b i n g v a r i a b l e s .
I t i s n o t pos-
s i b l e t o c o n s t r u c t a p r o b a b i l i t y measure (1 on A t h a t i s n o n t r i v i a l w i t h 2 2 r e s p e c t t o A ( X ) and A ( X ) f o r examole, such t h a t X and X a r e s t o c h a s t i c a l l y independent w i t h r e s p e c t t o t u r b i n g variable,
4.2.
then X+U,
X-U,
Q. Furthermore, i f U i s a D o t e n t i a l d i s f o r example, a r e n o t D o t e n t i a l v a r i a b l e s .
EXAMPLE
Consider a s t u d y on t h e e f f e c t i v e n e s s o f a r e t r a i n i n g propram on t h e r a t e o f r e c i d i v i s m ( f o r a more d e t a i l e d d e s c r i p t i o n o f t h e example see S t e y e r (1984)).
We m i g h t choose 0
where (a. ,bj,ck) 1
{(a.,b.,ck): 1 J
i,j,k
= Oyl},
r e p r e s e n t s one o f t h e e i g h t p o s s i b l e e l e m e n t a r y e v e n t s
t h a t a p a r o l e e i s male o r female (ai),
takes p a r t a t t h e r e t r a i n i n g program
o r n o t ( b . ) , and comnits a c r i m e o r n o t a f t e r t h e r e t r a i n i n g ( c k ) . I4e J choose t h e s i g m a - f i e l d A on n t o be t h e s e t o f a l l subsets o f n, which con-
@, the
e i g h t s e t s {(ai,b.,c ) } and a l l unions o f these s e t s . J k t h e r o n , we m i g h t choose t h e s t o c h a s t i c v a r i a b l e s
sists o f
Fur-
331
R. Steyer
1, i f t h e o a r o l e e i s female II =
0, o t h e r w i s e
1, i f t h e a a r o l e e takes n a r t a t t h e r e t r a i n i n g X = I
0, o t h e r w i s e 1, i f t h e r e i s no r e c i d i v i s m f o r t h e a a r o l e e y = '
0, o t b e w i s e , If
t o r e p r e s e n t t h e e v e n t s t h a t m i o h t be observed. A2 := A(1d.X).
: = {1,2,3'*, A1 : = A ( I J ) ,
then '4 i s o r e o r d e r e d t o b o t h X and Y , and
and A3 : = A('l,X,Y),
X i s nreordcred t c Y.
T
Furthermore, !.I t h e n i s a n o t e n t i a l d i s t u r b i n g v a r i -
i + l e w i t h r e s p e c t t o X and (At,
t
E
T ) , hecause
(1
i s enui- o r preordered t o
X and 4 . l b i s f u l f i l l e d , too, as i s e a s i l y seen, i fwe choose ')(X=x) = l ( l l = w ) = 0.5 and O(X=x,LI=w)
= 0.25,
f o r a l l combinations o f values o f X and
1.1 .
O b v i o u s l y , t h e f o r m a l i z a t i o n o f t h i s example, c o n s i s t i n a o f o n l y t h r e e d i chotomous v a r i a b l e s , i s v e r v r e s t r i c t e d . n a r t o f the r e a l i t y i n v e s t i g a t e d .
I t reoresents o n l v a very small
I t does n o t r e o r e s e n t a l l v a r i a b l e s t h a t
m i g h t be i m p o r t a n t f o r t h e Dhenomena o f r e c i d i v i s m o f o a r o l e e s .
I t does
n o t contain Variables l i k e the parolee's m a r i t a l s t a t u s , the k i n d o f crime committed, b e i n g r a i s e d o r n o t i n a complete f a m i l y , a l backaround, e t c .
the parolees education-
I t would be n o s s i b l e , o f course, t o f o r m u l a t e a new mo-
d e l i n c l u d i n g a l l these v a r i a b l e s .
However, such an e x p l i c i t enumeration
of v a r i a b l e s o c c u r i n o i n t h e D a r t o f r e a l i t y s t u d i e d , w o u l d n e v e r be complete.
T h e r e f o r e , an a l t e r n a t i v e k i n d o f a p o l i c a t i o n o f t h e f o r m a l t h e o r y
s h o u l d be c o n s i d e r e d . F i r s t , one may i n t e r n r e t t h e n r o b a b i l i t y space ( R , A , P )
t o renresent a speci-
f i e d e x n e r i m e n t , o r more a e n e r a l l y , t h e p a r t o f r e a l i t y t o be d e s c r i b e d . Second, one m i q h t d e f i n e T
:=lR and i n t e r p r e t each s i g w a - f i e l d At,
t E T,
t o r e p r e s e n t a l l e v e n t s w h i c h mav o c c u r up t o p o i n t t o f t i m e , i n c l u s i v e l y . F o r t h i s k i n d o f i n t e r p r e t a t i o n s o f t h e formal s t r u c t u r e s (n,A,P) (At,
t
5
T),
and
i t i s n o t necessary t o know a l l t h e e v e n t s i n each A t o r even
enumerate them e x p l i c i t e l y .
I f (n,A,P)
represents a s D e c i f i e d D a r t o f rea-
l i t y , t h e r e i s no a n b i g u i t y about t h e v a r i a b l e s and e v e n t s i n v o l v e d o r about
t h e i r temooral o r d e r i n g so l o n g as i t i s made c l e a r enough w h i c h p a r t o f r e a l i t y i s considered,
333
h r a l linear stochastic dependencies
I f we f o l l o w t h i s second g a t h o f i n t e r n r e t i n s t h e f o r m a l t h e o r y i n t h e s t u -
dy on t h e e f f e c t i v e n e s s o f t h e r e t r a i n i n g w i t h n a r o l e e s , a l l o r o p e r t i e s o f t h e n a r o l e e b e f o r e t h e r e t r a i n i n s a r e o o t e n t i a l d i s t u r h i n a v a r i a b l e s , e.a. t h e k i n d o f t h e p a r o l e e ' s crime, t h e k i n d o f exneriences i n j a i l , e t c . Obviously, a causal a r o n o s i t i o n i s i n f u l l accordance w i t h o u r i n t u i t i o n o n l y f o r t h i s second k i n d o f i n t e r n r e t a t i o n o f t h e s i g m a - f i e l d s At,
allo-
w i n g f o r a l l p o t e n t i a l d i s t u r b i n g v a r i a b l e s , d i s t u r b i n a v a r i a b l e s which a r e t h e r e , even though t h e y m i q h t n o t b e observed. However, so l o n g as one mentions
w i t h r e s p e c t t o which f a m i l y (At,
t E T)
a causal o r o p o s i t i o n i s made, causal o r o p o s i t i o n s as d e f i n e d i n t h e n e x t s e c t i o n a r e m e a n i n g f u l and n o t t r i v i a l , so l o n g as a t l e a s t one p o t e n t i a l d i s t u r b i n q variable i s allowed f o r .
The more o o t e n t i a l d i s t u r b i n g v a r i a b l e s
a r e i m n l i c i t l y o r e x p l i c i t l y a l l o w e d f o r by t h e i n t e r o r e t a t i o n o f t h e siGma f i e l d s At,
t
E
T, t h e more meaninoful a causal p r o n o s i t i o n i s .
At a r e i n t e r p r e t e d t o r e o r e s e n t a l l events t h a t mav o c c u r
UP
Only i f t h e
t o point t o f
t i m e i n c l u s i v e l y , i n t h e r e a l nrocess considered, w i l l causal o r o p o s i t i o n s f u l l y c o i n c i d e w i t h t h e i n t u i t i v e meaning o f t h e term ' c a u s a l ' .
I t should
b e n o t e d , however, t h a t t h e formal t h e o r y prooosed i s n e u t r a l w i t h r e s p e c t t o these q u e s t i o n s o f adequate a o n l i c a t i o n .
5. CAUS4L LINEAR STnCHASTIC DEPENDENCE U s i n g t h e concents o f p r e - and eouiorderedness and o f p o t e n t i a l d i s t u r b i n o v a r i a b l e s t r e a t e d i n t h e n r e v i o u s s e c t i o n s , as w e l l as t h a t o f a c o n d i t i o n a l e x o e c t a t i o n , we mqv now C o r m a l l y d e f i n e causal l i n e a r s t o c h a s t i c dependence and z t u d y i t s f o r m a l p r o p e r t i e s .
As n o t o n l y causal l i n e a r q t n c h a s t i c
dependence, h i i t a l s o independence i s o f i n t e r e s t , we f i r s t d e f i n e causal lin e a r s t o c h a s t i c in/deDendence,
which means dependence o r independence.
Tn-
dependence a d d i t i o n a l l y r e q u i r e s E(Y IC)as=E(Y), and deoendence t h e n e g a t i o n o f it.
5 . 1 . DEFINITION L e t Y be a r e a l - v a l u e d s t o c h a s t i c v a r i a b l e on t h e p r o b a b i l i t y sDace (n,A,P) w i t h f i n i t e e x p e c t a t i o n E(Y), l e t X be a ( n ' , A ' ) - s t o c h a s t i c be a ( n " , A " ) - s t o c h a s t i c l e t (At, A(uET
v a r i a b l e on (n,A,P),
v a r i a b l e and 1.1
l e t C C A be a s i g m a - f i e l d ,
t E T) be a monotonically increasing f a m i l y o f sigma-fields w i t h
At) c A , and l e t E(YIC,D)
e x p e c t a t i o n o f Y g i v e n A(C
U
D).
:= E[YIA(C
U
D)]
denote t h e c o n d i t i o n a l
We say t h a t Y i s c a u s a l l y l i n e a r l y
334
R. Sfeyer
t F: T ) , i f f t h e f o l -
s t o c h a s t i c a l l y in/denendent on C w i t h r e s p e c t t o (At, lowing conditions hold: (a)
C i s p r e o r d e r e d t o A(Y) w i t h r e s D e c t t o (At,
t E T) (nreorderedness).
(b)
For a l l o o t e n t i a l d i s t u r b i n ? sigma-fields 0 w i t h resnect t o (At, a r d c,
t E T)
h o l d s , where Ho1.1 denotes t h e comnosi t i o n o f an A"-measurable r e a l - V a l ued f u n c t i o n tJ w i t h a (V,A")-measurable
s t o c h a s t i c v a r i a b l e ',I ( i n v a r i a n c e ) .
IffC o n d i t i o n s ( a ) and ( b ) , as w e l l as ( c ) E ( Y / T ) as= E(Y) ( l i n e a r s t o c h a s t i c indeoendence) h o l d , we say t h a t Y i s c a u s a l l y l i n e a r l y s t o c h a s t i c a l l v independent from D w i t h r e s p e c t t o
(At,
t F T ) , and denendent on C, o t h e r w i s e .
1
If C o n d i t i o n s ( a ) and ( b ) h o l d , as we11 as ( d ) C = X- ( A ' ) , we a l s o say that
Y i s c a u s a l l y l i n e a r l y s t o c h a s t i c a l l y i n / d e p e n d e n t on X w i t h r e s o e c t
-
t o (At,
t t T).
I f no c o n f u s i o n i s n o s s i b l e , we mav o m i t t h e e x p l i c i t r e f e r e n c e t o (At,
t E
T).
There a r e a numher o f s i t u a t i o n s i n w h i c h E q u a t i o n (1) h o l d s .
Suppose f o r
example, t h a t t h e c o n d i t i o n a l c x n e c t a t i o n o f t h e Y q i v e n C and V i s t h e
sum o f a c o m p o s i t i o n FOX and a c o m n o s i t i o n HoU, E(Y[C,D) as= FOX + Hob1 where F i s an A'-measurable and H an A"-measurable r e a l - v a l u e d f u n c t i o n , whereas X,
111
s r e ( C , A ' ) - , (D.A")-measurable,
respectively.
T h i s w i l l be
r c f e r r e d t o as the a c ' d i t i v i t y c o n d i t i o n , a s p e c i a l case o f which i s t h e mu1 t i p l e r e g r e s s i o n e q u a t i o n E(Y[ X,IsI)
as= by" + byXX + byMW.
The a d d i t i v i t y c o n d i t i o n a l s o h o l d s , f o r example, i f
whereas i t does n o t h o l d , f o r example, i f
W a f finear stochastic dependencies
335
T h i s l a s t e q u a t i o n c o n t r a d i c t s t h e a d d i t i v i t y c o n d i t i o n , because o f t h e m u l t i n l i c a t i v e t e r m X-11, which i s n e i t h e r C- n o r D-measurable.
5 . 2 . TEEOREM L e t t h e Dresumptions and n o t a t i o n s o f 5 . 1 be v a l i d . (a)
If
f o r a l l p o t e n t i a l d i s t u r h i n g s i g m a - f i e l d s P, as= FOX t Ho\J,
E(YIC,P)
(2)
where F i s an A ' - and H an A"-measurable r e a l - v a l u e d f u n c t i o n , X i s (C,A')and If i s (PYA")-measurable ( a d d i t i v i t y ) ; i f (b)
!' and
111
a r e s t o c h a s t i c a l l y indenendent, and i f
( c ) C i s p r e o r d e r e d t o A(Y) w i t h r e s p e c t t o (At, t E T ) , then Y i s c a u s a l l y l i n e a r l y s t o c h a s t i c a l l y in/denendent on C . Proof. I n o r d e r t o prove t h i s o r o p o s i t i o n , we suDpose t h a t t h e e x n e c t a t i o n E(HoI.1) i s equal t o z e r o .
Note t h a t t h i s does n o t r e s t r i c t g e n e r a l i t y ,
because, b y s u b t r a c t i n g i t s e x p e c t a t i o n , any c o m n o s i t i o n H'oN w i t h an e x p e c t a t i o n unequal zero can e a s i l y be t r a n s f o r m e d i n t o a c c m p o s i t i o n
No:: w i t h e x p e c t a t i o n z e r o .
I t i s e a s i l y seen t h a t t h e c o n d i t i o n s ( a ) and ( b ) o f theorem 5 . 2 i m n l y F q u a t i o n ( l ) , because
(see A.17)
E(Y1X) as= E[E(YIC,D)IXl as= E( FoXtHoYI X)
(see 2)
as= ~ ( ~ 0 x t1 ~ E ()H O W / X ) as= FOX
t
E(Hold) = FOX
( s e e A.15)
,
(see A.9)
where E(Ho191) = 0 and t h e theorem i s used t h a t s t o c h a s t i c independence nf
and X i m p l i e s E(H0lJIX) as= E(HoY).
k'e may now i n s e r t FOX as=
E ( Y ] X ) i n t o Equation ( 2 ) , which y i e l d s E q u a t i o n ( l ) ,which proves t h e theorem. We now t u r n t o a n o t h e r s i t u a t i o n i n which Y i s c a u s a l l y l i n e a r l y s t o c h a s t i c a l l y in/dependent on C.
Sul)pose, E(YlX,W)
t o o , E q u a t i o n (1) i s f u l f i l l e d . w h i c h E(YIX,V)
as= E(Y1X).
I n this situation,
14 b e i n g a c o n s t a n t i s a s n e c i a l case i n
as= E(Y1X) i s t r u e .
Hence, i f t h e r e i s t h e p o s s i b i l i t y t h a t
a n o t e n t i a l d i s t u r b i n g v a r i a h l e i s an a c t u a l one, i t m i g h t be h e l d c o n s t a n t a t one o f i t s values.
Thus, t h e e x p e r i m e n t a l c o n t r o l t e c h n i q u e o f h o l d i n g
o o t e n t i a l d i s t u r b i n g v a r i a b l e s c o n s t a n t , can be based on t h e i n v a r i a n c e condition.
I t s h o u l d be noted, however, t h a t t h e v a l i d i t y o f a causal
R. Sreyrr
336
o r o n o s i t i o n i s then r e s t r i c t e d t o t h e case t h a t LI i s c o n s t a n t a t i t s v a l u e L1.
.
5 . 3 . THEOREr4 L e t t h e oresumntions and n o t a t i o n s o f 5 . 1 b e v a l i d . ( a ! C i s o r e o r d e r e d t o A ( v ) w i t h r e s n e c t t o (At, t
If C
T ) , and i f
( b ) f o r a l l n o t e n t i a l d i s t u r h i n q si?ma-fields P , E(V!C,D) (C-conditional
as= E ( Y ! C ) ,
(3)
l i n e a r s t o c h a s t i c indeoendence o f Y f r o m
D),
t h e n Y i s cau-
s a l l y l i n e a r l y s t o c h a s t i c a l l v i n / d e n e n d e n t on C. Proof. E n u a t i o n ( 3 ) i m n l i e s E n u a t i o n ( l ) , because we may d e f i n e H and I.1 such t h a t Hob1 = 0 . 5 . 4 . COROLL73Y L c t tk,e oresumDtions and n o t a t i o n s o f 5 . 1 be v a l i d . A ( Y ) w i t h r e s p e c t t o (At,
I f C i s preordered t o
t E T ) and
(4)
E(Y1C) as= Y
(comnlete dependence of Y on C), then Y i s c a u s a l l y l i n e a r l v s t o c h a s t i c a l l y in/deaendent on C. Proof. E(YIC) as= Y i m n l i e s F(vlC,?)
as= E I E ( v l C ) l C , f l
as= E ( Y I C ) ,
f o r a l l sivma-fields
DcA
,
(see A . 1 2 )
which i m D l i e s 5.3b. Hence, causal l i n e a r s t o c h a s t i c deoendence does n e i t h e r r e q u i r e a l l o f t h e v a r i a t i o n o f t h e dependent v a r i a b l e Y t o be determined h y C ( s e e E q u a t i o n (4)),
n o r i s i t necessary t h a t a l l i n f l u e n c i n g v a r i a b l e s t h a t a r e e q u i - o r
n r e o r d e r e d t o C a r e known ( s e e E n u a t i o n ( 3 ) ) . notential disturbing siama-fields
D
It i s only required that a l l
behsve as d e s c r i b e d by E n u a t i o n ( 1 ) .
This equation i s f u l f i l l e d already i f Y i s C - c o n d i t i o n a l l y l i n e a r l y stochast i c a l l y independent f r o m
D, i . e .
i f E q u a t i o n ( 3 ) h o l d s , o r i f X and
s t o c h a s t i c a l l y independent and a d d i t i v i t y can be assumed.
W are
Causal linear stochactic dependencies
337
G. SII.fP1.F: CPVSAL REG-LINEAR DEPEVCENCE I n t h e case o f a s i m p l e r e p - l i n e a r denendence o f Y on X which i s c h a r a c t e r i zed b y t h e s i m p l e r e g r e s s i o n e q u a t i o n E(Y1X) as= ayo
+
ayXX,
(5)
the invariance condition implies t h a t , f o r a l l Dotential d i s t u r b i n g variables
1.1,
Idhatever t h e tyDe o f t h e f u n c t i o n H01.1,
the regression c o e f f i c i e n t
aYX i s l e f t unchanged when t i i r n i n o from E n u a t i o n ( 5 ) t o ( 6 ) , o r o v i d e d t h e i n v a r i a n -
ce condi ti on h o l ds
.
Another remarkable p r o n e r t y o f t h e s i m o l e r e g r e s s i o n c o e f f i c i e n t a Y X i s t h a t i t i s equal t o t h e CorresDonding r e g r e s s i o n c o e f f i c i e n t ayX1,lq,l
of
t h e c o n d i t i o n a l s i m p l e r e g r e s s i o n o f Y on X g i v e n I.!=w:
i f t h e invariance condition holds.
T h i s i s e a s i l y seen f r o m E q u a t i o n ( 6 ) ,
because, f o r #=w, E(vlX,I.l=w) as= aYO + ayXX + H(w), where H(w) i s a c o n s t a n t so t h a t we may d e f i n e ayoIl,,=w
.- aYO + H(w)
and
._
aYXI,,,=w .- ayX. Thus, E q u a t i o n ( 5 ) and t h e i n v a r i a n c e c o n d i t i o n i m p l y , t h a t aYX i s a l s o t h e r e a r e s s i o n c o e f f i c i e n t o f t h e c o n d i t i o n a l s i m p l e r e p r e s s i o n o f Y on X g i v e n I J = w ,
where I d i s any p o t e n t i a l d i s t u r b i n g v a r i a b l e
This motivates the f o l l o w i n g 6 . 1 . DEFINITION
L e t t h e presumptions and n o t a t i o n s o f 5 . 1 be v a l i d .
We say t h a t Y i s s i m p l y
c a u s a l l y r e g - l i n e a r l y in/deDendent on X and t h a t aYX i s t h e s i m p l e causal r e g - l i n e a r e f f e c t o f X on Y , i f f Y i s c a u s a l l y l i n e a r l y s t o c h a s t i c a l l y i n / dependent on X and E q u a t i o n ( 5 ) h o l d s .
338
R . Stryer
7 , IVEAK CAUSAL LINEAR STOCHASTIC DEPENDENCE I t may be a r y e d t h a t t h e k i n d o f causal l i n e a r s t o c h a s t i c deoendence d i s cussed above i s u n r e a l i s t i c a l l y r e s t r i c t i v e .
L e t us c o n s i d e r once again
t h e i n v a r i a n c e c o n d i t i o n s a v i n q t h a t , f o r a l l p o t e n t i a l d i s t u r b i n g sigmaf i e l d s 0. E(YlC,P)
as= F ( Y 1 C )
+ HolJ,
where tio1.l denotes t h e c o m p o s i t i o n o f an A"-measurable r e a l - v a l u e d f u n c t i o n H w i t h a (D,/.")-measurable
stochastic variable
1.1.
I f , f o r examnle, C i s
generated b y t h e r e a l - v a l u e d N ( J ) - d i m e n s i o n a l s t o c h a s t i c v a r i a b l e
x
= (X1,*.
.,YN(J)),
i t i s e a s i l y seen t h a t t h e r e a r e no m u l t i o l i c a t i v e
terms i n v o l v i n g a P-measurable f u n c t i o n and one o r more o f t h e random v a r i a -
.
Therefore, the invariance c o n d i t i o n exclub l e s X . , j E J = (1,. . , N ( J ) l . J des i n t e r a c t i o n s i n t h e a n a l y s i s o f v a r i a n c e sense o f any o r d e r between t h e v a r i a b l e s X . and any o o t e n t i a l d i s t u r b i n g v a r i a b l e 1 J . This m n c ' i t i o n i s a J r a t h e r s t r i c t one t h a t cannot be guaranteed t o h o l d even i n randomized exq e r i m e n t s , where t h e t e r m ' c a u s a l ' seems t o be a D n r o n r i a t e , t o o , t o charact e r i z e t h e dependence o f Y on t h e t r e a t m e n t v a r i a b l e s X . i n d i c a t i n g memberJ s h i p t o t h e e x p e r i m e n t a l grouos. Therefore, a weak tync? o f causal l i n e a r s t o c h a s t i c denendence i s now i n t r o duced.
A l l t h a t has t o be done i s t o r e p l a c e t h e i n v a r i a n c e by t h e averaoe
c o n d i t i c n , which can he f o r m u l a t e d as f o l l o w s : f o r a l l o o t e n t i a l d i s t u r b i n g v a r i a b l e s 1.1, t h e f o l l o w i n c j e q u a t i o n h o l d s f o r PX-almost a l l E(YIX=x): E(YIX=x) =
E(Y/X=X,IJ=W) P , ~ ( ~ w ) ,
where Pb, denotes t h e d i s t r i b u t i o n o f
14.
I f 1.1 i s a d i s c r e t e random v a r i a b l e ,
t h i s e q u a t i o n can be w r i t t e n E(Y [ X=X) = I w E ( Y I:(=x,W=w) where t h e summation i s o v e r a l l values w of W .
P(I,rqf), T h i s average c o n d i t i o n sim-
p l y means t h a t PX-almost a l l c o n d i t i o n a l e x p e c t e d v a l u e s E(YIX=x) a r e t h e average o f t h e c o n d i t i o n a l e x o e c t a t i o n s E(YIX=x,IJ=w) o f Y g i v e n X=x and W=w across t h e values w of 14.
Together w i t h preorderedness, t h e averace c o n d i -
t i o n d e f i n e s weak c a u s a l l i n e a r s t o c h a s t i c deDendeqce o f Y on X .
339
Causal linear stochastic dependencies
7.1.
DEFIYITION
L e t t h e oresumptions and n o t a t i o n s o f 5 . 1 be v a l i d .
Ire say t h a t Y i s weak-
l y c a u s a l l y l i n e a r l y s t o c h a s t i c a l l y in/dependent on X, i f f
( a ) X i s o r e o r d e r e d t o Y w i t h r e s o e c t t o (At, ( b ) Fc-r a l l p o t e n t i a l d i s t u r b i n g v a r i a b l e s (At,
t
E
1.1
t
T) ( p r e o r d e r e d n e s q \ .
w i t h r e s o e c t t o X and
T) E( Y I X=X) =
f o r PX-almost a l l x
E
1 E (Y I X=X,IJ=w)
n' (averace c o n d i t i o n ) .
PI,,( &)
,
(7)
I f additionallv
( c ) E(Y1X) as= E(Y) ( l i n e a r s t o c h a s t i c independence), we say t h a t Y i s weakly c a u s a l l y ; i n c a r l y s t o c h a s t i c a l l y indenendent f r o m
X and, dependent on X, ot+em!ise. P s p e c i a l s i t u a t i o n i n which E q u a t i o n (7) i s f u l f i l l e d i s c h a r a c t e r i z e d by E(Y IX,L')
as= E(Y I X ) , which d e f i n e s X - c o n d i t i o n a l l i n e a r s t o c h a s t i c indepen-
dence o f Y from 14. 7 . 2 . THEOREM L e t t h e oresumptions and n o t a t i o n s o f 5.1 be v a l i d . ( a ) X i s p r e o r d e r e d t o Y w i t h r e s p e c t t o (At,
If
t E T) and
( b ) f o r a l l p o t e n t i a l d i s t u r b i n g v a r i a b l e s IJ,
then Y i s weakly c a u s a l l y l i n e a r l y s t o c h a s t i c a l l y in/deoendent on X. Proof. E q u a t i o n (8) i m p l i e s E ( Y )X=x,I\r=w) = E ( Y IX=x), f o r P(X,l,,)-almost (x,w) E n ' x 0 " .
1 E(YIX=x,lJ=w)
all
Hence,
P1,,(dw) = / E ( Y I X = x ) PY(dw) = E(YIX=x).
The importance o f t h e average c o n d i t i o n , however, stems f r o m t h e f a c t t h a t i t i s i m p l i e d by t h e s t o c h a s t i c indeoendence o f a l l o o t e n t i a l d i s t u r b i n g
v a r i a b l e s W a t one s i d e and t h e N(J)-dimensional v a r i a b l e X = (Xl,...,XN(J)) on t h e o t h e r s i d e . 7 . 3 . THEOREM
L e t t h e oresumptions and n o t a t i o n s of 5 . 1 be v a l i d .
If
R . Steyer
340
( a ) X i s n r e o r d e r e d t o Y and ( b ) X and a l l p o t e n t i a l d i s t u r b i n ? v a r i a h l e s a r e s t o c h a s t i c a l l y indeoendent, then Y i s weakly c a u s a l l y l i n e a r l y s t o c h a s t i c a l l y i n / d e n e n d e n t on X . Proof. -
A . 1 7 y i e l d s E[E(Y IX,l.l) 1x1 as= E ( Y I X ) , w h i c h i m n l i e s ( a ) € ( V / X = x ) = E[E(YIX,lJ)IX=x], f o r PX-almost a l l x 'toc':ast.ic indenendence o f X and ',I then i m p l i e s E(Y;X=x) = F:[E(YIX=x,l*!)]
=
E
9'.
E(Y!X=x,LJ=w) P,,(dP/)
\4iere t h e l a s t l i n e f o l l o w s f r o m ( a ) and 5.3.22 o f C l n s s l e r and S t u t e ((19771, P. 1 ' 9 ) .
I n a randomized e x p e r i m e n t , X = (Xl,
... ,XN(J))
may r e n r e s e n t t h e v e c t o r o f
t r e a t m e n t v a r i a b l e s i n d i c a t i n g t h e e x o e r i m e n t a l groun t o w h i c h an e x p e r i mental u n i t ( s u b j e c t ) b e l o n g s . disturbing variables
1.1
Randomization i m n l i e s t h a t a l l o o t e n t i a l
( r e n r e s e n t i n g any p r o p e r t i e s o f t h e e x n e r i m e n t a l
u n i t s before o r a t t h e t r e a t m e n t ) and t h e t r e a t m e n t v a r i a b l e
x
= (X
. ., X N ( J ) )
a r e s t o c h a s t i c a l l v independent.
c o n d i t i o n h o l d s i n randomized e x o e r i m e n t s .
Therefore, the a v e r a w
Thus, weak causal l i n e a r s t o -
c h a s t i c dependence ( o r independence, i f E(Y1X) as= E ( Y ) ) i s guaranteed by r a n d o m i z a t i o n ( f o r an examole see S t e y e r (1984)).
Matching, o f course,
serves the same nurnose, h u t o n l v f o r those o o t e n t i a l d i s t u r b i n g v a r i a b l e s L.1 w i t h r e s w c t t o which t h e e x n e r i m e n t a l groups a r e matched.
Thus, t h e
c o n t r o l techniques o f r a n d o m i z a t i o n and m a t c h i n g can he based on t h e t h e o r y nrooosed, t o o . However, and maybe more i m p o r t a n t l y , t h e t h e o r y o f c a u s a l l i n e a r s t o c h a s t i c deoendence i s a l s o r e l e v a n t f o r n o n e x ? e r i m e n t a l s t u d i e s , because bGth t h e i n v a r i a n c e and t h e averaoe c o n d i t i o n may be t e s t e d , once a n o t e n t i a l d i s t u r b i n a v a r i a b l e 1.1 i s observed.
I f n e i t h e r h o l d s , any causal h y p o t h e s i s
s h o u l d he r e j e c t e d f o r t h e model c c v s i d e r e d .
I f one o r b o t h o f them h o l d ,
one may m a i n t a i n t h e c a u s a l h y p o t h e s i s ( i n i t s weak o r i t s s t r i c t form) as l o n o as no p o t e n t i a l d i s t u r b i n g v a r i a b l e i s f o u n d f o r which t h e i n v a r i a n c e and t h e average c o n d i t i o n s do n o t h o l d .
T e s t i n g i f t h e averaqe c o n d i t i o n
( s e e E q u a t i o n ( 7 ) ) h o l d s i s a t e s t o f weak causal l i n e a r s t o c h a s t i c dependence and t e s t i n g i f t h e i n v a r i a n c e c o n d i t i o n (see E q u a t i o n ( 2 ) ) h o l d s i s a t e s t o f causal l i n e a r s t o c h a s t i c deoendence ( i n t h e s t r i c t e r sense).
O f
course, such t e s t s make sense o n l y if X i s assumed t o be n r e o r d e r e d t o Y .
I n t h e l a s t theorem, i t i s shown t h a t causal l i n e a r s t o c h a s t i c in/dependence
34 1
Grusal linear stochastic dependencies
i m n l i e s weak causal l i n e a r s t o c h a s t i c in/denendence.
7.4. THEOREFI I f V i s causally l i n e a r
L e t t h e presumptions and n o t a t i o n s o f 5.1 be v a l i d .
l y s t o c h a s t i c a l l y denendent on X, then Y i s a l s o weakly c a u s a l l y l i n e a r l y s t o c h a s t i c a l l y dependent on X. Proof. 'Je have t o show t h a t Eouation ( 1 ) i n p l i e s E n u a t i o n ( 7 ) .
Equation ( 1 )
imnl i e s : E(Y IX=X,H=W) = E[E(Y IX,sI)
I
X=X,M=W]
H(w).
) t = E [E(Y IX) !X=x,'J=w] t E ( H o ~ ~ I X = X , ~ J ==WE(YIX=x) Hence, E(Y1X.x)
= E(YlX=x,l.I=w)
E(YIX=x) = =
because
/
/
E(YIX=x)
E(YIX=x,l.'=w) P,,,(dw)
1 H(w)
PI,,(&)
E[E(YIX,I*l)]
-
H(w) and
p l , l ( ~= )
-I
/
[E(YIX=x,'J=w)
H(w) P,.l(cbr) =
-
H(w)I Pl,l(dw)
1 E(YIX=x,',l=w)
= E(HoIJ) = 0, which i s e a s i l y seen f r o m = E[E(V!X)] t E(HolJ),
which i m n l i e s E(HoIJ) = 0, because E[E(YIX,IJ)l
= E[E(YIX)l
= E(Y).
F o r examvles i l l u s t r a t i n g t h e a p p l i c a t i o n o f t h e the0r.y nroposed, as w e l l as f o r a d i s c u s s i o n r e 1 a t i r . g t h i s t h e o r y t o simultaneous e q u a t i o n modeling, the reader i s r e f e r r e d t o Steyer (1984). APPENDIX I n t h i s anpendix a r e o r e s e n t e d t h e d e f i n i t i o n o f , and some theorems p e r t i n e n t t o t h e c o n d i t i o n a l e x n e c t a t i o n E(YIC) o f Y n i v e n C t h a t a r e f r e q u e n t l y c i t e d i n t h i s chanter.
F o r an i n t r o d u c t i o n i n t o t h e conceot and i t s back-
around see e.g. Sauer (1974) o r LoPve (1977, 1978).
, Rreiman
(1968), Cannsler and S t u t e (1977),
A s h o r t e r and t h e r e f o r e more c o n v a n i e n t came f o r
E(Y I C ) i s C - c o n d i t i o n a l e x n e c t a t i o n of Y.
Y i s a v e r y general and u s e f u l concent.
The C - c o n d i t i o n a l e x n e c t a t i o n o f I t i s used, f o r examole, t o d e f i n e
t h e C - c o n d i t i o n a l n r o h a b i l i t y P(AlC) o f an e v e n t A , as w e l l as t h e C-condit i o n a l v a r i a n c e V(Y I C ) and c o v a r i a n c e C(Y,ZlC)
o f stochastic variables.
S n e c i a l C - c o n d i t i o n a l e x p e c t a t i o n s a r e a l s o o b t a i n e d , i f C i s t h e sigmaf i e l d generated by a s t o c h a s t i c v a r i a b l e X o r by a f a m i l y (X stochastic variables. e t c . may be used.
j'
j E J) o f
I n these cases, t h e n o t a t i o n s E ( y I X ) , E(YIX., j E J ) , J
342
R . Steyer
The mathematical d e f i n i t i o n g i v e n helow does n o t anneal v e r y much t o i n t u i tive insight.
T h e r e f o r e , i t m i g h t be h e l p f u l t o r e c a l l t h a t E(YIX) i s a
s t o c h a s t i c v a r i a b l e , t h e values o f which a r e i d e n t i c a l w i t h t h e c o n d i t i o n a l e x n e c t a t i o n s E(Y1X-x) g i v e n X=x.
Another way t o t h i n k a b o u t E(Y1X) i s t h a t
i t i s a s t o c h a s t i c v a r i a b l e c o n s i s t i n g o f the b e s t n r e d i c t i o n s o f Y piven
a v a l u e x o f X.
I n many t e x t h o o k s on a p p l i e d s t a t i s t i c s , t h e v a r i a b l e con-
s i s t i n g o f t h e b e s t o r e d i c t i o n s o f Y i s denoted by f . o f as t h e mathematical e q u i v a l e n t o f
f.
E ( y I X ) may be t h o u g h t
S i m i l a r l y , E(Y1C) c o n s i s t s o f t h e
b e s t p r e d i c t i o n s o f Y based on t h e s i g m a - f i e l d C. Definition 1 L e t Y: 1
(r,A,P)
+lR
be a r e a l - v a l u e d s t o c h a s t i c v a r i a b l e on t h e n r o h a b i l i t y space
w i t h f i n i t e e x p e c t a t i o n E(Y) and l e t C
s t o c h a s t i c v a r i a b l e E(Y1C): R
+
A be a s i g m a - f i e l d .
C
The
!R i s c a l l e d t h e C - c o n d i t i o n a l e x p e c t a t i o n
o f Y , i f f E(Y1C) i s C-measurahle and for all C
EIlCE(Y/C)] = E(lCV), Let A
6
E
A be an e v e n t and lA i t s indicator function.
(1)
C.
P ( A 1 C ) : R +!R i s c a l -
l e d t h e C - c o n d i t i o n a l n r o b a b i l i t y o f A, i f f
P ( A I C ) : = E( 1,lC). L e t X be a ( n ' , A ' ) - s t o c h a s t i c
(2)
v a r i a b l e on (n,n,P).
X - c o n d i t i o n a l e x p e c t a t i o n o f Y, i f f C = A(X,A')
E(YIX) i s c a l l e d t h e
i s t h e s i g m a - f i e l d genera-
t e d by X and A',and E(Y1X) := E(Y1C). L e t (Xj,
j E J ) be a sequence o f ( n . , A . ) - s t o c h a s t i c
E(YIXj, j E J ) i s c a l l e d t h e ( X j , C = APjEJ
A(X.,A.)I J J
J
J
(3) v a r i a b l e s on (n,A,P).
j E J ) - c o n d i t i o n a l e x p e c t a t i o n o f Y, i f f
i s t h e s i g m a - f i e l d generated by [(X.,A.), J J
j
E
J1, and
E ( Y ( X j , j E J ) := E(Y1C).
) i s c a l l e d t h e (X1,.,.,XN(J))-conditional 4(Xj,Aj)]
(4) e x p e c t a t i o n o f Y,
i s t h e s i g m a - f i e l d generated by [(X.,A.), J J ,N(J) 1, PI(J) E N = (1,2,. . I , and
j
E
J],
.
E(Y IX1,.
...XN(J))
:= E(Y I C ) .
(5)
Garsal linear stochastic dependencies
Note t h a t E(Y1C) i s the general concent.
343
Hence, if n r o o o s i t i o n s are t r u e
f o r E ( Y ( C ) , then they a l s o h o l d for P(PlC), E(YIX), E(Y\X., j E J ) , and J ,XN(J)). Through the d e f i n i t i o n above, the C-condi t i o n a l expecE(Y lX1,.. t a t i o n E( Y C) i s cniauely determined o n l y w i t h nrobabi lit y one. Therefore ,
.
there are g e n e r a l l y c ' i f f e r e n t versions o f E ( Y IC) , which are, however, equiv a l e n t , w i t h o r o b a h i l i t y 1. almost s u r e l y i n general
Equations on E(Y[C) are t h e r e f o r e t r u e o n l y
, which
w i l l be abbreviated by the symbol ' a s = ' .
N o t i c e t h a t t h e r e i s a d i f f e r e n c e between the C-conditional expectation E(Y1C) o f Y, i . e . the c o n d i t i o n a l expectation o f Y w i t h resaect t o the s i q m a - f i e l d C and the c o n d i t i o n a l e x o e c t a t i o n E ( Y 1 C ) o f Y given the event C, and correspondinqly, a d i f f e r e n c e between the c o n d i t i o n a l nrohabi 1it y P ( A ( C ) o f the event A w i t h r e s n e c t t o the s i g m a - f i e l d C and the c o n d i t i o n a l
p r o b a b i l i t y P(A1C) o f the event A qiven the event C.
E(YIC) and P(A1C) are
s t o c h a s t i c variables, whereas E ( Y IC) and P(A(C) are r e a l - v a l u e d constants. I n the f o l l o w i n g theorem, a number o f p r o p o s i t i o n s on E(Y1C) are gathered.
Some o f them are on c o n d i t i o n s under which E(Y\C) i s equal t o a constant function.
I n these and r e l a t e d contexts, the symbol f o r the f u n c t i o n and
the constant ( i t s values) w i l l be t h e same. does n o t o n l y denote a r e a l cnnstan:,
T k syrrkol ' a ' , f o r example,
b u t a l s o a f u n c t i o n a: n
t a k i n g the value a f o r a l l w E 8. The n o t a t i o n E(Y(C)
:0,
+R
almost sure, i n
P r o o o s i t i o n (11) i s an a b b r e v i a t i o n f o r E(YIC)(w) 2 O(W), f o r almost a l l w E
n, where 0: R + R i s a f u n c t i o n d e f i n e d by O(W)
= 0, f o r a l l w E R.
THEORCtI 1
L e t Y be a real-valued s t o c h a s t i c v a r i a b l e on the p r o b a b i l i t y space (n,A,P)
w i t h f i n i t e e x p e c t a t i o n E(Y).
I f E(Y1C) i s the C-conditional ex-
p e c t a t i o n o f Y, then the f o l l a w i n g p r o p o s i t i o n s are true: EK(Y1C)l = E(Y) E(Y1C) = E(Y), i f C = InJI, where E(Y) : n E ( Y ) ( w ) = E(Y), f o r a l l w E R E[Y-E(YIC)l
+lR i s
d e f i n e d by
= 0.
E ( Y ( C ) as= Y,
i f Y i s C-measurable
E(Y1C) as= a, i f Y as= a, where a: n +lR i s defined by a(w) = a, f o r a l l w E R E(Y1C) 2 0, almost sure, i f Y 5 0, almost sure, where
0: n +lR i s d e f i n e d by O(W) = 0, f o r a l l w 6 R
.
344
R . Stuyrr
Accordin? t o E q u a t i o n ( 6 ) , t h e e x i e c t a t i o n o f t h e C-condi t i o n a l e x p e c t a t i o n o f Y i s equal t o t h e e x o e c t a t i o n of Y .
conditional exnectatinn o f if C =
P r o n o s i t i o n ( 7 ) shows t h a t t h e C-
i s equal t o the constant f u n c t i o n [ ( Y ) :
V
fi + R ,
in,@). A c c o r d i n ? t o E q u a t i o n ( 8 ) . t h e e x n e c t a t i o n o f t h e d i f f e r e n c e
Y-C(Y!C) o f Y and i t s C - c o n d i t i o n a l e x n e c t a t i o n i s z e r o .
c a l l e d the error or r esidual o f Y w i t h resaect t o
difference V-E(Y!C),
n l a v s an i m n o r t a n t r o l e i n a n n l i c a t i o n s ,
E(’!!C),
analied.
Note t h a t t h e
Pronosition ( 9 ) i s often
i s C-measurable, f o r examale, i f C i s t h e s i m a - f i e l d Generated
Y
-
hv the ( ? . A , ) - s t o c h a s t i c v a r i a h l e s X j E !I, and V = Xi, V = X X k , or 1’ J i’ j v = x t Xk, where .i,k C (1. Hence, s o e c i a l cases o f P r n n o s i t i o n ( 9 ) a r e ,
i
f o r examnle, E ( X ! X ) as= X ,
xi ’ i f i E J , J ) as= X..X i f i,k F J , 1 k’ ,!) as= Xi+Xk, i f i , k E ,1.
c ( x . ~ x . , j c ? ) as= 1
J
E(Xi.%k\Xj,
j
E(Xi+XklXj,
,j F
F
The f o l l o w i n g theorem c o n t a i n s n r o n o s i t i o n s on E(VIC), where Y i s t h e weiqht e d sum o r t h e n r o a u c t c f o t h e r s t o c h a s t i c v a r i a b l e s . V
2
Again, t h e n o t a t i o n
Z, almost sure, which occurs i n P r o o o s i t i o n ( 1 6 ) , i s an a h b r e v i a t i o n f o r Z(u;,
Y(.J)
f c r almost a l l w E R.
THEOREK 2 Let
and Z he r e a l - v a l u e d s t o c h a s t i c v a r i a h l e s on t h e n r o b a b i 1it y space
V
(?,f!,P)
w t h f i n i t e e x n e c t a t i o n s E ( Y ) and E ( Z ) , r e s o e c t i v e l v , and l e t C,
r 0 C 1. be two s i a m a - f i e l d s . I f E ( Y \ C e x o e c t a t i o n s o f Y and Z, r e s o e c t i v e l v
and E( Z ! C) a r e t h e C-condi t i o n a l t h e n t h e f o l 1owing Dronosi t i ons a r e
true : E ( Y 1 C ) as= E ( Z / C ) , i f Y as= Z .
E(Y.ZlC)
as= Y.E(ZlC),
finite EIE(VICO).Z\CI as= E(VIC,)*E(ZIC), E(avt>ZIC)
RS=
?E(YIC)
(12)
i f V i s C-measurable and i f E ( Y * Z ) i s
+
bE(ZIC),
(13)
i f Co c C .
(14)
i f a, b ElR.
(15)
€ ( v I c ) i E ( Z / C ) , a l m o s t s u r e , i -Y 6 Z, a l w c s t s u r e .
(16)
THEOREU 3 L e t Y he a r e a l - v a l u e d s t o c h a s t i c v a r i a b l e on t h e o r o b a b i l i t y soace ( O , A , P ) w i t h f i n i t e exoectation E(Y).
I f E(Y1C) i s t h e (‘-conditional
exoectation
Grusal linear stochastic dependencies
o f Y and Co
c C i s a sirpa-field,
345
then the f o l l o w i n g equations are t r u e :
EIE(YIC)ICol as= E(YICO) as= EIE(YICO)ICl , EIY-E(YIC)ICoI as= 0. The s i q m a - f i e l d Co w i l l he a subset o f the s i q m a - f i e l d C, f o r example, i f C i s generated by the (n.,A.)-stochastic
J J v a r i a b l e s Xk,
(nk,Ak)-stochastic
variables X.,
,7
k E K, where K C J .
and Co by t h e
.j E
Hence, s n e c i a l cases
o f the Equations (17) are E[E(YIX1,X2) I X1l as= E(YIX1) as= E[E(YIX1)IX1,X21 , E[E(YIX., ,j E J ) I X k , k E K1 as= E ( Y I X k , k E K ) J as= E I E ( Y I X k , k E K)IXj, j E 51. Equation (18) reveals t h a t the Co-conditional e x n e c t a t i o n o f t h e e r r o r o r
-
residual F = Y
This n r o ? o s i t i o n on F i s much
E(Y1C) i s zero, i f Co C C.
s t r i c t e r than t h a t o f Equation (8)
, sccording
t o which the ( u n c o n d i t i o n a l )
e x p e c t a t i o n of F i s zero. REFERENCES Baaozzi, R.P., Bauer, H.
Causal models i n marketing, New York, Yi1ey (1980).
, Wahrscheinlichkei t s t h e o r i e
und Grundziige der r!asstheorie,
R e r l i n , de G r u y t e r (1974). Breiman, L .
, Probahilitv,
Reading, Mass., Addison-l,ksley (1968).
Cook, T.D. & Camohell , D.T.
, nuasi-experimentation.
Design and analy-
s i s issues f o r f i e l d s e t t i n g s , Boston, Houqhton M i f f l i n (1979 Ganssler, P. & S t u t e , H.,
IMahrscheinlichkeitstheorie, R e r l i n , Springer
( 1977). Halmos, P.R., Heise, D.R., Loeve, M. LoPve, tl.
Measure theory, P r i n c e t o n , N.J.,
, Probability , Probability
Simon, H . A . ,
theory I , B e r l i n , Snringer (1977). theory I 1 , B e r l i n , S o r i n g e r (1978).
On the d e f i n i t i o n o f t h e causal r e l a t i o n , The Journal o f
Philosophy 49 (1952) StegmGller,
Van Nostrand ( 969)
Causal a n a l y s i s , New York. U i l e y (1975).
, 517-523.
!J., Erklarung,
Begrhdung, Kausali t a t .
P r o b l e m und Resul-
t a t e der Wissenschaftstheorie und a n a l y t i s c h e n Phi losonhie, Band I , B e r l i n , Springer (1983). Steyer, R.,
S t r u c t u r a l equations, s t a b i l i t y , and e q u i l i b r i u m p o i n t s i n Paper read a t the 13th Euro-
mu1 ti v a r i a t e autoregressi ve Drocesses
.
pean Mathematical Psychology Group Weeting, B i e l e f e l d , SeDtemher (1982).
R . Steyer
346
I131 Steyer, R,
flodelle zur kausalen Erkllrung s t a t i s t i s c h e r Zusamnenhange
I n J . Rredenkamn 8 H. Feger ( E d s . ) , Enzyklonadie der Psychologie,
Serie Fnrsctluncpettloden der Psychologie, Rand 4 , Gottingen, Hoqrefe (1983). [ l a ] S t e v e r , R.,
Causal l i n e a r s t o c h a s t i c deoendencies: An i n t r o d u c t i o n , in
J , R . Nesselroade & A . von Eye ( E d s . ) , Individual develonment a n d
s o c i a l change: exnlanatorv a n a l y s i s , New York, AcaOenic Press (!O%). (151 Suones, P., A n r o b a b i l i s t i c theory o f c a u s a l i t y , Pmterdam. North Holland (1970).
TRENDS liV MATHEMATICAL PSYCHOLOGY E . Degreef and J. Van Bu errliaut (editors) 0 Ekevier Science P u b d r s B . V. (North-HolhndJ, 1984
347
FACTDRIZATInN AN[! ADDITIVE DECCIIV'IISITION OF A G A K ORDER Reinhard Suck Uni v e r s i t a t Osnabrlick
P o s s i O i l i t i e s o f d e r i v i n g a Cartesian nroduct s%ruct u r e f r o m a weak o r d e r a r e i n v e s t i F a t e d .
As an i n -
t e r m e d i a t e s t e p a "comnensatory s t r u c t u r e " i s i n t r o ducer! and a r e o r e s e n t a t i o n theorem f o r t h i s s o e c i a l t y n e proved.
C o n d i t i o n s a r e f o r m u l a t e d t h a t quaran-
t e e t h e d e s i r e d f a c t o r i z a t i o n and y i e l d an a d d i t i v e reoresentation.
The naner c l o s e s w i t h some remarks
on n o s s i b l e a o p l i c a t i o n s . 1. INTRnDUCTION C o n j o i n t measurement can be t h o u g h t o f as a d e v i c e f o r measuring e m i r i c a l e n t i t i e s e n t a i l i n g a weak o r d e r , f o r which no n a t u r a l form o f c o n c a t e n a t i o n o f t h e o b j e c t s i s d e f i n e d , b u t which, n e v e r t h e l e s s , s h o u l d f i g u r e a reasona b l e t y n e o f measurement s c a l e , i . e . an i n t e r v a l s c a l e a t l e a s t . D e n o t i n a t h e s e t o f ob.iects by A and t h e weak o r d e r by
5
the only other p r i -
m i t i v e i n a c o n j o i n t measurement system i s , t h a t A possesses a n r o d u c t s t r u c t u r e , t h a t i s t o say: t h e r e a r e two s e t s X1,
A = XI
x
X2.
X p such t h a t (1.1)
Thus, t h e o b j e c t s i n q u e s t i o n a r e i n f a c t p a i r s o f some k i n d o f a t t r i b u t e s . I n t h e oresence o f c e r t a i n c o n d i t i o n s one can show t h a t a r e w e s e n t a t i o n
exists, i.e.
a r e a l v a l u e d f u n c t i o n f. such t h a t b o t h comoonents o f a p a i r
c o n t r i b u t e a d d i t i v e l y t o i t s o v e r a l l measure. a, b
E
To be more n r e c i s e : f o r a l l
A we have a5 b
iff
f(a)
+
f(b)
(1.2)
and t h e r e a r e two r e a l v a l u e d f u n c t i o n s fl and f 2 on X1 and X 2 r e s p e c t i v e l y such t h a t f o r a = (x,y),
348
R. Suck
Examnles o f emnirica l y r e l e v a n t conjoint measurement s t r u c t u r e s are e a s i l y P r e v i f ' c t , see Krantz Luce, 5unnes, Tversky, Chap. 6 ' 1 9 7 1 ) . G u t there are a l s o nlerltv of i n s t a n c e s , where e i t h e r the n a t u r a l l y defined
c o m m e n t s do not jlield a c c n i o i n t measurement s t r u c t u r e , because the o t h e r conditions are not f u l f i l l e d , o r the emairical objects do n o t comprise a Cartesian oroduct. In both cases one would l i k e to knov whether i t f s nossihle t o f i n d s e t s XI, X2 such t h a t (1.1) holds and ( K l , X p , 5 ) i s a conioint measurement s t r u c t u r e . Ire note i n oassino t h a t in t h i s case the i d e n t i t y sinn i n (1.1) should he taken w i t h a crrain of s a l t ; as long as (1.1) holds with s o w kind of isotnornhism a r y m e n t involved, we a r e s t i l l satisfied. The present naper deals with conditions t h a t v i e l d the existence of s e t s X1, X p f u l f i l l i r o (1.1) and a renresentation of ( A , s), which e x h i b i t s the nronerties of an a d d i t i v e c o n i o i n t s t r u c t u r e , i . e . ( 1 . 2 ) and ( 1 . 3 ) . I t has enormous consequences t o dron tbe condition t h a t the s e t of o b j e c t s c o n s i s t s of a a i r s , s i n c c almost a l l of the usual axiors of conioint s t r u c tures rely beavi l y on the Cartesian nroduct rrorwrtv.
To annroach t h i s o b j e c t i v e we f i r s t have a c l o s e r look a t the s t r u c t u r a l imnlications t h i s condition has on the weak order ( A , .), when A s a t i s f i e s ( 1 . 1 ) a n d (Y1, X 2 , <) i s a svrnretric a d d i t i v e conjoint measurement s t r u c ture. The- wc nroceer' t o fornulate axioms, from which t h p f a c t o r i z a t i o n and a s u i t a b l e renresentation theorem can be derived. The paner closes w i t h sow remarks on anolications of this theory. The nroblem bears some resemblance t o multidimensional s c a l i n g o r even fact o r a n a l v s i s , because we are v i r t u a l l v e x t r a c t i n ? comoonents o r dimensions from I\, which - when nrooerly scaled - account f o r the emqirical findinqs. The main d i f f e r e n c e , however, i s t h a t we do not suqaest another algorithm f o r f i t t i n q data t o an a d d i t i v e model b u t r a t h e r assume an axiomatic aooroach. Furthermore, we a r e not s o e c i f i c a l l y dealing with a r o x i m i t i e s o r c o r r e l a t i o n matrices, although t h i s might be of minor imnortance. 2. P9ELIMINARIES
civen a s e t A a n d a hinary r e l a t i o n R on A . I+?. o f t e n use a R b i n s t e a d of (a,b) R . I f not ot.!terwise y t z t e d , r e l a t i o n s a r e always binary.
349
Factorkation and additive deconiposition of a weak order
L e t 9 be
P
s u b s e t o f A and R a r e l a t i o n on A .
R/R i s t h e r e s t r i c t i o n o f
R to S i.e. R/R = { ( a , h )
E R;
a
E
9, b E
3).
The s e t of o r d e r e d D a i r s o f t v o s e t s A, B i s t h e C a r t e s i a n w o d u c t A O f t e n , we use a b as a h b r e v i a t i o n o f ( a , b ) , w i t h domain A and ranoe f ( A ) c o n t a i n e d i n F. f ( A ) = 5.
The r e a l numbers a r e denoted by
8.
x
f: A + R denotes a f u n c t i o n f i s called surjective, i f
F? and a f u n c t i o n
f: A
+IR i s
A o u a s i o r d e r i s a r e f l e x i v e and t r a n s i t i v e r e l a t i o n ,
c a l l e d r e a l valued.
a weak o r d e r i s connected ( i . e .
a -< b o r b 5 a ) and t r a n s -
A
f o r a l l a, b
<.
i t i v e . ;leak o r d e r s a r e u s u a l l y denoted by
Ire unr!@rstan$
symmetric n a r t o f s and < t h e asymmetric p a r t .
-
t o be t h e
I t i s r e a d i l y shown t h a t
t h e symmetric o a r t o f a o u a s i o r d e r (and a f o r t e r i o r i or a weak o r d e r ) i s an e q u i v a l e n c e r e l a t i o n ( r e f l e x i v e , svrnmetric, t r a n s i t i v e ) .
-
If
L is a
q u a s i o r d e r , i t s symmetric n a r t i s denoted hy L . Cirlce 1.e deal w i t h s e v e r a l o r d e r r e l a t i o n s on t h e same s e t (and a t t h e same t i m e ) , we have t o d i s t i n g u i s h t h e d i f f e r e n t e q u i w l e s c e c l a s s e s . derives the equivalence
-
a,
and t h e c l a s s e s
from L derives
most cases we use t h e m r e o r e c i s e n o t a t i o n : " ( A , s ) "5
From
L and aL,
5
In
i s a weak o r d e r " f o r
i s a weak order"; t h e same a n p l i e s f o r a q u a s i o r d e r and any o t h e r r e l a -
tion. I n a weak o r d e r (A, a, a '
E
a s u b s e t R o f A i s c a l l e d o r d e r dense, i f f o r a l l
5)
A such t h a t a < a ' an element b E R e x i s t s s a t i s f y i n g a -< b
5
a'.
T h i s n o t i o n i s i m o o r t a n t i n t h e r e p r e s e n t a t i o n theorem f o r o r d i n a l measu-
rlow, we d e f i n e an a d d i t i v e
rement, see f o r example Roberts, n. 111 (1979).
c o n j o i n t s t r u c t u r e , u s i n g t h e n o t i o n i n t h e sense o f K r a n t z e t a l . ,
p. 256
(1971). L e t X1,
X 2 be nonernpty s e t s .
(Xl,
X2, s ) i s an a d d i t i v e c o n j o i n t s t r u c t u r e
i f f t h e f o l l o w i n g c o n d i t i o n s a r e s a t i s f i e d f o r a l l x,
XI,
xl,
x2, x3 E X1
and Y. Y'Y Y1' Y2' Y3 E x2. (1) I\reak o r d e r : s i s a weak o r d e r on X1
x
X2,
x ' y i m p l i e s x z 5 x ' z f o r a l l z E X 2 ( t h e o t h e r com-
(2)
IndeDendence: xy Donent 1 ikewi se)
(3)
Thomsen c o n d i t i o n : if xlyl
(4)
R e s t r i c t e d s o l v a b i l i t y : ifxoyl 5 xy s x2y1, t h e n t h e r e e x i s t s x1 such t h a t xy x p l ( t h e o t h e r component l i k e w i s e ) .
5
. -
x2y2 and x2y3
-
X3Y1 t h e n X1Y3
-
X3Y2. E
X1
-
(5)
Archimedean P.xiorr: e v e r y s t r i c t l y bounded s t a n d a r d sequence i s f i n i t e
3 SO
R. Suck
[ a sequence (xi),xi t h e r e e x i s t s y, v '
XI,
F
i=1,2,
... i s
a s t a n d a r d senuence on X1 i f f
-
X2 such t h a t xiy
E
X ~ + ~ Vf o' r
i=1,2,
... and y
and
q i m i l a r l y a s t a n d a r d senuepce on X 2 i s d e f i -
v ' are n o t i n d i f f e r e n t .
ned).
(6) E s s e n t i a l n e s s : e a c ' i c o n o n e n t i s e s s e n t i a l i.e. t t l e r e e x i s t s x , x ' ~ X ~ and y F X 2 such t h a t n o t xy
-
x ' y ( t h e o t h e r comnonent l i k e w i s e ) .
C f course " i f f " i s s h o r t f o r i f and o n l y i f .
3. SASIC RFSULTS O N 5V'VfETPIC CnNJPIPIT STRUCTUSEq I n t h i s oaraqranh we i n t r o d u c e some more o r l e s s ohvious D r o D e r t i e s o f an a d d i t i v e c o n j o i n t measurenent s t r u c t u r e .
!.le r e s t r i c t o u r a t t e n t i o n t o
symmetric s t r u c t u r e s f o r reasons b e s t exoounded b v s c a n n i n o Holman's (1971) p r o o f o f t h e r e p r e s e n t a t i o n theorem f o r a d d i t i v e c o n j o i n t s t r u c t u r e s . Furthermore, we assume ttle e x i s t e n c e o f an e l e m e n t
X2,z
x and
y s2 y.
xy F X1 defined:
x
r1(x,y)
i s t h e s o l u t i o n o f xy
xy
- c1
,v
provided the eouations are solvahle.
and 6 (x,y)
2
suc'l t h a t f o r a l l
si:
Then, two f u n c t i o n s
X1
x
X2
+
t % e s o l u t i o n o f xy
Xi
are
- _x c2
L.'ith t h e h e l n o f t h e s o l v a b i l i t y a x i m
we c o u l d s o e c i f y c o n d i t i o n s t h a t r e n d e r these e q u a t i o n s s o l v a b l e , h u t we do n o t b o t h e r t o do s o , f o r we a r e n o t s t r i v i n g f o r t h e most g e n e r a l t r e a t ment.
''henever we use ci(x,y)
i n t h i s c h a p t e r , r e t a c i t l y assume i t s e x i s -
tence. The nurr,ose o f t h e f o l lowinq r e s u l t s i s t o m o t i v a t e
t t e c o n d i t i o n s chosen
i n the n e x t c h a o t e r t o o b t a i n a f a c t o r i z a t i o n o f ( A , note t b a t
I(
b e i n g an indenendent o r d e r r e l a t i o n on X1
weak o r d e r i n g s f o r x,v
E
To b e g i n w i t h we
5 ) .
x
;J2 g i v e s r i s e t o
s1 and s 2 on X1 and X 2 r e s p e c t i v e l y by t h e d e f i n i t i c n :
X1 i f f f o r some z
x s1 v
E
X2,xz L y z
( s 2 i s d e f i n e d on X 2 s i m i l a r l y ) . Each o f these r e l a t i o n s can be used t o develor, a new weak o r d e r on X1 Ile d e f i n e f o r a l l xy, x ' y '
E
XI
x
X2
s1
xy L x ' y '
iff x
xy L ' x ' y '
i f f y s2 y'.
I t i s e a s i l y confirmed t h a t
x'
L and L ' a r e weak o r d e r s on X1
x
X2.
x
X2.
351
Factorization and additive decomposition o f 4 weak order
The c h i e f p o i n t o f i n t e r e s t ahout L and L ' i s t h a t t h e y d i s t i n g u i s h between e l e n e n t s from t h e same e n u i v a l e n c e c l a s s o f
- , i.e.
i f x,y
-
x'y'
t h e n xy
and x ' y ' a r ? q e n e r a l l v n o t i n d i f f e r e n t w i t h r e s p e r t t o L o r L ' . ' J i t h o u t l o s s o f g e n e r a l i t y we may s i m p l i f y t h i n g s a b i t try assuming t h e e q u i v a l e n c e c l a s s e s o f t h e i n d i f f e r e n c e r e l a t i o n a s s o c i a t e d w i t h si s i s t o f one element o n l y .
[ ( i ' )we
-
e a s i l y conclude t h a t xy
( t h e same h o l d s f o r L ' ) .
L
x ' y ' and x.y
x ' y ' imrJlies xy = x ' y '
To v e r i f y t h i s a s s e r t i o n assume xy
xy L x ' y ' , which i m p l i e s x
sl x ' and x ' s 1 x, so x - 1 x ' .
above s i m l i f i c a t i o n we have x = x ' .
So xy
- xy'
t o con-
w i t h r e s n e c t t o L ( L ' ) by
Cenoting i n d i f f e r e n c e
-
x ' y ' and
I n v i r t u e o f the
o r y s 2 y ' and y ' s 2 y,
which a o a i n y i e l d s y = y ' . NOW, we o i c k a f i x e d element f r o m X1
equivalence class o r d e r on t u r e of
m.
m.
x
X2,
xo yo say, and r e s t r i c t L t o i t s
T h i s r e s t r i c t i o n , a g a i n denoted b y L, i s a s i m o l e
P u t a more thorpugh i n v e s t i g a t i o n r e v e a l s much more s ' t r u c -
(w,
L ) , which i s due t o t h e o t h e r c o n d i t i o n s f u l f i l l e d by
Take, f o r example, two elements xy, xFv1 E Xoyo such t h a t (X1,Y2,s). Keeping i n mind t h e a d d i t i v e r e p r e s e n t a t i o n and t h e i n d i f f e r e n xy L xlyl. t f2(y1) ce o f xy and xlyl, we know fl(x) t f 2 ( y ) = fl(xl) and fl(x)
c fl(xl).
Thus, f 2 ( y ) 5 f2(y1).
L o o s e l y soeaking i n xy and xlyl
b o t h components c o n t r i b u t e d i f f e r e n t l y t o
t h e o v e r a l l v a l u e o f xy and xlyl
respectively.
Vhat i s d e f i c i e n t i n one
component i s compensated f o r i n t h e o t h e r . I n c h a p t e r 4 we a r e g o i n 7 t o d e f i n e a compensatory s t r u c t u r e w i t h o u t a l l u d i n g t o components.
To t h i s Durnose rre p o i n t o u t t h a t f o r a i v e n xy
t h e presupposed e x i s t e n c e o f x ' = s1(5,y') c i f i c element i n
Xoyo,
a l l , we a s s e r t x ' y '
and y ' = s2(x,y)
Xoyo
Vrovides a sve-
which i n a sense i s comnlementary t o xy.
E x,yo.
E
First of
T h i s i s , i n f a c t , t r u e by t h e Thomsen c o n d i -
t i o n a p p l i e d t o t h e e q u a t i o n s xy
- zy'
and 5y
- x'y.
I t seems t o be r e a -
sonable t o conceive o f t h e element x ' y ' j u s t d e f i n e d as a r e l a t i o n on
m,
which i n t h e c o n t e x t o f t h e comnonent f r e e t r e a t m e n t o f t h e n e x t c h a p t e r f e a t u r e s as a p r i m i t i v e . We d e f i n e xy K x ' y ' i f f x ' = cl(x,y),y' K enjoys the f o l l o w i n g properties
= .E2(x,y),
R . Suck
352
xy I' I f xv
-
~
K xi!';,
xlvl
'
xlvl,
1
i m' n l i e s x ' y ' K xv
(3.1)
~
x , ~K x ' Y ' , and
xv L xlyl,
L x'y'
tben x i v i
(3.7)
To n r o v e ( 3 . 2 ) we ob-
( 3 . 1 ) i s an easy consenuence o f t h e d e f i n i t i o n o f Ei. i m p l i e s x s1 xl,
serve t h a t xy L xlyl
:'.
1: a n d xyl 5 x v .
which t o o e t h e r w i t h xy
- c1 (x,:/,)
Trori t h i s and xyl
:w i n f e r t1 ( x , y l ) s 1 f,l ( x , y )
s1 x ' .
or x i
y an(-!
-
xlyl
yields
x\/ - E l ( x , . ~ ) y
So, try t h e d e f i n i t i o n o f L ,
The f o l l o v i n o D r n n e r t v , a1 thouch n o t used i n t'ie s c q w l , sheds a d d i t i o n a l
of t h e r e l a t i o n K I f x v 1: x ' y ' and x ' y ' K x " v " , t h e n xy L x " y " .
l i c l i t on t h e n a t u r e
can be r e p l a c e d h.v xy = x " v " , s i n c e
:)I
-
!n t h e n r e s e n t case xy L x"!f'
,"y".
Sn f a r we t r i e d t o uncover some n r o r e r t i e s of t h e i n d i f f e r e n c e curves o f an
a d d i t i v e c o n j o i n t s t r u c t u r e , t h e r e b v l i m i t i n g o u r a t t e n t i o n t o one e q u i v a lence c l a s s o n l y .
Yext, we c h a r a c t e r i z e XY
L ' xlvl
i f f xy
K
L' by L and
S.
'.!e have
K x i v i and
x'y', x p l
x'v' L xivi
(3.3)
T h i s i s proved hy t h e d e f i n i t i o n o f K and t h e f o l l o w i n q c h a i n o f e q u i v a l e n ces : xy L ' xlvl
i f f y s 2 yl,
i f fc1
( x , ~ )s l
E ~ ( x , Y ~ )i, f fX '
s1 x i ,
i f f x ' y ' L xi".
Iiow, we deal w i t h a c o n d i t i o n c o n c e r n i n g t h e c o n o a t i b i l i t y o f
on XI
x
X,.
L
s , L, and K
!e e x o l o i t t h e s i m n l e f a c t o f a d d i t i v e c o n j o i n t s t r u c t u r e s t h a t
an element xy, whic'l i s s u r l e r i o r t o a n o t h e r e l e i w n t xlvl s e o a r a t e l v , exceeds i t w i t h r e s o e c t t o 5 , t o o . f r e e treatment
i n h o t h components
I n view o f t h e component
we have i n mind, we m i s t n o t use such a d e s c r i o t i o n , b u t
c o n v e r t i t t o a s t a t e m e n t i n v o l v i n c . L and
I(.
only.
The c o n d i t i o n reads as
follows: i f xlyl
L xy, xlvl
x i v i L x'v',
K x i y i , xv
K x ' y ' , and (3.41
then x l y l s xy
I t s h o u l d be n o t e d t h a t ( 3 . 2 ) does n o t r e n d e r t h e a s s u m t i o n o f ( 3 . 4 ) emnty
( n o t even almost empty), because ( 3 . 2 ) assumes xy
-
xlyl.
The n r o o f o f (3.4) i s s i m o l e , i f we observe t h a t by (3.3) we have xlyl
Thus v1 c 2 y , b u t x1 s 1 x f o l l o w s f r o m xlyl
L
xy.
r e l a t i o n s K and L and i t s n r o n e r t i e s ( 3 . 1 ) - ( 3 . 4 )
L ' xy.
I n t h e n e x t naraqraph t h e c r o n un i n a d i f f e r e n t
353
Factorization and additive decumpositiofi ofa weak order
4. CflN,TnIHT STRUCTURES IfITHOUT CWPnNENTS I n t h i s n a r t o f t h e naper we n r e s e n t c o n d i t i o n s , f r o m which a f a c t o r i z a t i o n and a d d i t i v e r e n r e s e n t a t i o n o f t h e k i n d d e t a i l e d i n t h e i n t r o d u c t i o n can be derived.
The o r o n e r t i e s n f c o n j o i n t s t r u c t u r e s e m h a s i zed i n t h e a r e c e d i n q
chanter are c r u c i a l f o r t h i s e n t e r n r i s e . One o f t h e lessons we l e a r n e d from i t was t h e comoensatorv s t r u c t u r e o f t h e i n d i f f e r e n c e curves.
t o be.
DFF I N I T I rlti A t r i p l e ( A , L,
and
K), ;s
F
ccmnensatorv s t r u c t u r e i f f A i s a non emoty s e t
L and K b i n a r y r e l a t i o n s s a t i s f y i n o t b e f o l l o w i n o axioms.
(1)
( A , L ) i s a weak o r d e r
(2) (3)
F o r a l l a E li e x i s t s a '
(4) (5)
I f a L b, a K a ' , b
A such t h a t a K a '
K i s synmetric
I f a K a ' and a '
K b ' , then b ' L a ' b, t h e n a K b .
The r e a l numbers w i t h t h e usual
Q
r e l a t i o n and K = I ( x , y ) ; x t y = l ) p r o v i d e
an examnle o f a comoensatory s t r u c t u r e .
A c o m e n s a t o r y s t r u c t u r e i s a n o t h e r case o f fundamental w a s u r e m e n t ( f o r a comprehensive d i s c u s s i o n o f t h i s conceot see Roberts ( 1 9 7 9 ) ) , v e r y s i m i l a r t o o r d i n a l s t r u c t u r e s , b u t somewhat e n r i c h e d due t o t h e r e l a t i o n K .
To n i v e
i n s t u d y i n ? a u d i t o r y o e r c e o t i o n , sounds o f d i f f e r e n t i n t e n s i t y a r e D w s e n t e d t o t h e l e f t e a r ( s a y ) . The s u h j e c t i s asked t o adan e m o i r i c a l examnle:
j u s t a second sound, which can be o r e s e n t e d t o t h e r i C h t e a r , i n such a manner t h a t t h e combined loudness o f b o t h sounds matches f e r e n c e sound.
with a f i x e d re-
A s u i t a b l e r e p r e s e n t a t i o n theorem reads as f o l l o w s .
THEOREM 1 L e t ( A , L, K ) be a comoensatory s t r u c t u r e c o n t a i n i n q an order-dense c o u n t a b l e subset.
Then, t h e r e i s a f u n c t i o n f: A
+IR
satisfyina
a L b i f f f ( a ) ,< f ( b ) I f a K a ' and b K b ' , then
f(a)
+
f(a') = f(b) t f ( b ' ) .
B e f o r e e n t e r i n g t h e d e t a i l s o f t h e D r o o f , we s k e t c h i t s key i d e a .
354
R. Suck
i f i t were n o t f o r ( 4 . 2 ) .
l ? w i o u q i $ / (4.1) o f f e r s no d i f f i c u l t v , first
S ~ O Wt h a t
So, we
i t i s p o s s i h l e t o h a l v e A , t h u s g e t t i n o b!o e s s e n t i a l l y d i s -
j u n c t i v e suhsets, one c o n s i s t i n o o f i t s " s v a l l " elements, t h e o t h e r c o n t a i n i n a the " l a r o e r " objects.
The l a t t e r can he shown t o c o i n c i d e w i t h t h e
c o m e n s a t i n a elements o f t h e f o r m e r s e t .
Conseoucntly, an a r b i t r a r y o r d i -
n a l r e q r e s e n t a t i o n o f t h e " s m a l l " s e t , can he t r a n s f o r m e d and extended t o t b e " l a r o e " s e t i n such a wav t h a t (4.1) and ( 4 . 2 ) h o l d s i m u l t a n e o u s l y . The r e a d e r n o t p ? r + i -
The cnmnlete p r o o f i s s n l i t i n t o a s e r i e s o f lemmas.
r u l a r l y i n t e r e s t e d may s k i n t h i s D a r t and r e s u m r e a d i n g w i t ? T b e n r m 7 .
1: a K b and h K c i m p l i e s a
LF""P
i
c
Proof: F i rs t observe t h a t i n v i r t u e o f (4) (a,c)
9 L i m o l i e s ( h , h ) 9 L, w h i c h 9 L . So
The s a w c o n t r a d i c t i o n d e r i v e s f r o m (c,a)
i s impossible a L c and c L a .
LEP!!lA 2 : K = K
a L b } i s an e q u i v a l e n c e r e l a t i o n A .
a,b);
U
Proof: R e f l e x i v i t y and symmetry a r e o b v i o u s . Transitivity: a ( h K c or b tions
b and b
c).
k
a implies (a K h o r a
h ) and
U s i n g d i s t r i b u t i v i t v o f "and" and " o r " and c o n d i -
(l), (3), ( 5 ) and Lemma 1, we g e t a L c o r a K c .
LE!?'A 3: a K a and b K b i f f a K h and a L b. Proof: Assume a
K a,
Ll
K b anti a L b ; ( 4 ) i m p l i e s b L a.
t h i s assumotion b y b L a, y i e l d s a L b . a L h follows.
Replacing a L b i n
So, by t h e c o n n e c t i v i t y o f L ,
a K b i s now a consequence o f ( 5 ) .
The o p p o s i t e d i r e c t i o n i s o b v i o u s f r o m ( 5 ) . From t h i s L e m a we see t h a t a K b and a L b o c c u r s a t b e s t once.
So, gener-
a l l y , i t i s p o s s i b l e t o s e n a r a t e t h e s m a l l elements o f an e q u i v a l e n c e c l a s s
o f K from t h e l a r g e ones.
Denoting the equivalence class o f a w i t h respect
t o K by aK, we d e f i n e s f a ) = { b E aK; b
L
c for all c
E
aKl
aK: c L b f o r a l l c E a 1 K The p r o n e r t i e s o f s ( a ) and t ( a ) a r e d w e l t upon i n t h e n e x t few lemmas. t ( a ) = {b
F
Factorization and additive decomposition of a weak order
L F W A 4: s ( a ) #
B.
L
If b,c F s ( a ) , then b
c.
355
The same h o l d s f o r t ( a ) .
Proof: -
g L; then b K a.
Sunnose, b E aK e x i s t s such t h a t (a,b) f o r i ft h e r e i s h '
llow t
E
6
aK ( b , b ' ) % L, then b ' K b and b ' K a, u s i n a
L
h i s inferred,
s(a),
Lemma 2. B u t from Lemrna 1 a
c o n t r a d i c t i n o t h e choice o f b.
I f t h e r e i s n o such b, then a E s ( a ) .
Piow, if b,c E s ( a ) , t h e n b L c and c L b, p r o v i n g t h e second p a r t .
A
s i m i l a r prcof applies f o r t ( a ) . LEMMA 5: s ( a ) = t ( a ) = aK o r s ( a )
t(a) =
kl.
Proof: Assume b
E
Therefore, t ( a ) = aK.
s ( a ) n t ( a ) ; then f o r a l l c E a K y b L c L b, i m o l y i n g b L c. c E s ( a ) and c
t ( a ) , which i s e q u i v a l e n t t o s ( a ) = aK and
If no such b e x i s t s , t h e i n t e r s e c t i o n i s empty.
LEllFlA 6: s ( a ) u t ( a ) = aK Proof: Ry Lemma 5 t h e r e a r e o n l y two cases p o s s i b l e , one i s t r i v i a l .
s(a) n t(a) = We assume b
9
p
s ( a ) and show b E t ( a ) .
f u r t h e r m o r e (b,c)
4 L, f o r o t h e r w i s e b
For a r b i t r a r y d F aKy i f d If d
E
Now, l e t
and b E aK.
s ( a ) , t h e n d L b.
4
Since s ( a ) #
8,
t h e r e i s c E s(a);
s ( a ) ; so b K c .
E
s(a), then d K c.
L
By L e m a l , d
In any case we have d L b, showing b
b. E
t(a),
f r o m which t h e rrronosi t i o n f o l l o w s . LEMMA 7: I f b E s ( a ) , c E t ( a ) , then h K c. Proof: -
i
Since b,c E aK, b K c o r b
i
Lemma 5 t ( a ) = s ( a ) = aK.
B u t by (2) b ' E A e x i s t s such t h a t b K b ' ,
c.
But b
so b ' E aK = s ( a ) , whence by L e m a 4 b '
c would i m n l y c E s ( a ) and by
L
c and by ( 5 ) b K c.
LEMrlA 8: a K b i f f ( a E s ( a ) and b E t ( a ) ) o r ( a E t ( a ) and b E s ( a ) ) .
Proof: Assume a K b. s(a) = t(a),
O f course, a,b
i.e. b
L e m a 6, b E t ( a ) .
E
t(a);
E
aK; i f a,b E s ( a ) , t h e n a
b u t i f a E s ( a ) and b
i
b and
4 s(a), then by
356
R . Suck
P s i m i l a r arllument a o n l i e s . i f we assure a
E
t(a).
The o o q o s i t e i m n l i -
c a t i o n i s c o n t a i n e d i n Lemma 7 .
LFIT'P 9: I f s ( a ) = t ( a ) and s ( b ) = t ( h ) , t h e n a L b and a K b . proof: -
F r o m Lemma 7 b'e i n f e r a
I(
a and h Y, b and Lemma 3 then g i v e s t h e asscr-
:iOP . OLr
n e x t s t e o i s t o d e f i n e a r e l a t i o n on t h e s e t o f e o u i v a l e n c e c l a s s e s o f
K , i f h i c h t u r n s p u t -c be a s i m n l e o r d e r , i .e. a weak o r d e r n i t h i n d i f f e r e n c e imnlyinn i n d e n t i t y .
i f f there are c
E-
i. = faK; a c A ) K s ( a ) , d F s ( b ) such t h a t c L d
aK L t bK i f f t h e r e a r e c
C
t ( a ) , d C t ( b ) such t h a t c L d .
a
K Ls b K
LEV:A 10: I f a
E
s(x) ind b
E
t ( y ) , then a L b .
Proof: Sunoose h L a and a ' K a, b ' K b.
Condition (4) y i e l d s a' L b ' .
From a ' K a and a Y x we i n f e r a ' K x and a L a ' .
Similarly b' L b
Now a L a ' , a ' L b ' , and 0 ' L h g i v e s a L h.
I f n o t b L a, then o f course a L b h o l d s .
0 or
LEiI'lC, 11: a ) s ( a ) n s ( b ) = b) t(a)
c) x
E
s(a) = s(b)
P or t(a)
t(b) =
= t(b)
t(b1 imnlies s(a) = t ( a ) = s(b) = t ( b ) .
s(a)
Pboof: a ) SUPOOSP x y E by,.
F
s(a)
s(b).
9
Moreover x L y .
so y L z and y
Then a
K
b and aK = b K . I f y
s ( a ) , then
Take z E bK; by d e f i n i t i o n o f s ( b ) x L z,
s(b).
b ) The n r o o f i s s i m i l a r t o a ) . c) Obviously x '
F
s(h)
x ' E t ( a ) , so x L x L xi.
0
X I ,
t ( a ) , where x ' K x .
Rut x
E
s(a) yields
whereas x ' E s ( b ) q i v e s s ' L x; t o g e t h e r
By Lemma 3, x K x , by Lemma 5, s ( a ) = t ( a ) and s ( b ) = t ( b ) ,
and by Lemma 3, a
b.
Thus s ( a ) = s ( b ) and t ( a ) = t ( b ) .
LEMVA 12: (AK, L s ) and (AK, Lt) a r e s i m p l e o r d e r s
Factorization a i d additive decontposition of a weak order
351
proof: Since t h e elements o f s ( a ) are a l l L - e q u i v a l e n t , the d e f i n i t i o n o f Ls does n o t denend on the n a r t i c u l a r elements c, d chosen f r o m s ( a ) , s ( b ) resnective1.v.
C o n n e c t i v i t y , a n t i s v m e t r y , apd t r a n s i t i v i t y a r e r e a d i -
l y shown f r o v t h e d e f i n i t i o n . LE'I'lP 13: I f 0 i s an o r d e r dense s u b s e t o f ( A , L ) , then DK = {dK; d E DI i s an o r d e r dense subset o f AK. Proof: I f aK Ls bK, c E s ( a ) m d e more , (bK, aK)
9 L,
E
s(b) e x i s t s a t i s f y i n g c L e.
then ( e , c )
4 L.
D
Since
If,further-
i s o r d e r dense i n A ,
d G D e x i s t s such t h a t c L d and d L e . But, i f
NOW, i f d E s ( d ) , v!e i m m e d i a t e l y have aK Ls dK and dK-LS bK.
d F t ( d ) , by Lemma 10 e L d, s i n c e e E s ( h ) : hence d L e and d
s(b).
From Lemma l l c ) we have s ( d l = t ( d ) , s o apain d E s ( d ) . Since DK s o e c i f i e d i n
Now, t h e o r o o f o f Tlieorem 1 "s e e s i l ! l p r o v i d e d .
Lemma 13 i s c o u n t a b l e p r o v i d e d 9: AK
D
i s countable, we have a r e o r e s e n t a t i o n
+ R , m n o t o n e w i t h L s . Then, d e f i n i n a f o r a
f(a) = {
exo(-a(aK)) i f a exo(-g(a,))
0
9 P
E 4
t(a)
if a s(a) i f a E s(a)
'I
t(a),
where e x a ( x ) = ex, we e v e n t u a l l y o b t a i n t h e d e s i r e d r e o r e s e n t a t i o n . Lemmas 5 and 6 we see t h a t f i s w e l l d e f i n e d f o r a l l a v e r i f y (4.1).
F
By The r e a d e r may
A.
Concernina ( 4 . 2 ) we observe r e f e r r i n a t o Lemma 8:
f(a) t f(a') = 0 iff a
K a'.
The " i f " n a r t o f t h i s s t a t e m e n t b r i n o s about ( 4 . 2 ) , whereas t h e " o n l y i f " p a r t i s r e f o r m u l a t e d i n t h e f o l l o w i n g C o r o l l a r y o f Theorem 1. Corollary.
L e t (A, L, R) s a t i s f y t h e assumptions o f Theorem 1.
i s a r e a l number k and a r e o r e s e n t a t i o n f such t h a t (4.1),
iff ( a )
+
f ( b ) = k, t h e n a K b
Then t h e r e
( 4 . 2 ) and (4.4)
hold. The c o n s t a n t k i s zero i n t h e r e o r e s e n t a t i o n ( 4 . 3 ) . The n e x t theorem s n e c i f i e s t h e c o n d i t i o n s o f t h e f a c t o r i z a t i o n and a d d i t i v e decomoosi t i o n .
358
R . Suck
THEOREM 2
The system (A, 5 , L , K ) c o n s i s t i n c r o f a nonemotv s e t A and t h r e e b i n a r y r e lations
:,
L , K on A s a t i s f y i n g
(1)
( A , 5 ) i s a weak o r d e r
(2)
(A, L) i s a ouasiorder
(3)
If a
(4)
I f a K b , then a
(a,
L
b , then a L h o r b L a
-, K - )
-
b
i s a comensatory s t r u c t u r e
(5) /a /a ( 6 a ) F o r a l l a,b E A t h e r e i s a t l e a s t one element c such t h a t a
-
c and
i
c and
b i c ( 6 b ) F o r a l l a,b F A t h e r e i s a t l e a s t one e l e m e n t c such t h a t a h'
c ' , where b K b ' and c K c '
( 6 c ) There are a,b F A such t h a t a <- b (7)
I f a L b, a K a ' , b K b ' , a ' L b ' , then a
Then, t h e r e a r e s e t s X1,
5
b.
P
X 2 and s u r j e c t i v e m a m ni:
weak o r d e r 5 ' on X1
x
X2 such t ' l a t
a
5
b i f frll(a)
n 2 ( a ) 5 ' ql(b)
+
Xi
( i = 1 , 2 ) and a
(4.51
n2(b).
Eoreover, i f a s u i t a b l e Archimedean c o n d i t i o n h o l d s , then (X1,
X2,
5') is
an a d d i t i v e c o n j o i n t s t r u c t u r e . I n the l a t t e r case t h e r e a r e two f u n c t i o n s fi: a If a
L b then fl(a)
5
h i f ff l ( a l
A
t f 2 ( a ) s fl(b)
+ R ; i = 1 , 2 such t h a t + f2(b)
(4.6)
5 fl(b)
a K b i f f fl(a)
= f 2 ( b ) and f 2 ( a ) = f l ( b ) .
(4.7)
A few comnents on t h e c o n d i t i o n s a r e i n o r d e r . F i r s t , we want t o b r i n q t h e few n r o o e r t i e s d i s n l a y e d i n c h a n t e r 3 Snc t h e c o n d i t i o n s o f b o t h theorems i n t o r e l a t i o n .
An i n d i f f e r e n c e c l a s s o f a con-
j o i n t s t r u c t u r e i s a comoensatory s t r u c t u r e . ( 4 ) coincide.
An e q u i v a l e n t t o
E s n e c i a l l y (3.2)
and c o n d i t i o n
L' of a c o n j o i n t s t r u c t u r e can e a s i l y be de-
fined i n ( A , 5 , L, K) a v a i l i n o oneself o f ( 3 . 3 ) , i . e . a L ' b i f f a K a ' , b K b ' and a ' L b ' .
C o n d i t i o n ( 7 ) o f Theorem 2 seems i r n o r t a n t as i t t a k e s
c a r e o f t h e c o m p a t i b i l i t y between
5,
L, and K.
I t o r i g i n a t e s from (3.4).
Second, we do n o t i n s i n u a t e t h e n e c e s s i t y o f o u r c o n d i t i o n s w i t h t h i s comnarison.
This problem i s n o t y e t d e a l t w i t h Droperly.
t e n t i a l axioms ( 6 ) a r e f a r f r o m b e i n g necessary.
F o r example t h e e x i s -
On the c o n t r a r y t h e y f o r c e
359
Factorization and additive decomposition ofa weak order
the s t r u c t u r e t o be so b u l q y t h a t t h e r e p r e s e n t a t i o n cannot r e s u l t even i n a bounded r e c t a n g l e , l e t alone an i n c o m p l e t e s t r u c t u r e , which a r e t r e a t e d b y Doi anon , Falmapne (1974), J a f f r a y (1974)
, Narens , Luce
(1976)
, Suck
(1963) and which a r e v e r y d e s i r a b l e from t h e n o i n t o f view o f a o o l i c a t i o n s . T h i r d , an e x t e n s i o n t o m r e than two components would be e x p e d i e n t .
Provi-
s i o n a l l y , one can imaoine t h e i n d i f f e r e n c e c l a s s e s o f 5 t o c o n s i s t o f a s t r u c t u r e t h a t s a t i s f i e s Tbeorem 2.
Somehow, an i t e r a t i v e a o n l i c a t i o n o f
Theorem 2 s h o u l d Droduce a m u l t i d i m e n s i o n a l s t r u c t u r e , b u t as y e t we cannot p r e s e n t a f i n a l answer. F o u r t h , because o f t h e d e f i c i e n c i e s j u s t mentioned and t h e amount o f t e c h n i c a l lemmas needed, t h e o r o o f o f Theorem 2 i s o m i t t e d .
5 . REMARKS ON APPLICATIONS The d i f f i c u l t i e s t h a t a r i s e i n a o p l y i n g o u r t h e o r e m t o e m n i r i c a l work i n osycholooy must n o t be shoved under t h e r u g .
The d a t a have t o p r o v i d e !n-
f o r m a t i o n about an o r d e r r e l a t i o n w i t h i n d i f f e r e n c e s e t s t h a t a r e s u f f i c i e n t l y l a r g e t o d i s c l o s e a f i n e r s t r u c t u r e , which i s a c a n d i d a t e f o r t h e and !t r e l a t i o n s .
L
One s h o u l d a t l e a s t have a good guess about one o f t h e
I n t h i s case t h e e x n e r i m e n t can n o s s i b l y be olanned s o p h i s t i -
components.
c a t e d l y enough t o nroduce t h e necessary i n f o r m a t i o n , Although t h e p o s s i b i l i t i e s o f a n p l i c a t i o n s a r e r a t h e r poor, t h e r e a r e sever a l problems o f t h e o r e t i c a l i n t e r e s t , which can be r e f o r m u l a t e d i n terms o f t h e p r e s e n t dt\velopment. One examnle i s t h e q u e s t i o n o f t h e f o u n d a t i o n s o f m u l t i d i m e n s i o n a l s c a l i n q . C a s t i n t h e framework o f t h e n r e s e n t paper i t t u r n s o u t somewhat l i k e t h i s : g i v e n a s e t o f s t i m u l i A and a p r o x i m i t y r e l a t i o n P; assume P i s a weak o r d e r on A (A
x
x
A.
Do s e t s X1,...,Xn
A, P) and(X1
x
... x
Xn,
and a weak o r d e r P ' e x i s t such t h a t
P ' ) a r e h o m m o r p h i c and i s t h e r e a represen-
t a t i o n , which i s c o m p a t i b l e w i t h a m e t r i c ?
An analogue o f Theorem 2 f o r n
components c o u l d s p e c i f y s u f f i c i e n t c o n d i t i o n s .
The aporoach o f B e a l s ,
K r a n t z , Tversky (1968) (see a l s o Beals, K r a n t z (1967) and Tversky, K r a n t z (1970)) s i d e s t e p s t h e q u e s t i o n o f t h e e x i s t e n c e o f t h e f a c t o r s , because i t s t a r t s w i t h t h e f a c t o r i a l s t r u c t u r e (Xl,
X2,
..., Xn,
P').
The papers o f
Dr'dsler (1979) u s i n q axioms from a b s t r a c t geometry t o c h a r a c t e r i z e a p l a n e and O r t h (1980) d e t e r m i n i n q o n e - d i m e n s i o n a l i t y i n ( A tune w i t h a component f r e e a x i o m a t i z a t i o n .
x
A , P) a r e more i n
The f o u n d a t i o n a l n r o b l e m o f
360
R . Suck
l n u t t i d i w n s i o n a l s c a l i n g has r a i s e d a q a r t i a l l v c o n t r o v e r s i a l debate (see f o r examole Roskam : l ? S l ) cnd Schonemann, Roro ( ! ? S l ) ) ,
and a s ~ l i s ~ ~ c t n r y
s o l u t i o n i s n o t y e t achieved. Another l i n e o f t h e i n v e s t i n a t i o n o f s i m i l a r i t y d a t a focused f e a t u r e matchin?.
Tversky (1977) develoned a f e a t u r e t h e o r v based on t h e assumntion t h a t
each o b j e c t o f t h e donain under s t u d y can be r e n l a c e d hy a s e t o f f e a t u r e s taken from a s u f f i c i e n t l y l a r 9 e r e s e r v o i r .
n e n o t i n n t h e f e a t u r e s e t assc-
c i a t e d w i t h t h e o b j e c t a b v A and t h e s i m i l a r i t y
r d e x by s ( a , h ) , i t i s
shown t h a t a s i m i l a r i t y s c a l e S and a n o n n e g a t i v e f u n c t i o n f e x i s t s s a t i s fyinn s(a,bl S(a,b)
=
o
s(c,d)
5 5 c,d)
i f f S(a,h)
-
r ( ” f ~ ” ) af(A-P)
-
8f(B-A)
f o r some 0 , a , R 2 0
(5
T h i s r e n r e s e n t a t i o n does c o n s i d e r a b l y b e t t e r i n case o f asvlnmetric s i m i r i t i e s and o t h e r p c c u l a r i t i e s t h a t cannot e a s i l y he r e c o n c i l e d w i t h t h e
q r n n e r t i e s o f a m e t r i c soace. The r e o r e s e n t a t i o n ( 5 . 1 ) i s d e r i v e d f r o m a c o m n l i c a t e d s e t o f axioms i n c l u d i n q t h e e x i s t e n c e o f t + e f e a t u r e s e t and t h e f o l l o w i n n m a t c h i n q c o n d i t i o n s(a,bl
= r(Ar8, A-S,
5-C)
(5.2)
where F ( X , Y , Z ) i s a f u n c t i o n o f t h r e e arguments, each a s u b s e t of t h e feature set.
P l t h o u q h i n t u i t i v e l y a n o e a l i n q , t h i s seems t o be a s t r o n q con-
d i t i o n s u g a e s t i n g i n a sense t h e d e c o m p o s a b i l i t y o f t h e s i m i l a r i t y i n t o t h e r e l e v a n t f a c t o r s , which a r e t h e common f e a t u r e s h P , t h e f e a t u r e s a has and b not, A-R,
and b has and a n o t , ? - A ,
’re c o n j e c t u r e t h a t o u r f o r n u l a t i o n o f t h e DrOblem o f n w l t i d i m n s i o n a l scal i n g j u s t e x p l a i n e d and an a n n l i c a t i o n o f a D r o o e r v a r i a n t o f Theorem 2 could y i e l d a representation (5.1) w i t h o u t using (5.2). F u r t h e r a D p l i c a t i o n s a r e c o n c e i v a b l e i n t h e i n v e s t i a a t i o n of t h e s t r u c t u r e
o f i n t e r a c t i o n e f f e c t s i n f a c t o r i a l desians ( K r a n t z e t a l . (1971), p.248 and F i s h b u r n ( 1 9 7 5 ) ) . B u t i n t h i s as w e l l as i n t h e a f o r e m e n t i o n e d nroblems f u t u r e r e s e a r c h has t o show, whether o u r e x n e c t a t i o n s o f f r u i t f u l a D p l i c a t i o n s w i l l b e a r o u t .
361
Factorization and additive decomposition of a weak order
REFERENCES Beals, F., K r a n t z , D.H.,
M e t r i c s and geodesics induced
by o r d e r r e l a -
t i o n s , Flathematisches Zei t s c h r i f t , 101 (1967) , 285-298. Beals, R.,
K r a n t z , D.H., Tversky, A.,
Foundations of m u l t i d i m e n s i o n a l
s c a l i n g , P s y c h o l o o i c a l Review, 75 (196E), 127-142. Doignon, J;P.,
F a l m a y e , !I.C.,
D i f f e r e n c e measurenent and s i m o l e s c a l a -
b i l i t y w i t h r e s t r i c t e d s o l v a b i l i t y , J o u r n a l o f 'lathematical Psycholocly
1'. (LP74), 473-499. D r i i s l e r , J . , Foundations o f m u l t i - d i m e n s i o n a l m e t r i c s c a l i n g i n C a y l e r K l e i n geometries, R r i t i s h Journal o f rv'athematical and S t a t i s t i c a l Psychology, 32 ( 1 9 7 9 ) , 185-211. Fishburn, P.C.
, Nondecomposable
c o n j o i n t measurement f o r b i s y m m e t r i c
s t r u c t u r e s , J o u r n a l o f Mathematical Psycholoqy, 12 (1975), 75-89. Holman, E.W.,
A n o t e on c o n j o i n t measurement w i t h r e s t r i c t e d s o l v a b i l i -
ty, J o u r n a l o f P a t h e m a t i c a l Psychology, R ( 1 9 7 1 ) , 439-494. J a f f r a y , J.Y.,
On t h e e x t e n s i o n o f a d d i t i v e u t i l i t i e s t o i n f i n i t e s e t s ,
J o u r n a l o f Mathematical Psychology, 11 (1974), 431-452. K r a n t z , D.H.,
Luce, R.D.,
Sunneg, P.,
Tversky, A.,
Foundations o f mea-
surement, V o l . l , New York, Academic Press ( 1 9 7 1 ) . Narens, L., Luce,
P.D., The a l g e b r a o f measurement, J o u r n a l o f P u r e and
P p o l i e d Algebra, 8 ( 1 9 7 6 ) , 197-233. O r t h , B.,
On t h e f o u n d a t i o n s o f m u l t i d i m e n s i o n a l s c a l i n g : An a l t e r n a t i v e
t o t h e Beals, Krantz, Tversky a m o r a c h , I n Lantermann,
E.D., Feger, H .
( C d s . ) , Z i m i l a r i t y and Choice, Paners i n honour o f Clyde Coombs, Bern,
Huber ( 1 9 8 0 ) . Roberts, F.S.,
b a s u r e m e n t t h e o r y w i t h a p p l i c a t i o n s t o decisionmakina,
u t i l i t y and t h e s o c i a l sciences, Readina, Plass., PddisonJfesley ( In7f)). Roskam, E . E . ,
Contributions o f multidimensional s c a l i n q i n s o c i a l
s c i e n c e r e s e a r c h , I n B o w , I . (Ed.),
M u l t i d i m e n s i o n a l Data Reprcsenta-
t i o n : when and why, Ann Arhor, ' l a t h e s i s Press ( 1 9 8 1 ) . Schonemann, P.H.,
Boro, I . , rleasuwment, s c a l i n g and f a c t o r a n a l y s i s ,
I n Borcl, I . , M u l t i d i m e n s i o n a l Data R e p r e s e n t a t i o n : when and why, Ann P r b o r : Mathesis P r e s s (1981). Suck, R.,
On t h e i n t e r r e l a t i o n between e x t e n s i v e measurement and a d d i -
t i v e c o n j o i n t measurement, I n Flicko, C.,
Schulz, U . , Formalization o f
p s y c h o l o g i c a l t h e o r i e s . Proceedings o f t h e 1 3 t h Eur. I l a t h . Psycho1 . Group Meeting, B i e l e f e l d : M a t e r i a l i e n des U n i v e r s i tXtsschwerpunktes llathernatisieruno XXXX, (1983).
362
(151
R. Suck
Tversky, A.,
T e a t u w s o f s i m i l a r i t v , P s y c h o l o n i c a l k v i w . 84 (1977),
317-352. (161
Tverskv, A . ,
K r a n t z , D.H.,
The dimensional r e n r e s e n t a t i o n and t h e me-
t r i c s t r u c t u r e of s i r n i l a r i t v d a t a , J o u r n a l o f t'atbematical Psycholooy,
7 (1970), 572-596.
PARTIII DA TA ANAL YSIS
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TRENDS IN MATHEMATICAL PSYCHOLOGY E . Degreef and J. Van Bu enhmct (editors) @Elsevier Science Acbl&rs B. V. (North-Holland), 1984
365
THE PROPLEFl nF REPRESENTATION BASED UPON TI.10 C R I T E R I A
G. De Weir,
f4.
Cassner, X . Hubaut
U n i v e r s i t e L i b r e de B r u x e l l e s
The problem c o n s i d e r e d h e r e i s t h a t o f e l e c t i n q a del e g a t i o n when t h e D o n u l a t i o n o f e l e c t o r s i s seDarated i n t o c a t e q o r i e s , t h e d i v i s i o n b e i n g based uoon one o r s e v e r a l c r i t e r i a.
A two c r i t e r i a - b a s e d s i t u a t i o n c u r r e n t l v e x i s t s i n the Netherlands.
Using a s l i g h t l v more general v e r -
s i o n o f t h i s system, we oroved t h a t t h e o n l y t y n e o f s i t u a t i o n a v o i d i n o i m D o s s i b i l i t i e s i s one where each c r i t e r i o n o n l y p r e s e n t s two a l t e r n a t i v e s .
Further-
more, s e t t i n o a number of c o n d i t i o n s , as weak as DOSs i b l e , which must be r e s p e c t e d i n o r d e r t o ensure f a i r r e o r e s e n t a t i o n , t h e same r e s u l t remains
-
which
i s n o t so s u r o r i s i n c i as i t c o u l d have been suggested by Arrow's famous theorem.
1. INTRODUCTION A D o o u l a t i o n must e l e c t a c e r t a i n number o f delegates who w i l l f o r m an assembly such as t h e P a r l i a m e n t o r t h e Congress,
...
The n r o h l e m o f t h i s r e p r e s e n t a t i o n b e i n g f a i r o r n o t a r i s e s because t h e pop u l a t i o n i s d i v i d e d i n t o c a t e g o r i e s , t h e d i v i s i o n b e i n g based unon one c r several c r i t e r i a .
These c r i t e r i a a r e u s u a l l y geooraphic ( p r o v i n c e s i n B e l -
gium, denartements i n France, s t a t e s i n t h e U.S .,...) and p a r t i s a n (when t h e n o p u l a t i o n i s d i v i d e d i n t o o o l i t i c a l c a t e g o r i e s ) , b u t o t h e r types o f m o t i v e s can o c c u r ( a n d have a l o n g t h e y e a r s ) such as economic, s o c i a l o r even r a c i a l c r i t e r i a. The n e r f e c t and i d e a l s o l u t i o n would be t o e l e c t a d e l e g a t i o n n r o p o r t i o n a l l y ( i n t h e mathematical sense) t o t h e nur&er o f persons i n each o f t h e d i f f e r e n t
C . De Meur, hi. Gaimer and X. Hubaut
366
categories: b u t s i n c e i n t e o e r s o l u t i o n s are, o f course, r e q u i r e d , the problem i s n r a c t i c a l l y always i m p o s s i b l e .
I t i s thus necessary t o f i n d a d i s -
t r i b u t i o n o f seats w i t h as l i t t l e v a r i a t i o n as q o s s i b l e from the d i f f e r e n t u n i t s ' quotas ("number o f deleqates deserved by t'le s t r i c t o r o n o r t i o n a l r u l e ) and whicb does n o t transcress our n a t u r a l sense o f f a i r n e s s . For t h i s ournose, one must f i r s t e s t a b l i s h a number o f r u l e s , which, when h e i n q followed, should ensure
fair r e n r e s e n t a t i o n .
The second step i s t o
f i n d avnortionment methods which obey the given i m n e r a t i v e s . when i t i n v o l v e s one c r i t e r i o n , ha:
The nrohlem,
heen thoroughly discussed, and a general
review of the research made i n t h a t f i e l d can be found i n R a l i n s k i an< Youno
[21
.
To our b e s t knowledge, excent f o r a paper by Anthonisse [11, examining t w o c r i t e r i a czses, the same k i n d o f problem i n v o l v i n r r more than one c r i t e r i o n has never been considered though these cases are frequent i n many c o u n t r i e s . I+le focussed o u r a t t e n t i o n on t h i s tvne o f oroblem exnanding i t t o a somewhat more general frame.
2 . THE ONE-CRITERION PROSLEI4 Generalizing the o n e - c r i t e r i o n oroblem, p a r t o f i t s s t r u c t u r e remains; i t
will thus be i n t e r e s t i n g and u s e f u l t o review some o f i t s s n e c i f i c asDects. What can we expect t o hannen avoid
-
-
o r r a t h e r what can we reasonably hope t o
i n a o n e - c r i t e r i o n based s i t u a t i o n ?
Several " d i f f e r e n t " naradoxes can occur and thus b r i n g s t o the mind some " n a t u r a l " questions (and d e r i v e d r u l e s ) such as: ( i ) Can we accent an annortionment method which allows a o a r t y t o r e c e i v e more seats than i t s nuota rounded un? ( i i ) For a determined number o f delegates, i f one class of the Dooulation increases w h i l e another decreases, can we l e t the f i r s t ' s aoportionment decrease w h i l e the second's increases? ( i i i ) S h o u l d a method " s y s t e m a t i c a l l y f a v o r " small u n i t s a t the expense of l a r o e ones o r vice-versa? ( i v ) Can we a l l o w t h e c r e a t i o n o f a new u n i t t o change the r e l a t i v e p o s i t i o n s o f the o t h e r s ? ( v ) I f the t o t a l r e n r e s e n t a t i o n s i z e increases, can a u n i t l o s e any delegates ( i f , o f course, no o u t s i d e reason f o r t h i s should appear)? Answering the f i r s t question, some systems do accent such methods f o r a very
361
Problem of representation based upon two criterio
l o g i c a l reason; as a m a t t e r o f f a c t , one o f them i s c u r r e n t l y used i n the Belgian " l e $ s l a t i v e "
e l e c t i o n s [31
.
S i m p l i f y i n g s l i g h t l y , t h i s i s how i t works. Supnose a p o p u l a t i o n o f
must e l e c t 100 r e n r e s e n t a t i v e s .
~~~~~~
c a s t and n a r t i e s A, B and C r e c e i v e r e s p e c t i v e l y 2920,
The vote i s
95220 and 1860 votes.
The f i r s t stage i s t o e s t a b l i s h each n a r t y ' s quota and t o g i v e o u t " d i r e c t " seats (whole numbers contained i n quotas) ( f i g . l a ) . IJhich p a r t y ( i e s ) should g e t the
This way, 98 seats have been assigned. two remaining seats?
I n a way t o t h e ? a r t y ( i e s ) " o f f e r i n g the most".
By
t h i s , we mean the f o l l o w i n q : f i c t i v e l y , one e x t r a delegate i s added t o each p a r t y ' s t o t a l and, f o r each Darty, we compute the"va1ue" o f a delegate (numb e r o f voters p e r d e l e g a t e ) .
The n a r t y whose r e n r e s e n t a t i v e s are worth the
R (fig. lb).
most v o t e r s receives the f i r s t e x t r a seat; here, i t i s n a r t y One more s e a t has n o t y e t been assioned.
The same system i s used and i t i s
aqain B who receives i t , and thus finds i t s e l f w i t h 97 seats, the s u r p r i s i n g prediction ( f i g . l c ) . PARTY -
VOTES -
QUOTA -
DIRECT SEATS
A
29 20
2.92
2
B
95220
95.22
95
1860
C
-1
1.86
100000
100
98
Figure l a
PARTY -
DIRECT SEATS t 1
A
3
B
96
C
2
"VALUE" OF EACH DELEGATE 973.3
m 930 Figure l b
SITUATION 1 2 96
-1 99
C. De Meur, hf. Cmnrer o r d X. Hubmi
368
PARTY -
(SITUATION 1) + 1
3
"VALUE" OF EFCH CELEGATE
F! NPL REPRECENTATION
973.3
p-7.q
q7 2
930
Fiqure l c Usinc, one o f t h e f i r s t methods used i n t h e U n i t e d S t a t e s (naned a f t e r i t s c r e a t o r , Alexander H a m i l t o n ) , t h e t v q e o f s i t u a t i o n d e s c r i h e d i n n o i n t ( i i ) car, c c c u r .
Hamilton's rethod consists i v rcun''iq
.
'7
u p i t s ' quotas, o i -
v i n q t c each u n i t t h e whole number c o n t a i n e d i n i t s n u o t a and then a s s i q n i n q any r e m a i n i n q s e a t s t o those h a v i n q t h e l a r g e s t remainders. A s i m o l e examnle o f t h e s i t u a t i o n can he found i n f i w r e 2.
-
HFt'ILTnFI'S '1ETHPD
17 SEPTS P.VAILABLE
t'
t
PWVLPTION
nUOTP.
2 75
4.6C
5
DELFGPTES
POPULATIPN 304
nIlOTA 4.35
DELEWTES 5
177
3.01
3
190
3.09
3
333
6.6E
7
395 (1.)
6.44
6
155 -
2.64
2 -
154 ( + )
P.51
17
1043
1000
(+I
3 (4) 17
Figure 2
3. THE DUTCH FRnFLFM '.lhat can we hone t o hannen i f t h e p r o b l e m i n v o l v e s more t h a n one p o o u l a t i o n divisiov critericn?
P t w o - c r i t e r i a hased s i t u a t i o n c u r r e n t l y e x i s t i n ?
in
t h e Netherlands i s t h e m o t i v a t i o n o f t h i s s t u d y . Voters o f a r e a i o n a r e c h a r a c t e r i z e d by t h e commune i n w h i c h t h e y l i v e and b y the n a r t y o f t h e i r c h o i c e . The f i r s t s t e p o f t h e e l e c t i o n i s f o r t h e p o o u l a t i o n t o v o t e f o r commune counci 1l o r s
.
The second s t e p i s f o r a l l c o u n c i l l o r s o f a r e g i o n t o f o r m a r e q i o n a l
369
Problem of representation based upon two criteria
c o u n c i l comDosed o f a number o f t h e i r members. The Putch law s t a t e s : "Each communal c o u n c i l assiqns one o u a r t e r o f i t s members t o t h e r e g i o n a l c o u n c i l , f r a c t i o n s of .5 o r more b e i n n rounded un, f r a c t i o n s under .5 b e i n q rounded down". The l e n i s l a t o r a l s o f o r e c a s t s t h a t an e f f o r t s h o u l d he made so t h e n o l i t i c a l c o m p o s i t i o n o f t h e r e g i o n a l c o u n c i l s h o u l d match t h a t o f t h e r e 9 i o n ' s t o m u n a l counci 1 s . The s i t u a t i o n can be m a t h e n a t i z e d as f o l l o w s : consider a m a t r i x i n commune C . .
i s t h e n u r b e r of o a r t y P.'s c o u n c i l l o r s
J
The r e q i o n a l c o u n c i l i s d e s c r i b e d b y a m a t r i x I?,
1
frcm
where element sij
"derived"
i n t h e sense t h a t elernent rij i s always s m a l l e r o r eoual t o t h e c o r -
r e s o o n d i n g element s
ij'
T h i s m a t r i x R must a l s o r e s o e c t i m n e r a t i v e s o f po-
l i t i c a l and r e q i o n a l balance.
1) The sum ri
To be more p r e c i s e :
o f t h e elements o f l i n e i must c o i n c i d e w i t h t h e number o f
r e g i o n a l delegates assianed t o commune Ci,
as e s t a b l i s h e d by t h e l e g a l r u l e
mentioned above.
2) The sum r
.j
o f t h e elements o f column j must be equal t o t h e number c ?
r e n r e s e n t a t i v e s assigned t o n a r t y
P.
J
as determined b y H a m i l t o n ' s method
( f i n . 3a and 3 b ) . How can one e s t a b l i s h t h i s m a t r i x R? The f i r s t i d e a e n t e r i n g i n mind would be t o t r e a t each commune s e p a r a t e l y , as i f i t was t h e o n l y one, assicjning t h e r e p r e s e n t a t i v e s by H a m i l t o n ' s method.
One c a l l s t h i s t h e i n i t i a l s i t u a t i o n ( f i q . 3 c ) .
By means o f c o n s t r u c -
t i o n , each commune r e c e i v e s it s f a i r share o f r e n r e s e n t a t i ves
.
Nevertheless
t h i s o m i t s t o t a k e q l o b a l " o a r t y b a l a n c e " i n t o c o n s i d e r a t i o n because n o t h i n o i n t ? e r o n s t r u c t i o n guarantees t h a t t h e t o t a l number o f d e l e g a t e s - p e r - p a r t y s h o u l d be e x a c t l y t h e one e s t a b l i s h e d beforehand b y H a m i l t o n ' s method. i t i s n o s s i b l e t h a t these numbers c o i n c i d e .
i s t h e m a t r i x R.
Yet,
I n t h a t case, t h e i n i t i a l m a t r i x
I n o e n e r a l , however, these do n o t c o i n c i d e .
How can t h e
i n i t i a l m a t r i x be a l t e r e d i n o r d e r t o o b t a i n one t h a t s a t i s f i e s t h e two concii t i o n s simultaneously? Because t h e t o t a l number o f r e p r e s e n t a t i v e s b y commune i s p r e d e t e r m i n a t e d , t r a n s f e r r i n g s e a t s can o n l y be a u t h o r i z e d between p a r t i e s w i t h i n t h e same commune.
O b v i o u s l y i t i s b e t t e r t o s t a y as c l o s e as D o s s i b l e t o t h e i n i t i a l
370
G. De Meur, M. Gllcnierrrnd X . Hubaut
setting comnuna 1 P.
J
i n i t i a1
5*l...s .
.J‘
..s
.m
JT -! y i j
F i g u r e 3a
xi;
Regi ona 1
= r5ij 1
/
’i ri,
=!-I
j c’ r. 1
Ham
F i g u r e 3c
.5
“quota” P
.j
F i g u r e 3b I n c e r t a i n cases, t h e a l t e r a t i o n s can he p e r f o r m e d w i t h o u t t o o much ambiquit,y as shown by t h e example o f f i g u r e 4.
37 1
Problem ofrepresentation bared upon two critenk
communal
c c c c
1 2 3 4
p1
p2
p3
5 0 7 4
2 1 2 0 3
3 1 0
p4 9 5 3
0
3
7
8
11
C 5 0 x+ 16
3
19 7 12 7 13
5 2 3 2 3
23
58
15
initial
\ . c 2 1 0
J
reg i on a 1
1 0 2 1
1 0 0 0
1 0 0 0
2 2 1 1
4
i
l
l
2
5 2 3 2 3
o 0 0 1
o 0 0 1
2 1 1 1
2
2
7 1 5
5 2 3 2
O 1 r n P I 3 4
2
-3
-6 '
15
Figure 4 The apportionment method seems to be well established now. e v e r , sow unsolvable cases, as can be seen in f i g . 5 .
There a r e , how-
The i n i t i a l s e t t i n g gives two s e a t s t o party P1 and only one t o Darty P2. Global p o l i t i c a l balance, however, imposes two t o P2 and one t o P1. P1 should give up one s e a t t o Darty P 2 , which i s c l e a r l y impossible.
372
C.De Meur, M. Casrm-r and X. Hubaui
lk o e n o r a l i z e d t h e n r o h l e m usincl d i v i s o r F i n s t e a d o f 4 and p r o v e d t h a t t h e only s i t u a t i o n -
a v o i d i n c l i m n o s s i h i l i t i e s i s one where t h e r e a r e o n l y two
n a r t i e s and two communes. I'e t r i e d t o see i f t h e i r n n o s s i h i l i t i e s i n t h e o t h e r cases w r e due t o t h e n a r t i c u l a r i t y o f t h e system used i n H o l l a n d .
Thev a m e a r e d t o he more f u n -
damental.
S = Communal
c3
O
2 / /
/
R = regional
C1 c2
c3
?
?
Fiaure 5
1
1
313
Problem of representation based upon two criteria
4. CRITICISfl O F
TPF
SYSTEM
One c o u l d t h i n k t h a t t h e i m n o s s i b i l i t v d i s c o v e r e d h e r e i s t h e d i r e c t r e s u l t o f t h e method used i n r o u n d i n ? t h e c a l i t i c a l quotas ( H a m i l t o n ' s method).
Indeed, why choose t o me a method which has been n r w e n t c r a i s e v a r i o u s naradoxes? Furthermore, b y a c c e n t i n g t r a n s f e r s o f s e a t s from p a r t y t o n a r t y i n the same commune, t h e d i s t r i b u t i o n r e s u l t i n q from t h e method used i n t h e f i r s t Dlace can be d e s t r o y e d .
% i n n back t o t h e h u i l d i n q o f t h e m a t r i x R , we asked o u r -
selves : a ) which o f i t s c h a r a c t e r i s t i c s are oredetermined? b ) which requirements (as weak as n o s s i b l e ) s h o u l d he q r e i n o o s e d uoon t h e elements o f R? Answering t h e f i r s t q u e s t i o n , i t i s ohvious
1) t h a t t h e sum o f t h e elements o f each l i n e can be c o n s i d e r e d n r e d e t e r m i ned: furthermore, these sums a r e n o t determined a r h i t r a r i l y : t h e o r d e r creat e d hy t h e number o f communal c o u n c i l l o r s o f each commune n u s t he respected. P l s o , t h e c ' i f f e r e n c e between t h e numbers o f r e g i o n a l delegates cannot e x ceed t h e d i f f e r e n c e between the numbers o f t h e i r communal c o u n c i l l o r s ( e x c e n t i n case o f a t i e ) . Symhol ic a l l y : Vi,j
si.<s.
j.
*
ri. "j.
and si.-si.
3 rj.-ri.
.
2 ) t h a t t h e sum o f t h e e l e n e n t s o f cacb column o f R i s nredetermined, t h i s t i m e b y some aoportionment n e t h o d ( n o t n e c e s s a r i l y H a m i l t o n ' s ) . Ire must have: Vi,j
s . I. < s . i * r .i ,< r . j and s . ~ - s.i 3 rei-rai
i n o r d e r t o r e s n e c t o l o b a l p o l i t i c a l balance. Concerning t h e requirements t o he imnosed uoon R ' s elements, t h e f o l l o w i n g c o n d i t i o n seems n a t u r a l : V i ,,j,kyl
sij
< skl
* r1.j ..
' 'kl
P. i n commune C J s h o u l d send a t l e a s t as many d e l e a a t e s as P . t o t h e r e g i o n a l c o u n c i l . The J Drevious c o n d i t i o n , however, seems t o be v e r y s t r o n g and Derhans a r t i f i c i a P a r t y P1 i n commune C k y h a v i n a more c o u n c i l l o r s than p a r t y
i n t h e sense t h a t i t i s based uoon a comparison between a b s o l u t e numbers
,
374
G. De Meur. M. Gawner a
d
X. Hubmi
w h i l e t h e oroblem f u n d a r e n t a l l y denands a r e l a t i v e h a l a n c e o f those numbers based unon b o t h t h e communes and t h e n a r t i e s .
L e t us r e o l a c e i t by t h e f o l -
lotwin? weaker and m r e n a t u r a l c o n d i t i o n Vi,j,k
s 1.J . < sik
-
rij 5 rik and s . -s
2 r. - r . .
ik i j i k 1.1 r e q u i r i n o t h a t t h e o r d e r c r e a t e d hv t h e number o f covmunnl c o u n c i l l o r s Der nart::
w i t h i n t h e same commune w i l l n o t he r e v e r s e d i n t h e r e g i o n a l c o u n c i l .
P,v n o t r e s n e c t i n q a s i m i l a r c o n d i t i o n on t h e columns o f I?, we a r e l e s s i n c o n t r a d i c t i o n w t h common n o l i t i c a l m o r a l s thar! hv v i o l a t i n ? t h e above ment i oned cnndi t i on iwnosed unon t h e communes.
l h e x n e c t e d l y t h e e x a n n l e , shown i n f i w r e 5 , does n o t s a t i s f y these weak demands. "hich c o n l u s i o n s can be d r a w ? rossjhl!!
t h i s examyle mav be t o o n a r t i c u l a r :
I " ) i t i s n o t 'connected' i n the sense t h a t t h e s e t o f communes can be senarat e d i n t o two qroups IC1,C2) and {C31 i n such a way t h a t n o n a r t y has a s e a t i n b o t h grouqs a t t h e same t i m e . 2 " ) t h e oresence o f zeroes, e s o e c i a l l y w h i l e d e a l i n g w i t h votes seems v e r y
imorobah 1e
.
These r i n h t f u l o b , j e c t i o n s can be oonosed by t h e f o l l o w i n g examoles ( f i o . 6 ) . An easy, h u t t e d i o u s , v e r i f i c a t i a n w i l l a l l o w us t o show t h a t t h e e s t a b l i s h ment o f m a t r i x F! i s alwavs f e a s i b l e i n t h e e l e m e n t a r y case o f two comnunes and two n a r t i e s , and t h a t these a r e t h u s t h e o n l y f e a s i b l e cases.
To end, i t i s f i n a l l y n o t so s u r p r i s i n g t h a t e v e r y t h i n n works b e t t e r when each c r i t e r i o n o n l y p r e s e n t s t v o a l t e r n a t i v e s , as c o u l d have been sugaested by A r r o w ' s famous theorem and a l s o by n o l i t i c a l r e a l i t y , t h e l a s t round o f
many e l e c t i o n s o f t e n b e i n q h e l d between two onponents.
375
Problem ofrepresentation based upon two criteria
A = 1 and B = 0 C = 0 and D = 1
E = 0 and F = 1
which leads t o a c o n t r a d i c t i o n s i n c e A+C+E = 1 and B+D+F = 2 .
( 2 , 3 ) EXAIIPLE
o r 1
0
1
o r 2
0
0
with
C 5 A 6 B
and
E s D s F
=z
B = l a n d A = C = O F = 1 and D = E = 0
which leads t o a c o n t r a d i c t i o n s i n c e
A+D = 0 and B+E = C+F = 1. Fioure 6 REFERENCES
[11 '
[2]
h t b o n i s s e , J.M.,
Ceoprafische en p o l i t i e k e s a m n s t e l l i n g van de Ge-
westraad, t o appear, Dresented i n Brussels (1961). B a l i n s k i , M.L., and Young, H.P., F a i r r e o r e s e n t a t i o n meeting the i d e a l
o f one man, one vote, Yale U n i v e r s i t y Press, New Haven and London (1982). [3]
Code E l e c t o r a l , Moniteur du 19/8/1928 e t h n i t e u r du 28/4/1929.
This Page Intentionally Left Blank
TRENDS IN MATHEMATICAL PSYCHOLOGY E. Defreef and J. Van Bu enhaut (editors) 0 Elsevier Science Publisgrs B. V. (North-Holland), 1984
377
TREE REPRESENTATIOPS OF RECTANGULAR PROXIVITY P'P-TRICES
C e e r t De Soete L n i v e r s i t y o f Shent, Zslcjim I,!ayne S. DeSarbo, Ceorce !J. Furnas. and J . Pouqlas C a r r o l l Be1 1 L a b o r a t o r i e s , Vurray H i 11
P, l e a s t - s q u a r e s a l g o r i t h m f o r f i t t i n g u l t r a m e t r i c and n a t h l e n g t h o r a d d i t i v e t r e e s t o two-way, p r o x i m i t y data i s nresented.
bro-mode
The a l g o r i t h m u t i l i z e s
a nenal ty f u n c t i o n t o e n f o r c e t h e u l t r a m e t r i c inequal i t y g e n e r a l i z e d f o r asymmetric, and g e n e r a l l v r e c t a n g u l a r ( r a t h e r than sauare) p r o x i m i t y m a t r i c e s i n e s t i m a t i n g an u l t r a m e t r i c t r e e .
T h i s s t a o e i s used i n an
a1 t e r n a t i n ? l e a s t - s o u a r e s f a s h i o n w i t h c l o s e d - f o r m formulas f o r e s t i m a t i n g p a t h l e n g t h c o n s t a n t s f o r der i v i n g path length trees.
1. INTRODUCTION One o f t e n encounters two-way,
two-mode n x m asymmetric r e c t a n o u l a r m a t r i c e s
( A = 1 6 . .I) o f p r o x i m i t y d a t a i n t h e b e h a v i o r a l (and o t h e r ) s c i e n c e s .
They
4J
i n d i c a t e t h e r e l a t i o n s h i n s between two d i f f e r e n t c l a s s e s o f e n t i t i e s ( e . g . , o b j e c t s and v a r i a b l e s , s u b j e c t s and s t i m u l i
, stimuli
and resnonses).
The a n a l y s i s o f such d a t a o f t e n seeks u n d e r s t a n d i n g o f t h e s t r u c t u r e c f t h e s e s e t s o f o b j e c t s and t h e i r i n t e r - r e l a t i o n s h i p s .
F o r examnle, p r o f i l e
d a t a o f a s t i m u l u s by s c a l e t y n e a r e o f t e n f o r m d when one e v a l u a t e s o r r a t e s a number o f d i f f e r e n t s t i m u l u s o b j e c t s on v a r i o u s a t t r i b u t e s c a l e s , e.g.
a number o f d i f f e r e n t s o f t d r i n k s a r e r a t e d on v a r i o u s p r e s o e c i f i e d
scales r e l a t i n g t o taste, l e v e l o f carbonation, etc.
I f these r a t i n g s a r e
i n t e r p r e t e d as judaments o f closeness t o an i d e a l exemolar o f t h e a t t r i b u t e b e i n g r a t e d , t h e r e s u l t i n g m a t r i x can be viewed as a r e c t a n g u l a r p r o x i m i t y matrix. Another t y p i c a l example o f such r e c t a n g u l a r d a t a occurs where one c o l l e c t s s u b j e c t x s t i m u l u s data. F o r i n s t a n c e , v a r i o u s s u b j e c t s r e n d e r p r e f e r e n c e j u d g w n t s o v e r a number o f d i f f e r e n t s t i m u l i (e.9. s u b j e c t oreference
378
G . De Soetr
et
01.
r a t i n g s F o r v a r i o u s brands of t e l e p h o n e s ) . S t i l l another t y p i c a l t y q e o f nonsymmetric d a t a encountered i n t h e behavio-
r a l s c i e n c e s a r e conCusicns d a t a where
H =
m (so t h a t
A
row and column e l e n w t s corresnond t o t h e same o b j e c t s .
i s snuare) and t h e Yet, 6 .
4j
i s not
n e c e s s a r i l y equal t o 6... F o r examnle, t h e row elements m i g h t be v a r i o u s t4
messanes s e n t and t h e column elements t h e v a r i o u s messages r e c e i v e d .
The
main d i a g o n a l elements w o u l d r e c o r d t h e accuracy o f t h e communication p r o cess, w h i l e t h e o f f - d i a n o n a l elements w o u l d r e f l e c t t h e e r r o r o f the p r o cess,
The famous t!i 1 l e r - H i c e l v (1955) d a t a on c o n f u s i o n s o f consonants
nhnneiws i n a v a r i e t v o f n o i s e c o n d i t i o n s p r o v i d e an e x c e l i e n t example o f such n r o x i m i t i e s .
A l s o , one c o u l d c o n s i d e r brand-swi t c h i n g m a t r i c e s such
as those o r e s e n t e d i n ?eSiirke (1982) where D r o h a b i l i t i e s o f s w i t c h i n g f r o m hrand/oroduct
!,.
i i n p e r i o d .t t o j i n o e r i o d ttl r e n r e s e n t t h e
Here t h e 6&
6
G
e n t r i e s r e o r e s e n t b r a n d l o y a l t y and t h e 6 . . ( i
r e f l e c t t h e degree o f b r a n d s w i t c h i n g .
4J
entries i n
# j ) entries
S i m i l a r t y p e s o f m a t r i c e s a r e found
i n t h e s o c i o l o g i c a l l i t e r a t u r e i n t h e f o r m o f s o c i a l i n t e r a c t i o n and m o b i l i ty
matrices.
I n summary, t h e r e a r e a number o f d i f f e r e n t tyoes o f r e c t a n g u l a r d a t a found across many d i v e r s e d i s c i D l i n e s .
We nropose h e r e a new methodology f o r
f i t t i n g u l t r a m e t r i c and a d d i t i v e t r e e s t r u c t u r e s t o such d a t a . w i t h a r e v i e w o f t h e r e l a t e d l i t e r a t u r e i n two-way,
'de b e s i n
two-mode c l u s t e r i n g .
The new nronosed model and a l o o r i t h m a r e t h e n d e s c r i b e d .
An a p o l i c a t i o n o f
f i t t i n ? an u l t r a r e t r i c t r e e t o b r a n d s w i t c h i n g d a t a f o r v a r i o u s m a j o r brands o f s o f t drinks i s presented.
F i n a l l y , f u t u r e research o p n o r t u n i t i e s i n
t h i s area are described.
2. PRFVIOUS
LlOQK
'I! T''fl-''?PE
CLUSTERING
There e x i s t a number o f q u i t e d i f f e r e n t aporoaches t o c l u s t e r i n q b o t h row and column elements i n a two-way,
two-mode r e c t a n a u l a r d a t a m a t r i x .
Tryon
and B a i l e y (1970) d e s c r i b e a h e u r i s t i c f o r examining b o t h v a r i a b l e s and i n dividuals i n a clusterina setting.
Here,a " V - a n a l y s i s " ,
o r "clustering"
o f t h e v a r i a h l e s (columns), i s p e r f o r m e d i n i t i a l l y v i a a f a c t o r a n a l y s i s on t h e v a r i a b l e by v a r i a b l e c o r r e l a t i o n m a t r i x o r on a s u b s e t o f t h e most c o l l i n e a r variables.
Then, an " 0 - a n a l y s i s "
o f the i n d i v i d u a l s i s performed
where s u b j e c t s a r e i n i t i a l l y s c o r e d on t h e s e v e r a l c l u s t e r s / f a c t o r s o b t a i ned i n t h e " V - a n a l y s i s " and a c l u s t e r i n g o f t h e sub.iects i s t h e n performed.
3 79
Tree representations of r e c t a n p l a r p r o x i m i t y matrices
One f i n a l l y o b t a i n s a reduced "snace" f o r v a r i a b l e s and one f o r s u b j e c t s based on t h e i r scores from t h e " V - a n a l y s i s " . n r o f i l e scores o f t h e v a r i o u s c l u s t e r s .
The u s e r can then examine
Thus, t h e Tryon and B a i l e y (1970)
procedure does n o t r e a l l v n r o v i d e a j o i n t c l u s t e r i n c r o f s u b j e c t s and v a r i a b l e s , b u t r a t h e r a c l u s t e r i n g o f s u b j e c t s whose r e s u l t a n t c l u s t e r s v a r y cont i n u o u s l y on a s e t o f f a c t o r scores. H a r t i a a n (1975, 1976) and H a r t i o a n and Engelman (1983) develop a b l o c k c l u s t e r i n g method f o r c a t e g o r i c a l r e c t a n g u l a r d a t a .
Each b l o c k i s d e f i n e d t-y
a c l u s t e r o f cases and a c l u s t e r o f v a r i a b l e s such t h a t each v a r i a b l e i n t h e b l o c k i s c o n s t a n t o v e r t h e cases i n t h e b l o c k , e x c e o t f o r cases t h a t a l s o b e l o n g t o o t h e r b l o c k s ( b l o c k s may o v e r l a p ) .
The c o n s t a n t v a l u e taken
by a v a r i a b l e i n a b l o c k i s c a l l e d t h e "modal v a l u e " f o r t h a t b l o c k .
The
goal o f t h e a n a l y s i s i s t o n r o v i d e a s u c c i n c t r e n r e s e n t a t i o n o f t h e d a t a b y a few l a r g e b l o c k s w i t h c o r r e s o o n d i n g b l o c k modal v a l u e , t o q e t h e r w i t h r e s i d u a l s i n g l e b l o c k s c o n s i s t i n g o f s i n g l e values d e v i a t i n g from t h e appron r i a t e b l o c k modal value.
An i t e r a t i v e a l g o r i t h m i n H a r t i g a n (1975, 1976)
i s u t i l i z e d t o c l u s t e r b o t h o b j e c t s and v a r i a b l e s s i m u l t a n e o u s l y . VcCormi ck , Schwei t z e r , and LJhit e ( 1972) prooose t h e i r "bond energy a l g o r i t h m " t o p r o v i d e a j o i n t c l u s t e r i n g o f row and column elements i n r e c t a n n u l a r data.
T h i s i s accomolished by p e r m u t i n g t h e rous and columns o f an
i n n u t d a t a a r r a y i n such a way as t o nush t h e n u m e r i c a l l y l a y e r a r r a y e l e ments t o g e t h e r ( o r as t h e a u t h o r s mention, t o maximize t h e summed bond enerqy o v e r a l l row and column n e r m u t a t i o n s i n t h e i n n u t a r r a y
sure o f e f f e c t i v e n e s s " ) .
-
t h e " b e t a mea-
An i t e r a t i v e s e q u e n t i a l - s e l e c t i o n h e u r i s i t i c i s
u t i l i z e r to p r o v i d e an a t l e a s t l o c a l l y optimum s o l u t i o n . DeSarbo (1982) has develooed t h e GENNCLUS methodolooy f o r t h e simultaneous c l u s t e r i n g o f b o t h rcw and column elements.
He g e n e r a l i z e s t h e ADCLUS
(Shenard and A r a b i e ( 1 9 7 9 ) ) model, r e o r e s e n t i n g i n t e r s t i m u l u s n r o x i m i t i e s 2s
ccnibinations o f d i s c r e t e and p o s s i b l y o v e r l a p o i n q p r o n e r t i e s , t o t h e
case o f asymmetric o r o x i m i t i e s .
The GENNCLUS procedure a l l o w s f o r t h e e s t i -
mation o f e i t h e r overlaooing o r nonoverlapping c l u s t e r s .
It utilizes a
s e r i e s o f g r a d i e n t based procedures and c o m b i n a t o r i a l o p t i m i z a t i o n methods i n an a l t e r n a t i n g l e a s t - s q u a r e s framework.
S a r l e (Note 6) has r e c e n t l y de-
veloped a s i m i l a r procedure f o r t h e GENNCLUS "dual-domain'' case where sep a r a t e c l u s t e r s would be d e r i v e d f o r b o t h row and column o b j e c t s , and be subsequently r e l a t e d t o each o t h e r v i a an e s t i m a t e d w e i o h t s m a t r i x .
380
G. De Sorte ef a/.
:lone o f t h e above ITy?ntic,rcr' re?hcc's f o r j o i p t l v c l u s t e r i n g row and column e l e n e n t s i n r e c t a n p u l a r d a t a have d e a l t w i t h f i t t i n o t r e e q t r u c t u r e s ( u l t r + m e t r i c a n d / o r a d d i t i v e t r e e s ) t o such d a t a .
" h i l e such metliods ( c f . F a r t i -
rlan (19G7); C a r r o l l and Chang ( 1 9 7 3 ) : Cunninchar, ( 1 9 7 4 ) ; S a t t a t h and Tversky
( 1 ? 7 7 ) ; C a r r o l l an6 "ruzanskv (1975) ; C a r r o l l (1076) ; C a r r o l l and Pruzansk!! (:!LP):
r e 5oete. l ! o t e 2 ) have been developed f o r f i t t i n n such s t r u c t u r e s
t o t h e one-node symmetric q r o x i m i t i e s (see C a r r o l l , C l a r k , and DeEarbo, Yote 1, f o r t h e three-i.1a.y case), o n l v Furnas (Note 3 ) has i n v e s t i g a t e d t h e "tree-untnldin?" nroblem of f i t t i n o
J
t r e e t o rectanqular proximi t i e s , or@-
vic'inr! a j o i n t r e p r e s e n t a t i o n o f b o t h row and column o h , j e c t s .
He develons
an " u l t r a m e t r i c i n e q u a l i t y " c o n d i t i o n f o r such r e c t a n c u l a r d a t a ( h e a l s o c o n s i d e r s a d d i t i v e t r e e s ) and nronoses ar, a y l o r e r a t i v e c l u s t e r i n q n e t h o d t o f i t an u l t r a n e t r i c t r e e . l.!e s h a l l e x t e n d t h e v o r k o f Turnas (Note 3) t o accommodate b o t h u l t r a m e t r i c and a d d i t i v e t r e e s h y u t i l i z a t i o n o f a m a t h e m a t i c a l -rcic1rawing aooroach based on a n e n a l t y f u n c t i o n a l g o r i t h m .
The aooroach t o be d e s c r i b e d h e r e
i s aimed a t e x n l i c i t l y o p t i m i z i n o a l e a s t - s q u a r e s loss f u n c t i o n , whereas F u r n a s ' (Note 3) e a r l i e r an?roach was h e u r i s t i c i n n a t u r e and o n l y a p p r o x i mately least-squares.
The u n d e r l y i n o model and t h e o r y f o r u n f o l d i n o v i a u l t r a m e t r i c and Dath l e n g t h o r a d d i t i v e t r e e s was worked o u t by Furnas (Note 3 ) .
He der;ved
re-
cessary r n d s u f f i c i e n t c o n d i t i o n s , and uniqueness n r o n e r t i e s , f o r such r e o r e s e n t a t i o n s and d e v i s e d some d a t a a n a l y t i c methods.
Those methods were
c n l y h e u r i s t i c s ( a m l o m e r a t i v e a l o o r i t h m s f o r U l t r a m e t r i c t r e e s ; ad hoc methods f o r e r r o r - f r e e A d d i t i v e t r e e d a t a ) .
The nurnose o f t h e c u r r e n t work
i s t o d e v i s e and e v a l u a t e e x n l i c i t l e a s t squares methods f o r f i t t i n o these models, and q i v e examnles o f t h e i r u s e f u l n e s s . I n i t s s i m o l e s t form, usual n r o x i m i t y a n a l y s i s b e p i n s w i t h a s i n g l e s e t o f o b j e c t s and a square symmetric m a t r i x o f a l l t h e p a i r w i s e measuremen's between them.
The
Teal
i s t o f i n d a s i n q l e g l o b a l s t r u c t u r e (e.g.,
a spatial
c o n f i g u r a t i o n o r t r e e s t r u c t u r e ) r e o r e s e n t i n a a l l t h e o b j e c t s and, as f a i t h f u l l y as n o s s i b l e , t h e o r o x i m i t i e s between them. Unfolding analysis i s a s l i g h t v a r i a n t .
I t b e g i n s w i t h I14w s e n a r a t e c l a s s e s
Tree representotions of rectawplur proximity uiatrices
381
of objects (e.!., peonle a n d c a r s ) and w i t h the r e c t a n r u l a r p a t r i x of measurements only between n a i r s of opposite classes (e.?, hcw much each person l i k e s each c a r ) . I t s noal i s s t i l l the same - t o find a sinflle global renrescniation, Dlacino in i t both s c r t s of objects such t h a t the distances between them reoresent, as f a i t h f u l l y as oossible the oriflinal between-class c'ata. As in the usual one-class a n a l y s i s , the resultinrl s t r u c t u r e renres e n t s the d a t a w i t h fewer, honefully i n t e r p r e t a b l e o a r a w t e r s . An useful by-nroduct of an unfolr'iccl analysis i s t h a t the global s t r u c t u r e provides 5 reoresentation f o r the i m n l i c i t within-class Droximities, in addition t o the between-class relationshins i t e x n l i c i t l y t r i e d t o f i t . Thus f o r examnle i t becomes evident which c a r s a r e s i m i l a r , by v i r t u e of being liked b v the same neoole, and which neoole are s i m i l a r by v i r t u e of l i k i n p t i e same cars.
Unfolding theory has been well elaborated f o r multidimensional models, b u t there i s qood evidence t h a t the s i m i l a r i t y s t r u c t u r e of many f a m i l i a r domains i s better f i t b y t r e e s t r u c t u r e s (Sattzt! fi Tversky ( 1 9 7 7 ) ; C a r r p l l , Pruzansky R Tverskv (1982)). One would expect unfolding analyses in such domains t o require t r e e reoresentation as well ( s e e f o r examole F u r n a s , Note 3 ) . As background t o the work nresented here, we review the necessary and suf-
f i c i e n t cnnditicns f o r , a n d the r e s u l t i n g uniqueness of the two-mode t r e e reoresentations. I\ b r i e f s u m a r y i s Dresented here ( f o r the e r r o r l e s s case) - f u r t h e r d e t a i l s m a y he found in Furnas (Note 3 ) . Ultrametric distances ( i .e.,
distances associated rri t h u l t r a m e t r i c t r e e s )
must obey the Ultrametric Inequality:
In the case of rectangular distance matrices, however, i t i s not possible to test s a t i s f a c t i o n of the ultrametric i n e q u a l i t v since one of the three distances w i l l be missinp f o r every t r i p l e . Furnas showed t h a t , f o r rectanpular matrices with a distance measure defined only between items of d i f f e r e n t classes ( A , w i t h elements reDresented bv l e t t e r s e a r l y in the alohabet and 2 represented by l e t t e r s l a t e i n the alphabet) the following Two-Class Ultrametric condition i s necessary and s u f f i c i e n t f o r representat i o n as an ultrametric tree:
C . De Soere et al.
381
!.!hen t h i s i n e q u a l i t y i s s a t i s f i e d , the r e p r e s e n t a t i o n i s unique i n t e r n a l s t r u c t u r e o f o u r e l y one-class subtrees.
un t o the
For examnle, the three
trees i n Figure 1 are ecluivalent i n t h e i r two-class s t r u c t u r e .
m-m-m
a b c v d w x y z e
F i n u r e 1.
a b c v d w x y z e
a bcvdwxyze
Indeterminacies i n Tvro-Class Trees: I n t e r n a l ( t r u c t u r e o f he-Class Clusters.
For examole, the ( w d ) and ( z e ) suhtrees are w e l l determined, b u t the ( a h c ) and ( x u ) subtrees are n o t .
By d e f a u l t , such two-class t r e e s w i l l be shown
i n t h e i r l e a s t s t r u c t u r e d form
-
t h a t i s w i t h no s t r u c t u r e f o r the oure
subtrees and the fewest n o s s i b l e number o f e x t r a nodes, as i n the middle e x a n l e above.
Note t h a t t h i s indeterminacy i s u s u a l l y small, i n t h a t most
suhtrees have mixed membership, n a r t i c u l a r l y once thev a e t 3 bp o f reasonable s i z e .
I n the examnle above, o n l y f o u r o f the 45 distances i n the f i n a l
t r e e were n o t coomletely determined, and even those f o u r were s t r i c t l y bounded. Necessarv and s u f f i c i e n t c o n d i t i o n s f o r u n f o l d i n o n a t h l e n q t h o r a d d i t i v e trees are much more comolicated (Furnas, Note 3, Dresented a bounded determ i n i s t i c a l a o r i t h m f o r t h i s ) and w i l l n o t be qiven here.
I t suffices i n -
stead t o n o t e t h a t o v e r a l l distances i n a p a t h l e n g t h o r a d d i t i v e t r e e can be decomoosed i n t o the sum o f an u l t r a m e t r i c o a r t and an a d d i t i v e l y decomoosable o a r t (see C a r r o l l and Pruzansky (1980)).
The same i s t r u e f o r r e c -
t a n g u l a r submatrices o f distances from a path l e n o t h o r a d d i t i v e t r e e , and the a n a l y t i c t e c h n i o w s nrooosed here make use o f t h i s decomnosition. I n oath l e n g t h o r a d d i t i v e t r e e s , the unfolded r e n r e s e n t a t i o n s have two s o r t s o f non-uniqueness.
-
One i s e x a c t l y e q u i v a l e n t t o the u l t r a m e t r i c case
n u r e l y one-class subtrees have indeterminable i n t e r n a l s t r u c t u r e .
The
second i s a s i n g l e continuous f r e e narameter, s h i f t i n g r e l a t i v e lengths o f the t e r m i n a l branches o f the two classes.
By adding a constant t o a l l
Tree representations of rectangular proximity matrices
383
t & r r i n a l branches o f one c l a s s and subtracting i t frot? t ? e terlninal ':ranc4es o f tne o t n e r c l a s s , a l l oeween-class distances have a net clianqe of zero. :>us, For exapple, t w tnrce trees disol?yed in Fioure 2 are comoatible wit6 the same two-class matrix.
Figure 2.
Indeterminacy i n Two-Class Additive Trees: Adding a C o n s t a n t t o Terminal Qranches of one clFss a n d subtracting i t from the o t h e r .
A convention i s used in t h i s naner t h a t attemots t o halance the length of
the terminal branches of the two classes i n a Fanner to he described l a t e r . These indeterminacies require a degree of caution so as not t o "over-interw e t " the tree reoresentations r e s u l t i n g from an unfolding a p a l y s i s , just as the o r i e n t a t i o n of axes cannot be determined in a two-wav multidimensional s c a l i n g analysis. Ope i r n o r t a n t n o i n t t o note i s t h a t the approach discussed i n this paper makes a very strong assumotion about the data ( A ) - n a w l y t h a t of comoarab i l i t y of the data from row to row ( a n d / o r column t o column) of the data matrix. In the terminoloay coined by Cwmbs (1964). the matrix i s regarded as an uncondLtLond ( r a t h e r than a row o r column conditional) proximity matrix. 6. TEE /?l.rCt?ITHfJ
1. For estimating an Ultrametric Tree The algorithm consists of the following phases. a ) Transform the data matrix A i n t o a matrix T b e s t annroximating A in a least-squares sense where T s a t i s f i e s the two c l a s s ul trametric inequality:
f o r i , j = 1,... ,n ( i f j ) and k . L = 1,... ,m ( k f l ) . fin a l t e r n a t i v e statement of the two-class u l t r a m e t r i c i n e q u a l i t y isrbich can be shown t o be equivalent
t o e x n r e s s i o n ( 3 ) i s t h a t f o r ever\’ o u a d r u o l e o f ? r ’ n L c comnrised o f two from each c l a s s , t h e two l a r q e s t of t h e f o u r d e f i n e d d i s t a n c e s must be e q u a l , i . e . q i v e n t c t , t i h , t l l ,and -t.
rk
I
( t h e o n l y f o u r d i s t a n c e s amcnfi i and
i n c l a q s m e and / and l j n c l a s s two d e f i n e d hv t h e r e c t a n o u l a r n r o x i r i -
t v d a t a ) , the two l a r o e s t o f those f o u r d i s t a n c e s must be e n u a l .
The p r o -
blem czn he r e f o r m l a t e d a s t h a t o f s o l v i n a t h e o n t i m i z a t i o n nrohlem: n
m
n
subject. t o t h e c o n d i t i o n t h a t T s a t i s f i e s t h e t w o - c l a s s u l t r a m e t r i c ineouaTo do t h i s , an e x t e r i o r n e n a l t v f u n c t i o n armroach (Rao ( 1 9 7 2 ) ) i s
lity.
u t i l i z e d t o convert the constrained nroblem i n t o a s e r i e s of unconstrained ones.
with
The auqmented f u n c t i o n :
0, i s m i n i m i z e d f o r an i n c r e a s i n g sequence o f values o f
p
p,
where
t h e p e n a l t y D a r t o f exDression ( 5 ) P ( T ) , i s d e f i n e d as
i - 1 m k-1 X I z (uijkP i = 2 j=1 k = 2 1=1 n
P(T) =
Z
n
-
vijh4
and
The s o e c i f i c s t e p s o f t h e p e n a l t v f u n c t i o n a l g o r i t h m a r e g i v e n i n De Soete, DeSarbo, Furnas and C a r r o l l ( N o t e 4 ) .
t,) C o n s t r u c t a square (ntm) b y (ntm) m a t r i x P w h i c h s a t i s f i e s t h e o r d i n a r y one-class u l t r a m e t r i c i n e q u a l i t y . D = ( ( d ) ) , f o r a , b = 1, ...,n+m i s symab m e t r i c (dab = Aha) and i s d e f i n e d f o r a f b . Because o f symmetry, we need o n l y d e f i n e dab f o r a > b . D can be thouSlst of as t h e m a t r i x h a v i n g T as the n
x
m s u b m a t r i x c o n s i s t i n g o f t h e l a s t n rows and t h e f i r s t m columns.
The p r o b l e m i s t o fill i n t h e ( l o w e r h a l f o f t h e symmetric) m
x
m and n
x
n
385
Tree representations of rectangularproximity matrices
submatrices c o m p r i s i n g t h e f i r s t columns r e s p e c t i v e l y .
M
rows and columns and t h e l a s t n rows and
T h i s i s accomolished by use of t h e f o l l o w i n g cqua-
t i o n s (Furnas, Note 3 ) :
i f m+l i a 6 a+n
'(a-m) b
and 1 min fmax
rf
= l i = l...n nb
<
b 5 m
i f l s a s m
(-tia,tib)l
and 1 6 b d m
Ifnecessary, a p o s i t i v e c o n s t a n t i s added t o t h e dab so t h a t t h e y s a t i s f y the t r i a n g l e i n e q u a l i t v . c ) Using s t a n d a r d h i e r a r c h i c a l c l u s t e r i n g methods (see Johnson, (1967)), t h e u l t r a m e t r i c t r e e r e n r e s e n t a t i o n o f b o t h row and column elements i s obt a i n e d f r c n C.
2. F o r E s t i m a t i n g a Path Length o r A d d i t i v e Tree As discussed i n C a r r o l l (1?76), C a r r o l l and Pruzansky (1980). C a r r o l l , C l a r k and DeSarbo (Plcte 1) , based on t h e work o f F a r r i s (1972) and H a r t i g a n (1975). given
an u l t r a m e t r i c t r e e , i t i s n o s s i b l e t o c o n v e r t i t i n t o an a d d i t i v e
t r e e by adding a t r i v i a l " s t a r " o r "bush" t r e e ( i .e.,
an a d d i t i v e t r e e ha-
v i n g o n l y one i n t e r i o r node) t o i t . The a l q o r i t h m h e r e i s t h u s based on t h e f a c t t h a t any s e t o f o a t h l e n g t h t r e e d i s t a n c e s can be decomaosed i n t o a s e t o f u l t r a m e t r i c d i s t a n c e s p l u s a s e t o f a d d i t i v e c o n s t a n t s f o r each o f t h e row and column elements.
The numerical problem can be s t a t e d as:
n
m
0
s u b j e c t t o t h e c o n d i t i o n t h a t T s a t i s f i e s t h e two-class u l t r a m e t r i c inequa-
lity.
Once T i s e s t i m a t e d v i a t h e a l g o r i t h m d e s c r i b e d e a r l i e r , t h e a d d i t i -
ve c o n s t a n t s a r e e s t i m a t e d i n c l o s e d f o r m v i a : m
n
m
n
n
m
C.L)c Socte cr a / .
3 Hh
as p e n e r a l i z e d f r o m C a r r o l l and P w z a n s k y (1980).
t i o n o f t h e .i. and
t,: Gives
them b o t h z e r o means.
P!ote t h a t t h i s comoutaT h i s i s t h e e x a c t con-
v e n t i o n used f o r f i x i n q t h e i n d e t e r m i n a t e c o n s t a n t i n t h e r e l a t i v e l e n g t h s o f t h e t e r m i n a l branches o f t h e row and column oh,jects.
Pnce t h e
Xi and E k
con5tants a r e e s t i m a t e d , t h e a l g o r i t h m c y c l e s back t o t h e u l t r a m e t r i c t r e e e s t i m a t i o n phase i n e s t i m a t i n c l T Qiven fi, and 5,:. 4
This a l t e r n a t i n g l e a s t -
squares procedure (IJold, ( 1 9 6 6 ) ) continuer. c y c l i n s t.ack and f o r t h o v e r t h e -
t4.b vaand cb a r e ob-
se two m a j o r nhases u n t i l conver?ence i n t h e loss f u n c t i o n and/or l u c s i s reached.
Once t h e f i n a l e s t i m a t e s of T and t h e
tained, D i s reconstructed from
hi,
T u s i n a t h e method i n e x o r e s s i o n ( 9 ) , and
t h e a n n r o n r i a t e a d d i t i v e c o n s t a n t s a r e added. r e d frolr: t h i s t r e e (see Dobson ( 1 9 7 4 ) ) .
The a d d i t i v e t r e e i s recove-
T h i s can he done by s i m p l y conver-
t i n g t h e u l t r a m e t r i c t r e e i n t o an a d d i t i v e t r e e , b y d e f i n i n a t h e l e n g t h o f e v e r y branch t o be t h e d i f f e r e n c e i n h e i g h t v a l u e s o f t h e two nodes connect e d by t h a t branch ( t h u s d e f i n i n g t h e h e i g h t s o f t e r v i n a l nodes t o be z e r o ) , and then adding t h e c o n s t a n t s A; and Ek t o t h e l e n o t h s o f t h e " l e a v e s " o f
the t r e e ( t h e branches c o n n e c t i n g 'ck,e t e r r i n a l nodes t o t h e f i r s t n o n t e r m i n a l or i n t e r n a l n o d e ) .
Because o f t h e i n d e t e r m i n a c y mentioned e a r l i e r , how-
e v e r , an a d d i t i v e c o n s t a n t can be added t o t h e l e a v e s c o r r e s n o n d i n g t o t h e elements o f one c l a s s and t h a t same c o n s t a n t s u b t r a c t e d from those o f t h e other set.
F i n a l l y , i f d e s i r e d , t h e r o o t node o f t h e u l t r a m t r i c t r e e can
be removed s i n c e t h a t node i s redundant f o r a Dath l e n g t h o r a d d i t i v e t r e e , where t h e two branches i s s u i n q f r o m t h a t r o o t node (assuming t h e t r e e s t a r t s a t t h a t node), b e i n g r e n l a c e d b y a s i n g l e b r a n c h whose l e n g t h i s Recall t h a t a nath length o r a d d i t i v e
t h e sum o f t h e l e n g t h s o f t h e two.
t r e e i s , i n a fundamental sense, unrooted; a l s o t h a t t h e u l t r a m e t r i c t r e e c o r r e s n o n d i n g t o i t i s n o t unique
-
so t h a t t h e r o o t i s h i g h l y a r b i t r a r y ,
and c o u l d i n f a c t be o l a c e d between any two Podes o r even a t any node o f the additive tree.
However, f o r most purposes, i t i z c o n v e n i e n t t o r e p r e -
s e n t t h e n a t h l e n g t h t r e e as a h i e r a r c h i c a l t r e e , i m D l y i n g a r o o t .
While
t h e r o o t i s n o t unioue, t h e D r e s e n t f j t t i n g p r o c e d u r e w o u l d t e n d t o p l a c e i t a t the r o o t o f t h e "dominant" o r b e s t f i t t i n g u l t r a m e t r i c t r e e , which
seems i n t u i t i v e l y a reasonable c h o i c e .
T h e r e f o r e , i n manv cases, i t may
he d e s i r a b l e t o r e t a i n t h e r o o t node and t h e l e n g t h s o f i t s two branches, d e s o i t e t h e f a c t t h a t i t i s redundant.
387
?'Tee representotions of rectongular proximity matrices
4. APPLT5ATIOK
F,ass, Pessemier, and Lehmann (1972) conducted an e x o e r i m e n t w i t h 280 s t u dents and s e c r e t a r i e s .
These s u b j e c t s were r e a u i r e d t o s e l e c t a 12-ounce
can o f s o f t d r i n k f o u r days a week f o r t h r e e weeks from among t h e f o l l o w i n g brands: Coke, 7-Up. Tab, L i k e , Pepsi , S p r i t e , D i e t Peosi
, and
Fresca.
Table 1 o r e s e n t s t h e aSYI!Itn!?triC brand s w i t c h i n a m a t r i x f o r t h e two p e r i o d s measured. Pedal t +1
MaPcDdFms 612 ,186
,107 ,448
D10 M)5
nm .im ,160 n87 .m D87 .in .m nos ,114 093 226
M9 ,185 ~ 4 7 ,186 n53 n93
n33 n64
.IN
n55
5113
.I40
n99
Dl2
n36 0-46
WIO
nm
MO
nm
,152
239
n30
31s
M1
.157 ,116 .147
n93 ,107
wo 043 n75 329 093 ,107
,131
,103
m6
n37 D86 ,116
a 9
256 067
m
DeSarbo (1982) a p p l i e d h i s GENNCLUS Drocedure t o a n o r m a l i z e d v e r s i o n o f t h i s d a t a s e t and found t h r e e s e t s o f o v e r l a p o i n g c l u s t e r s f o r t h e row and column s o f t d r i n k s : a c o l a c l u s t e r , a d i e t d r i n k c l u s t e r , and
5
lemon-lire
c l us t e r . 13e a D f l l i e d o u r methodology t o f i t an u l t r a m e t r i c t r e e t o t h e d a t a i n Table
1 and t h e t r e e i s shown i n F i g u r e 3. The t r e e accounts f o r 87.6% o f t h e Note, i n F i g u r e 3, t h e u n d e r l i n e d s o f t d r i n k s r e p r e -
v a r i a n c e o f t h e data.
s e n t t h e row o b j e c t s and t h e n o n - u n d e r l i n e d s o f t d r i n k s r e n r e s e n t t h e column o b j e c t s .
One i m m e d i a t e l y n o t e s t h e d i f f e r e n c e s i n t h e h e i g h t s a t
w h i c h d i f f e r e n t row brands o f s o f t d r i n k s a r e j o i n e d w i t h t h e i r same respect i v e column c o u n t e r o a r t s .
F o r exanivle, Coket i s j o i n e d w i t h Cokez+l
at a
low h e i g h t i n t h e t r e e i n d i c a t i n g a o r o p e n s i t y of Deople i n t h e s t u d y t o r e p e a t t r i a l o f Coke
-
an i n d i c a t o r o f s t r o n g b r a n d l o y a l t y .
7-Up a l s o d i s p l a y s t r o n g b r a n d l o y a l t y i n a s i m i l a r manner.
Pepsi and However, some
s o f t d r i n k s (e.g. Tab and L i k e ) do n o t have such s t r o n g b r a n d l o y a l t y , as
C . I)e Soete e t al.
388
r-----l 1
7 C
m Im
F i g u r e 3.
13
U l t r a m e t r i c Tree Representation o f the S o f t D r i n k Brand S w i t c h i n g Data.
demonstrated by the l a r g e r h e i g h t a t which t h e i r row and column o b j e c t s j o i n i n r i g u r e 3, as w e l l as t h e i r r e l a t i v e l y small main diagonal elements
in Table 1. Thus, the h e i g h t a t which the AUme row and column o b j e c t j o i n i s i t r v m e l y r e l a t e d t o the brand l o y a l t y o r main diagonal elements o f Table 1. I n one c l u s t e r o f the t r e e , we f i n d Tabx+l,
C i e t Pepsif,
D i e t Pepsif+land
Tree represen totions of rectangular proximity matrices
close t o t h i s , L.ikef+l and Tab,.
389
These comprise a d i e t s o f t drink c l u s t e r .
I n another c l u s t e r , S o r i t e f t l , S p r i t e f , and close t o t h i s , 7-Ueftl 7-Unt, comorise a lemon-line c l u s t e r of s o f t drinks.
and
F i n a l l y , one finds LikeR, Pensif, PensiR+l in one group a n d Fresca f' Frescab+l, CokeR and Cokef+l in a proup next to i t . I t i s a mixed c l u s t e r w i t h non-diet colas and Frescaf, FrescaR+l a n d Likeb. Fresca i s placed here because of i t s s u b s t a n t i a l switching t o CokeRtl, Pensiftl, a nd i t s e l f . LikeA i s olaced here because of the high switchinp probability t o Pepsi
R+l*
Thus, the t r e e graphically rearesents the brand swi tchincj phenomena presented i n Table 1. Here, too, switching can be hough.!?!r categorized according t o non-diet colas, lemon lime drinks, and d i e t drinks. 5 . DISCUSSION
b!e have oresented a methodological description o f the model and algorithm which nrovi des an ul trametri c o r addi ti ve t r e e renresentati on f o r row and column o b j e c t s i n a two-way, two mode rectangular Droximity matrix. The example presented in the previous section i l l u s t r a t e s how the Drocedure can render i n s i g h t i n t o the s t r u c t u r e o f a rectangular nroximity matrix.
There are a number of possible extensions f o r t h i s procedure. One obvious extension i s to redesign the alaorithm(s) in order t o estin8t.e multiple t r e e s t r u c t u r e s as done i n Carroll and Pruzansky (1980) and C a r r o l l , Clark, and DeSarbo (Note 1 ) . This oronosed model can be formally expressed i n the u l t r a m e t r i c case as: A
*
T1 t T 2 t . . . t T R ,
(12)
In
where T1 through TR a r i s e from R d i f f e r e n t ultrarnetric t r e e s t r u c t u r e s . the case of additive t r e e s , expression ( 1 2 ) becomes: A s T, t T2 t . . . +TR t A ,
where A
=
(13)
((a..)) and aii = hi t c . from expression (11)). J
4f
S i m i l a r l y , a procedure can be generalized t o accomodate "hybrid models" where both d i s c r e t e t r e e s t r u c t u r e s and continuous s p a t i a l dimensions a r e estimated. Here, f o r the ultrarnetric case, one can express this as: A
p
T1
+ T2 +...+ TR
t H,
(14)
where H i s a proximity matrix qenerated from, say, an S-dimensional nons y m t r i c multidimensional s c a l i n g orocedure ( e . g . , Harshman, Note 5 ) . One merely has to add A from exoression (13) t o expression (14) t o express the hybrid model f o r the a d d i t i v e t r e e case. Another logical extension, a n d o w t f - a t i s c u r r e n t l y being oursued by the authors , i s t o generalize t h i s nrocedure to handle three-way rectangular proximi t i e s arrays where, as i n INDTREES ( C a r r o l l , Clark, and DeSarbo (Note l ) ) , a c o m n t r e e tooolocp i s estimated across s l i c e s of the a r r a y , allowing f o r d i f f e r e n t i a l branch lennths c r node h e i g h t s . REFERENCE NOTES Carroll , J . D . , Clark, L.A., and DeSarbo, 'J.S., The renresentation of three-way oroximities data by s i n g l e and multiole t r e e s t r u c t u r e mod e l s , Unpublished Paoer, Pel1 Laboratories, llurray H i l l , N.J. (1983). De Soete, G., A least-squares alqorithm f o r f i t t i n g a d d i t i v e t r e e s t o nroximity d a t a , Unnublished Paner, University of chent, Belgium (1982) i n Dress, Psychome t r i ka ( 1983) . Furnas, G.bJ. , Objects a n d their f e a t u r e s : T3e metric representation of two c l a s s d a t a , Vnnublished Doctoral D i s s e r t a t i o n , Stanford University (1980). De Soete, G., DeSarbo, I-I.2., Furnas, G . ! d . , and C a r r o l l , J.D.,
The e s t i -
mation o f u l t r a m e t r i c and nath length t r e e s from rectangular proximity data, Unnublished Paper, Fell Laboratories, Murray H i l l , N.J. (1983). Harshman, R , Models f o r analysis of asymmetrical r e l a t i o n s h i p s among
I! o b j e c t s o r s t i m u l i , Unnublished baDer, I h i v e r s i t y of Nestern Ontario, Canada (1978). S a r l e , Iof.S., personal communication (1982). REFERENCES Bass, F.M.,
121 [31
Pessimier, E . A . , and Lehmann, D . R . ,
An exnerimental study
of r e l a t i o n s h i n s between a t t i t u d e s , brand nreference, and choice, Behavioral Science, 17 (1972), 532-541. C a r r o l l , J.D. , S o a t i a l , non-sDatial a n d hybrid models f o r s c a l i n g , Psychometrika, 41 (1976), 439-463. C a r r o l l , J.D., an d Cham, J . J . , A method f o r f i t t i n g a c l a s s o f h i e r a p chical t r e e s t r u c t u r e models t o d i s s i m i l a r i t i e s d a t a , and i t s application t o some body Darts d a t a of M i l l e r ' s . I n Proceedings o f the 8 1 s t
391
Tree representatwns of rectangularproximity matrices
Pnnu?l Ccnvention o f t h e Am. Psych. Ass.
[41
C a r r o l l , J.D. and Pruzansky, S.,
, Vol.
8 ( 1 9 7 3 ) , 1097-1098.
F i t t i n g o f hierarchical tree struc-
t u r e (HTS) models, m i x t u r e s o f HTS models, and h y b r i d models, v i a mat h e m a t i c a l p r o y a m m i n p and a1 t e r n a t i n ? l e a s t squares. a t t h e U.S.-Japan
Paoer p r e s e n t e d
Seminar on F h l t i d i w n s i o n a l s c a l i n g , U n i v e r s i t y o f
C a l i f o r n i a a t San Dieoo, La J o l l a , C a l i f o r n i a , August 20-24 ( 1 9 7 5 ) .
[51
C a r r o l l , J.D. and Pruzanskv, S. i n E.D.
, Discrete
Lantermann and P . Feper (eds.),
and h y b r i d s c a l i n g models,
S i m i l a r i t y and choice, Bern,
Hans Huber (1980). Combs, C.H.,
A Theory c f Data, New York, b l i l e y (1964).
Cunningham, J.P.
, Free
t r e e s and b i d i r e c t i o n a l t r e e s as a r e p r e s e n t a -
t i o n o f p s y c h o l o o i c a l d i s t a n c e , J o u r n a l o f Mathematical Psychology,
17 (1978), 165-188.
181
GENNCLUF: New models f o r q e n e r a l n o n h i e r a r c h i c a l c l u s -
DeSarbo, W.S., t e r i n g analysis Dobson, A.G.,
, Psychometrika,
47 (1982) , 449-475.
Unrooted t r e e s f o r numerical taxonomy, J o u r n a l o f A o P l i e d
P r o b a b i l i t y , 11 ( 1 9 7 4 ) , 32-42. F a r r i s , J.S., E s t i m a t i n a p h y l o g e n e t i c t r e e s f r o m d i s t a n c e m a t r i c e s , American N a t u r a l i s t , 106 (1972) , 645-668. [111 H a r t i g a n , J.P., C l u s t e r i n c A l o o r i t h m s , New York, W i l e y ( 1 9 7 5 ) . [121 H a r t i g a n , J.A., lqodal b l o c k s i n d e f i n i t i o n o f west c o a s t mammals, S y s t e m a t i c Zoolooy, 25 (1976), 149-160. H a r t i g a n , J.A. and Engelman, L., B l o c k c l u s t e r i n g , i n t h e BMDP S t a t i s t i c a l S o f t w a r e tlanual
, revised
D r i n t i n g , U n i v . o f Cal. Press, B e r k e l e v
CA (1983).
H a r t i g a n , J.A.
, Renresentation
o f s i m i l a r i t y m a t r i c e s by t r e e s , J o u r -
n a l o f t h e American S t a t i s t i c a l A s s o c i a t i o n , 62 (1967)
, 1140-1158.
H i e r a r c h i c a l c l u s t e r i n g schemes, Psychometrika, 32
Johnson, S.C.,
(1967) , 241-254. McCormick, W.T.,
Schweizer, P.J.,
and White, T.Y.,
Problem decomposi-
t i o n and d a t a r e o r o a n i z a t i o n b y a c l u s t e r i n q technique, O p e r a t i o n s Research , 20 (1972) , 993-1009. M i l l e r , G.A.,
and N i c e l y , P.E.,
An a n a l y s i s o f p e r c e p t u a l c o n f u s i o n s
among some E n g l i s h consonants, J o u r n a l o f t h e A c o u s t i c a l S o c i e t y o f America, 27 (1955) , 338-352.
1181
Pruzansky, S . , Tversky, A.,
and C a r r o l l , J.D.,
S o a t i a l versus t r e e
r e p r e s e n t a t i o n s o f p r o x i m i t y data, Psychometrika, 47 (1982) , 3-24.
[191
Rao, S.S.,
(1979).
O p t i m i z a t i o n : Theory and a p p l i c a t i o n s , New York, GIiley
G . De Soere e t 01.
392
[?01
Sattath, S.,
1211
Shenard, R.N.,
4 2 (1977),
and Tversky, A . ,
A d d i t i v e s i m i l a r i t y t r e e s , Psychometrika,
319-345. P d d i t i v e c l u s t e r i n g : Renresentation o f
and A r a b i e , P.,
s i m i l a r i t i e s as c o m h i n a t i o n o f d i s c r e t e o v e r l a n n i n q p r o n e r t i e s , Psychol o o i c a l Review, 86 ( 1 9 7 9 ) , 87-123.
I221
Sonquist, J . P . , r l u l t i v a r i a t e
m t l r ~ ' ':Kjl(!ir,o:
the v a l i d a t i o n o f a
search s t r a t e c y , I n s t i t u t e f o r s o c i a l r e s e a r c h , U n i v e r s i t y o f M i c h i oan, Ann Arbor, t.lichi?an (1971).
[231
Tryon, R.C.,
1241
'.lold, H.,
and B a i l e y , D.E.,
C l u s t e r a n a l y s i s , New York, Mc Graw-Hi11
(1970). E s t i m a t i o n o f n r i n c i q a l comoonents and r e l a t e d models by
i t e r a t i v e l e a s t squares, I n P.R. K r i s h n a i a h ( e d . ) ,
V u l t i v a r i a t e Analv-
s i s , Kew York, bcademic Press (1966).
Address COrreSDondence t o : G e e r t De Soete k n a r t m e n t o f Psycholooy U n i v e r s i t y o f Ghent H e n r i Dunantlaan 2
8-9000 Ghent B e l q i um
6 . De Soete i s " A s p i r a n t " o f t h e B e l g i a n " N a t i o n a a l Fonds v o o r !.letenschapp e l i .jk Onderzoek"
.
1.1 .I;. DeSarbo, C.V
. Furnas , and
J .D. C a r r o l l a r e a1 1
Members of T e c h n i c a l S t a f f a t the B e l l L a b o r a t o r i e s i n h r r a v H i l l , New Jersey.
TRLh'DS IN MATHEMATICAL PSYCHOLOGY E. B g r e e f andJ Van Bufenhaut (editors) 0 Elsevier Science Pub& ers B.V. {Xxth-lfottand),1984
393
UEAK AND STRONG flflDELS IN ORDER TO DETECT ANL! MEASURE POVERTY Paul Dickes U n i v e r s i t y o f Nancy
The naner o r e s e n t s t h e a a n l i c a t i o n of c l u s t e r - a n a l y t i c methods (weak n o d e l ) and t h e l o @ i s t i c model o f Pasch ( s t r o n p m d e l ) on a l i s t o f d i c h o t o m i z e d i t e m s cover i n a the l i v i n ? Conditions o f f a m i l i e s w i t h c h i l d r e n i n f i v e d i f f e r e n t c o u n t r i e s o f t h e Euronean Community. Clus t e r - a n a l y t i c methods r e v e a l t h e nresence o f t h r e e t y n o l o a i c a l Frourrs: one o f them has n r a c t i c a l l y nc! disadvantanes.
The o t h e r two a r e marked hy vany d i s -
advantapes and c o n t a i n , on t h e one hand, sincrle-nar e n t f a m i l i e s , on t h e o t h e r hand, f a m i l i e s o f f o r e i g n origin.
The Rasch model shows t h a t a s u b s e t o f i t e m s ,
b e l o n g i n a t o d i v e r s e l i v i n g c o n d i t i o n s , a r e model-conform.
Thus t h e measurement o f p0vert.v t h r o u g h t h e
accumulation o f disadvantaqes has a sense.
The t y p o -
l o g i c a l prouos c h a r a c t e r i z e d by many disadvantanes a r e t o be c o n s i d e r e d as f a m i l i e s a t r i s k o f p o v e r t y .
1. IliTRPDlJCTInN The d i s t i n c t i o n between weak and s t r o n q models i n t h e methodoloqy o f s o c i a l s c i e n c e s hecare more obvious as f a r as l a t e n t t r a i t models ( L o r d and Novick
(1966)) and methods o f s c a l i n g (Van d e r Ven ( 1 9 8 0 ) ) were develooed. S t r o n n models c o n s i s t o f o r o D o s i t i o n s , d e r i v e d f r o m a o s t u l a t e s o r axioms, and l e a d i n g t o e m p i r i c a l o n e r a t i o n s t h r o u g h w h i c h assumntions can be f a l s i fied.
Weakmodels a r e those f o r which f a l s i f i c a b i l i t y i s reduced.
The c l a s -
s i c a l t e s t t h e o r y i s c o n s i d e r e d as weak, whereas t h e modern t e s t t h e o r y o r t h e t h e o r i e s o f l a t e n t t r a i t s a r e g e n e r a l l y r e c o g n i z e d as b e l o n g i n g t o s t r o n g models ( F i s c h e r (1974); K r i s t o f (1983)).
A second d i s t i n c t i o n , p a r t i a l l v l i n k e d w i t h t h e f o r m e r one, i s a t t a c h e d
394
P. Dickes
t o t h e d i f f e r e n c e hetween c o n f i r m a t o r y and e x D l o r a t o r y l e v e l o f a n a l y s i s . The c o n f i r m a t o r y avoroach, b e s t exemnl if i e d hv ,loreskoo ( 1 9 7 4 )
, a1 lows
the
a n a l y s t t o u t i l i z e t h e o r e t i c a l assumptions ahout d a t a f o r t h e s e l e c t i o n o f n a r a r e t e r s , and t o t e s t t h e f i t hetween t h e o r e t i c a l and e r n n i r i c a l r e s u l t s . The e x o l n r a t n r v v q r o a c h , l a r r e l y used i n t h e F u l t i v a r i a t e e x o l o i t a t i o n o f d a t a (such as t h e f a c t o r a n a l v s i s , t h e c l u s t e r a n a l v s i s ,
nression, e t c . ) , data.
the m u l t i o l e r e -
Fakes no d i r e c t assumntions a b o u t t h e s t r u c t u r e o f these
The a n a l y s t t r i e s t o reduce t h e c o r n l e x i t y and m u l t i n l i c i t y o f d a t a
t o n o r e common a s n e c t s . reduc ti on techn in ue
The e x o l n r a t o r y a n a l y s i s i s m a i n l y u w d as a data-
.
S t r o n n models and c o n f i r m a t o r v a n a l y s i s s a t i s f v F n r e t o e a i s t e m l o n i c a l cons i d e r a t i o n s than the a o n l i c a t i n n o f veak methods.
They h e l o n o t o t h e hypo-
t h e t i c o - d e d u c t i v e nrocess o f s c i e n t i f i c o r o o r e s s and a r e s i n o l e d o u t by t h e n o s s i h i l i t i e s o f f a l s i f i c a t i o n thev o f f e r
, which a r e c o n s i d e r e d c r u c i a l i n
t h e f o r r w l a t i o n and d e m o n s t r a t i o n o f s c i e n t i f i c laws ("onner (1968)).
'Ji th
them an a t t e m n t i s made t o reach t h e s t a n d a r e s o f f o r m a l i z e d sciences as a r e the n a t u r a l sciences. The i n t e r e s t o f e x o l n r a t o r v methods l i e s i n t h e develonment o f h y n o t h e t i c o i n d u c t i v e grocesses o f s c i e n t i f i c d i s c o v e r y .
C o n f r o n t e d w i t h a l a r g e amount
o f d a t a v i t h unknown s t r u c t u r e , t h e r e s e a r c h e r has t h e n o s s i h i l i t y , w i t h these methods, t o d e t e c t sow r e q u l a r i t i e s leadinyc t o bynotheses which then a r e t o be c o n f i r m e d . The o h j e c t i v e o f the n r e s e n t naner i s t o d e m n s t r a t e t h e c o m p l e m e n t a r i t y o f t h e two k i n d s o f anoroaches i n a s n e c i f i c f i e l d o f s o c i a l r e s e a r c h concerned w i t h t h e s t u d y o f n o v e r t y .
C l u s t e r a n a l y t i c - m e t h o d s and l a t e n t t r a i t
lrodelswere a p p l i e d t o a same d a t a m a t r i x , d e s c r i h e d f u r t h e r on.
The r e s u l t s
o f these two, r a t h e r a n t a o o n i s t i c annroaches, l e a d t o a b e t t e r u n d e r s t a n d i n o o f t h e ohenomena, and t h i s w o u l d n o t have been reached w i t h o u t t h e use of t h e two m e t h o d o l o a i c a l i n s t r u m e n t s . 2 . C0NPITIr)FIS OF THE PEqEARCH The f o l l o w i n g r e s e a r c h n r o j e c t b e l o n q e d t o t h e Euronean Program t o Combat P o v e r t y which took o l a c e i n t h e l a t e s e v e n t i e s (Schaher e t a l , (1980:). The objectives o f t h e r e s e a r c h were t o d e t e c t and s t u d y o e r s i s t e n t p o v e r t y
i n seven r e q i o n s o f f i v e ! k n h e r Y t a t e s (France, R e l q i u r , Germany, Nether1ands and Luxemburcr)
.
Weak and strong modelr
395
The i n v e s t i o a t e d o o n u l a t i o n c o n s i s t e d o f a l l f a m i l i e s who had a c h i l d b o r n i n 1970.
T h i s r a t h e r homoneneous grouo was chosen i n o r d e r t o s t u d y n e r -
s i s t e n c y o f n o v e r t y f r o m one g e n e r a t i o n t o t h e o t h e r . I n each r e g i o n 200 f a p i l i e s were randomly s e l e c t e d f r o m e x h a u s t i v e l i s t s o f onnulation.
The t o t a l samnle c o n s i s t e d o f 1400 f a m i l i e s .
were foreseen f o r t h e r e s e a r c h .
Two stapes
The f i r s t one s t r e t c h e d o v e r t h e f i r s t
months o f 1979: t h e 1400 f a m i l i e s were i n t e r v i e w e d w i t h a 20-30 m i n u t e s ' Q u e s t i n n n a i r e i n o r d e r t o n a i n a cleneral knowledge o f t h e i r l i v i n q c o n d i r.:
-.crs.
The nurpose o f t h i s r a t h e r i n s t r u m e n t a l stage was t o d i s c o v e r f a -
m i l i e s a t r i s k o f n o v e r t y and t o develor, i n s t r u m e n t s t o measure i t .
It
was a n r e n a r a t o r y s t a v e t o t h e second one, which s t a r t e d one y e a r l a t e r .
There, i n t e n s i v e i n t e r v i e w s were oerformed, t h e o r i c i n a l samnle b e i n g oversannled w i t h f a m i l i e s a t r i s k o f noverty.
The a n a l y s i s and t h e r e s u l t s we
n r e s e n t h e r e a r e based s o l e l v on t h e f i r s t - s t a v e d a t a . 3. RPTIONALE AND DATA 'YATQIX Income i s used as t h e n r i v i l e p e d c r i t e r i a i n saxon, t o measure v o v e r t v .
most researches, m a i n l y anplo-
4 l t h o u a h one can deduce o u t o f t h e income, e s -
o e c i a l l y i n w e s t e r n i n d u s t r i a l i z e d c o u n t r i e s , many asoects o f l i v i n g cond i t i o n s , i t a m e a r s as n a r t i c u l a r l y u n s u i t a b l e i n comnarative r e s e a r c h . Comoarisons between c o u n t r i e s w i t h d i v e r s e economical and n o l i t i c a l c o n d i t i o n s a r e q u i t e d i f f i c u l t and o f t e n i n a o p r o o r i a t e .
Even w i t h i n t h e same
r e g i o n o r c o u n t r y , i f income can be chosen as main p o v e r t y c r i t e r i a , i t s t i l l remains d i f f i c u l t t o s t u d y t h e a c t u a l e f f e c t o f income on p o v e r t y . The s t a r t o o s i t i o n o f t h i s r e s e a r c h i s a t a f i r s t onen and n o t c o n s t r a i n i n o . P o v e r t y concerns d i f f e r e n t dopains o f l i v i n g c o n d i t i o n s .
Thus we s e l e c t e d
86 v a r i a b l e s o u t o f t h e s c i e n t i f i c l i t e r a t u r e and/or o r a c t i c a l e x p e r i e n c e s w i t h p o o r oeoole, t h a t were c o n s i d e r e d as c h a r a c t e r i s t i c s o r c o r r e l a t e s o f poverty.
These v a r i a b l e s cover d i v e r s e domains o f t h e l i v i n g c o n d i t i o n s o f
p o p u l a t i o n s as f o r examnle: f a m i l y s t r u c t u r e , n a t i o n a l i t y , e t h n i c o r i g i n , j o b s i t u a t i o n , e d u c a t i o n a l background, economic s i t u a t i o n , h o u s i n g c o m d i t i e s , h e a l t h c o n d i t i o n s , c u l t u r a l and r e l a t i o n a l o r a c t i c e s .
Post variables
were s i t u a t i o n a l b u t some were s u b j e c t i v e as f o r examnle f e e l i n g s o f w e l l b e i n g , f e e l i n g s of d e o r i v a t i o n , e t c . The answers ic t h e o u e s t i o n s were dichotomized.
I n t h i s wav a l i s t o f 86
b i n a r y i t e m s a c t u a l i z e d t h e disadvantages o f each f a n i l y .
The f i n a l d a t a
396
P. Dicbes
m a t r i x c o n s i s t e d o f a v e c t o r o f C6 i t e m s and a v e c t o r o f 1400 f a m i l i e s (nersons)
.
The coded m w e r s t o each itern were e i t h e r :
1: i f t h e f a m i l y i s n a r k p d hv a o i v e n disadvantaoe,or 0: i f the f a m i l v i s n o t marked hv t h e d i s a d v a n t a a e .
Tile h y b r i d n a t u r e o f n e a s u r e r p n t l e v e l s d i r e c t e d o u r c h o i c e c f h i n a r v resnonse s t r u c t u r e .
%me o r i a i n a l v a r i a b l e s a r e n o p i n a l , o t h e r s c r d i n a l , o f
ertual i n t e r v a l o r even r a t i o n a l .
The t r a n s f o r m a t i o n o f h i o h e r l e v e l s c a l e s
t o a l o w e r l e v e l one has no d i r e c t m a t h e m a t i c a l l i m i t a t i n n , h u t t h e r e v e r s e i s not necessarily true. 4. TYPnLnGICAL INVTIFICATIfiN nF QPnUP? AT R I q K n F PnVEPTY The u t i liz a t i o n o f c l us t e r - a n a l y t i c nrocedures doesn ’ t r e o u i r e anv t h e o r e t i c a l assumotion a b o u t d a t a .
A l l t h e s u h j e c t s i n t h e sample a r e s i t u a t e d
i n a snace which has as vany dimensions as t h e r e a r e i t e m s .
O u t o f t h e r e l a t i o n s between t h e nersons i t i s o o s s i b l e t h a t c l u s t e r s e w r ne, c b a r a c t e r i z e d by t h e f a c t t h a t t h e s i m i l a r i t i e s w i t h i n t h e qrouos a r e o r e a t e r than t h e s i m i l a r i t i e s hetween t h e grouns.
The c l u s t e r - a n a l y t i c t e c h n i n l r s ? r e based on weak models i n t h e sense t h a t t h e y have n o a l a o r i t h m u s a b l e f o r d i s c a r d i n o i t e n s o r p e r s o n s .
The o b t a i n -
e d r e s u l t s a r e e x c l u s i v e l v deoendent on i t e m s and s u h j e c t s s u b m i t t e d t o andlysis.
The anproach i s an e x o l o r a t o r v one, i n t h e sense t h a t unknown s t r u c -
t u r e o r o r d e r have t o be d i s c o v e r e d and i n t e r n r e t e d .
IJe c o n s i d e r t h a t t h e c l u s t e r - a n a l v s i s reaches t h e e x p e c t a t i o n s i f : a)
t h e t y p o l o a i c a l a t t r i b u t i o n i s s t a b l e ( o r r e l i a h l e ) , and
h)
i f a t l e a s t one o f t b e i d e n t i f i e d grouns can be r e c o o n i z e d as marked
b y more disadvantaoes than t h e o t h e r ( s ) . The nurnose o f t h e a n a l y s i s i s t o i d e n t i f y nroun(s) a t r i s k o f n o v e r t v , w i t h o u t oresunnosino an e x n l i c i t d e f i n i t i o n o f what n o v e r t v o r n o v e r t i e s r e a l l y i s o r are.
‘ale w o u l d succeed i f t h e knowledqe we a a i n e d o u t o f t h e
s a v n l e c o u l d be a p n l i e d t o t h e n o p u l a t i o n .
I t w o u l d t h e n he o o s s i b l e t o
d e t e c t f a m i l i e s a t r i s k o f a o v e r t y and t o develop oversamDlinn schemes. The r e l i a b i l i t y o f t h e c l a s s i f i c a t i o n was e s t i v a t e d t h r o u a h t v o t y p e s o f procedures.
F i r s t , t h e o r i a i n a l samole was D a r t i t i o n e d i n t o odd and even
Subsamoles i n o r d e r t o develoo c r o s s v a l i d a t i o n c a l c u l a t i o n s .
Then, t h r e e
397
Weak and strorig models
d i f f e r e n t c l u s t e r - a n a l y t i c methods were a m l i e d t o each suhsamnle.
Thus
i n v a r i a n c e o f r e s u l t s was exnected o v e r s a r n l e s and m t h o d s . There e x i s t s a l a r p e range of c l u s t e r - a n a l y t i c methods, among which a r e s e a r c h e r has t o choose a m r o n r i a t e ones t o h i s s n e c i f i c nurnoses. r e s e a r c h o n l y h i e r a r c h i c a l c l u s t e r i n ? technioues were used.
For t h i s
They a r e b e s t
s u i t e d f o r l a r a e data m a t r i x and d o n ' t o r e c l u d e an!/ numher o f c l u s t e r s , as o t h e r techniques do. T h r r e c l u s t e r i n g - t e c h n i q u e s were used h e r e : t h e comnlete l i n k a g e method, t h e Uards method o f c l u s t e r - a n a l y s i s ,
and an a s s o c i a t i v e method.
The f i r s t b / o methods a r e a n r l c m r a t i v e and t b e l a s t one i s d i v i s i v e .
The
e u c l i d e a n d i s t a n c e was emnloyed as d i s s i m i l a r i t y c o e f f i c i e n t i n t h e a p c l o merative techniques.
The Chi-square W J S t h e d i s s i m i l a r i t y c o e f f i c i e n t o f
the d i v i s i v e technique. The a n a l y s i s was performed u n t i l 8 c l u s t e r s were o h t a i n e d . eight-clusters solutions,
To each o f t h e
r e a l l o c a t i o n t e c h n i n u e s , hased on e u c l i d e a n d i s -
The o n t i m a l number o f c l u s t e r s was reached when con-
tance, were a p o l i e d .
veraence was o b t a i n e d between t h e r e a l l o c a t e d c l u s t e r s s o l u t i o n s .
This
occured a t t h e l e v e l o f t h r e e - c l u s t e r s s o l u t i o n s f o r odd and even subsamnles o f subjects. C r o s s v a l i d a t i o n was nerformed i n t h e f o l l o w i n g ways:
-
f o r each subsample, t h e +nvo d i s c r i m i n a n t f u n c t i o n s were c a l c u l a t e d f o r the three c l u s t e r s
-
.
these e q u a t i o n s were a n n l i e d across t h e s a m l e s .
T h i s means t h a t t h e d i s -
c r i m i n a n t f u n c t i o n s o f t h e odd-subsample were a v n l i e d t o t h e even-suhsamo l e as w e l l as t h e d i s c r i m i n a n t f u n c t i o n s o f t h e even-subsamole t o t h e odd-subsample.
-
t h e o b t a i n e d f u n c t i o n s , oroduced v e r y s i g n i f i c a n t c m o n i c a l c o r r e l a t i o n s : odd-suhsample Rcan = 0.90 and 0.06 and even-subsamnle Rcan = 0.92 and
0.86. P r a c t i c a l l y 943: o f c o r r e c t c l a s s i f i c a t i o n s were o h t a i n e d . 84% and 87% o f t h e groups 2 and 3 were c o r r e c t l y c l a s s i f i e d . Thus, t h e groups appeared as l a r g e l y comoarable o v e r t h e samnles. The t h r e e groups can be d e s c r i b e d as f o l l o w s :
-
t h e f i r s t group c o n t a i n s 7 Y o f t h e t o t a l samole and i s c h a r a c t e r i z e d by t h e absence o f disadvantages.
-
t h e second group (19% o f t h e t o t a l sample) shows an accumulation o f d i s advantages i n many domains as f o r example: housing, h e a l t h , income.
398
P. Dickes
There t h e f a r i i l i e s a r e c o e n l e t e ( f a t h e r and mother n r e s e n t and p a r r i e d ) and m a i n l y o f f o r e i g n o r i n i n .
-
t h e t h i r d qrouo (6" o f t h e t o t a l samole) n r e s e n t s t h e q a w a c c u ~ r u l a t i o n o f disadvantanes as t h e second one.
Vowever, i n t h i s aroun t h e f a t h e r
i s seldom head o f t h e household, t h e main income c o w s f r o m s o c i a l t r a n s f e r o r s o c i a l a s s i s t a n c e : t h e f a m i l i e s a r e n a t i v e o f t h e r e a i o n s where they l i v e and a r e v a i n l y broken o r i n c o w o l e t e hnmes. Petween these t h r e e s t a b l e grouns, two o f them can he r e c o q n i z e d as b e i n o a t r i s k o f poverty.
I f t h e f o l l o w i n c . s t a t e m e n t i s t r u e : o o v e r t v aooears
when f a m i l i e s o w s e n t disadvantanes i n m u l t i D l e domains, t h e n t h e two orouns " a t r i s k " must i n c l u d e t h e " q o o r e s t " .
P u t t h e v mav a l s o i n c l u d e "non-noor",
which means t h a t these f a m i l i e s a r e a t r i s k o f w v f r t v , h u t t h a t t h e y d o n ' t manifest ooverty ( n o t y e t o r n o t l o n a e r ) . A t t h i s s t a a e o f t h e research, we d i d n ' t know i f t h e a c c u m u l a t i o n o f d i s -
advantaoes Dresented b y t h e arouns " a t r i s k " , had a s i n n l e and s o l e s i o n i fication.
A t t h e e x n e r i m e n t a l l e v e l , t h e o n l y n o s s i t r l e i n d i c a t i o n we had,
was t h e f o l l o w i n q : t h e two orouos " a t r i s k " had t o he more heteroqeneous t'ian t h e non-disadvantaged group, and 5sr' :c
c r e s e n t more disadvantages.
The t y p o l o q i c a l a n a l v s i s shows t h a t t h e l i f e s i t u a t i o n s o f b e i n q " a t r i s k
o f n o v e r t y " v a r y i n a q u a l i t a t i v e way.
Even i f we b a v e n ' t y e t v o r k e d o u t
a c l e a r and D r e c i s e d e f i n i t i o n o f n o v e r t y , o u r n r e - s c i e n t i f i c n o t i o n s have a l l o w e d us t o d e s c r i b e t y v i c s l n r e c a r i o u s f a m i l y s i t u a t i o n s . 5. OBJECTIVE MEbSllEVENT
pr
rCWPTY
I n the e x n l o r a i x r j s t a c e of d a t a a n a l y s i s , t h e a c c u m u l a t i o n o f d i s a d v a n t a oes has h e l o e d t o d i f f e r e n t i a t e rtrouns a t " r i s k " and a t " n o n - r i s k " o f poverty.
A more e l a b o r a t e h y n o t h e s i s about d a t a can now be r e a s o n a b l y f o r m a l i z e d : noverty i s the accumulation l i v i n q conditions.
of disadvantages i n d i f f e r e n t domains o f t h e
T h i s h y n o t h e s i s i s i n agreement w i t h t h e t r a d i t i o n a l
research, where D o v e r t y i s s o l e l y d e f i n e d i n income terms. s i d e r e d as a one arld a l o n e continuum.
P o v e r t y i s con-
The h y o o t h e s i s a b o u t unidimension-
a l i t y needs a s t r o n o nodel i n o r d e r t o be t e s t e d .
The c l a s s i c a l t e s t t h e o r y ( G u l l i k s e n , (1950)) assumes u n i d i m e n s i o n a l i t y o f the c o n s t r u c t s , b u t i s n a r t i c u l a r l y weak i n m o d e l - t e s t i n q ( N o v i c k (1966); F i s c h e r (1974), K r i s t o f (1983)).
399
Weak andstrong models
L a t e n t t r a i t models armear r u c l i more s a t i s f a c t o r y .
They n o s t u l a t e t h e u n i -
d i r e n s i o n a l i t y o f t h e c o n s t r u c t ( l a t e n t t r a i t ) , have e x n l i c i t narameters about i t e m s and nersons and l e a d t o n r o n o s i t i o n s f r o r which assumntions c o u l d be t e s t e d ( F i s c h e r ( 1 9 7 4 ) ) . Petween d i f f e r e n t l a t e n t t r a i t models (Lawley (1943); l m - d (1952), (1953): L o r d and Novi ck ( l O G G ) : F i s c h e r (1374)) t h e Qasch model (Rasch (1960),
(1968)) anqears as n a r t i c u l a r l y a n n r m r i a t e t o t h e nursued nurnose ( c u s t a f f son ( 1 9 7 7 ) ) .
He i s o h i e c t i v e , narsimonious ( o n e - n a r a w t e r model), e a s i l y
i n t e r o r e t a b l e , has s a t i s f a c t o r . meters and i s easil!!
alnorithms i n order t o estimate the nare-
testable.
The l o o i c o f t h e model and t h e nrocedures t o r e s o l v e t h e aarameters need r l o t to !e
nresented here.
S u f f i c i e n t l i t e r a t u r e i s a v a i l a b l e (Rasch (1966):
Fischer (1974)). The d a t a m a t r i x o f 86 i t e m s and 1400 f a m i l i e s c o n s t i t u t e d the s t a r t i n g user! a comouter orovram implemented b y G u s t a f f -
D o i n t o f the analysis. son (1377).
The e s t i m a t i o n 0ft.t.e ?arameters was o b t a i n e d t h r o u a h c o n d i t i o n -
a l 1ik e l i h o o d a1 g o r i thms
.
I n a f i r s t s t e o , v e r v s t r o n o c r i t e r i a were a n n l i e d i n o r d e r t o t e s t t h e f i t between model and d a t a . F i v e examoles were c o n s t r u c t r e d :
-
Samnle number 1: t o t a l samnle
-
Sample n u h e r 3: t y r m l o o i c a l oroun 2
-
Samole number 5: t y n o l o p i c a l arouos 2 and 3.
Samnle number 2: t y n o l o o i c a l proua 1 S a m l e number 4: t y n o l o o i c a l proun 3
From t h e 06 o r i c i n a l i t e m s , 11 were r e c o g n i z e d as model-conform.
The e s t i -
m a t i o n o f item-oarameters remained r l r n c t i c a l l v i n v a r i a n t o v e r t h e f i v e subsamoles.
T h i s i t e m - s e t was c o n s i d e r e d as " a n c h o r - s e t " .
A t f i r s t glance
t h e r e t a i n e d i t e m s covered d i f f e r e n t domains: h e a l t h , incolne, e d u c a t i o n , etc. I n t h e second s t e o , t h e n r e v i o u s s e l e c t i o n c r i t e r i a was l i c h t e n e d .
I t was
e x n e c t e d t o f i n d an i n v a r i a n c e o f i t e m s ' e s t i m a t i o n s o v e r t h e samples numb e r 1, 2 and 5 . model.
From t h e o r i a i n a l i t e m l i s t , 31 remained conform t o t h e
T h i s new s e t c o n t a i n e d t h e "anchor" i t e m s , a l r e a d y t e s t e d , b u t a160 The remained i t e m s
o t h e r i t e m s , which had been found as model-conform.
400
P. L)irkes
l a r r l e l v covered t h e d i f f e r e n t d o r a i n s o f l i v i n o c o n d i t i o n s o f t h e f a m i l i e s . Pecause t h e b,yqothesis o f u n i d i n e n s i o n a l i t y o f t'7e 3 1 i t e m s c n u l d r o t he r e i e c t e d , we concluded t h a t t h e i t e m s o f t h i s s e t m a s u r e d a s o l e and s a w thinn.
n u t o f t h e obvious h e t e r o n e n e i t y o f t h e + m a i n s t o whic'l helonped
%he i t e m s , we concluder! t h a t o n l v a r e f e r e n c e t o nnvertl! c n u l d h e s t i n t e r o r e t t h e homoneneity of th,e 1 P t e n t t r a i t .
T h i s was c n n f i r w d hv t k f a t t
t h a t t h e " h a r d c o r e v a r i a h l p s " , t r a e i t i o n a l l v a s c o c i a t e d w i t h a o v e r t v , as f o r examnle income, s o c i a l t r a n s f e r , d e a r i v a t i o n s - , helonned t o t h e r e t a i ned l i s t . Thus, o u r f i r s t h y p o t h e s i s s e e w d confirmed and n o v e r t y anneared as an acc u m u l a t i o n o f d i f f e r e n t disadvantanes, i n d i f f e r e n t d c r a i n s o f t h e l i v i n o conditions.
T h i s h / a o t h e s i s would have heen r e j e c t e d i f t h e r e t a i n e d i t e m s
belnnned t o one s p e c i f i c dopain ( l i k e h o u s i n n o r h e a l t h ) .
I n t h i s case
the l a t e n t continuum would have o n l v r e v e a l e d one s a e c i f i c accumulation o f i i q a ~ : v a ~ . + a c c s .I f we 'lad heen a h l e t o f i n d two o r more c p n t i n u a o f t h i s k i n d , we c o u l d have t a l k e d ahout a o v e r t i e s b u t n o t a h n u t n o v e r t v .
I n our
case however, we n r a s u r e d a l a t e n t continuum, i n t e r n r e t e d as d e s c r i b i n o
. Fach i t e m has noverty
a s n e c i f i c a o r i t i o n on t h i s t r a i t .
Tbe h i o h e r t h e o o s i t i o n
( e s t i m a t i o n o f item-Parameter), the h a r d e r < s t h e m v e r t v i t r e v e a l s .
Items
w i t h lo\., values a o i n t o u t t o m i n o r disadvantanes ( a s f o r exam?le overcrowd i n ? , h a v i n a '10 brcks i n t h e house, e t c . ) .
I t e n s w i t h h i n h values r e v e a l
s e r i o w plnverty s i t u i i t i o n s ( a s f o r examnle t h e denendance on c n c i a l a i d , being suh-tenant, etc.!. Each nerson has a l s o a s p e c i f i c o o s i t i o n on t h i s s a w continuum.
Rasch-mo-
del has n r o a r i e t i e s t h a t a l l o w t o make t h e s u m a t i o n o f a l l t h e disadvantanes.
I n t h i s way t o eacb nerson corresnonds a " v a l u e " , which i s a s u f -
f i c i e n t s t a t i s t i c t o make n e r s o n ' s e s t i m a t e on t h e s c a l e .
Thus t h e more
disadvantaoes a f a m i l v has, n o o r e r i t i s and t h e h i g h e r t h e v a l u e i t r e a c h es
.
The measurement i s o b l e c t i v e i n t h e sense t h a t 1) t h e n e r s o n ' s narameter i s indenendent o f t h e i t e m - s e t and 2) t h e i t e w ' s narameter t h e s a m l e s o f nersons.
i s indeoendent o f
Thus, t h e measurement o f t h e n e r s o n ' s narameter
can be r e a l i z e d w i t h any r u b - s e t o f v o d e l - c o n f o r m i t e m s and cound be apn l i e d t o any s u b s a m l e s o f nersons.
401
Weak andstrorrg models
These 9 r o w r t . i e s
o f t h e Rascb model w i l l a l l o w s e v e r a l o n e r a t i o n s , as f o r
examnle:
-
t o comnare t h e r e s u l t s o f d i f f e r e n t i n v e s t i a a t i o n s w i t h t h e same s c a l e i f a t least
-
sow model-conform i t e m s ;re cormon:
t o c r e a t e a bank o f i t e m s where n o v e r t y i t e m w ~ u l dbe c h a r a c t e r i z e d by t h e i r narameter:
-
t o c o n s t i t u t e u s e r - s u i t e d s c a l e s f o r s D e c i f i c r e s e a r c h nurnoses; t o cumulate knowledoe f r o m one s t u d v t o t h e o t h e r , i n o r d e r t o n u r i f y o r t o enlarcre t h e ranne o f t h e o r i n i n a l i t e m - n o o l ;
-
t o s t u d y chanae.
6 . R I S K AND MEPSllPE’FNT n F P W E P T V The exn1orator.y a n a l y s i s o f t h e d a t a m a t r i x i n t h e h y D o t h e t i c o - i n d u c t i v e s t a n e o f t h e research r e v e a l s t h r e e t y p o l o p i c a l nrouns. a n a l y s e d as b e i n g a t r i s k o f p o v e r t y .
Two o f those were
I n o u r n a r t i c u l a r samnle, i n t h e
chosen European c o u n t r i e s , i t a m e a r s t h a t disadvantaCes a r e more f r e q u e n t i n one-narent f a m i l i e s and i n f a m i l i e s b e l o n g i n g t o e t h n i c prouns d i f f e r e n t f r o m t h e one o f t h e c o u n t r y the,v l i v e i n .
The o h t a i n e d resu1,ts a r e deoen-
d e n t on t h e n a t u r e o f t h e samnle: i f o t h e r n o o u l a t i o n s had been s t u d i e d , o t h e r groups a t r i s k would o r o h a b l y have emerged.
These r e s u l t s a r e i n
themselves i n t e r e s t i n ? , and a l l o w p r a c t i c a l a o n l i c a t i o n s t o be made.
One
example c o u l d be t o d e t e c t qrouns a t r i s k o f D o v e r t v , i n o r d e r t o oversamnle them i n a n e x t r e s e a r c h . This f i r s t analysis doesn’t of
noverty.
however, say a n y t h i n p about t h e measurement
I s one Derson n o o r e r than t h e o t h e r ?
Do t h e disadvantapes
o f one proun corresnond t o t h e disadvantaoes o f t h e o t h e r flroun?
Can t h e
a c c u m u l a t i o n o f disadvantaoes, as we observed i t i n t h e h i g h - r i s k groups, be l i n k e d t o a uniaue concent?
A l l these q u e s t i o n s needed t o he answered,
so a D r e c i s e measurement i n s t r u m e n t can be made a v a i l a ! ) l e . I n t h e c o n f i r m a t o r y s t a o e an e x n l i c i t hyDothesis about t h e d a t a m a t r i x was t e s t e d : p o v e r t y has a uninue d i w n s i o n and i s c h a r a c t e r i z e d by t h e accumul a t i o n o f disadvantanes i n d i f f e r e n t domains.
The u t i l i z a t i o n o f a s t r o n g
model a l l o w e d us t o c o n f i r m t h e i n i t i a l h y p o t h e s i s , and t o c o n s t r u c t a measuring i n s t r u m e n t which had t h e c h a r a c t e r i s t i c s o f o b j e c t i v i t y . !Je e x o e c t t h a t i n o t h e r samoles o r o t h e r p o n u l a t i o n s t h e h v n o t h e s i s w i l l l i k e w i s e be confirmed.
In t h i s war/, t h e h y o o t h e t i c a l s t a t e m e n t c o u l d be
402
I? Dickrs
t e s t e d aqain and t h e w a s u r i n o i n s t r u m e n t i m r o v e d . F i n t i r e 1 shows tCle d i s t r i b u t i o n o f tl7e t v n o l o o i c a l ornuns i n r e l a t i o n w i t h the numhnher o f disadvantanes t h w accumulate. Percent
1
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Sigm6=3.33
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o
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2
3
4
5
6
7
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tlumber o f disadvantages F i g u r e I : Number o f disadvantanes and t y n o l o c l i c a l n r o u n s . The two qrouns a t r i s k q r e s e n t mre disadvantaoes and a r e p o r e h e t e r o o e neous than t h e non-disadvantaned n r o u n . Thus the c o n f i r m a t o r y a n a l y s i s a l l o w s an i n t e r p r e t a t i o n o f t h e d e s c r i p t i v e statements o f c l u s t e r - a n a l y t i c t e c h r \ i q u e s : t h e nrouns a t r i s k o f D o v e r t y Dresent more disadvantanes and t h e accumulation n f disadvantanes i s o o v e r t y .
REFE P E K E S 111
F i s c h e r , G.H.,
Finfijhrunn i n d i e Theorie nsvcholooischer Tests, Fern,
Huher (1974). Theory o f mental t e s t s , New York, ' > I i l e y ( 1 3 5 0 ) .
!2]
C u l l i k s e n , H.,
131
Gustafsson,
403
Weak andstrong models
Jnresko?, K.G.,
Pnalyzinn osychological data by s t r u c t u r a l analysis o f
c o v a r i a n c e m a t r i c e s , 1~ K r a n t z , D.H. e t a l . (Eds.),
Contemnorarv deve-
lonments i n mathematical nsvcholooy, V o l . 11, San F r a n c i s c o , Freeman
( 1374) , 1-56.
I51
K r i s t o f , ',I.,
K l a s s i s c h e T e s t t h e o r i e und T e s t k o n s t r u k t i o n , jrt Feger, H .
& Rredenkamr, , (1. (Eds - 1 , rlessen i!nd Tes ten, Fnzy!/.lopTdie d e r Fsycholo-
o i e , C d t t i n y e n , !!oorefe, Ed, 3 ( 1 W 3 ) , 594-603. Lawley, D.N.,
on nrohlems connected w i t h i t e m s e l e c t i o n and t e s t con-
s t r u c t i o n , Proceedinas o f t h e 9o.val S o c i e t v o f Edinburph, 61-P (1943)
,
273-287.
A t h e o r v o f t e s t scores, Psychometric rlonoyranhs, 7 ( 1 9 5 2 ) .
Lord, F.M.,
Lord, F.Y., The r e l a t i o n o f t e s t scores t o t h e t r a i t u n d e r l y i n g the t e s t , E d u c a t i o n a l and p s y c h o l o y i c a l Measurement, 13 (1953) , 517-548. Lord, F.M.,
and Novick, F4.R.,
r e s , Readin?, "ass., 101
Novick, M.R.,
I111
Ponner, K.R.,
ry,
S t a t i s t i c a l t h e o r i e s o f m n t a l t e s t sco-
Addison-',lesley (1968).
The axioms and n r i n c i D a 1 r e s u l t s o f c l a s s i c a l t e s t theo-
J o u r n a l o f f'athematical Psvcholoay, 3 (1966), 1-18. The l o n i c o f s c i e n t i f i c d i s c o v e r v , London, Hutchinson
(1968). Rasch, C., P r o b a b i l i s t i c models f o r some i n t e l l i n e n c e and a t t a i n m e n t t e s t s , CoDenhaflen, Wielsen and L-vdiche ( 1 9 6 0 ) .
[I31
Rasch, C., A mathematical theor.
o f o b j e c t i v i t y and i t s consequences
f o r model c o n s t r u c t i o n , Paper d e l i v e r e d a t t h e Eurorlean I ' e e t i n a on S t a t i s t i c s , Econometrics and Planaaement ( 1 9 6 8 ) . Schaber,
e t a l . , Pauvrete o e r s i s t e n t e - arande r e g i o n , P r o j e t 28,
Commission des Comunautes Euroneennes, Croune d ' e t u d e n o u r l e s n r o blPmes de l a n a u v r e t e ( 1 3 C O ) . Van der Ven, P , . ,
E i n f l j h r u n g i n d i e S k z l i e r u n ? , s e r n , Huber ( 1 9 7 9 ) .
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7RIi\'DS IA'hf.4TIIEMATICAL PSYCHOLOGY E. lkareef arid J. Van Ru enhaict (editurs) 0 E k e v i e r Science Publisf% 0. V (Ilorth-Holland), f 984
405
APPLICATIONS OF A RAYESIAN POISSON HODEL FOR MISREADINGS Fargo C.H. Jansen U n i v e r s i t y o f Groningen
A p a r t f r o m t h e w i d e l y known model f o r b i n a r y s c o r e d
iterns, Rasch, has developed s e v e r a l o t h e r models f o r t h e a n a l y s i s o f achievement t e s t d a t a .
The model we
c o n s i d e r h e r e i s t h e s o - c a l l e d m u l t i p l i c a t i v e Poisson model f o r m i s r e a d i n g s .
T h i s model assumes t h a t t h e
t o t a l number o f m i s r e a d i n q s on a t e x t f o r a c e r t a i n i n d i v i d u a l i s apDroximately Poisson d i s t r i b u t e d w i t h an i n t e n s i t y narameter
w h i c h depends on t h e r a t i o
o f two o t h e r p a r a n e t e r s , cne x r t a i n i n g t o t h e a b i l i ty o f t h e i n d i v i d u a l and one t o t h e d i f f i c u l t y o f t h e text.
A Bayesian v e r s i o n o f t h i s model was develooed
b y Owen (1969).
I n t h i s Daper we adont a s l i g h t l y
d i f f e r e n t formulation i n the t r a d i t i o n o f the general aoproach b y L i n d l e y ( L i n d l e y and Smith (1972)).
The
method i s i l l u s t r a t e d by an e m p i r i c a l examnle.
1. INTRODUCTION A p a r t f r o m t h e s o - c a l l e d one-narameter l c g i s t i c model f o r b i n a r y s c o r e d i t e m s , Rasch has developed s e v e r a l o t h e r l a t e n t t r a i t models f o r t h e a n a l y s i s o f achievement t e s t d a t a .
Among these a r e t h e models f o r o r a l r e a d i n g
speed and f o r misreadings i n a t e x t .
These models a r e d e s c r i b e d i n t h e mo-
nograph " P r o b a b i l i s t i c models f o r some i n t e l l i g e n c e and a t t a i n m e n t t e s t s " , Rasch (1960), and summarized by L o r d and Novick (1968)
.
Compared t o t h e l o g i s t i c model these models have a t t r a c t e d much l e s s a t t e n tion.
As a r e s u l t these models a r e much l e s s developed and p r a c t i c a l a p p l i -
c a t i o n s a p a r t from those o f Rasch h i m s e l f a r e r a r e .
T h i s seems u n f o r t u n a t e
because t e s t s t h a t would l e n d themselves e x c e l l e n t l y t o analyses based on Poisson process models a r e w i d e l y used i n e d u c a t i o n a l o r a c t i c e .
406
M.G .II. Jansen
Eayesian v e r s i o n s o f t h e P o i s s o n model f o r m i s r e a d i n g s were a l r e a d y devel o o e d i n 1969 by Owen.
R e l a t e d models were develoned by Van d e r Ven ( 1 9 6 9 ) .
2. THT POISSON MODEL FOR FlI%READIPGS I n c o n t r a s t t o t h e one-narameter l o g i s t i c model, t h e P o i s s o n model f o r misr e a d i n g s i s a l a t e n t t r a i t model f o r t e s t s r a t h e r t h a n i t e m s .
Rasch how-
e v e r , b e g i n s h i s develonment o f t h e l a t t e r w i t h t h e assumotion t h a t a t e x t t o be r e a d can be c o n s i d e r e d as a s u c c e s s i o n o f m u t u a l l y i n d e n e n d e n t Bern o u i l l i t r i a l s , c o r r e s o o n d i n g t o t h e words i n t h e t e x t , where t h e chances of an e r r o r a r e r e l a t i v e l y s m a l l .
A second, b u t e x t r e m e l y i m p o r t a n t as-
sumotion i s t h a t t h e o r o b a b i l i t y o f a m i s r e a d i n g i s a r a t i o o f two f a c t o r s , one p e r t a i n i n g t o t h e a b i l i t y o f t h e n u p i l and one p e r t a i n i n g t o t h e d i f f i c u l t y o f t h e words. Then, g i v e n t h a t t h e e r r o r P r o b a b i l i t i e s a r e i n d e e d s m a l l and t h e number
o f words i n t h e t e x t ( = number o f i t e m s ) i s l a r c e , t h e d i s t r i b u t i o n o f t h e t o t a l number o f e r r o r s i n t h e t e x t ( = t e s t s c o r e ) may be approximated by a Poisson d i s t r i b u t i o n w i t h i n t e n s i t y Darameter equal t o t h e sum o f t h e
“ernoulli
oarameters.
Rasch assumed t h a t t h e e r r o r p r o b a b i l i t i e s w i t h i n
a g i v e n t e s t and p e r s o n c o m b i n a t i o n were c o n s t a n t b u t t h i s i s n o t necessary, a p o i n t a l s o made b y L o r d and N o v i c k , ch. 2 1 (1968).
The i n t e n s i t y parame-
t e r has t h e same m u l t i p l i c a t i v e s t r u c t u r e as t h e narameters on i t e m - l e v e l . I n f a c t , we c o u l d t a k e t h e P o i s s o n model f o r t h e t o t a l scores as o u r s t a r t i n g p o i n t , w i t h t h e i t e m - l e v e l model as one p o s s i b l e u n d e r l y i n g s t r u c t u r e . L o o k i n g a t t h e e x a m l e s g i v e n b y Rasch, i t i s c l e a r i t i s t h e model a t t h e t e s t - l e v e l f o r w h i c h we have e m p i r i c a l e v i d e n c e .
The t e s t - l e v e l model i s
completed by t h e a s s u m t i o n t h a t we have two o r more t e s t s f o r which a Poisson model h o l d s , and t h a t t h e u s u a l c o n d i t i o n s o f l o c a l independence and independence accross o u n i l s h o l d . F o r m a l l y , c o n s i d e r a sample o f N p u p i l s t a k i n g k t e s t s , where t h e d i s t r i b u tion
cif
t h e number o f e r r o r s made b y Derson i on t e s t j i s g i v e n by
t4e supoose t h a t t h e i n t e n s i t y p a r a m e t e r uij ters, the d i f f i c u l t y 6
j
i s a r a t i o o f two o t h e r parame-
o f t h e t e s t and t h e a b i l i t y
si
o f the person.
A Bayesian Poisson modelfor misreadings
407
The e x p e c t e d number o f e r r o r s w i l l be h i g h e r i f t h e t e s t i s more d i f f i c u l t a n d / o r t h e person i s l e s s a b l e and l o w e r i f t h e t e s t i s l e s s d i f f i c u l t and/ o r t h e person more a b l e . O b v i o u s l y n e i t h e r , t h e 6 . n o r t h e E~ a r e u n i q u e l y determined, f o r t h e k t e s t s and N persons.
J
T h i s s t a t e o f a f f a i r s can be remedied by imposing an
a r b i t r a r y r e s t r i c t i o n on t h e s e t o f parameters.
Rasch proposes t a k i n g t h e
most d i f f i c u l t t e s t as a p o i n t o f r e f e r e n c e by choosing a v a l u e o f one f o r t h e c o r r e s p o n d i n g d i f f i c u l t y narameter.
N o t i c e t h a t , because a l l Poisson
parameters have t o be nonnegative, s e t t i n g one d i f f i c u l t y parameter t o be p o s i t i v e , i m p l i e s t h a t a l l d i f f i c u l t y parameters 6 meters E~ are 2150 r o n n e g a t i v e .
j
and a l l a b i l i t y para-
I n t h e Bayesian framework, i d e n t i f i a b i l i ty problems a r e s o l v e d d i f f e r e n t l y , b y choosing s u i t a b l e n r i o r d i s t r i b u t i o n s which t e l l us how t o i d e n t i f y t h e parameters, as we w i l l show i n t h e n e x t s e c t i o n . From a t h e o r e t i c a l p o i n t o f view, one v e r y d e s i r a b l e c h a r a c t e r i s t i c o f t h i s model, which i t shares w i t h t h e " o r d i n a r y " Rasch-model, i s t h e s e p a r a b i l i t y o f t h e parameters.
T e s t parameters can be e s t i m a t e d i n d e p e n d e n t l y o f t h e
person parameters and v i c e v e r s a .
F o r a d i s c u s s i o n o f what i s a l s o known
as " s p e c i f i c o b j e c t i v i t y " , see Rasch (1980) and W r i g h t and Stone (1979). 3. BAYESIAN VERSIONS OF THE MULTIPLICATIVE POISSON MODEL
A f u l l Bayesian model f o r t h e m u l t i p l i c a t i v e Poisson model f o r m i s r e a d i n g s has been developed by Owen (1969).
He t o o k t h e f o r m u l a t i o n i n Eqns ( 1 ) and
( 2 ) as s t a r t i n g p o i n t b u t adopted a n o t h e r d e f i n i t i o n of t h e parameters. The person parameter he used i s i n v e r s e l y p r o p o r t i o n a l t o i t s c o u n t e r p a r t i n Rasch's f o r m u l a t i o n .
T h i s change was b r o u g h t about t o p r o v i d e a com-
p l e t e symmetry between t h e person and t e s t p a r a n e t e r s .
P r i o r information
was s p e c i f i e d b y choosing m u t u a l l y independent gamma d i s t r i b u t i o n s f o r t e s t and person parameters, and s e v e r a l p o s t e r i o r d e n s i t i e s , i n c l u d i n g t h e condit i o n a l and m a r g i n a l p o s t e r i o r d e n s i t i e s , were d e r i v e d ,
A s i m i l a r approach was used b y Van d e r Ven (1969), i n o r d e r t o f i n d s u i t a b l e s t r o n g t r u e - s c o r e models f o r t h e a n a l y s i s o f t i m e - l i m i t t e s t s . Our own f o r m u l a t i o n i s more s i m i l a r t o t h e approach o f Leonard (1971) and t h a t of L i n d l e y (1970).
We adopt t h e f o l l o w i n g t r a n s f o r m a t i o n o f t h e o r i g i n a l
408
M . G . H . Jai~seii
a b i l i t y and d i f f i c u l t y Darameters A; = -1nE.
b j = -1n6
and
1
j
and l n ( u . .) = 8 . - b . 1.l 1 J The
i k e l i h o o d f u n c t i o n i n terms o f 9 and b L(e,b)
Treating
el,.
. . ,oN
= exn (zy. ,(ei-b.) 1J
r e s n e c t i v e l y bl,.
J
-
(3) s civen by
I:exD ( e i - b .J) }
. . ,bk
/!)..!(!I
(4)
1J
as exchanqeahle w i t h i n each s e t
we assume t h e f o l l o w i n g m u t u a l l y indenendent m a r p i n a l n r i o r d i s t r i b u t i o n s
and
w h c - ~u t , , $ 8 and $b a r e suooosed t o be known c o n s t a n t s . A more c o n v e o i e n t f o r m u l a t i o n i s based on t h e f o l l o w i n g .
Let
Then t h e p r i o r d i s t r i b u t i o n s can be f o r m u l a t e d as f o l l o w s .
The p r i o r v a r i a n c e s a r e assumed known, as i n t h e n r e v i o u s f o r m u l a t i o n .
The
mean uo can a l s o be assumed known o r he r e t a i n e d as an e x p l i c i t parameter. I n t h a t case we assume t h a t
u, i s d i s t r i b u t e d u n i f o r m l y o v e r t h e r e a l l i n e .
All n r i o r d i s t r i b u t i o n s a r e c o n s i d e r e d m u t u a l l y i n d e n e n d e n t . posterior distribution i s proportional t o
The j o i n t
409
A Bayesian Poisson modelfor misreadings
E s t i m a t e s f o r t h e parameters can be o b t a i n e d by t a k i n g t h e l o S a r i t h m o f t h e
*
b y c ' i f f e r e n t i a t i n g w i t h r e s p e c t t o u,e:
p o s t e r i o r d e n s i t y p ( 0 ,b,uelY), bj,
and
s e t t i n g t h e i r d e r i v a t i v e s e q u a l t o z e r o and s o l v i n g t h e r e s u l t i n g equa-
t i ons :
-
X . yij iYJ
- jz y 1. .J
+
I: yij i
+ z
*
I: exp(ei+u-b.)J i,j
*
I: exp(ei+u-b.) J j
= 0,
*
ei / be = 0,
t
and
-
*
exp(e.+p-b.) 1
i
J
-
b . / 4 = 0. J b
As i s obvious f r o m e q u a t i o n s (11) through ( 1 3 ) an i t e r a t i v e nrocedure i s r e q u i r e d t o s o l v e them. Under a D p r o p r i a t e c o n d i t i o n s , t h e r e s u l t w i l l be t h e mode o f t h e j o i n t p o s t e r i o r d e n s i t y .
4 . EXAMPLES The f i r s t example i s based on an a r t i f i c i a l d a t a s e t c o n s i s t i n g o f t h e r e s ponses o f 100 i n d i v i d u a l s on 3 t e s t s , f r o m a s t a n d a r d normal d i s t r i b u t i o n . o f r e s p . bl = -0.5
The a b i l i t i e s were randomly s e l e c t e d F o r t h e t e s t s , we choose d i f f i c u l t i e s
b 2 = 0.0 and b3 = +0.5.
T h i s corresponds t o an e r r o r
p r o b a b i l i t y o f r e s p . 3, 2 and 1 p e r c e n t f o r a " s t a n d a r d " person w i t h a b i l i t y e = 0, g i v e n a t e s t l e n g t h o f 50 i t e m s .
The p r i o r v a r i a n c e s were assu-
med t o b e e q u a l t o a v a l u e o f 1. The r e s u l t s can be f o u n d i n t a b l e 1.
Both t h e Bayesian and t h e o r d i n a r y
maximum l i k e l i h o o d e s t i m a t e s f o r t h e narameters e and b ( - l n ( E ) and - l n ( 6 ) ) a r e shown. The Bayesian e s t i m a t e s show t h e t y p i c a l r e g r e s s i o n e f f e c t compared t o t h e maximum l i k e l i h o o d e s t i m a t e s .
The r e g r e s s i o n e f f e c t i s more marked f o r t h e
a b i l i t y t h a n f o r t h e d i f f i c u l t y e s t i m a t e s , due t o t h e asymmetry i n t h e sample i n f o r m a t i o n (100 nersons and 3 t e s t s ) . These r e s u l t s a r e s i m i l a r t o r e s u l t s o b t a i n e d by u s i n g Bayesian methods f o r e s t i m a t i n g t h e parameters o f t h e b e t t e r known one-parameter l o g i s t i c model (Jansen & Lewis (1983)).
410
M.C.H. ]amen
Raw Score
Theta E s t i m a t e
ML
T)e
= 0
*
-1.19 -0.49 -0.09 0.20 0.42 0.61 0.76 0.89 1.01 1.12 1.21 1.30 1.38 1.45 1.52 t
!J
Beta E s t i m a t e
BAY
F:L
- _ _ _ -1.16
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Test Nr.
1 2 3
-0.72 -0.38 -0.10 0.13 0.33 0.50 0.64 0.77 0.89 1.00 1.09 1.18 1.27 1.34 1.41 etc.
-0.507 -0.028 0.535
BAY
-0.503 -0.027 0.530
e Tahie i
Maximum L i k e l i h o o d and Bayesian E s t i m a t e s f o r t h e Person and T e s t Parameters o f t h e A r t i f i c i a l Cata S e t ( N = 100, k = 3 ) . Another i n t e r e s t i n g f e a t u r e i s t h a t , w i t h t h e Bayesian approach, we a l s o have f i n i t e e s t i m a t e s f o r nersons w i t h a p e r f e c t s c o r e ( e r r o r s c o r e o f zero).
T h i s i s i m p o r t a n t s i n c e , w i t h easy t e s t s , a r e l a t i v e l y l a r g e p r o -
p o r t i o n o f t h e sample c o n s i s t s o f . j u s t these persons. F o r t h e second examole we used an e m n i r i c a l l y o b t a i n e d d a t a s e t .
This data
s e t c o n s i s t e d o f t h e responses o f p u p i l s on a s p e l l i n g t e s t t h a t was admin i s t e r e d t w i c e under d i f f e r e n t f o r m a t c o n d i t i o n s .
The p a r a m e t e r e s t i m a t e s
f o r t h e person and t e s t s can be f o u n d i n T a b l e 2, on t h e n e x t page. The r e s u l t s i n Table 2 a r e cowoarable t o t h o s e i n t h e f i r s t T a b l e ,
41 1
A Bayesian Poisson modelfor rnisreadingr
Raw Score
Theta E s t i m a t e B Av
A 'L
----
0
-n.70 -c.o1 0.40 0.69 0.91 1.09 1.25 1.33 1.50 1.50 1.70 1.79 1.87 1.94 2.01
1.
2
3 4 5 6 7 8 9 10 11 12 13 14 15
Test N r .
Beta E s t i m a t e
-
P'L
1 2
-n.41
-0.02
0.116 -0.116
0.28 0.53 n.73 0.91 1.07 1.21 1.33 1.44 1.54 1.63 1.72 1.30 1.87 1.94
PAY
0.115
-n.m
etc. Table 2 thximum L i k e l i h o o d and Pavesian Estimates f o r t h e person and T e s t Parameters o f t h e Data Obtained by c i v i n p t h e Same T e s t under Two D i f f e r e n t Format C o n d i t i o n s ( N = 716, K = 2 ) . 5. DISCUSSION The r e s u l t s show t h a t Bayesian methods a r e n o t e n t i a l l v u s e f u l f o r e s t i m a t i n g t h e narameters o f t h e m u l t i n l i c a t i v e Poisson model.
Vore r e s e a r c h how-
e v e r , i s necessarv t o s u b s t a n t i a t e t h i s c l a i m . R t FE RE N CES
[l]?.nr!ersen, F.?., D i s c r e t e s t a t i s t i c a l models w i t h s o c i a l s c i e n c e a n p l i c a t i o n s , N o r t h H o l l a n d , Amsterdam (1.982). [2]
Leonard, T.,
Pavesian methods f o r m u l t i n o m i a l data, ACT T e c h n i c a l B u l -
l e t i n 4 , The American C o l l e q e T e s t i n g Propram, Iowa City ( 1 9 7 1 ) . 131
L i n d l e y , D.V.,
P Rayesian s o l u t i o n f o r some e d u c a t i o n a l p r e d i c t i o n p r o ETS, P r i n c e t o n ( 1 9 7 0 ) .
blems, 111, Research B u l l e t i n RR-70-33, [4]
L i n d l e y , D.V.,
8 Smith, P.F.I!.,
Rayesian e s t i m a t e s f o r t h e l i n e a r PO-
d e l , J o u r n a l o f t h e Royal S t a t i s t i c a l S o c i e t y ( S e r . B ) , 34 (1972), 1-41. [5]
Lord, F.M.,
R Novick, PT.R.,
Addison-Wesley,
S t a t i s t i c a l t h e o r i e s o f mental t e s t scores,
Readinq. Mass.
(1968).
41 2
[hl
M . C . H . Jamen
Rasch, C., P r o b a b i l i s t i c models f o r sow i n t e l l i c e n c e and a t t a i n m e n t t e s t s , Conenhacen (196O), ( R e n r i n t e d by t 4 e I l n i v e r s i t y n f ChicaTo Press (1020)).
171
Oven, R . J . ,
P Ra.vesian a n a l y s i s o f Rasch's m u l t i n l i c a t i v e Po'scon r o d e l
f o r w i s r c a ~ i F T s , Res. R u l l
181
Ven, A . H . C . S .
.
E.T.S.,
Vrincetnn (1369).
van d e r , The ? i n o r r i a l e r r o r m d e l a n n l i e d t o t i m e - l i p i t
t e s t s , Unnuhlished d o c t o r a l d i s s e r t a t i o n , Hijmepen ( 1 9 G R ) . [ql
I . ; r i ? h t , R.D., ( 1379).
R Stone, ".H.,
R e s t t e s t d e s i g n , WSn P r e s s , Chicapo
?'RE,\'DS I N MATHEMATICAL PSYCIIOLOGY E . D e p e f a n d j . Van Bu genllaut (editors) 0 Elrevier Science Publisferr B. V. (Nordi-Holland), 1984
413
THE POLYCt'OT@YCL'S RASCH 11nDEL AND DICHOTOFIIZATIPN OF CRbDED RESPONSES
Paul I;. ' , I . Jansen R i j k s Psycho1 o g i sche Diens t , Den Haao , and Edw. E . Roskam Department o f Psycholooy, U n i v e r s i t y o f Plijmerjen
T h i s oaper discusses t h e c o m o a t i b i l i t y o f t h e o o l y chotomous Rasch model w i t h d i c h o t o m i z a t i o n o f t h e resnonse continuum.
I t i s argued t h a t i n t h e case
o f graded resnonses , t h e response c a t e q o r i e s present e d t o t h e s u b j e c t a r e e s s e n t i a l l y an a r b i t r a r y DOl y c h o t o m i z a t i o n o f t h e resoonse continuum, r a n g i n q e.9.
f r o m t o t a l r e j e c t i o n o r disaoreement t o t o t a l
accentance o r aoreement o f an i t e m o r s t a t e v e n t , and s h o u l d n o t i n t e r f e r e w i t h s u b j e c t measurement.
A d i s t i n c t i o n i s made between p o l y c h o t o m i z a t i o n o r d i c h o t o m i z a t i o n ' b e f o r e t h e f a c t ' , i.e. i n c o n s t r u c t i n g t h e resnonse format, and p o l y c h o - o r d i c h o t o m i z a t i o n ' a f t e r t h e f a c t ' , e.g. i n d i c h o t o m i z i n g e x i s t i n g polychotomous data. I t i s shown t h a t , a t l e a s t i n case o f d i c h o t o m i z a t i o n
' a f t e r t h e f a c t ' t h e oolychotomous Rasch model i s n o t c o m o a t i b l e w i t h such d i c h o t o m i z a t i o n .
I f data s a t i s -
f y t h e u n i d i m e n s i o n a l oolychotomous Rasch model, t h e y
w i l l n o t do so a f t e r d i c h o t o m i z a t i o n , unless a r a t h e r s p e c i a l c o n d i t i o n o f t h e model parameters i s met.
In
so f a r as d i c h o t o m i z a t i o n ' b e f o r e t h e f a c t ' i s n o t e s s e n t i a l l y d i f f e r e n t from dichotomization ' a f t e r the f a c t ' , t h e v a l u e o f t h e unidillrensional polychctcmous Rasch model i s q u e s t i o n a b l e . The i m p a c t o f o u r c o n c l u s i o n on r e l a t e d models i s b r i e f l y discussed.
414
P.G.W. Jamen and E.E. Roskam
1. INTRODUCTION Various models e x i s t f o r t h e r e n r e s e n t a t i o n and a n a l y s i s o f d a t a where t h e resnonse f o r m a t c o n s i s t s o f two o r more a1 t e r n a t i v e s , c a t e g o r i e s o r o p t i o n s . These models t y o i c a l l y ccrlcern a t t i t u d e q u e s t i o n n a i r e s ,
h u t may a l s o a n o l y
t o antitude items.
bone t h e o r o h a h i l i s t i c models, t h e m u l t i d i m e n s i o n a l nolychotomous Rasch model (Rasch (1961)) a m e a r s t o be t h e o n l y model f o r i t e m 1 4 t h m u l t i n l e resaonse o p t i o n s , which i s i n a sense m u l t i d i m e n s i o n a l (we w i l l comment l a t e r OF t h e m l t i d i m e n s i o n a l c h a r a c t e r o r t h i s m c d e l ) . Unidimensional l a t e n t t r a i t models r e a r e s e n t t h e suh.iect by a s i n C l e s c a l a r pFraveter.
F o r polychotomous data, t h e r e a r e t h e L i k e r t ( ( 1 9 3 2 ) , c f . Roskam
(19E3))model, t h e u n i d i m e n s i o n a l nolychotomous Rasch model, and Samejima's
(1969) nolychotomous
l o c i s t i c d i f f e r e n c e model.
R e l a t e d t o t h e unidimen-
s i o n a l qolychotomous Rasch model a r e A n d r i c h ' s (137E) P a t i n q Rasch model and Plasters' (1922) P a r t i a l C r e d i t model.
I t s h o u l d be n o t e d t h a t t h e L i -
k e r t model i s n o t a o r o h a h i l i s t i c resaonse model, h u t a c t u a l l y a p o l y c h o t o IIOUS
Guttman s c a l e model w i t h t h e added assumntion t h a t t h e s u h j e c t s ' p a r a -
meters a r e n o r m a l l y d i s t r i b u t e d i n t + e D o p u l a t i o n o f s u b j e c t s . The u n i d i m e n s i o n a l models a r e n o t n e c e s s a r i l y meant t o he a l s o models f o r graded responses, h u t R o t h i n g aooears t o p r e v e n t one from a m l y i n g them t o qraded resDonse data, as u s u a l l y o b t a i n e d f r o m a t t i t u d e q u e s t i o n n a i r e s , where resoonse o o t i o n s a r e t y o i c a l l y o f t h e ' d e g r e e o f agreement' k i n d .
An
e x a m l e o f t h e a n p l i c a t i o n o f t h e u n i d i m e n s i o n a l oolychotomous Rasch model t o a t t i t u d e d a t a i s a i v e n by Andersen (1973). The mere f a c t t h a t a model does n o t aooear t o p r e v e n t i t s a n n l i c a t i o n t o graded resoonses does n o t mean t h a t i t i s c o m o a t i h l e w i t h p s y c h o l o g i c a l assumntions and/or m e t h o d o l o q i c a l r e q u i r e m e n t s w h i c h a n o l y t o a oraded response format.
T h i s p a p e r addresses i t s e l f t o t h e c o m p a t i b i l i t y o f t h e p o l y c h o -
tomous Rasch model w i t h such assumptions and/or r e o u i r e m e n t s i n t h e case o f qraded resoonses.
I n n a r t i c u l a r , we address o u r s e l v e s t o t h e i r c o m p a t i b i l i -
t y w i t h d i c h o t o m i z i n g the resoonses, e i t h e r i n c o n s t r u c t i n g t h e response format, o r a f t e r havinq c o l l e c t e d m l i l t i p l e choice data.
1.1. GRADED RESPONSES 'ale assume t h a t we have a o r i o r i knowledge c o n c e r n i n g t h e o r d e r i n g o f resDonse
Thepolycliotomous RPFCIImodel
415
catey;ories o r o p t i c n s whenever t h e i r n h r a s i n g c o n t a i n s a t l e a s t o r d i n a l n u a n t i f i e r s such as ' s t r o n ? l v ' , ' ~ o d c r a t e l y ' , ' s o m e t i m e s ' , ' n o t a t a l l ' , and t h e l i k e , o r when t h e r4sr)ense c a t e r l o r i e s a r e marked by symhols h a v i n a a n a t u r a l semantic o r d e r , l i k e ' a ' , viousl:/,
'b',
'c',
..., o r
'I1, ' Z ' , '3')...
.
Ob-
resnonse o p t i o n s l i k e " d o n ' t know" do n o t n a t u r a l l y f i t i n t o an o r -
d e r i n g o f resnonse c a t e g o r i e s . The g e n e r a l n r i n c i r ? l e of u n i d i m e n s i o n a l graded resoonse models f o r a t t i t u d e q u e s t i o n n a i r e s can be exnressed as f o l l o w s .
I t i s assumed t h a t s u b j e c t s
can be r e a r e s e n t e d a t l e a s t o r d i n a l l y as p o i n t s i n a l a t e n t u n i d i m e n s i o n a l continuum.
' t l i t h a n i v e n i t e m , t h e resnonse catecrories correspond t o a n a r t i -
t i o n i n n o f t h a t c o n t i n u u p , w a n i n q t h a t , i n a llranner o f sneakinq, t h e subj e c t l c c a t e s h i m s e l f on t h a t continuum by p i c k i n ? a resnonse o o t i o n accordi n ? t o where he s t a n d s .
The f o l l o w i n p i s an example:
I am a b l e t o jumo h i g h e r than 1.GO meter:
Each i t e m
-
o r o b a b l y yes
-
c e r t a i n l y yes
certainly not probably n o t j u s t about yes o r no
response c a t e g o r y e x m e s s e s a depree o f F o s i t i v e judgment
tovards i t s object.
T h i s i s most c l e a r l y so i n t h e d e t e r m i n i s t i c p o l y c h o -
tomous Guttrnan s c a l e model, as a r e s e n t e d e.?. by Coombs ( 1 9 6 4 ) .
The same
concent o f c a t e g o r i e s n a r t i t i o n i n p a l a t e n t soace i s a l s o c e n t r a l t o m u l t i dimensional scaloclram a n a l y s i s (Cuttman (1967); Linqoes ( 1 9 6 0 ) ) . I n d e t e r n i n i s t i c models, t h e correspondence between resnonse c a t e g o r i e s and segments o f t h e l a t e n t continuum i s onc-to-one. case i n a r o b a b i l i s t i c models.
This i s obviously n o t the
' ! i t h o u t s p e c u l a t i n g on t h e n a t u r e o f t h e s t o -
c h a s t i c mechanism u n d e r l y i n p a n o n - d e t e r m i n i s t i c resoonse s t r u c t u r c , w
can
s t i l l adoDt t h e concept of resoonse continuum which i s p a r t i t i o n e d i n t o succ e s s i v e c a t e g o r i e s , such t h a t t h e n r o b a b i l i t y o f a s u h i e c t ' s n i c k i n g a c e r t a i n resnonse orJtion i s a o r o b a b i l i s t i c f u n c t i o n o f h i s n o s i t i o n on t h e l a t e n t continuum.
I n f a c t , we assume t h a t t h e s u b j e c t ' s exoressed resnonse
corresoonds t o an i n t e r v a l on a resnonse continuum which i s i t s e l f i m e r f e c t l y c o r r e l a t e d w i t h t h e l a t e n t continuum.
Here, l i k e i n t h e d e t e r m i n i s t i c case, each i t e m
3 resnonse c a t e g o r y i s an
i n d e x o f t h e degree o f t h e s u b j e c t ' s n o s i t i v e a t t i t u d e towards y o r e o b j e c t ,
416
P.G.W. Jonseii oiid 1.1:. Rosklriii
b u t i n t h e v r o h a h i l i s t i c case t h i s i n d e x i s i m n e r f e c t . nnnse podel s t a t e s , i n an e x n l i c i t a l r l e b r a i c f o r v ,
A n r o h a l i i l i s t i c res-
the n r o h a h i l i t y t!iat the
r ; u h i e c t v i 1 1 n i c k a n a r t i c u l a r resnonse o ? t i o n , n i v e n h i s l a t e n t t r a i t n o s i t i o n , and g i v e n c e r t a i n o a r a w t e r s s n e c i f i c t o t h e i t e m and i t s response catenories.
','e c o n s i d e r the resnnnse c a t e g o r i e s as i n t e r v a l s a l o n n a resnonse continuum, o r , as a n o l y c h o t o n i z a t i o q o f t h a t continuum. !n i t s s i m q l c s t form, !,.re have a dichotomizatica i n t o , e.n.
1.P. "LYCHD-
' y e s , a n r e e ' , and ' n o , d i s a y r e e ' .
OR DICti'?T"'~IZPTIQP; 9EFOEE THE FACT nr! nFTT?
For the q r e s e n t d i s c u s s i o n r e must d i s t . i n g u i s h qnlJ!chotc.F!iati@? ( o r i t s
2-
e x t r w e : d i c h o t w i i a t i o n ) b e f o r e t h e f a c t , f r o v n o l y c h o - d i chotomi z a t i o n t e r thc f a c t . -
I n c o n s t r u c t i n ? t h e resnonse o u t i c n s , t h e resnonse continuum
i s n a r t i t.ioned i n t o s u c c e s s i v e i n t e r v a l s b e f o r e t h e f a c t , i .e. b e f o r e t h e resnnnses are a c t u a l l v n i v e n arc' c o l l e c t e d .
I f , on t h e o t h e r hand, we con-
h i n e d i f f e r e n t resnonses i n t o a s v a l l e r s e t o f c a t e 9 o r i e s a t t h e t i m e o f dat a a n a l y s i s , we ( r e - ) n a r t i t i o n t h e resoonse continuum a f t e r t h e f a c t , i .e. a f t e r t h e resnonses a r e a c t u a l l y n i v e n and c o l l e c t e d .
The l a t t e r i s c o m o n
n r a c t i c e when graded resvonses a r e d i c h o t o v i z e d a f t e r t h e d a t a have been cn:l e c t e d , e.g. i n o r d e r t o s i m l i f y t h e i r a n a l i s i s . feepiny
t4iq
+ i s t i n c t i o n i n mind, t h e q u e s t i o n a r i s e s whether o r n o t t h e
nolycho- o r d i c h o t o v i z a t i o n o f resnonses i n t e r f e r e s w i t h s u b j e c t measurement. The nroblem o f n o n - i n t e r f e r e n c e ' b e f o r e t h e f a c t ' r e f e r s t o t h e e f f e c t o f the s t r u c t u r e o f r r a d e d resnonse a l t e r n a t i v e s on s u h l e c t measurement.
It
would seem o n l y r e a s o n a h l e t h a t i t does n o t m a t t e r whether t h e s u b j e c t must choose h i s resnonse f r o m one s e t o f nraded resnonses o r f r o m a n o t h e r s e t o f oraded responses ( b o t h n e r t a i n i n p t o t h e same resnonse continuum), a t l e a s t n o t so t h a t t h a t w o u l d i n t e r f e r e w i t h whatever i t i s t h a t i s h e i n g measured.
I n o a r t i c u l a r , t h i s concerns resoonse f o r m a t s w h i c h d i f f e r o n l y w i t h r e s o e c t t o t h e q u a n t i f i e r s used t o exnre;s
and the l i k e .
deqrees o f e . g . accentance, agreement,
Exchanoing one s e t o f q u a n t i f i e r s i n t h e resnonse f o r m a t f o r
a n o t h e r s e t o f q u a n t i f i e r s s h o u l d be i m m a t e r i a l and n o t i n t e r f e r e w i t h suhi e c t neasurement.
A s i t u a t i o n where one can r e a s o n a b l y e x o e c t and a c c e n t
such i n t e r f e r e n c e , i s one v h e r e channing t h e resnonse a1 t e r n a t i v e s w o u l d a c t u a l l y a f f e c t t h e meanin? o r c o n t e n t o f t h e i t e m ( e . 7 .
... r a r e l y
' d o you a t t e n d
/ sometimes / o f t e n ' v s . ' d o you a t t e n d f o o t b a l l w i t h l i t t l e D l e a s u r e / ... / w i t h much n l e a s u r e ' ) . football
...
7%e polychotomous Rascli model
417
The nroblem o f ( n o n - ) i n t e r f e r e n c e ' a f t e r t h e f a c t ' concerns t h e q u e s t i o n i!hether o r n o t c o h i n i n c . answers i n d i f f e r e n t resnonse c a t e o o r i e s would subs t a n t i a l l y a f f e c t suh j p c t m a s u r e n e n t . Ire n i v e a formal d e f i n i t i o n o f 'non-interference' l a t e r .
The s i m o l e s t wav
t o study i t i s t o i n v e s t i n a t e the e f f e c t o f dichotomization.
"ost r e s u l t s
a r e e a s i l y a e n e r a l i z e d t o any ( r e - ) n a r t i t i o n i n ? o f t h e resnonse continuum.
1.3. LOSq (IF Ii4FQR''ATInbl VS. SIF?ST~VITIVEEFFECTS OF RE5PnNSE FORtWT There has been some t h e o r e t i c a l d i s c u s s i o n as w e l l as e r n q i r i c a l r e s e a r c h on t h e e f f e c t s o f t h e resnonse f o r l n a t on s u b j e c t m e a s u r e x n t , r c ; t z h l ~Ip.:ith r e s q e c t t o t h e number of c a t e f l o r i e s .
There i s a l s o some evidence t h a t t h e d i s -
t r i b u t i o n o f t h e resnonses i s a f f e c t e d by t h e f o r m u l a t i o n o f t h e res?onse c a t e q o r i e s ( c f . Bern 8 Repa?ort (1954) ; Van Feerden II H o o n s t r a t e n (1979) : r u o t e d i n H o o p s t r a t e n (1970), 60-62).
8 0 t h t h e t h e o r e t i c a l d i s c u s s i o n and
t h e e m n i r i c a l r e s e a r c h Itrere m o s t l y concerned w i t h s t a t i s t i c a l asnects, e.q. r e l i a b i l i t y and ( l o s s o f ) i n f o r m a t i o n , and n o t so much v i t h t h e q u e s t i o n ivhether o r n o t t h e numher and n h r a s i n r l o f resnonse a l t e r n a t i v e s i n t e r f e r e s w i t h s u b j e c t measurement n e r se.
R.v t h i s we mean whether o r n o t one i s
measurinq t h e same c o n s t r u c t when d i f f e r e n t k i n d s and numbers o f graded resnonse a l t e r n a t i v e s a r e used w i t h t h e same i t e m s .
For the nresent discus-
s i o n we assume and a c c e p t t h a t t h e n h r a s i n g and number o f praded resnonse a l t e r n a t i v e s i n q u e s t i o n n a i r e s w i l l a f f e c t t h e n r e c i s i o n and r e l i a b i l i t v o f reasurement, !which can, i n o r i n c i o l e , he compensated f o r by t h e number o f items.
I f , however, t h e o h r a s i n n and number o f nraded resnonse a l t e r n a t i v e s
a f f e c t s what i s b e i n a measured, r e a r e c o n f r o n t e d w i t h a s u b s t a n t i v e nrobleln
o f i t e m response t h e o r v . !re t a k e t h e o o s i t i o n t h a t t h e o h r a s i n g and number o f nraded resnonse catepor i e s s h o u l d n o t i n t e r f e r e w i t h what i s b e i n g measured. t i o n as a m e t h o d o l o g i c a l w o r k i n g h v n o t h e s i s . s t r u c t u r e on o r e c i s i o n , d i s c r i m i n a b i l i t y ,
','e t a k e t h i s n o s i -
E f f e c t s o f resnonse c a t e o o r y
r e l i a b i 1 i t . v and r e l a t e d n r o n e r t i e s
o f measurement devices have no s u b s t a n t i v e imoortance.
1.4. THE !lODEL APPPnACH I N THE STUDY r)F INTERFERENCE S i n c e we a r e concerned w i t h t h e e f f e c t o f polycho- o r d i c h o t o m i z a t i o n on s u b j e c t measurement, r e must a t l e a s t s t a r t t h e a n a l y s i s on t h e b a s i s o f
some i t e m response model by which s u h j e c t parameters can be e s t i m a t e d , and
P.G.W.J a w e n and E.E. Roskam
41 8
v h i c h can be t e s t e d e m n i r i c a l l y .
Statements ahout subjects' l a t e n t t r a i t
scores v o u l 6 h e o r a t u i tous I 4 t h c u t a t e s t a h l e i t e m resnonse model. f r o r such a mndel, two a m r o a c h e s a r e onen.
?tartino
One aonroach i s s t . r i c t l y e w i -
r i c a l : !.re cap a n a l y s e d a t a h e f n r e and a f t e r d i c h o t n m i z a t i m and see how s u b j e c t measurerent i s a f f e c t e d .
T h i s anproacti i s q u e s t i a n s h l e
or1
'oc-ical
Croimds: e s t i m a t i o n o f s u h . i e c t n a r a w t e r s i s a l v a \ t s c o n d i t i o n a l unon t h e asstined v a l i d i t y o f t!ie model, arld i f a model w o u l d n c t he c o m n a t i b l e w i t h d i c h o t n v i z a t i o n i n t h e f i r s t n l a c c , t $ e r e s u l t of i t s a n n l i c a t i o n t o d i c h o tomized d a t a w o u l d 5e n e a n i n 9 l e s s .
a roach.
p,v
T h i s l e a d s us t o a d o n t i n q t h e model an-
t h i s !.le w a n t h a t we i n v e s t i r a t e t h e q r e d i c t i n n s o f a model i f
d a t a were tiicliotoir.ized,
and see under w h i c h c o n d i t i o n s , i f any, s u b j e c t
w a s u r e m n t i s i n v a r i a n t under d i f f e r e n t no1,ycJlo- o r d i c h o t n n i z a t i o n s . Tbe j o i n i n q assumntion.
The n r o ' 3 a h i l i ty o f s u h i e c t s ' r e s n o n d i n ? w i t h c a t e -
?or!/ j fir c a t e a o r y k i s e n u a l t o t h e n r o h a h i l i t v o f r e s n o n d i n o 1,:itl.l r v 11 i f c a t e g o r y h r e n l a c e s t h e c a t e c l o r i e s ,i and k .
cateyo-
I n o t h e r words: i f i n
one resncnse f o r m a t , t h e sut).iect can cfioose between e . 7 .
'disaorce'
,
'aflree'
and ' s t r o n q l i a C r e e ' , and i n a n o t h e r response f o r m a t he t a r ckccse between 'disaCree'
, and
' a c l r e e ' , t h e n r o h a b i l i ty o f resnonrlin? ' a v r e e ' i n t h e l a t t e r
case 1411 e o u a l t h e sun of t h e n r o h a h i l i t i e s o f resnondino ' a q r e e ' o r ' s t r o n o l y a o r e e ' i n t h e f o r n e r case. t i o n ria!/
V i l d and o b v i o u s thouph t 3 i s assumn-
seen, i t has t h e s t r n n r i n n l i c a t i o n t h a t t h e r e i s no d i f f e r e n c e
between d i c h o t o m i z a t i o n b e f o r e t h e f a c t and d i c l i o t o m i z a t i o n a f t e r t h e f a c t . T!iis i o i n i n g assumntion i s o f course t r i v i a l l v t r u e i f resnonses a r e d i c h o tomized a f t e r t h e f a c t . The n r i n c i n l e o f n o n - i n t e r f e r e n c e .
The p r i n c i p l e o f n o n - i n t e r f e r e n c e can
be f o r n u l a t e d i n terms o f w h e t h e r t h e s u b j e c t oarameter, F ; , i n some measurew n t n o d e l , '!, w i t h a d d i t i o n a l narameters, R , r c f e r r i n a t o i t e m s and/or cat a q o r i e s , i s a f f e c t e d by t h e o a r t i t i o n i n ? o f t h e resnonse continuum. P and P ' be two o a r t i t i o n i n m o f t h e resqonses t o a s e t o f i t e m s .
Let
Nan-inter-
ference can he d e f i n e d i n a forlrral !.ray as f o l l o w s .
I f P satisfies
' " ( c , ~ )and ,
Pi satisfies V'(<',e'),
non-interference i s
d e f i n e d by E = t ( F ; ' ) , i n v h i c h t i s an admissable t r a n s f o r m a t i o n o f t h e s c a l e o f F;. T h i s d e f i n i t i o n does n o t e x n l i c i t l y r e q u i r e t h a t r! and V ' a r e t h e same rrodel. !!owever,
< - F ix: c c n s i d e r t h e i t e m response model as a f o r m a l i z e d t h e o r y o f
t h e d a t a o e n e r a t i n o process, i t w o u l d r u n c o u n t e r t o t h e v e r y p r i n c i o l e o f
419
Thepolychotomous Rasch model
tlieorizinq, i f we vere forced to assum t h a t d i f f e r e n t resnonse models ( a n a r t from -ere ct-snyr i r the i tem- and catenorv-narameters) waul+ aonlv t o an!, n a r t i c u l a r p a r t i t i o n i n g of the response continuum. I n t h a t case, t'ie q t r u c t u r e of the m o d ~ l srl a n d ! I ' would denend i n an -a d hoc way on the n a r t i c u l a r resqonse f o r r a t . %,i f 1; i s the unic'imensional nolychotomous ?as& model '?oldin0 for P , a n d D' i s a dichotomization, we exqect the dichotomous 9asch model t o hold f o r I". Fornal d e f i n i t i o n of dichotomization. Qorlse rode1 of the forp:
Cunnose
have a nolychotomous r e s -
r(nlv,i) = f(v,i,o) where v = 1 , . .., II r e f e r s t o s u b j e c t s , i = 1,. . . , k r e f e r s t o i tens , a n d a = 1,. .., m r e f e r s t o resccmse categories and
n ( n l v , i ) denotes the nrobability t h a t s u b j e c t v chooses cateoory g a t
item i . a dichotomization D(h]is meant a j o i n i n s of resnonse a l t e r n a t i v e s such t h a t a l t e r n a t i v e s g=l,. . , h - l , are coded as ' n o ' o r 'negative' , and a l t e r natives o=h , . . . , m y are coded as ' v e s ' o r ' D o s i t i v e ' .
Ry
.
Under t!ie Joining Assumntion, non-inter-erence i s s a t i s f i e d i f a n d only i f
*
P ( t l v , i , n [ h l \ = f (E;,i,h)
with
'.;hein?
=
c f(cv,i,c)
(g=h,..,,m)
(2)
an admissable transformation of E V .
Furthermore , we use the fol l~ol~linn notation: A =
- 1, i f subject v resoonds w i t h cateaory h a t item i
avih
5 0,
a D[h] vi I
= =
otherwise,
1, i f s u b j e c t v resnonds w i t h e i t h e r catecory o = h , . a t item i , 0 , otherwise.
. . ,m
420
P.C. W.Jansen and E . E . Ruskdm
1. TtlF VLYCtlllTO!'"IIS 1 . 1 . TIiE
PA5CV "nPEL (P?")
5pFTIFIC O~,lFCTT\/TT?
9 P I Y C I P L . F !V
The Rasch p o d e l i s based on t h e q r i n c i q l ? o f s n e c i f i c o h ! e c t i v i t \ t
i n coma-
r i s o r s . . l t a n v a r s d i f f i c u l t t o F i v e an e x a c t d e f i n i t i o n o f t ! i i s Q r i n c i q l e . I t i s concerned 1 4 t h c n m a r a t i v e n u a n t i t a t i v e s t a t e w n t s ahotit Y r o o e r t i e s o f
ohjects.
}!ere, !.re n r e s e n t i t f i r s t i n o r d i n a l t e r m : i n d e t e r m i n i n g whether
e . a . orle s u b j e c t ' s e t t i tude t o v a r d s s o c i a l i s m i s more q o s i t i v e tl7an a n o t h e r
s u b j e c t ' s , v e ask f o r t h e i r a o s i t i v e o r n e g a t i v e resnonses towarr% a samnle
o f i s s u e s from the n e r t i n s n t u i i v ? r s p o f c o n t e n t . i s said t o
IT
The c o m a r a t i v e i4r;:;mt
s n e c i f i c p l l v n t l i e c t i v c ? !.!hen i t i s i n d e n e n d e n t o f t ? e i a r t i c u -
l a r sanole of issues.
Tn t e r n s o f n r o l > a b i l i s t i c i t e v resnnnse t $ e o r v , sne-
c i f i c o h i e c t i v i t y weans t h a t t h e e s t i m a t e d d i f f e r e n c e ( o r r a t i o , deoendinn on t h e s c a l e t v n e ) o f s u b i e c t s ' l a t e n t t r a i t scores i s i n d e n e n d e n t o f t!ie (unknown) i'tern ? a r a m t e r s . eni;iv3lrlrlt
t3
I n t h i s f n r w l a t i o n , s n e c i f i c o 4 , j e c t i v i t v i:.
saipt.? i . i + v n
'~\nc.o. i + <'.wild he
v o t e d t + a t samrlle indenen-
dence i s n o t m e r e l y an e m i r i r a l m a t t e r , as i t seems t o he c o n s i d e r e d hy I.or.I((:?X?), :;ure,
34 senoJ, b u t a t h e o r e t i c a l q r o a e r t v o f t h e i t e m resnonse s t r u c -
a n d as such exnressed i n t h e f o r m
o r n o t data c o r r 3 s n n n d t o t ' i e m d e l i s !q
nrob;l'.iiis ;ic
i n r A . i \ r i*:'r
i13:i:
r?s.?on<+
j l:p-.i ?,',a$
;ii:$ip::s'
0.1.
qf
t h e i t e m ri!sncnse noJe1.
'4hether
ccwrse a r l a t t e r .If e m i r i c ? l t e s t .
i;'i?,?r,v,z . ' ) ! ? i.jr!.?~:?q&?:?
7r j , ? _ "ii 4,-
qarzp-.t(ys c.?~: hn est.ip;lticl i i & o e n - ! e l i t
of
3 e i t e r ? o a r a n e t e r s , and v i c e versa, and t 3 i s i n t u r n m a n s t h a t -there e x i s t s u f f i c i e n t s t a t i s t i c s f o r t h e it e v o a r a w t e r s ~ d l i ca r~p ipiieoiind,.nt suL..ject parat;eters, an:! vicrt v e r s a .
F ~ o m.:+is
'ro,;' t h e
re?iiirernnnt f n l l n v s t ? a t tl:e
i t e m rps?nnse model must helcmn t n t ' i e ; ? x i n n e r l t i a l f v i i l y v i . t h s ? a a r i ' ! l P l a ravtsrs
(cF. C i s t h e ? !l?!4!,
Vivsn a r l i Ffer?n: + r i v a t i m
>.!i-??O\.
? o ~ i . a i . (1,?:?3\ a n + . i ? . ~ s ? n(1.-..,.3) ' 1 a v ~
o f t l i e ! m i d i v s n s i o n a l dichotonnus r?ascil model,
n o t based on s t a t i s t i c a l c o n s i d e r a t i o n s ( s e e Roskar! R .Tansen (1933)).
2.::. T!lE "!ILTI31 'F!i%IQ.IAL A!lO i?E ,ill1 lI:!~VSlW'l. P...'' ( P V ' Pnci IIP .'..:)
: asch ( l 9 5 l ) h.is d p r i v e d t h e f o l io:.ring sr,Eci f i c a l l v n h . i e c t i ve itern r9soonss model f o r aol,yc$ntornus d a t a . ROUS
!?ascl7 model (f!Pn''),
bv a aarameter v e c t o r .
I t i s c a l l e d t 3 e mu1 ti d i n e n s i o n a l m l y c h o t o -
s i n c e each i t e m and eac? suh.iect i s c h a r a c t e r i z e d
42 I
The polycliotomous R u c h model
(4)
where
v = 1 , . . . , I! refers t o suhjects, i = 1, ... , k refers t o items, a n d q,!I = 1,..., m rcfnrs t o resqo'ise ca.:-nCrrips.
I t should be noted t l l a t the nodel assumes nominally the sane resnonse catepories with a l l items. Each resnonse cateflory reflects a l a t e n t t r a i t of i t s own. Therefore, .!e feel t h a t this model should rather he called a m u l t i t r a i t model t h a n a rul ticXmensicna1 model. The W R ! ! i s estimable vri t h the follovinp identifying restrictions (Fischer ( 19741, 432) :
c(m' v
= u(m) =
i
o
for a l l v a n d i (5)
for a l l h
I:P) = 0
i i
=
1, ...,P
The s u f f i c i e n t s t a t i s t i c s for the suhiect aarareters, E L h ) , in the rTRr,! are : avih
=
'vh
vhere Avh i s the frerruency of suhject v resoonding with cateaory h . Ifthe
cih)
and o i h ) are linearlv deaendent, the following unidimensional reduction i s vossihle (Kasch (lq[il), 33;)):
P.C.U'. Jmueii oitd E.E. Roskain
422
?rl t'ip
unidjvensioniil as
1.11.1
1
2s
in
?!I$?
r v l t i r l i r l r n s i n r l a l rinr'rtl, t l i e r?s?(.rn-
si? c a t c y r i r s a r p t i c s a w I.iit$ a l l it w s . 1% s h o u l d he n o t e d t ! ? a t t+r! r i i c h o t o m u s Oasc!i model i s r e r e l v a s n e c i a l ca-
se o f t h e I P P " .
I f n = i , en. ( 3 ) and ( 3 ) reduce t o t h e f a m i l i a r d i c h o t o -
mnus Rasch podel,.
The s u f f i c i e n t s t a t i s t i c f o r t h e s u h i e c t n a r a w t e r i n t h e L ' P W i s :
T'lerefore,
29).
The
t h e Q narameters a r e a l s o c a l l e d s c o r i n g parameters (Rasch (1961) I!,
narameters can he c o n s i d e r e d l o c a t i o n q a r a m e t e r s .
o f t h e n a r a r w t p r s i s c l a r i f i e d bv F i s c h e r ( ( l 9 7 4 ) , 4371,as
The meanino
follows.
The q r e f e r e n c e o f s u b j e c t v f o r t b e resnonse e x n r e s s c d hv c a t e q o r y h i s a f u n c t i o n o f t h e s u h i e c t n a r a m t e r , and two o t h e r parameters w h i c h c h a r a c t e r i z e t h e resoonse c a t e g o r v .
The n a r a w t e r $ ( h ) w e i n h t s t h e s u b j e c t narameter
d i f f e r e n t l y f o r d i f f e r e n t catenories.
The parameter $ ( h ' i n c r e a s e s o r de-
creases t h e s u b j e c t ' s tendency t o resqond by c a t e q o r v il. 50, i t exoresses t h e ' e a s i n e s s ' o r t h e anoeal o f t h a t categorl/ i n i t s e l f .
A s n e c i a l case a r i s e s when t i l e s c o r i n o o a r a w t e r s a r e o f t h e f o r m o ( h ) = h - l , rwanina e q u i d i s t a n t scorino. A n d r i c h ' s (1978) P a t i n ? Rascll Yodel i s a s o e c i a l case o f t ? e WW1. c o n t a i n e q u i d i s t a n t s c o r i n q as a sub-case.
Roth
llhether o r n o t data s a t i s f y
t $ e e n u i d i s t a n t s c o r i n q c o n d i t i o n i s a n e m n i r i c a l m a t t e r , b u t Aiidersen (1977) has shown t h a t e n u i d i s t a n t s c o r i n a p a r a w t e r s can be d e r i v e d as a m e t h o d o l o q i c a l recluirement, i f we want a s u f f i c i e n t s t a t i s t i c f o r t h e subi e c t narameter w h i c h i s a l s o n u a s i - c o n t i n u o u s . T'ie c a t e g o r y c h a r a c t e r i s t i c resqonse curves (CCC's) o f t h e U P W , f o r t h e s q e c i a l case o f e a u i d i s t a n t s c o r i n o parameters, a r e shown i n F i q u r e 1.
The polycliotowious Rasclr model
423
Fiaure 1 Category c h a r a c t e r i s t i c curves o f an UPRt1 w i t h $ ' s equal t o 0, 1.5,
2, 1.5 and 0 and w i t h $ I s equal t o 0, 1, 2, 3
and 4 ( e q u i d i s t a n t s c o r i n q ) ; ui i s equal t c
@.
2.3. THE UPRM AND DICHOTCIf?IZATION OF A G I V E N DATA Sunpose some m u l t i c a t e g o r y d a t a s e t , A , s a t i s f i e s t h e UPRM. d i c h o t o m i z a t i o n o f A, where avih
and A [ h l a r e d e f i n e d by e l . ( 3 ) .
Theorem: I f A s a t i s f i e s t h e UPRM f o r some $ ( 9 ) 2 $(q-l), i . e .
dered ~ ~ ~ ~ , g = l , . . , , h , . . . , m ,
L e t A[h] be a
f o r weakly o r -
then t h e dichotomized d a t a s e t A[hI w i l l s a t i s -
f y a dichotomous Rasch model i f and only i f = a for
1 5 g 6 h-1
$(d = b for h s g f m (where a=O and b = l as i d e n t i f y i n g c o n s t r a i n t s ) and t h e d i chotolni zecl d a t a w i l l s a t i s f y :
,
414
where
proof, I'e assume nolychotomous d a t a v h i c h s a t i s f y t h e I'PRV.
A f t e r dichotonli-
z a t i o n , these d a t a w i l l s a t i s f y t h e dichotomous Rasch model, if and o n l y i f ( c f . m . 2): P(tlv,i,Q[hl)
=
z p(qlv,i),
?=h,
...,rn
(14)
0
whew P(glv,i) as e n . Sufficiency.
( e ) with
i s as i n e l . ( 8 ) , and P ( + ( v , i ,D[h])
i s o f t h e same f o r m
m=2, e x c e n t f o r an adwissable s c a l e t r a n s f o r m a t i o n .
The n r o o f o f s u f f i c i e n c y i s s i m l e : s u b s t i t u t i n ? ( 1 1 ) i n ( E )
leads i m m e d i a t e l y t o ( 3 2 ) and ( 1 3 ) . Uecessity.
The o r o o f o f n e c e s s i t y i s more c o m o l i c a t e d .
I n o r d e r t o show
t h a t (11) i s necessary i t s u f f i c e s t o show t h a t i t i s necpssarv i f t v = o (P,).
i I n t h a t case, w . ( 1 4 ) i m d i a t e l y l e a d s t o ( 1 3 ) ; so t h e l a t t e r i s
necessary.
O b v i o u s l y , (11) i s necessary f o r m=2.
in
Next assume t h a t a new
c a t e g o r y i s added t o t h e ' n o ' - s e t o r t h e ' y e s ' - s e t o f a d i c h o t o m i z e d d a t a s e t , and assume t h a t ( 1 4 ) h o l d s .
I t can then be shown t h a t t h e s c o r i n a para-
meter o f t h e added c a t e g o r y i s e i t h e r equal t o a, o r enual t o b ( t h e s c o r i n q parameter o f t h e ' n o ' - c a t e n o r v ,
o r o f the 'yes'-cateoory,
resg.).
F o r de-
t a i l s o f the p r o o f , t h e r e a d e r i s r e f e r r e d t o Jansen ( 1 9 8 3 ) .
The c a t e g o r y c h a r a c t e r i s t i c curve5 f o r t h e case o f a d i c h o t o n i z a b l e IPRV a r e shown i n F i g u r e 2 . If t h e c o n d i t i o n o f t h e t'ieorem i s met, t h e parameter nh-l able.
i s not identifi-
The d i c h o t o m i z a t i o n e f f e c t s a s h i f t o f t h e l a t e n t s c a l e o f s u b j e c t s
and i t e m s .
T h i s s h i f t i s i l l u s t r a t e d i n F i g u r e 3, f o r two h y a o t h e t i c a l d i -
chotomi z a t i o n s .
425
The polychotoinous Rascli model
a -1
a
1
I
1
Figure 2 Catecory c b a r a c t e r i s t i c curves of an UPRI! with ( J ' S equal t o 0, 1.5, 2 , 1.5 and 0 , and $ ' s equal t o 0 , 0 , 1, 1, 1. u i s eaual t o 0 . i
Figure 3 ResDcnse curves f o r item i i n dichotomizations D[hl
, and
D[h+ll
426
P.C.W. /amen and E.E. Roskam
A c l o s e y r e l a t e d theorerr can b e f o r v u l a t e d w i t h r e s n e c t t o t h e . j o i n i n ? o f I f d a t a s a t i s f y a UPR!'
any two c a t e g o r i e s .
f o r m c a t e g o r i e s , and two (se-
man ti ca l v a d j a c e n t , o r n o t ) c a t e g o r i e s , sav j and 4 , a r e j o i n e d ' a f t e r t h e f a c t ' t o make a new c a t e g o r v , say h, and i f a ~J?!Y f o r ri-1 c a t e n o r i e s and w i t h t h e same s u b j e c t and i t e m narameters i s t o h o l d f o r t h e d a t a a f t e r j o i n i n ? .j and F , then t h e n a r a w t e r s s h o u l d s a t i s f y :
where
l,!e w i l l show t h a t en. ( 1 5 ) can o n l y h o l d w i t h $ ( h ) indenendent o f t v and 3.
if , ( j )
= 4(k).
Suaoose eq. ( 1 5 ) h o l d s , t h e n , t a k i n g 5, = ai,
i t follows that:
U s i n g eq. ( 1 7 ) , we f i n d t h a t en. ( 1 5 ) i m n l i e s :
+ exn +(k)) e x n i $ ( h ) ( t v - a i ) j
(exo
=
(18) ex9 ~ ( j ) ( e x o ( ~ ( j ' ( ~ v - ~ i )+) )exn $ ( k ) ( e x o ( o ( k ) ( t v - 01. ) )
The l a t t e r m i g h t be t r u e if$ ( h ) i s some f u n c t i o n o f 6(k).
Snecifically,
t h e values o f NOW l e t 5,
fcr short.
-
,(<'I, $(k),
+ ( h ) s h o u l d be i n d e p e n d e n t o f 5, and ui,
and
regardless o f
and 0( k ) . ai
= x,
$ ( j ) = a, $(k)= h , and ( J ( J ) / ( $ ( J ) + $ ( ~ ) ) =
D,
and q = l - n
Then en. (18) can be w r i t t e n as:
The r i g h t member o f (19) s h o u l d be independent o f x, i n n a r t i c u l a r i f p=?=.5 D i f f e r e n t i a t i n g w i t h r e s n e c t t o x, and n u t t i n g t o zero, we f i n d t h a t ( 1 5 ) = $ ( k ) . r n d i f t h i s c o n d i t i o n i s met,
h o l d s i f and o n l y i f ? ( h ) = ,(jl = 6(k.).
Consequently, j o i n i n g any two c a t e g o r i e s ' a f t e r t h e f a c t ' w i l l s a t i s f y t h e UPRM w i t h one c a t e g o r v l e s s and w i t h t h e same s u b j e c t and item oarameters,
i f and o n l y i f t h e s c o r i n a narameters o f t h e j o i n e d c a t e g o r i e s a r e e q u a l .
Repeatedly j o i n i n g c a t e p o r i e s , we f i n d t h a t d i c h o t o m i z e d d a t a s a t i s f y t h e Rasch model w i t h t h e same s u b j e c t and itern parameters i f and o n l y i f t h e S c o r i n g o a r a w t e r s t a k e on o n l y two d i f f e r e n t v a l u e s .
421
The polychotomous Rasch model
Obviously, c o n d i t i o n (11) imnoses s t r o n g r e s t r i c t i o n s on t h e p o s s i b i l i t y o f d i c h o t o m i z i n g resoonses which s a t i s f y t h p UPV?. D i c h o t o m i z a t i o n does n o t i n t e r f e r e w i t h s u b j e c t measurement i f and o n l y i f t h e s c o r i n g parameters t a k e on o n l y two d i f f e r e n t values, and i f so, t h e r e i s o n l y one d i c h o t o m i z a t i o n which does n o t i n t e r f e r e w i t h s u b j e c t measurement.
I n fact,. condi-
t i o n (11) means t h a t t h e resnonse c a t e q o r i e s b e l o n g t o two orouos, w i t h i n each o f which t h e r e i s no d i s c r i m i n a t i o n , i.e. t h e s u b j e c t ' s o r e f e r e n c e f o r one c a t e g o r y o v e r a n o t h e r i s indenendent o f t h e s u b j e c t , b u t s o l e l y d e t e r mined by t h e a t t r a c t i o n o f these c a t e a o r i e s i n themselves.
2.4. THE I M P L I C A T I D N S OF THE POLYCHOTflrlnUS
RASCM llnDFL
As can be seen f r o m t h e f o r m u l a t i o n o f t h e polvchotomoirs rasch model, i t i s f o r m a l l y analogous t o R r a d l e y
-
Terry
-
L u c e ' s c h o i c e model f o r t h e c h o i c e
o f one a l t e r n a t i v e f r o m a s e t o f resoonse a l t e r n a t i v e s .
It implies that
the n r o b a b i l i t y o f a c e r t a i n choice i s a f u n c t i o n o f the r e l a t i v e strengths o r u t i l i t i e s o f each a l t e r n a t i v e .
Vhen a l t e r n a t i v e s a r e combined, e.g. by d i c h o t o m i z i n g t h e s e t o f a l t e r n a t i v e s , t h e u t i l i t y o f a combined s u b s e t o f a l t e r n a t i v e s would be equal t o t h e sum o f t h e u t i l i t i e s o f a l t e r n a t i v e s i n t h e subset.
L e t h r e f e r t o t h e u n i o n o f two a l t e r n a t i v e s , j and k .
Eaua-
t i o n (15) expresses o u r i n t u i t i v e i n t e r p r e t a t i o n o f r e n a r t i t i o n i n g t h e r e s oonse s e t , and i t j u s t i f i e s i t s d i c h o t o m i z a t i o n : when we c o l l a p s two o r more resnonse a l t e r n a t i v e s i n t o a s i n g l e one, we t a k e i t t h a t t h e p r o b a b i l i t y o f choosing t h e combined one i s equal t o t h e sum o f t h e p r o b a b i l i t i e s o f choosing any o f t h e a l t e r n a t i v e s e n t e r i n g i n t o t h e combination. t h e r e s u l t i n g Xivh n a l xiV,
However,
i s no l o n g e r decomposable i n t h e same way as t h e o r i q i -
unless t h e s c o r i n g parameters a r e e q u a l .
A d i f f e r e n t problem concerns t h e e f f e c t o f r e n l a c i n g a s e t o f resDonse
OD-
t i o n s by a d i f f e r e n t s e t o f resoonse o p t i o n s , f o r i n s t a n c e by o m i t t i n g one response o p t i o n , e.g.
' d o n ' t know'.
A t f i r s t glance, i t would seem reason-
a b l e t o assume t h a t t h i s would n o t a f f e c t t h e parameters d e t e r m i n i n g t h e response D r o b a b i l i t i e s o f t h e r e m a i n i n g o p t i o n s ; i t would mean t h a t response P r o b a b i l i t i e s a r e r e d i s t r i b u t e d o v e r t h e o p t i o n s i n accordance w i t h t h e c o n s t a n t r a t i o r u l e i m o l i c i t i n t h e S T L s t r u c t u r e o f t h e polychotomous Rasch model.
T h i s r u l e would be v i o l a t e d , however, i f t h e sub.ject would
r e a c t t o t h e omission of an o D t i o n by choosing an o p t i o n n e x t t o i t . But a d d i n g o r d e l e t i n g response a l t e r n a t i v e s i n t h e resoonse f o r m a t o f com-
mon a t t i t u d e q u e s t i o n n a i r e s does n o t i m p l y addina o r d e l e t i n g p o s s i b l e
42 a
P.C. W.J a m e n and E . E . Rosbam
resnonses, h u t r a t h e r m a n s t h a t sepnents ; n t k . ~resoonse continuum a r e s o l i t uo o r a r e combined. An obvious, b u t e a s l l y i g n o r e d o r o n e r t y o f t h e UvRrd i s t h a t i t does n o t i n any
WSY,
i m l y an a p r i o r i o r d e r i n o o f t h e resoonse a l t e r n a t i v e s .
{Jhile
t h i s i a n o r i n a resnonse o r d e r i s o n l y n a t u r a l i n t h e m u l t i d i m e n s i o n a l model, the u n i d i m e n s i o n a l model, i n t + i s r e s o e c t , i s n o t d i f f e r e n t f r o m t h e m u l t i d i w n s i o n a l model.
Due t o t h e u n i d i m e n s i o n a l r e d u c t i o n i n eq. ( 7 ) , t h e o r -
derinc. o f resoonse a l t e r n a t i v e s i s e s t a b l i s h e d t h r o u o h t h e a n a l v s i s o f t h e data.
I n vie):, o f t h i s , t h e model cannot t a k e i n t o account anv a o r i o r i
e s t a b l i s h e d o r o p e r t i e s o f t h e response a l t e r n a t i v e s , and would thereCcr'e n c t n e c e s s a r i l y be c o m n a t i b l e w i t h j o i n i n g c a t e g o r i e s on t h e b a s i s o f s o r e
a n r i o r i arooerty. I n o u r assumption of j o i n i n o resoonse c a t e g o r i e s we have assumed t h a t r e -
p a r t i t i o n i n q o r dichotomization 'before the f a c t ' i s n o t d i f f e r e n t from r e o a r t i t i o n i n q o r dichotomization a f t e r the f a c t .
The J o i n i n n AssumDtion
i s t r i v i a l l y s a t i s f i e d i f o i v e n d a t a a r e dichotomized. UPP" i s n o t c o m o a t i b l e w i t h i t .
Ve found t h a t t h e
I t may, however, be t h a t t h e UPRM i s comna-
t i h l e w i t h d i f f e r e n t o a r t i t i o n i n n s o f t h e resnonse continuum, i f t h e y a r e Dresented ' b e f o r e t h e f a c t ' , a n d
the subject i s free t o d i s t r i b u t e h i s resOh-
ponses d i f f e r e n t l y when a d i f f e r e n t resDonse f o r m a t i s o f f e r e d t o him.
v i o u s l y , when resnonse o n t i o n s a r e d e l e t e d o r new ones added, t h e UPRW o r e d i c t s t h a t t h e o r o b a b i l i t i e s o f resnonses f a l l i n g i n t o c a t e q o r i e s w h i c h a r e ccmFron t o two resoonse formats, s a t i s f y t h e c o n s t a n t r a t i o r u l e , u n l e s s a p a r t i a l change i n t h e resnonse f o r m a t would a f f e c t t h e narameters o f a l l resnonse c a t e g o r i e s .
2.5. CASES '+IERE UPRfl MICHT APPLY From t h e f o r e c o i n g , bee conclude t h a t t h e UPRIf may he v a l i d i n those cases where resDonse a1 t e r n a t i ves a r e n o t oraded, b u t a r e d i f f e r e n t q u a l i t i e s o f resnonse,
r e d u c i b l e t o a unidimensional l a t e n t t r a i t .
This i n t e r p r e t a t i o n
i s d i r e c t l y i n l i n e w i t h t h e d e r i v a t i o n o f t h e UPR3 f r o m t h e HPRY.
By i n -
t u i t i v e reasoning, we would b e l i e v e t h a t t h e UPRV o n l y a n o l i e s t o response a l t e r n a t i v e s which do n o t s e m a n t i c a l l y o r l o g i c a l l y e x c l u d e one another, o r ,
-
i n o t h e r words, where t h e i t e m cum resnonse c a t e g o r i e s m i g h t be seen as t h e c o m b i n a t i o n o f s e v e r a l yes/no q u e s t i o n s i n t o a f o r c e d c h o i c e f o r m a t . f o l l o w i n q m i g h t be an example:
The
429
The polychotonious Rarcli model
I f you were asked unexoectedly t o speech on a s o c i a l occasion, would y o u
. ..
t r y t o evade t h e anpeal
f e e l nervous feel flattered feel challenged
.
Each o f t h e p o s s i b l e responses can be a b i n a r y i t e m i n i t s e l f , which i s l o q i c a l l y i n d e p e n d m t f r o r the.oiher-5. (e.g.
Ii'hether o r n o t a u n i d i m e n s i o n a l t r a i t
'noise' o r 'assertiveness') underlies the n r o b a h i l i t v o f a a o s i t i v e
response i s an e m p i r i c a l m t t e r .
Resides, one would n o t f e e l j u s t i f i e d i n
combinin? resDonse c a t e g o r i e s because o f t h e i r obvious o u a l i t a t i v e d i f f e r ences.
I n t h i s example, i t can a l s o be e x o e c t e d t h a t o m i t t i n q one a l t e r n a -
t i v e f r o m t h e m u l t i n l e choice f o r m a t would l e a d t o a r e d i s t r i b u t i o n o f t h e resoonse p r o b a b i l i t i e s a c c o r d i n q t o t h e c o n s t a n t r a t i o r u l e . 3. CWCLUSIONS ANP C n W E O U E N C E Z Itre have shown t h a t d a t a s a t i s f y i n g t h e u n i d i m e n s i o n a l oolychotonous Rasch model w i t h a c e r t a i n number o f c a t e o o r i e s , w i l l n o t do so when .responses categories are joined.
I n o t h e r words: t h e c o m p o s i t i o n o f t h e response
f o r m a t a f f e c t s what i s b e i n g m a c u r e r ! i r a l i o n - t r i v i a l manner.
I n so f a r
t h e c o m v o s i t i o n o f t h e resnonse f o r m a t i s a r k , i t . r a r y , t h i s i m n l i e s t h a t U P W c a n n o t be a m e t h o d o l o g i c a l sound i t e m response model.
The same i s t r u e f o r
models which a r e s p e c i a l cases o f t h e U P R M , such as A n d r i c h ' s (1978) R a t i n g Rasch Model, and b l a s t e r ' s (1982) P a r t i a l C r e d i t Model.
The o n l y p o l y c h o t o -
mous model we have f o u n d i n t h e l i t e r a t u r e which. appears t o be i n v a r i a n t under j o i n i n g o f graded response c a t e g o r i e s , i s Samejima's (1969) L o c i s t i c D i f f e r e n c e Model.
Furthermore,. t h e well-known e q u i d i s t a n t L i k e r t - s c o r i n g
of araded responses s u f f e r s f r o m a s i m i l a r drawback.
These and r e l a t e d
i s s u e s w i l l be more f u l l y discussed i n a f o r t h c o m i n g paper. REFERENCES
[l] Andersen, E.B.,
Conditional inference f o r m u l t i o l e choice questionnai-
r e s , B r i t i s h J o u r n a l o f Mathematical and S t a t i s t i c a l Psychology, 26 (1973) [2]
, 31-57.
Andersen, E.B.,
S u f f i c i e n t s t a t i s t i c s and l a t e n t t r a i t models, Psycho-
m e t r i k a , 42 (1977)
, 69-81.
P.C.W. Jawen und E X . Koskam
430
Andricb, P . , F r a t i n q formulation f o r ordered resnonse c a t e g o r i e s , F s y c k v e t r i k a , 43 (10781, 561-573. Rerq, I .P. ?, Pananort, c . ! . ,
Pesaonse bias i n a n unstructured question-
n a i r e , The Journal of PS\Icholoov, 3s (1754), 475-?81. Coomhs, C1 . I t . , Theory o f d a t a , t k w Vcrk, ';:lev (19541, r e n r i n t e d , h n nrhor: !.!athesis Dress (1976). Fischer, C.H., Einfiuhrun? in d i e Theorie nsvcholovischer t e s t s , Eern tiuher (1974).
I8
I91 110
[
13
Ila 113
I16
C u t t m a n , L., The develoament o f nonmetric saace a n a l y s i s ; a l e t t e r t o John Ross, !ktl t i v a r i a t e Rehavioral Research, 2 (1?67), 71-82, Renrinted in J . C . Linnoes, Edw. E . "lskam, R I . Roro ( E d s . ) , ceometric reoresentations o f Relational D a t a ; Peadings i n "ultitimensional Scalinn, Pnn Arbor: b t h e s i s Press, 237-248. Heerden, (1. ;Jar & Hccqstraten, J . , Resnonse tendency in a questionnaire wi t h n u t q u e s t i o n s , Anolied rs.vchologica1 I'easurement, 3 (1979). Hoonstraten, J . , De machteloze nnderzoeker, !!eonel, Boom (1979). Jansen, P.G.!.!., R.asch analysis o f a t t i t u d i n a l d a t a , Doct. d i s s . Univers i tv o f Ni jmeaen/Pi.iks Psycholoaische Dienst, Den Haaq (1983). L i k e r t , R . , I\ technique f o r the measurement of a t t i t u d e s , Archives of Psychology, 140 (1932). Linqoes, J . C . , The r a t i o n a l of the Guttman-Linpoes nonmetric s e r i e s : a l e t t e r t o Ooctor P h i l i a Runkel, t l u l t i v a r i a t e qehavioral Research, 3 (1968), 495-508, Penrinted i n ,1.C. Lingoes, E d w . E . Roskam, & I . Borq ( E d s . ) , Geometric Reoresentations of Relational Data; Readinas in l?ul-tidimensional Scalinn, Ann Arbor: Vathesis F r e s s , 249-261. Lord, F.Y., hnalications o f item response theory t o ? r a c t i c a l t e s t i n ? n r n b l e m , H i l l s d a l e , Lawrence Frlbaum (1980). !lasters, G.N., A Rasch model f o r p a r t i a l c r e d i t scorinq, Psychometrika, 47 (1982), 149-174. On general laws and the meaning of measurement i n psycholony, ?roceedinns of the Fourth Rerkeley Symnosium on Vathematical S t a t i s t i c s and P r o b a b i l i t y , June 20 - July 30, 1960, Rerkeley and Los Anpeles, \ h i versi t v of C a l i f o r n i a Press (1961) ,321-333. Rnskam, E & . E . , Allaemeine Datentheorie, In ti. Feger b H . Bredenkamo, Hessen und Testen, Sand 3 der s e r i e Forschungswthoden der Psychologie der Enzyklopadie der Psvcholoqie, Gottingen, Hogrefe, 1-135.
Rase'?,
?.,
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TRENDS IN MATHEMATICAL PS YCHOLOG'Y E. D e p e f arid J. Van Bu enlrnut (editors)
B. V. (North-Holland), 1984
@Elsevier Science Publirf%
433
C 0:1P nF' F NT ANAL V T I S 0F T PC NS I T I P PI
P R f~P B I LIT I E I T S ApPLICATInN Tn PTISnPJFR'q nILFHF'A
P FIE
Kullervo Qainio llniversit\i o f Helsinki
The m a t r i x o f t r a n s i t i o n n r o h a h i l i t i w i s e x m i n e d as a w e i o h t e d sum o f c o m o n e n t m a t r i c e s .
A least
square method f o r d e t e r m i n i n a t h e w e i n h t s i s shown. The method o f c(rmonPnt a n z l y s i s i s a n n l i e d t o t h e r e s u l t s o f P r i s o n e r ' s Dilemma saw.
The n l a y e r ' s
exnectations f o r the o t h e r n l a y e r ' s stratefly choices are analyzed b y u s i n a 5 m o t i v a t i o n a l comnonents:
1) randov e x n e c t a t i o n , 21 e x p e c t i n n c o m n e t i t i v e beh a v i o r , 3 ) r e p e a t i n q t h e e x p e c t a t i o n s , 4) e x n e c t i n q "tit - f o r - t a t "
behavior.
b e h a v i o r , and 5 ) e x o e c t i n n r e n e a t e d
The comnonents e x o l a i n 55-82w o f t h e va-
r i a t i o n o f exnectations.
-
The n l a v e r ' s s t r a t e n v
c h o i c e s a r e analyzed by u s i n g 5 m o t i v a t i o n a l comnon e n t s : 1) random c h o i c e , 2) c o n e t i t i v e c h o i c e , 3 ) r e o e a t i n o t h e choice, 4) choosing hv t h e " t i t - f o r t a t " w t h o d , and 5) choosing i n an ' e n u a l i t a r i a n " way a c c o r d i n o t o t h e e x n e c t a t i o n s .
The comnonents
e x p l a i n 54-793 o f t h e v a r i a t i o n o f t h e s t r a t e q y choices
.
The w e i g h t s o f comnonents o f i n d i v i d u a l t r a n s i t i o n p r o b a b i l i t i e s a r e a l s o determined and t h e i r r e l i a b i l i t i e s comnuted. o v e r .80.
The h i g h e s t r e l i a b i l i t i e s a r e
The means and s t a n d a r d d e v i a t i o n s o f t h e
i n d i v i d u a l w e i a h t s o f comnonents a r e a l s o n i v e n . The general s u i t a b i l i t y o f c o m o n e n t a n a l y s i s i s b r i e f t y discussed.
-A
l i s t o f c o m u t e r nroorams
a v a i l a b l e i s also aiven.
434
K . Rainiu
1. CnllPnNENT ANALYSIS nF T!?PNSITIW P V S , V I L I T I F 4 1.1. AFI 1NTPrDIICTO"v FXAr'PLF L e t us assume t h e f o l l o w i n o t r a n s i t i o n n r o b a h i l i t i e s f r o m s t a t e s sl,
s?, s 3
and s o a t t i m e n o i n t t t o t h e s t a t e s x1 and x 2 a t t i w n o i n t t+l: t+l
x1
x2
.83 I t i s n o s s i b l e t o t h i n k o f t h i s m a t r i x as t h e w e i o h t e d
SUP
o f s e v e r a l ma-
The f o u r comnonent m a t r i c e s t o be used i n
t r i c e s , c a l l e d c o m o n e n t s helow. t h i s examole a r e t h e f o l l o w i n n :
.
cnrw 1 c o m ,2 s1
:: s4
1: i '. x1 .5
x2
comn .3
cr)no. 4
x1
.5
0
.5
0
1
0
0
: $ ;] 0
1-
w3=?
o u r t a s k i s t o determine t h e w e i o h t s wl-w4
4-'
1.1-7
i n such a way t h a t t h e w i G h t e d
sum o f t h e covponent m a t r i c e s (see above) s h a l l he as oood an e s t i m a t e as oossible for the t r a n s i t i o n n r o h a b i l i t v m a t r i x oiven i n the heqinninq. The comoonents have been f o r m a l l y chosen a c c o r d i n q t o t h e n r i n c i p l e o f o r thoqonality.
Such o r t h o o o n a l components a l s o seem t o he easv t o i n t e r n r e t .
I f t h e s t a t e s a r e i n t e r n r e t e d as b e h a v i o r a l s t a t e s t e t v e e n w h i c h an aoent can choose, then t h e t r a n s i t i o n n r o b a h i lit i e s r e n r e s e n t tendencies t o choose between t h e a l t e r n a t i v e s (xl and x 2 i n o u r e x a m l e ) .
The comoonents g i v e n
ahnve can be i n t e r p r e t e d :
1) Comnonent 1 -tendency t o choose a t random between x 1 and x2 r e g a r d l e s s o f s t a t e a t time n o i n t t. 2) Comnonent 2 -tendency t o choose x? r e g a r d l e s s o f s t a t e a t t i m e o o i n t t . 3 ) Somoonent 3 -tendencv t o choose x1 when t h e a o e n t i s i n s t a t e s1 o r s 2 a t
t i m e n o i n t t b u t x 2 i f she i s i n s t a t e s3 o r s 4 .
4) Comonent 4 -tendency t o choose x1 when t h e a n e n t i s i n s t a t e s 1 o r s3 b u t x 2 i f she i s i n s t a t e s 2 o r s 4 .
435
Component analysis of transition probubilities
?-;e tenrlenci.?
, ' P s c r i I i e d ahove mav hecome more r e a v i n n f u l ift h e s t a t e s a r e
o o e r a t i o n a l l y defined.
n c t u a l l v , t h e f i n u r e s o i v e n ahove a r e e m o i r i c a l r e -
l a t i v e t r a n s i t i o n frenuencies Fame
-
-
r e s u l t s o f a m u l t i - n l a y P r i s o n e r ' s Pilemra
d e s c r i b e d i n more d e t a i l i n t h e second n a r t o f t h i s r e n o r t .
!n t h e P r i s o n e r ' s Vilemva name r e n o r t e d h e r e , t h e n l a v e r was even asked t o F i v e h i s e x n e c t a t i o n a t each t r i a l . s h a l l use t h e synbol
f o r t h e n a r t n e r ' s c o o o e r a t i v e c h o i c e i n t!ie c a m
and R f o r h i s n o n c o o p e r a t i v e c h o i c e .
Corresoondin?l:/,
A'
means t h a t t h e
n l a v e r e x o e c t s t h e n a r t n e r t o choose A and R ' t h a t t h e n l a y e r exoects
I? t o
be + i s p a r t n e r ' s c h o i c e . I n o u r exanole s t a t e s have tclr! f o l l o c l i n g meaning:
s1 = P ' ? ( i . e . ,
a t t i m e n o i r l t t t'le o l a y e r has answered t h a t she exnects t h e
p a r t n e r t o choose I\ on t h e f o l l o w i n ? t r i a l a t t i w o o i n t t t l and t h e p a r t n e r has chosen P a t t i r e m i n t t )
s 2 = P.'R
s3
= R'I!
s4 = 3'R. The column s t a t e s x1 and x2 have t h e meanina: x1 = .I' a t t t l ( i . e . ,
t,he a l a y e r ' s e x n e c t a t i o n a t t h e f o l l o w i n g t i m e n o i n t
ttl i s A ' ) . x2 = 6 ' . Thus, t h e component 2 means t h e tendenc,y t o choose n o n c o o p e r a t i v e l y (compet i t i v e l y ) i n a l l situations. exnectation.
Comoonent 3 i s t h e tendency t o r e o e a t t h e save
Component 4 corresnonds t o t h e tendency t o e x n e c t t h e p a r t n e r
t o choose i n t h e s a w wav as i n t h e e a r l i e r t r i a l .
I f we were a b l e t o determine t h e w e i q h t s t o be used i n s u m i n a t h e comnonent m a t r i c e s we c o u l d g e t i n f o r m a t i o n about t h e r e l a t i v e s t r e n g t h o f t h o s e t e n dencies d e s c r i b e d ahove. I t i s easy t o s o l v e t h e nroblem i n o u r examnle.
f o l l o w i n q f o u r l i n e a r equations: .5w.
= .17 ( f r o m t h e 4. row)
w,+.5wl
= .44 ( f r o m t h e 1. row)
w,t.5wl
= .49 ( f r o m t h e 2. row)
<.
w t.5wl = .26 ( f r o m t h e 3 . row)
3
Itre have o n l y t o s o l v e t h e
436
K . Rabrkr
Tt-e weinLits s r e : I’ll=
.34
.?7
I”*=
\‘I3=
.3?
w4= .0’7
The s o l u t i o n o f e n u a t i o n s was a n o l i c a h l e t o o u r exalnnle ~ , ~ h r ? rvie e had j u s t
a s rlanv nrthoclonal cnlnionents as we had rows i n t h e m a t r i x ( i .P., 4 ) . l:, lie h a w l e s s e n u a t i o n q t h a n r o w . t n use a l e a s t s o u a r e r e t h o d
Usual-
4s a c e n e r a l s c l u t i o n , i t i s c n n v e n i e n t
(I’ t o d e t e r p i n e t h e w e i n h t s .
I n doing t h a t ,
a l l tbr? i n f o r m a t i o n the t r a n s i t i o n p r o b a h i l i t v m t r i x c o n t a i n s i s used. 1 . 2 . LFPST W U P P F qnLIITI”F! rnr comutatinns
t
VIP
f o r p t h e f o l l o w i n n v e c t o r s and m t r i c e s :
= vector of t r a n s i t i o n n r o h a h i l i t i e s (corresqondinn t o emoirical r e -
lPtive transition frenwncies). t i o n o r n h a h i 1 i t . v v a t r i x i s used.
...
2, 4 , .O, C
7
0nl.v t h e l a s t ( r i c l h t ) c o l u n n nf t h e t r a n s i D i m n s i o n s o f F a r e n and 1, n h e i n a
poterlcy o f 2, F heincr a p o s i t i v e integer).
(IT-tb
c c r t , i n e d c o m o n e r l t n a t r i x where t h e columns a r e t h e l a s t ( r i p h t )
columns o f each cormorlent m a t r i x . i s not included.
T4e colnoonent m a t r i x 1 w i t h e l e m n t s .5
Thus, t h e d i n e n s i n n s o f C a r e n arld c - 1 where c i s t h e t o -
t a l nulnher of components.
r’ 1’
= t h e transpose o f C . = the vector
of i v e i ? h t s .
I n the beoinnincl, t h o v e c t o r of t r a n s i t i o n n r o h a h i l i t i e s i s reduced hy t h e minimun (m) o f t h e elements o f t h e t r a n s i t i o n n r o h a b i l i t y m a t r i x . lnum i s the s r l a l l e s t e l v e n t o f E o r 1-E).
t o r hv 1.
(The m i n i -
‘,‘e s h a l l i n d i c a t e t h e reduced vec-
Thus,
p = r - v The tvreiaht o c comnonent 1 i s :
w1 = 2 m The o t h e r w e i o h t s a r e d e t e r p i n e d m a t r i x - a l a e h r a i c a l l v i n t h e f o l l o w i n a wav: (1) ‘re mean by ‘ l e a s t snuare method’ d e t e r p i n i n a t h e e s t i m a t e i n such a way t h a t t h e sum o f spuares of d i f f e r e n c e s between t h e e s t i m a t e and t h e e m n i r i c a l t r a n s i t i n n n r o b a b i l i t i e s i s vinimum.
I w i s h t o thank o r o f . Anders Ekholn, Deot. n f S t a t i s t i c s , I l n i v e r s i t y o f P e l s i n k i , fnr h i s v a l u a h l e h e l n i n t h e mathematical s n l u t i o n .
Component amlysis of tramition probabilities
1.1
437
(Clcl-'1c ' p
=
where C' i s t h e transnose of C and -1 i n d i c a t e s t b e i n v e r s e o f t h e mat r i x C'C. (11
c o n t a i n s t h e w e i a h t s o f c o m o n e n t s f r o m 2 t o n . (The v e i q h t v0 was
deternined i n the he9innino). EXl\WLE
The l e a s t square s o l u t i n n o f o u r examnle i n 1.1 i s shown i n t h e f o l l o w i n ? ,
[:;I
s t e n hy s t e n :
F = .44
.23
t q i n i m m = 1-.C3=.17
0
1
1
The i n v e r s e o f C'C = .75 -.50 -.50
!
-.50
1
0
-.50
0
1
(clC)-lcl
=
-75 .25 -.50 - . 5 0 -.50 .50
.25 .50 -.50
- .25 .50 .50
X. Roinio
438
The, wei Flits
1,'
=
(Compare t o t h e s o l u t i n r l s i n 1.1)
I f \I = the w e i n h t s f r o n 7 t o n then the e s t i r a t e F = .5-nl
+
C\/ =
!n t h e second n a r t o f t h i s r e n o r t we s b a l l a n q l v t h e connonent a n a l y s i s o f t r a n s i t i o n n r o b a h i l i t i e s t o our Prisoner's D i l e m a r e s u l t s .
(The i d e a o f
t + i s c o m o n e n t a n a l y s i s was a c t u a l l y b o r n i n dealinc. w i t h those r e s u l t s ) . Qnerallv,
t h e r e s u l t s nroduced by e x n e r i w n t a l games seem t o he s u i t a h l e
f o r ccvnonent a n a l y s i s . Another f i e l d o f n s y c h o l n o i c a l s t u d y where i t i s easy t o fnrm i n t e r n r e t a h l e c o m o n e n t s and, t h u s , a n n l v o u r a n a l v s i s i s t b e s t u d v of l e a r n i n g and f o r n e t t i n ? i n t e r w o f rlarkov c h a i n s (see, e . ? .
2 . Cr)flPnP!ENT f i N A L F I S OF "PISnNFR'C
, Cno~bs-Darrc!s-T\:ersky,
291-306).
DILEW'A
2.1. EX"EP1"ENTAL SET-l!p Several s t u d i e s on p r i s o n e r ' s Dilemma have been made a t t h e Denartment o f S o c i a l Psycholooy o f t h e l l n i v e r s i t v o f H e l s i n k i i n t h e y e a r s !PEl-P3,
the
r e s u l t s o f which have heen a n a l y z e d by t h e comnnnent a n a l v s i s r e n o r t e d h e r e . f i l l tbose t e s t s have heen made h v p i c r o - c o m u t e r s ( P e t Commodore) b u t t h e set-un o f the studies varies. settin?'
, because
through a s n e c i a l n e t w o r k . PP-Net.
Two o f them c o r r e s n o n d w e l l t o t h e ' n a t u r a l
' 3 ~ f n f o r r a t i o n o f b o t h p l a y e r ' s c h o i c e s was t r a n s m i t t e d
These s t u d i e s a r e a b b r e v i a t e d b y PDN
-
i .e.
In the o t h e r s t u d i e s (WW.1, F ' W N , and FFR83) t h e choices o f t h e
o t h e r n l a y e r were s i m u l a t e d by t h e nroqram.
4 detailed descrintion o f the
s i w l a t i o n i s g i v e n i n an e a r l i e r r e n o r t ( R a i n i o ( 1 9 8 1 ) ) .
Gimponrnt analysis of transition probabilities
Each q a i r o l a y e d 40 clams bfer e n 1a,yed
-
439
e x c e n t i n t h e NnV81 s t u d v where o n l v 10 p a w s
.
I n t h e WN-FER83 and i n t h e FD-FFPR3 s t u d i e s t h e oG!!-G?f
m a t r i x was shown
i n a r e y u l a r f o r m w i t h o a v - o f f s v a r v i n o from flame t o p a w .
I n the o t h e r
s t u d i e s t h e m a t r i x was shown i n a decomoosed form.
fin e s s e n t i a l f e a t u r e i n c u r e x o e r i m n t a l s e t - u n ( i n a l l s t u d i e s ) was t h a t h e f o r e each c h o i c e o f s t r a t e c y t h e n l a v e r had t o answer t h e q u e s t i o n : '!.\hat do y o u e x a e c t t h e o t h e r n l a v e r t o choose?'
A f t e r t h a t she o o t t h e o n o o r t u -
n i t y t o choose h e r own s t r a t e r r v . The suh,iects were s t u d e n t s n a r t i c i n a t i n g i n t h e elementarv cnurse o f s o c i a l nsycholooy a t the u n i v e r s i t v .
Thev d i d n o t know tbe P r i s o n e r ' s P i l e m a
I n t h e 5 d i f f e r e n t s t u d i e s a l t o s e t h e r 136 s u b j e c t s were t e s t e d .
?am.
The o l a y e r ' s s t a t e on each p a w i s d e s c r i b e d by t h r e e c h a r a c t e r i s t i c s .
1) The e x p e c t a t i o n , A ' o r B ' , A ' i n d i c a t i n g t h e p l a y e r e x p e c t s t h e o t h e r t o ctioose c o o p e r a t i v e l y and B ' b e i n g t h e n o n c o o n e r a t i v e [ c o m e t i t i v e ) c h o i c e . 2) The c b o i c e o f t h e s t r a t e n y , X o r Z, X i n d i c a t i n n t h e c o o n e r a t i v e and 7 t h e n o n c o o n e r a t i ve (colnneti t i ve) c h o i c e ,
3) The i n f o r m a t i o n abnut t h e simultaneous c h o i c e o f s t r a t e q y hy t h e o t h e r glaver,
P i n d i c a t i n o a c o o o e r a t i v e and 9 a n n n c o o n e r a t i v e c h o i c e by t h e o t h e r .
Thus, t h e r e a r e 8 d i f f e r e n t s t a t e s on each qame numhered as f o l l o w s : 0) A ' X A
-
t h e o l a y e r exoects t h e o t h e r t o choose 4, chonses h e r s e l f X,
and i s then i n f o r m e d t h a t t h e o t h e r o l a y e r choose A. 1) A ' X B
2) b'ZA 3) A'ZB
4) B ' X A 5) S'XB 6) R'ZA
7 ) S'ZB. 2 . 2 . COVPONENTS. Iele
ANALVSIS O N THE CROUP LEVEL
can e a s i l y a m l y
have 2 columns.
our c o m o n e n t a n a l y s i s t o such t r a n s i t i o n m a t r i c e s which
I t i s n a t u r a l t o analyze
1) tb,e e x o e c t a t i o n s (columns A ' and B ' ) u s i n q a l l t h e 8 s t a t e s ( f r o m A ' X P t o R ' B Z ) o f t h e e a r l i e r name as rows o f t h e t r a n s i t i o n m a t r i x , and
440
K. Raiflio
2 ) t h e choices o f t h e s t r a t e r i e s (columns X and 2) u s i n ? 8 s t a t e s ( f r o m X,P.tAi+l
t o 7tRtPi+l)
as row o f t h e t r a n s i t i o n w a t r i x .
Thus we have two 2 x 7 m a t r i c e s t o he analyzed, one f o r e x n e c t a t i o n s , shown i n Table 1 as ' t r a n s i t i o n n r n b a h i l i t i e s , e x p e c t a t i o n , e m i r i c a l ' , and a n o t h e r f o r t h e s t r a t e g y choi ces ( T a h l e 1, ' c h o i ce , emvi r i c a l l )
.
Pecause we have c o m u t e d t h e f r e n u e n c i e s o v e r t h e t o t a l p r o m o f subjects i n
a studv, o u r cnmnonent a n a l v s i s w i l l v i e l d a n a t h e m a t i c a l model f o r t h e r e a n tendencies o f i n d i v i d u a l s t o form t h e i r e x n e c t a t i c n s choices, corresqondinol:!
-
i n each d i f f e r e n t s t a t e .
-
a s w e l l as t h e i r
( L a t e r we s h a l l a n a l v -
ze the i n d i v i d u a l t e n d e n c i e s ) . The cornonents we s h a l l use i n o u r a n a l y s i s annear i n m a t r i x forms i n Table 3.
tlote t h a t t h e v have d i f f e r e n t r e a n i n a f o r e x n e c t a t i o n s and f o r c h o i c e s .
T9e n s v c h o l o r i c a l i n t e r n r e t a t i o n s a r e a l s o o i v e n i n Tahle 3 . The r e s u l t s o f t h e c n m o n e n t a n a l v s i s a m e a r i n Table 1 where t h e w e i g h t s a r e shown ( ' w e i o h t s , ex?.' and the r e s i d u a l s .
and ' w e i n h t s , c h o i c e ' ) as w e l l as t b e e s t i m a t e s
The Chi-snuPres have a l s o been cnmauted.
I n those cases
where t h e n-values show n o s t a t i s t i c a l l v s i ? n i f i c a n t d i f f e r e n c e s between t h e e s t i m a t e s and t h e e m n i r i c a l v e c t o r s t k e r e a r e n o t n e c e s s a r i l y any more comnonents t o be taken i n t o account. from t h e e x n e r i m e n t a l m a t r i x (e.n.,
"hen t h e e s t i m a t e d i f f e r s s ' o n i f i c a n t l y PnnN-FFR82, ' c h o i c e s ' ) t h e r e s i d u a l s mav
c o n t a i n some a d d i t i o n a l source o f v a r i a t i o n of t h e t r a n s i t i o n f r e q u e n c i e s and cannot he i n t e r n r e t e d as random v a r i a t i o n . The exn1anator.v Dower of t h e colnnoneot model t i v e method
-
-
c a l c u l a t e d by an a m r o x i m a -
a s s u m s t h e values shown i n Table 4.
-
k c o r d i n n t o our studv
t h e comnonent model can e x n l a i n a h o u t 70-857 o f t h e d e v i a t i o n o f t h e t r a n s i t i o n n r o b a b i l i t i e s f r o m t h e .5 values ( i . e . ,
from t h e values t h e y s h o u l d
assulne a t r a n d o v ) . R v comnarino t h e w e i n h t s o f comnonents we can draw sow a s y c h o l o g i c a l con-
c 1us ions .
1) The tendencv t o r e n e a t own c h o i c e seems t o he c o n s i d e r a b l y h i g h ( c h o i c e s , v3=.25, -37,.18, . 2 2 , .25) b u t t h e tendency t o e x a e c t t h e o t h e r n l a v e r t o r e n e a t h e r c h o i c e r a t h e r l o w ( e x n e c t . , rr5=.01,.15,.20,.17,.05).
Thus, t h e n l a y e r
behaves i n a more n e r s e v e r i n g n a y t h a n she assumes t h e o t h e r t o behave.
2 ) The tendency t o r e n e a t o n e ' s own e a r l i e r e x n e c t a t i o n i s c o n s i d e r a b l y hioh, t o o ( e x n e c t . , w3=.17,.35,.26,.13).
T h i s can he i n t e r o r e t e d as a r a t h e r h i o h
44 1
Component anufysis of transition probabilities
tendency t o oersevere, t o o .
3) The tendency t o choose i n a n o n c o o a e r a t i v e ( o r c o r i n e t i t i v e ) way v a r i e s much ( c h o i c e s , w2=.09,.13,.56,.24,.2),as w e l l as t h e tendency t o e x o e c t such b e h a v i o r ( e x o e c t . , w2=.28,.18,.20,.23,.01). 1,Jhen a n a l y z i n p t h e r e s u l t s on t h e clroun l e v e l we alrravs f a c e a nroblem i n tbe i n t e r n r e t a t i o n o f o u r mean r e s u l t s : a r e t h e suh.iects s i n i l a r t o each o t h e r , a l l behavinn i n t h e "mixed" way t h e model d e s c r i b e s , o r a r e t h e r e s u l t s s i m o l y means i n an unhomoenenus groua o f i n d i v i d u a l s who a c t u a l l v behave i n t h e i r o w , c!i f f e r i n o v a v s . To f i n d answers t o t h i s k i n d o f q u e s t i o n s we have t o analyze t h e r e s u l t s on the i n d i v i d u a l l e v e l
-
t r y i n ? t o f i n d o u t t h e comnonent w e i ? h t s c h a r a c t e r i s -
t i c t o each i n d i v i d u a l .
2.4. CntVWNENTS.
A H A L V S I S CN THE INnIVIDUbL LEVEL
There were u s u a l l y 40 nares f o r each s u b j e c t i n o u r s t u d y .
Recause t h e r e
a r e , thus, 39 t r a n s i t i o n s o e r i n d i v i d u a l i n t h e E x 2 c e l l s o f a t r a n s i t i o n t a b l e , t h e c e l l f r e q u e n c i e s become o u i t e s m a l l .
I n n r i n c i n l e , we can a m l y
o u r comnonent a n a l y s i s t o t h e t r a n s i t i o n f r e q u e n c i e s o f one i n d i v i d u a l , too, b u t t h e w e i g h t s o f c o m m e n t s we g e t , may v a r y c o n s i d e r a b l y by chance. To measure t h e r e l i a 5 i l i t i e s o f t h e i n d i v i d u a l component w e i g h t s , we need t o d i v i d e o u r t r a n s i t i o n s i n t o two s e r i e s , odd and even t r a n s i t i o n s . The c o r r e l a t i o n between t h e odd and even s e r i e s f o r each comoonent w e i g h t
v a s cornouted and c o r r e c t e d f o r l e n n t h a c c o r d i n q t o Snearman-Srown f o r m u l a . The r e l i a b i l i t y c o r r e l a t i o n s y i e l d e d hy t h i s s n l i t - h a l f method a r e shown i n Table 5 . The r e l i a b i l i t i e s a r e m a i n l y c o n s i d e r a b l y h i g h i f we t a k e i n t o account t h a t t h e w e i g h t s a r e colnouted f r o m r a t h e r s m a l l f r e a u e n c i e s o f t r a n s i t i o n s . On t h e b a s i s o f o u r r e s u l t s we way conclude t h a t comnonent a n a l y s i s opens a method f o r f i n d i n p o u t measures i n terms o f comoonent w e i g h t s which charact e r i z e t h e b e h a v i o r o f a n l a y e r i n Pn game q u i t e r e l i a h l y . !t i s n o s s i b l e t o see i n b o t i i t a b l e s same c h a r a c t e r i s t i c change d u r i n p t h e
nrocess.
1) The tendency t o r e n e a t own e x n e c t a t i o n s (exn., w3) i n c r e a s e s slowly (Table 6).
The tendency t o e x p e c t t h e o t h e r t o choose i n a r e n e a t e d way
(exp., w5) aopears on t h e l a t e r ahase (exD., w5 i s s1iahtl.v n e g a t i v e i n t h e
442
K.
Rditiio
b e q i n n i n o as one can see f r o v Tahle 7 ) and i n c r e a s e s f r o r i b l o c k 1 t o b l o c k
7 ( s r e Table .-). ses c l o w l y .
-
-he tenr'encv t n r e v a t own c h o i c e ( c h o i c e s , w3) i n c r e a -
f i l l these t e n d e n c i e s seep t o he s v m t n v s n f i n c r p a s i n o f a t i o u e
of t h e suhjects durino the t r s t .
7 ) ? n t h t h e tendencv t o e x p e c t ' t i t - f o r - t a t '
b e h a v i o r ( e x n . , w4) and t o
choose a c c o r d i n o t o t h i s s t r a t e n y (choices, w 5 ) a r e v e r v s t r o n n i n the ben i n n i n q (see p a r t i c u l a r l y Table 7 ) .
To be ' e q u a l i t a r i a n ' and t n e x n e c t the
p t h e r t o behave i n t h a t n a y , t o n , seems t o he, t h u s , t h e s t r o n n e s t ( o r p o s t corsnnn) tendencv i n t h e v e r v h e o i n n i n o .
I t seems t o n e t e x t i n v u i s h e d v e r y
n u i c k l v i n the course o f t h e f i r s t h l o c k ( w 4 o f exn. decreases from .42 t o
.03 and w5 of choices from .35 t o . 0 6 \ .
-
It i s interestinq t o notice that
t h e e x t i n c t i o n n f t h e ' e n u a l i t a r i a n tendency' does n o t l e a d t n c o m n e t i t i v e t w h a v i o r b u t simn1.v t n t i r e d n e s s
-
t h e s u b j e c t s t a r t s C.C. r e n e a t h i s exnec-
t a t i ons and c h o i c e s . ?.', Roth t'le tendency t o e x o e c t t h e o t h e r t o hchave c o m e t i t i v e l y (exn., w 2 )
and t o choose own s t r a t e n y i n t h a t rmv ( c h o i c e s , rr2) remain v u i t e c o n s t a n t durino the t e s t . The c o m o n e n t a n a l y s i s does n o t l e a d t o any v e r v s u r n r i s i n ? r e s u l t s . qecause i t i s a h l e t o n i v e measures o f those b e h a v i o r t e n d e n c i e s i n P r i s o n e r ' s Pilemma a l r e a d y known, i t makes t h e n i c t u r e p o r e c l e a r .
2, n i s c i i ~ ~ r m i n u r s t u d y on P r i s o n e r ' s Dilemma has t o be seen m e r e l y as an examole o f t h e a o n l i c a t i o n o f t h e c n m o n e n t a n a l y s i s i n t r o d u c e d i n t h i s r e n o r t , n o t as an a t t e m t t o analyze deeoly t h e human m o t i v a t i o n i n t h a t vame.
However, the
comnonent a n a l y s i s may o f f e r a oood b a s i s f o r makino comnarative a n a l y s i s hetrreen r e s u l t s n o t from d i f f e r i n q e x n e r i m e n t a l oames.
I n the future, i t
would be a l s o i n t e r s s t i n a tr f i n d o u t how t h e cormonent w e i n h t s may chanae i f some v a r i a t i o n i s s y s t e r n a t i c a l l v Droduced i n s u b i e c t ' s m o t i v a t i o n i n q a w s i t u a t i o n s by d i f f e r i n a t h e i n s t r u c t i o n s and t h e o a v - o f f s . The creneral a o p l i c a b i l i t v o f colnoonent a n a l y s i s was a l r e a d v b r i e f l y d i s c u s sed i n Thanter 1.
q u n m a r i z i n g we mav c h a r a c t e r i z e t h e comnonent a n a l y s i s
as a ' f a c t o r a n a l y s i s o f t r a n s i t i o n o r o b a b i l i t i e s ' , because i t o i v e s a h o l i s t i c n i c t u r e of the f a c t o r s 'behind' a s e t o f t r a n s i t i o n n r o b a b i l i t i e s a s f a c t o r a n a l y s i s does i n r e l a t i o n t o s e p a r a t e c o r r e l a t i o n s t u d i e s .
-
443
Cunipunerrt analysis of transition prubabilities
RFFFFEVCFS Coomhs, C.H.,
Dawes, R . k 4 . , Tversky, A . , ''atbematical psvchology,
"rentice-Val1
(117C).
K e w n y , J.G., S n e l l , J.L.,
Thomson, C.L.,
m a t i cs , Pren ti c e - H a l l ( 1957)
n.'!.,
Kuhlman,
I n t r o d u c t i o n t o " i n i t e rathe-
.
l . ~ i m b e r l e v , D.L.,
E x n e c t a t i o n s o f choice b e h a v i o r h e l d
by c o o p e r a t o r s , c o m e t i t o r s , and i n d i v i d u a l i s t s across f o u r c l a s s e s o f e x g e r i m e n t a l ~ a n e s ,J o u r n a l o f P e r s o n a l i t v Snc. Psycho1
. 34
(1976)
,
69-81. 'liller,
D.T.,
Folmes, J . R . ,
The r o l e of s i t u a t i o n a l r e s t r i c t i v e n e s s on
s e l f - f u l f i l l i n c oroohecies: A t h e o r e t i c a l and e m m i r i c a l e x t e n s i o n o f K e l l e y and 5 t a h e l s k i ' s t r i a n n l e h y c o t h e s i s , J o u r n a l of P e r s o n a l i t y Yoc.
Psyhcol. 3 1 (1975), 661-673. r l i s r a , S.,
T r i a n a l e e f f e c t and the c o n n o t a t i v e meanino o f t r u s t i n P r i -
s o n e r ' s Pilernna:
c r o s s c u l t u r a l s t u d y , I n t e r n a t . Journal o f Psychol
.,
14 (1979), 255-263. P r u i t t , D.G.,
Kimnel,
tique, synthesis
M.kl.,
, ar7d
Twentv y e a r s o f e x n e r i m e n t a l gaminq: C r i -
sun?est.ions f o r t h e f u t u r e , Fnn. Rev. Psychol.,
28 (1!177), 363-392. R a i n i o , K.,
S i m u l a t i o n of r u l t i - o l a y P r i s o n e r ' s D i l e m a a c c o r d i n p t c a
o u a n t i t a t i v e c o a l / e x o e c t a t i o n t h e o r y , Research Renorts, Den. o f SOC. Psychol
., l l n i v e r s i t v
o f Y e l s i n k i , 2 (1981).
RamoDort, A . , Chamah, A . P I . ,
Prisoner's Dileima, U n i v e r s i t y o f V i c h i -
pan Press (1965). Schulz, U., Hesse,
IpJ.,
A model f o r gaming b e h a v i o r i n sequences o f P r i -
s o n e r ' s Dilemma claws i n c o n s i d e r a t i o n o f e x n e c t a t i o n s , Psvchologische B e i t r a o e 20 (1978)
, 551-570.
434
K. Ruinio
TP.3LE 1.
C c m a n e n t a n a l y s i s . PD
Procram: c ~ F ' P A W D - Y ~ ~ V MT?: , ? ~ ~ PDNnnPslRL-FR2 '.leiohts, e w . :
.27
.22
'.leiflhts, c h o i c e :
.24
.ll . 3 1
.27
.2
.OG
.13
.14
Transition n r o b a b i l ities:exnectation Emni r i c a l
*'
rstimate
Resi d u a l s
I '
urn
.67
,33
40
P'XA
.64
5' .36
Ytrm
A ' M
40
.O?-
- .03
A'XO
.59
.41
64
A'XR
.58
.42
64
.02
-.02
A'ZA
.39
.61
2P
A'ZA
.44
.56
28
-.@5
F'ZB
.3P
.63
56
.6?
56
0
.37
.63
52
D'ZR B'Xn
.38
R'XA
.37
.63
52
-.01
S'XR
.27
.73
E3
9'XP
.3!
.63
83
-.04
R'ZA
.lO
.21
77
B'ZA
.17
.83
77
.02
-.02
B'ZB
.14
.66
248
R'ZS
.11
.89
748
.03
-.03
A'
P'
R'
.05 0
.n1 .04
Transition nrohabili ties:choice Emi ri c a l
Estimate
s urn
x
2
XI\!'
.76
.24
46
XnP'
.67
.33
XRA'
.62
.3C
Pesiduals
5urn
x
.23
46
-.01
.63
.37
45
.04
-.04
.64
.36
60
-.03
.03
X
Z
XA4'
.77
45
XAA'
60
XPD'
Z .01
X9P'
.49
.51
26
XRR'
.5
.5
26
-.01
.01
ZA4'
.38
.62
26
ZPP'
.46
.54
26
-.07
.07
ZAB'
.35
.65
79
ZAS'
.32
.68
79
.04
ZRA'
.44
.56
55
254'
.33
.67
55
.ll -.11
ZRR'
.12
.88
249
LRR'
.19
.8l
249
Chi -Souares
Exnectati ons : 3.64
DF = 3
Choices: 12.17 DF = 3 .
-.07
-.04
.07
C o m p o w n t mulysis of t m w i t i o n probubilities
!/‘I
L-
7.
Component w e i g h t s .
445
summary
A
E x n e c t a t i ons I;t u h :~
Corlnonen t s :
c1
c2
c3
c4
c5
Bloc4 1
.34
.28
.17
.27
- .01
Block 2
.lo .PI
.1n
.21 -.oi
.2n
.23
.35 .26 .13
.OP
.17
‘.a1
.30
22
P DFI-FER
PPFI-FER83 P P-rrA R8 1
-45 .4n .25
PP-FER83 (P w w R i )
.2n
.XI .2n
.15
. i ~ .05 .34 .n4
Choices
s t udv :
Comonents:
c1
c3
c2
c4
c5
PDN-FERE2 Block 1
.26
.nc)
.25
.i7
.19
Block 2
.13
.37
.17
.08
PDFI- FEE 83
.?.a .oe
.56
.02
.30
.08
.1e
PD-FER83
.06
.24 .42
.18 .22
.07
P D-”’4 981
.25
.I4
.20
.01
.20
.35
( P D - F ! ~ ~ ) .22 I n the study Pfl-NN81, TAqLE 3.
.20
t h e r e were o n l y 10 games n e r s u b j e c t .
Components and t l l e i r o s y c h o l o g i c a l i n t e r o r e t a t i o n s Cornponen t
1
2
3
Row
0
.5
1 2 3 4
.5 .5
.5
1 .5 .5 .5 .5 .5
5
.5
.5
6
.5 .5
.5
0 0 0
.5
0
0
7
.5
0 0 0 0
0
1 1 1 1 1 1 1 1 1
4
5
n
i
0
1
0
1
1
0
1
0
1
0
1
0
1 0 0
0 1 1
0
1
1
0
0 1 0 1 0
1
1
0
1 0
0 1
0
1
1
0
0
1
0
1
1 0 1 0 1
0
1
0
1
0
446
Analysis o f exaectations: The rows a r e s t a t e s A ' X A
...R ' Z R .
Tkc colunns a r e :
0 ) Fxnectation F ' i n the follnwing o a w . 1) F x q e c t a t i o r - q ' i n t h e follor.linq oame. The i n t e r n r e t a t i o n o f c o m o n e n t s : COPQ.
1) tendencv t o choose t h e e x D e c t a t i o n a t r a n d n r .
C o m ~ . 2 ) tendencll t o e x q e c t n n n c o o n e r a t i ve ( c n m e t i t i ve) c h o i c e - r e nardless o f the s t a t e . Coma. 3 ) tendencv t o r v c z t o n e ' s e a r l i e r c x n n c t t t i n n .
Corn?. 4 ) t e n d m c v t o e x n e c t t h a t t h e othc?r chooses i n accordance w i t ? one's e a r l i e r choice, i.?. i n, t h e saw way as t h e p l a y e r h e r s e l f has . j u s t cqosen ( e x n e c t i n n a ' t i t - f o r - t a t '
r u l e from the o t h e r ' s n a r t ) .
Corn. 5 ) tendencv t o e x n e c t t 3 a t the o t h e r s i n v l y c o n t i n u e s h e r c h o i c e
strateny . A=l!ysis
o f choices :
The rows are t h e f o l 1owinn s t a t e s :
0) Y M ' ( i .e.,
1) Y A R '
, 2)
P' on t h e flame t + l ) . 4 ) Z A A ' , 5 ) Z A R ' , 6) Z R A ' , 7 ) ZRS'
V and I\ on t h e pame t and
3) XSR',
KRB',
.
The columns a r e : 0 ) Y on t h e caw t + l
1) 7. on the ?a?
t+l.
The i n t e r a r e t a t i o n o f comnonents: C o m . 1) tendencv t o choose a t random Coma . 2) tendency t o choose n o n c o o o e r a t i velv (cnmpeti t i v e l v )
.
C o m . 3 ) tendency t o r e n e a t own c h o i c e . Como. 4) tendencv t o choose i n an ' e c h o i n n ' w a v , i .e.,
i n t h e same:Mav
as the o t h e r has j u s t chosen ( ' t i t - f o r - t a t ' ) . C o n . 5 ) tendencv t o choose i n an ' e q u a l i t a r i a n ' way i n accordance t o
one's exnectation ( i . e . ,
t o choose X i f e x o e c t a t i o n i s
P and
Z i f i t i s R).
447
Component analysis oftransition probabilities
TARLE 4.
E x n l a n a t o r v nower o f comonents Ctudv:
Fxoect
.
TEB82N
1 PLF 5.
Reliah
Rlnck 1
73"
Slocl. 2
C??'
TF P 834 I
553/
r4m1
74"
FFE87
734
f?wJl
64"
ities o f comments
qet-un o f the studies:
FEPE2N and FFBR3N 1161FR2 and NOV81
-
-
communication n e t Partner simulated.
(Correlations are corrected f o r lenoth)
.
Exnectations:
s tudv Cornon en t
FEPR2N
FES83N
M81F83
Nr)\l81
Comn. 1
.6G
.77
.65
.75
Comn. 2
.63
.84
.69
.47
Comn. 3
.71
.56
.61
.63
Corn. 4
.37
.44
Corn. 5
.12
.66
.50 .17
-.44
Comn. 1
.6P
.G7
.50
.56
Comn. 2
.67
.71
.69
Com. 3
.54
.53
.72
.28 .30
Comn. 4
.61
.12
.36
.44
Corm. 5
.a2
.60
.77
.53
0
Choices :
448
R. Rubriu
TPRLE 6 .
F l o c k s 1 and 2
Constancy o f c o m o n e n t s .
Yeans o f w e i p h t ( s t u d i e s FER871.1, FERE3N, and W ? ? , l )
Rloc!: 1: f i r s t 7n t r a n s i t i o n s . Block 7: l a s t 29 t r a n s i t i o n s .
F r - e c t a ti ons
v!l
w3
\'I*
w4
\.I
5
Plock 1
.43
.20
.14
.13
.lo
P,loCk 2
.31
.2G
.2h
.O?
.1E
Chni ces
T f l P L F 7.
Cmstancv o f c o m m e n t s .
4 f i r s t transitions
( r o w u t e d frnm d a t a 'jPR81, F:nI/Rl, and TFR82b!). I . t r a n s i t i o n . fror, name 1 t o 2 ; 2 . t r a n s i t i n n : f r o p 2 t o 3;etc. F x v c t a t i ons: w1
"I2
w3
w4
w5
1. t r a n s i t i o n
.21
.19
.16
.42
-.01
2. transition
0
.37
.23
.33
-.07
3. t r a n s i t i o n
9
.52
4. transitinn
0
.38
.ll .17
-.14
-.?I
.32
.16
Choi ces : I'
1. t r a n s i t i o n
C
w2
w3
w4
w5
.20
.1Q
.35
?. t r a n s i t i o n
0
.26 .38
.7?
.04
.36
3. t r a n s i t i o n
0
.19
-.01
.4R
.37
4. t r a n s i t i o n
0
.31
-.03
.34
.33
449
Frank Van Ovenqalle V r i j e U n i v e r s i t e i t Rrussel, RelPiun
The m t h o d we nrooose i s s u i t a h l e f o r l a r g e s e t s o f v a r i a b l e s , which have t o be prouned i n s m a l l e r u n i t s , i n order t o nredict a c r i t e r i o n .
Contrarv t o the
more c o n v e n t i o n a l w t h o d s , such as f a c t o r a n a l q i s , t h a t do n o t account f o r the c r i t e r i o n t n he n r e d i c tec! w h i l e q r o u o i n o t h e data, we nronnse an a n n l i c a t i o n o f multidimensional scaling, i n order t o f i n d a RADEX i n t h e d a t a .
Since a RPDEX c o n s i s t s o f (1)
an h i e r a r c h y o f circumnlexes whit+ d i v i d e low vs. h i g h D r e d i c t o r s , and ( 2 ) vJeak c o n t i n q u i t y r e f l i o n s which grour,
t o ? e t h e r homoaene s e t s o f o r e d i c t o r s
i n each c i r c u m n l e x , i t thus r e c o n c i l e s b o t h g r o u n i n o and n r e d i c t i ve nurqoses a t once.
.
1 INTR(1DUCTInN The aim i n D s y c h o l o n i c a l r e s e a r c h o f t e n i s t o o r e d i c t a c r i t e r i o n f i c b e h a v i o u r o r outcome
-
throuoh a number o f v a r i a h l e s .
-
a sneci-
One hopes t h a t
t h e o r e d i c t i o n o f t h e c r i t e r i o n t h r m n h these v a r i a b l e s i s as h i g h as possible.
This o h j e c t i v e i s o f t e n called: p r e d i c t i v e v a l i d i t v .
used s t a t i s t i c a l method t o f u l f i l lysis.
A frequently
t h i s purnose i s : m u l t i o l e r e g r e s s i o n ana-
A o a r a l l e l Duroose, which i s o f t e n added t o t h e f i r s t method above,
i s t o c l a r i f y how t h i s o r e d i c t i o n i s n o s s i b l e .
One t r i e s t o b u i l d up a
s t r u c t u r a l model which e x n l a i n s more c l e a r l y how e v e n t s a c t u a l l y take Dlace, i n o t h e r words, a model which e x o l a i n s t h e causal r e l a t i o n s between t h e o r e d i c t o r s and t h e c r i t e r i o n . s t r u c t u r a l models (e.q.
F o r t h i s ourpose o a t h - a n a l v s i s o r o t h e r l i n e a r
L i s r e l ) a r e used.
One can s u m a r i s e b o t h p r e d i c t i v e
methods by c o n s i d e r i n g r e p r e s s i o n - a n a l y s i s a f i r s t - l e v e l
aonroach, w h i l e
s t r u c t u r a l models a l s o i n c l u d e f a c t o r s on a deener causal l e v e l .
'.'hen the n r e d i c t o r s a r e well-defined d i s t i n c t concents, there !dill he l i t t l e nrohlem t n apnly the s t a t i s t i c a l methods mentioned. Pn the c o n t r a r v , when a larne number of d i f f e r e n t variables are gathered (which i s often the case i n nsyc+olo?ical research of an e x n l o r a t i v e e a t w e ] i t i s n o t very c l e a r +ow these data should he s e l e c t e d i n o r d e r t o annly renression-analysis o r causal rwdels. r i n c e a l l variables a r e s t i l l i n t h e i r o r i n i n a l forni, i t often wans t h a t t h e i r numher i s too l a r F e , or i t means t h a t t h e i r i n t e r c o r r e l a t i o n s a r e too hinh, which leads t o unre1iat)le reqression-coefficients. The researcher i s often forced, f i r s t of a l l , t o reduce the larne numher of variables to fewer u n i t s . These units must have a nsvcholonical meaninn a q d must he j u s t i f i e d b\i s t a t i s t i c a l means. This a c t i v i t v in which smaller senarate units are sounht f o r i s c a l l e d : discriminant v a l i d i t y (Denison ( 1 3 8 2 ) ) . For t h a t nurnose f a c t o r - o r c l u s t e r a n a l \ l s i s i s often used. B u t these c l a s s i c a l methods share an imoortant disadvantaae. 'Jhile s t r u c t u r i n o t h e i r d a t a , they do not account f o r the c r i t e r i o n t ' l a t h a s t o he predicted. T+e grounino of the d a t a i s not only f a r from ootimal f o r n r e d i c t i o n , somet i w s i t even c o n f l i c t s with i t . I n t b i s t e x t , t h i s c o n f l i c t between d i s c r i pinant a n d n r e d i c t i v e v a l i d i t y i s f u r t h e r elaborated on. I t i s , indeed, o o s s i b l e t h a t poor and s t r o n g p r e d i c t o r s f a l l t o a e t h e r i n one
f a c t n r o r c l u s t e r , such t h a t the f i n a l accuracv of t h i s f a c t o r i n Dredictior, becomes moderate ( f i g u r e 1 A ) . !he has l i t t l e control over the occurance o f such events. The nrohlem becows even more acute when several s t r o n n p r e d i c t o r s a r e av a i l a b l e . Since s t r o n g n r e d i c t o r s c o r r e l a t e verv h i g h l v w i t h the c r i t e r i o n , t h e i r i n t e r c o r r e l a t i o n s must be r a t h e r h i g h , too. The d a n y r t h a t a l l these
strong p r e d i c t o r s w i l l be nrouoed tooether in one f a c t o r o r c l u s t e r becomes very r e a l i s t i c . ' s f h a t should have been a multiole nrediction/exnlanation o f the c r i t e r i o n , i s now concealed i n one construct ( f i g u r e 1 R ) . Denison (1982) s t a t e s the problem in a s i m i l a r way: " ... a dilemna which i s a t the root of much ofbeliavioursl a n d a t t i t u d i n a l research: i f two measures a r e t o be taken as o n e r a l i s a t i o n s of conceotually d i s t i n c t c o n s t r u c t s , thev pust he a t l e a s t somewhat unrelated. However, i f B c l c s e n r e d i c t i v e r e l a tionshin a c t u a l l y e x i s t s between these two c o n s t r u c t s , the two measures must he hiahly r e l a t e d " . 1,'e w i l l now n r e s e n t a method, which t o a l a r g e e x t e n t can reconcile the conf l i c t between discriminant and o r e d i c t i v e v a l i d i t y .
45 1
The Radex structure
2
. N L T I I71 WFlS I OFIAI.
Our annroach, i n which h o t h divercrent r)urr)oses a r e s o l v e d a t once, i s based on m u l t i d i m e n s i o n a l s c a l i n q ( r ' I 3 ) .
T h i s apnroacb a c t u a l l v c o n s i s t s o f two
s tens.
cluster-
FiCure 1A Poor and s t r o n a n r e d i c t o r s rrrouaed t o o e t h e r i n one c l u s t e r / f a c t o r
ci: c 1uster
' c r
i tcr ion
F i o u r e 1B D i f f e r e n t s t r o n g D r e d i c t o r s grouped t o a e t h e r i n one c l u s t e r / f a c t o r .
1) Vonotone d i s t a n c e a n a l y s i s (MDA), which enables us t o l o o k f c r a w a n i n a f u l s t r u c t u r e i n the data. sential role.
I n t h i s s t r u c t u r e t h e c r i t e r i o n Dlays an es-
3 ) C o n f i r m a t o r y monotone d i s t a n c e a n a l y s i s (WIV) which enables t o t e s t
the h w o t b e t i c a l s t r u c t u r e aoainst the o r i c i n a l ?-CG,
I n t.his t e x t , v e w i l l f i r s t n i v e a s h o r t i n t r o d u c t i o n t o t h e h a s i c o r i n c i 7 1 ~ so f h o t h t w t h o d s o f '!nS.
Secondlv, we w i l l e x o l a i n o u r s n e c i f i c an-
r i v c a b used i n n r e d i c t i o n - r e s e a r c h .
P r k f i n a l l y , we s h a l l i l l u s t r a t e o u r
annroach hv an a n n l i c a t i o n i n t h e f i e l d o f e d u c a t i o n a l os!/cholony. ? . 1. rVl"TVlE 3lCTPl!CT P f ! A L \ T I <
l i o n - r e t r i c o r monotme d i s t a n c e anal!/sis
(HPP.) was f i r s t develoned b,y
chenard (1967 ah) and f u r t h e r worked o u t hy K r u s k a l (1964 a h ) .
L a t e r on,
t h e qheaard-Kruskal anproach i n i t i a t e d a v a s t r e s e a r c b i n o t h e r v a r i e t i e s
c.f & t a - a n a l y s i s ,
r f h i c b a r e now nrouned t o o e t h e r under t h e h e a d i n o o f non-
w t r i c ( o r m n o t o n e l m u l t i d i w n s i o n a l cca:inq. The h a s i c ournose o f Y!A i s t o r e p r e s e n t ( d i s ) s i m i l a r i t i e s ( e . ? . c o r r e l a ':ions) hePlesn v a r i a h l e s i n a s w t i a l v d e l .
I n t h i s model, v a r i a h l e s he-
cow p o i n t s and ( r ' i s ) s i m i l a r i t i e s become d i s t a n c e s . the h i v h e r t h e s i v i l a r i t v between t h e n . lower t h e i r s i m i l a r i t y .
The c l o s e r two n o i n t s ,
The more d i s t a n t two n o i n t s , t h e
li s n a t i a l r e o r e s e n t a t i o n r e n d e r s t h e i r l t e r o r e t a t i o n
and a n a l y s i s o f c o m l e x s t r u c t u r e s between v a r i a b l e s much e a s i e r . s t r u c t u r e s can he f a c t o r s ,
These
c l u s t e r s or o t h e r forms o f hovopene qrouos, b u t
o t h e r y o r e complex s t r u c t u r e s a r e a l s o o o s s i h l e .
I n o u r method we w i l l
aonl:/ such a more connlex s t r u c t u r e . I n o u r e x a n l e v e used ' V N I 5 V - I o f h t t r r t a n , Lingoes and Roskam (Guttman (1!?6S),
L i n r o e s (1973), Linnoes and Roskam ( 1 9 7 3 ) ) .
ferences w i t h t h e o r i s i n a l method o f ?kepard,
Althouqh apparent d i f -
t b e i r method i s a l s o based
on a monotone model. %notone means t h a t t h e r a n k o r d e r o f t h e s o a t i a l d i s t a n c e s s h o u l d be i d e n t i c a l t o the rankorder o f the o r i g i n a l ( d i s ) s i m i l a r i t i e s . never r e a l l v nossible.
!!o~.?ever, t 3 i s i s
The d e v i a t i o n from m o n o t o n i c i t y i s expressed by t h e
a l i e n a t i o n c o e f f i c i e n t , v h i c h can t a k e values between :
4
and 1.
The a l i e n a -
t i o n can thus be c o n s i d e r e d as a measure o f s o n - e x n l a i n e d vari.ance by t h e s o a t i a l model. ? . 2 . COflFIRf1ATOSY '1nrdOTWF DIqTFNCF: ANALYSIS ' J h i l e l o o k i n o :or
I:
s t r u c t u r e i n t h e d a t a , one s h o u l d n o t always he l e d bv
the distances i n the s n a t i a l reqresentation.
can a l s o s t a r t f r o m a sne-
c i f i c h y n o t h e s i s a h o u t t h e r e l a t i o n s hetween t h e d a t a , and then t r y t o f i n d
The Rodex structure
453
them back i n t h e s n a t i a l r e n r e s e n t a t i o n o f t h e data.
Fven i f t h e chosen
h v a o t h e s i s i s a v e w reasonable o r t e s t e d one, i t hanpens f r e n u e n t l y t h a t a m i n o r i t y o f points doesn't F i t tbe hyaothetical s t r u c t u r e .
T h i s i s so
because PDA i s a " h l i n d " method which o n l y accounts f o r t h e ( d i s ) s i m i l a r i ties.
Is i t a o s s i h l e t o f i n d a s n a t i a l r e D r e s e n t a t i o n t h a t i s oDtimal w i t h
reoard t o the constraints of the hypothesis?
And can t h i s new r e p r e s e n t a -
t i o n be t e s t e d f o r i t s ( d i s t a a r e e m n t w i t h t h e o r i n i n a l data?
These two
q u e s t i o n s can be s o l v e d by a c o n s t r a i n e d / c o n f i r m a t o r \ / monotone d i s t a n c e a n a l y s i s (CFDA)
.
The CFlDA-proaram t h a t we used i n o u r example was develoned by Lingoes and Rora (Linqoes and Rorn (1976), B o r n and Lingoes (1979), (193fl)). The p r o nran s t e r t s from the o r i o i n a l s o a t i a l confiquration ohtained during the p r e v i o u s 'IDA.
Then t h i s o r i o i n a l c o n f i g u r a t i o n i s s u b m i t t e d t o t h e con-
s t r a i n t s derived from the hynothesis.
4 0 t h t h e c o n s t r a l n t s and t h e d i s t a n -
ces s h o u l d be monotone i n v a r i a n t w i t h r e g a r d t o t h e o r i o i n a l ( d i s ) s i m i l a r i ties.
I f t h e o r i g i n a l c o n f i q u r a t i o n was n o t c o m l e t e l y s a t i s f y i n g t h e hy-
n o t h e t i c a l s t r u c t u r e , t h e a l i e n a t i o n o f t h e newer, c o n s t r a i n e d c o n f i g u r a t i o n
w i l l increase.
I n o t h e r words, t h e c o n s t r a i n e d c o n f i a u r a t i o n w i l l d e v i a t e
even more f r o m t h e o r i g i n a l ( d i s ) s i m i l a r i t i e s .
Is t h i s d i f f e r e n c e s i o n i f i c a n t o r n o t ? Should we a c c e n t t h e chosen hynothesis o r not?
Linaoes and Roro developed two annroaches t o t e s t t h e d i f -
f e r e n c e between b o t h c o n f i o u r a t i o n s , w i t h and w i t h o u t c o n s t r a i n t s .
The
f i r s t s t a t i s t i c a l anoroach i s an a d a n t i o n o f H o t e l l i n o ' s t - t e s t (Lingoes and Rora (1980)).
L a t e r on, Lingoes and 9 o r o (1982) have c r i t i c i z e d t h e
a n a l i c a t i o n of t h i s t e s t and i n 1983, t h e y develooed an a l t e r n a t i v e "mathem a t i c a l d e c i s i o n model" t o t e s t t h e e q u i v a l e n c e between b o t h c o n f i g u r a t i o n s . I n o u r example, we w i l l d e s c r i b e these approaches i n some more d e t a i l . 3. CONSTPAINTS I N PREDICTION-RESEARCH I n t h i s D a r t we w i l l d e s c r i b e o u r p a r t i c u l a r a n o l i c a t i o n o f tIDS i n p r e d i c tion-research.
T h i s a m l i c a t i o n w i l l enable us t o combine two Durposes a t
once:
1) s t r u c t u r i n g o f t h e v a r i a b l e s i n s e n a r a t e r e p i o n s , e.o. d i s c r i m i n a n t v a l id i ty;
2) s e l e c t i n g poor, moderate and s t r o n g p r e d i c t o r s o f t h e c r i t e r i o n , e.g. p r e d i c t i ve v a l i d i ty
.
F. I'm Ouerudle
454
'.'e a l r e a d y mentioned t h a t o u r anaroach c o n s i s t s o f two nhases: 1) l o o k i n g f o r a s t r u c t u r e o r h y n o t h e s i s a f t e r a " b l i n d " Wfl;
2) c o n f i r m a t i o n o r r e j e c t i o n o f t h e h y o o t h e s i z e d s t r u c t u r e by means of a PlW. The o a r t i c u l a r i t y o f o u r annroach l i e s i n t h e s t r u c t u r e we a r e l o o k i n g f o r : t h e '?.adex (Guttman ( 1 9 5 4 ) ) .
A r a d e x - s t r u c t u r e v i l l s e r v e o u r nurnose, be-
cause I!- cil: bave b o t h d i s c r i m i n a n t and p r e d i c t i v e v a l i d i t v .
'.'e w i l l now
e x o l a i n t h e c h a r a c t e r i s t i c s o f t h e r a d e x i n m r e d e t a i l , and show how these cap be t r a n s f o r m e d i n t o o n e r a t i o n a l c o n s t r a i n t s f o r t h e CVV-arooram.
3.1. THE RP!?EX-ST'?l'CTIJRF: The HNl i s hased on a ( d i s ) s i m i l a r i t y - m a t r i x between t h e v a r i a b l e s . f
r;r.nle,
In
our
ve used s w a r r a n - c o r r e l a t i o n s , b u t o t h e r association-measures a r e
also nossible.
R e f o r e e n t e r i n ? t h e a n a l y s i s , two o r e c e d i n o t r a n s f o r m a t i o n s
a r e necessary: 1) v a r i a S l e s w i t h a n e g a t i v e c o r r e l a t i o n w i t h t h e c r i t e r i o n a r e i n v e r s e d . 2) based on t h e s i g n i f i c a n c e - l e v e l o r s i m i l a r c r i t e r i a , v a r i a b l e s w i t h t o o low c o r r e l a t i o n s w i t h t h e c r i t e r i o n a r e e x c l u d e d f r o m t h e a n a l y s i s . should unnecessarily comnlicate the s o a t i a l renresentation. above t h e s e l e c t i o n a r e i n c l u d e d .
Thev
Variables
tJe s h a l l tic+/ c a l l them n r e d i c t o r s .
4 monotone d i s t a n c e a n a l y s i s (HDP) i s t h e n p e r f o r m e d on t h e c o r r e l a t i o n ma-
t r i x o f n r e d i c t o r s and c r i t e r i o n .
F o r rcpscns o f v i s i b i l i t y and i n t e r p r e t a -
b i l i t y o f t h e s o a t i a l c o n f i o u r a t i o n , a o l o t o f 2 dimensions i s o r e f e r e d . Because o f t h e f i x e d s e l e c t i o n - l e v e l o f t h e n r e d i c t o r s and t h e i n v e r s i o n o f neaative oredictors,
t h e c r i t e r i o n t o o r e d i c t w i l l l i e around t h e c e n t r e o f
the s o a t i a l p l o t . I n t h i s s o a t i a l r e n r e s e n t a t i o n t h e r a d e x - s t r u c t u r e i s souoht: 1) the i n t e n s i t y o f t h e c o r r e l a t i o r : w t t t h e c r i t e r i o n can be seen f r o m t h e distance t o the c r i t e r i o n .
To s e o a r a t e low, moderate, and s t r o n g pre-
d i c t o r s , one draws c i r c l e s w i t h t h e c r i t e r i o n as t h e i r comnon c e n t r e . These c i r c l e s c u t t h e s n a t i a l D l o t i n c i r c t i m n l e x e s (Guttman (1954), see fiaure 2A).
2 ) t o s e n a r a t e grouns o f h i g h l y i n t e r c o r r e l a t i n a and c o n c e n t u a l l y s i m i l a r n r e d i c t o r s , each c i r c u m l e x i s c u t i n t o r e g i o n s , l i k e a n i e i s c u t i n t o nieces ( f i g u r e 2 8 ) .
The Radex structure
455
A c o n f i g u r a t i o n w i t h t h e ahove mentioned p i e o r w h e e l - f o r m i s a radex (httrnan (1954)).
The n a r t s o f t h e radex a r e s o - c a l l e d r e n i o n s (Linqoes
(1Wl)). L o o k i n o f o r a radex i n a n l o t of 3 dimensions i s o f cnurse much more d i f f i cult.
So ur) t i l l now we have n r e f e r e d a h i q h e r a l i e n a t i o n i n two dimensions,
i n s t e a d o f a l o v e r a l i e n P t i o n i n t h r e e o r more d i w n s i o n s .
3.2. TFST OF HYPOTHTTICFL RAnFX-STRIICTURE There a r e , of course, s e v e r a l wavs f o r c u t t i n g un t h e snace i n a radex. nnce
a r a d e x - s t r u c t u r e found, i t i s i u s t one o f manv n o s s i h l e radexes t h a t
can he h y o o t h e s i z e d
or can be hidden i n t h e d a t a .
t h e D o i n t s w i l l f i t i n t o t h e radex.
I n manv cases, n o t a l l
'alith t h e CMnF,-nroaraln, a l t e r n a t i v e hy-
qotheses can he t e s t e d , and r e i e c t e d o r accented.
1 ci rcump 1 ex
F i a u r e 2A ?eripheral /mdial/central nredictors
.
c i rcumplexes o f noor/medium/strong
F. Van Ouerwolle
456
Finure 2R Radex w i t h d i f f e r e n t circumnlexes, each o f them c u t i n several reoions. Since CflDh i s hased on a n o n - m t r i c model, t4e radex s t r u c t u r e must be converted i n o r d i n a l r e n i o n a l c o n s t r a i n t s as i n o u t f o r the CFVA-prooram. 3.2.1. Constraints f o r circumnlexes. Points o f ce.;+ril
circumnlexes are c l o s e r t o the c r i t e r i o n than n o i n t s o f
pore nerinhere circumnlexes (see f i o u r e 3A).
i ( i n the c i r c u m l e x , Ci)
The distances hetween p o i n t s ,
and the c r i t e r i o n , c, r e s u l t i n the f o l l o w i n g
cons t r a i n t: i f : Ci
then : d ( i , c ) 3.2.2.
<
C. < 1
... <
> d(j,c)
>
C1
<
... >
c d(1,c)
Constraints f o r renions i n circumolexes.
Linooes (1981) develooed a method t o convert several forms o f r e g i o n a l hynotheses i n o r d i n a l c o s n t r a i n t s .
The keystones i n h i s anriroach are the
o u t e r - and i n n e r o o i n t s o f a r e o i o n .
These o o i n t s are t o he d e f i n e d f o r
each n a i r o f regions, R and R*.
P m i n t o f r e g i o n R i s an o u t e r n o i n t , 0, i f i t has the s h o r t e s t d i s t a n c e w i t h a n o i n t o f r e g i o n R*.
Goinn throuoh a l l n c i n t s o f R*,
o f them t a k i n o the c l o s e s t q o i n t o f
p
and f o r each
(which then becomes an o u t e r n o i n t ) ,
45 7
The Radex structure
one can d e f i n e a l l o u t e r m i n t s o f r e o i o n R i n r e q a r d t o r e g i o n R* ( s e e f i oure 3R).
The o u t e r m i n t s o f P can he c o n s i d e r e d as c o n s t i t u t i n s a bound-
*.
arv o f P acainst the onnosite rerlion P
Consequentlv, the i n n e r o o i n t s o f
a r e a l l n o i n t s o f P which a r e no o u t e r p o i n t s (see f i p u r e 3 3 ) . I n a s i m i l a r way, a l l o u t e r o o i n t s and i n n e r n o i n t s o f R
*
can be d e f i n e d .
Pnd t h i s can he d o w f o r a l l n a i r s o f r e c i o n s i n a c i r c u n n l e x .
To o r e v e n t
a s i t u a t i o n where a l l v o i n t s a r e o u t e r n o i n t s , one must a t l e a s t i n c l u d e more than 2 n o i n t s i n each r e o i o n . Since r e g i o n s i n a c i r c u n n l e x a r e n n t always convex, t h e y must comply w i t 4 t h e c o n s t r a i n t s o f a "weak c o n t i n o u i t v t y p e P " (Linooes ( l q 2 1 ) ) .
A repion
-
R s a t i s f i e s t h i s weak ( t v n e q ) c o n t i n w i t y c o n s t r a i n t " i f (each) i n n e r o o i n t
sow o u t e r n o i u t s o f P, t b s n i t i s t o anv o u t e r o o i n t o f o f P i s closer t o a n o t h e r r e o i o n R*"
(Linnoes ( 1 9 3 l ) , 0 1 9 0 - 2 9 1 ) .
For our radex-structure,
...
if each i n n e r p o i n t o f
t h i s d e f i n i t i o n o f Linooes i s s i i o h t l y d a r i t e d :
F! i s c l o s e r t o one o u t e r n o i n t o f P., than i t i s
t o any o u t e r n o i n t o f a n o t h e r r e o i o n R
*.
One can check o u r s l i g h t l y adaoted d e f i n i t i o n i n f i g u r e 31:
* i ,i n
hv t a k i n g a t a r -
r e n i o n R and s e e k i n a t h e s m a l l e s t d i s t a n c e w i t h one y e t inneraoint, o u t e r n o i n t ( t h e t a r q e t o u t e r n o i n t , o ) o f t h e S a m r e g i o n P,. T h i s d i s t a n c e d(i*,
*
o*) must be s m a l l e r t h a n t h e d i s t a n c e s het\.,een t h i s t a r n e t i n n e r n o i n t
i*o f r e g i o n R, and a l l o t h e r o u t e r n o i n t s o f R*.
T h i s i s t r u e f o r each i n -
n e r p o i n t o f P chosen as t a r n e t . To e x D l a i n o u r a r o u w n t s f o r t h e a d a n t i o n of L i n g o e s ' d e f i n i t i o n , we t a k e f i g u r e 3C once a g a i n . d(i*,
As can he seen i n t h i s f i g u r e , o n l y t h e d i s t a n c e
o*) ?s always s m a l l e r than t h e d i s t a n c e s hetween t h e t a r o e t i n n e r o o i n t
i*o f R and a l l t h e o u t e r n o i n t s cf ?*: o3 t h e i*o f !? and a n o t l i r r o u k r p o i n i d(i*,
02) >
d(i*,
03)
G f
i n f i p u r e 3C).
, 04,
05.
The d i s t a n c e between
2 can Ls l a r g e r ( e . o . T h a t ' s why o n l y
outernoint o f R
i s a c t u a l l y chosen f o r comnarison w i t h t h e o u t e r o o i n t s o f a n o t h e r r e g i o n R*. B u t i n c o n t r a s t w i t h Lingoes ( s e e examole n.297 (193l.\,.'q a radex s t r u c t u r e
each i n n e r p o i n t
o f R must s a t i s f y t h e weak c o n t i n g u i t y c o n s t r a i n t s .
To com-
n u t e a l l t h e r e g i o n a l c o n s t r a i n t s o f a radex o u t l i n e d above, a REGION-oronram was w r i t t e n i n F o r t r a n 4 by Van O v e t t r a l l e (1983).
458
,
1 c i r c u m p 1e x
F i g u r e 3A Reaionsl c o n s t r a i n t s f o r c i r c u m p l e x e s ( s e e t e x t )
F i q u r e 3P D u t e r o o i n t s and i n n e r n o i n t s o f r e g i o n R (see t e x t )
459
The Radex structure
F i g u r e 3C Regional c o n s t r a i n t s f o r r e a i o n s o f weak c o n t i n y i t y t y n e A (see t e x t )
f.
AF,
EYAV'LE
To i l l u s t r a t e o u r a m l i c a t i o n o f t h e r a d e x - s t r u c t u r e i n n r e d i c t i o n , we take an e x a m l e f r o m o u r r e s e a r c h
CP
student-achievement.
T h i s examnle i n c l u d e s
d a t a o f 2 p i l o t s t u d i e s from f i r s t y e a r s t u d e n t s a t u n i v e r s i t v , o a t h e r e d a f t e r t h e i r f i r s t h a l f year a t u n i v e r s i t y . Ue w i l l now i l l u s t r a t e o u r annroach s t e p by s t e n .
4.1. bl(lN0TONE DISTANCE ANPLYTIS Only 46 v a r i a b l e s were s e l e c t e d f o r an MDA-nlot,
by a s i o n i f i c a n c e - l e v e l o f
The s p a t i a l c o n f i y r a t i o n a f t e r a W N I S S A i s shown i n f i a u r e 4 .
a = 05.
One can r e a d i l y see t h a t t h e f i n a l e x a m i n a t i o n r e s u l t s , o u r c r i t e r i o n , l i e s i n the centre o f the n l o t . contact". vation".
The most c e n t r a l c i r c u m n l e x i s l a b e l e d " s o c i a l
I t i s surrounded by two medial r e g i o n s "self-esteem"
Finally,
one p e r i n h e r a l
and " m o t i -
r e a i o n "secondary school ", s connected
rri t h t h e o r e v i o u s s e l f - e s t e e m r e g i o n . Ve have drawn o u r h y n o t h e t i c a l s t r u c t u r e by means o f
-
c i r c l e s t o s e n a r a t e low, medial and h i q h circumnlexes,
-
s t r a i g h t l i n e s t o s n l i t uo t h e "weak c o n t i n g u i t y " r e g i o n s .
Although o u r h y p o t h e t i c a l radex g e n e r a l l y seems t o f i t w e l l i n t h e p l o t , s i x m i n t s c o n t r a d i c t our hycothesis.
To be i n l i n e w i t h o u r nroDosed
radex, thev p-ust l i e in another re?iion, indicated !w the 2 r r o m . Since m s t violations nccur i n the centre of the bllt!I%P-olot, these vio1at.ions can h e a consenuencp o f a reducer! accuracv o f a s n a t i a l renresentat i n n , d u e t o too w c h i r r e l e v a n t noints i n the oerinbery o f the n l o t . This a t - n u m e n t s e e m to be confirmed h v a sPcond analvsis where the inclusion l e vel i,,ias increased t o a c o r r e l a t i o n of 0.40 ( s e e f i o u r e 5). hl!/ 2 Pcints in this second s n a t i a l n l n t v i o l a t e the radex-structure, h u t our nerinherr71 I-enim o f secondar!, school i s now c o m l e t e l y l c s t .
4.2. ? ~ r ~ n : ,-:~“sT~?E!P’T~ :p~
F, F ? P ~ . X
Another, more simnle method to check a n a r t o f t5e hvnothesized radex-struct u r e , the circumnlex, i s s h w n i n f i o u r e 6 . 1,’e .just l i s t e d a l l o r i g i n a l c o r r e l a t i o n s tvi t b the c r i t e r i o n , c l a s s i f i e d accorrlinq t o t h e i r hypothetical imsition in a renion. Once aoain, there are several v i o l a t i o n s , h u t i n nen n r a l , the circumilex i s c o n f i r e d . r’oreover, we have now an e x a c t idea o f t’le houndaries hetween the circumolexes. The l e v e l s a t which tee c i r c l e s in our q l o t are drawn, coincide wit!] sn?prman-cnrrelations c f 0.40 a n d . X . Tlie radex orooosed in the f i r s t s n a t i a l confiauration ( f i y r e 4 ) , confirmed b t t a second s o a t i a l n l o t (ficlure 5 ) a n d hy the o r i v i n a l c o r r e l a t i o n s (fiTure C ) , i s en:ered i t - the PECIPII-nrorram (Van Overvalle (1983)) a n d r e s u l t s in ; 4 reclinnal c o n s t r a i n t s . These c o n s t r a i n t s serve now as i n n u t f o r t?e L-lIiyoec-pcrg Cf”1A-nrograF. ’
4.3. CfWFIri’!PTQ?Y F’WIPTWF DIqTPYCE PWLYSI5
Firtire 7 reoresents a s p a t i a l n l o t o f a c o n f i r v a t o r v / c o n s t r a i n t s a n a l y s i s . f i l l qoints are nor’ i n t h e i r expected Position accordinrl t o the radex-structure o u t l i n e ? above, excent f o r one p o i n t . I t i s not our i n t e n t i o n t o sneculate on the causes o f t h i s excention + n t h i s t e x t , s i n c e we nresent t h i s analysis merely f o r i l l u s t r a t i v e nurooses. nn the contrary, we w i l l now discuss
some anoroaches t o evaluate the r e s u l t s
o f a c o n f i r n a t o r y l c o n s t r a i n t s a n a l y s i s , beyond the visual i n t e r n r e t a b i l i t v
o f the s o a t i a l confiauration.
46 1
I
-
N OF %OIClORS MIENATXW .29
-
46
Figure 4 111N I S S A
S o a t i a l o l o t of sDearPan c o r r e l a t i o n s and c r i t e r i o n .
(a
0.5) between nredictors
F. Van Otarwalle
,, '
i
/
/
i
.
1.3-t w a r at 5 . 5
t1.fsctlrn
"lth
0
Figure 5 MINISSA 5 n a t i a l n l o t o f snearman c o r r e l a t i o n s and c r i t e r i o n .
(2
.40) between nredictors
?he Radex structure
rho = 2 0 I
463
30
48
50
60
70
80
90
I
I
I
I
I
I
I
100
1
. . ... ..
SOCIAL CONTFICT
Fiqure 5 Snearman c o r r e l a t i o n s o f p r e d i c t o r s w i t h c r i t e r i o n .
The p r e d i c -
t o r s a r e l i s t e d a c c o r d i n g t o t h e i r o o s i t i o n i n t h e weak c o n t i n q u i ty r e g i o n s
4.4.
.
TEST FOR RADEX STRUCTURE
Table 1 shows t h e a l i e n a t i o n c o e f f i c i e n t s o f t h e VINISSA and CPlDA c o n f i o u rations.
Imposinq r e a i o n a l c o n s t r a i n t s f o r o u r h v D o t h e t i c a 1 r a d e x - s t r u c t u r e
i s naved o f f by a l i t t l e i n c r e a s e o f t h e a l i e n a t i o n c o e f f i c i e n t . H o w v e r , t o t e s t whether t h e f i n a l CPlDA-plot i s e q u i v a l e n t w i t h t h e i n i t i a l
MINISSA o l o t , Lingoes-14oodroofe c o r r e l a t i o n s , based on t h e ranks o f t h e c o r r e l a t i o n s / d i s t a n c e s a r e used, i n s t e a d o f t h e a l i e n a t i o n c o e f f i c i e n t s . I n t h e i r f i r s t approach, Linooes and Sorg (1980) adaoted H o t e l l i n g ' s t - t e s t i n o r d e r t o t e s t whether t h e d i f f e r e n c e o f t h e c o r r e l a t i o n between d a t a and NINISSA-plot on t h e one s i d e , and t h e c o r r e l a t i o n between d a t a and CMDA-plot on t h e o t h e r s i d e , i s l a r g e / s m a l l enough t o r e j e c t / a c c e p t t h e e q u i v a l e n c e between b o t h c o n f i g u r a t i o n s .
A c c o r d i n g t o t h i s s t o c h a s t i c model
, we
464
Finure 7 CIlDA
S n a t i a l p l o t o f saearman c o r r e l a t i o n s ( a 5 .05) w i t h r e a i o n a l constraints
.
465
The Radex structure
must r e . i e c t t h e C I T A c o r i f i y r a t i o n (see Table 2) a1 i e n a t i on
VTSqp
(no cons t r a i n t s )
C W P ( w i t'l c o n s t r a i n t s \ i n c r e a s e o f a1 i e n a ti on Tz'-IG I
ChttIran-Linooes a l i e q a t i o n C o e f f i c T e r t s o f i n i t i a l (VINISSfl) and f i n a l (ClrPA) c o n f i o u r a t i o n . Linnces-l'oodroofe c o r r e l a t i o n between data data
-
' 7 N I S V : rhol CtlIV : rho2
.783 .767
Probahi lit y of d i f f e r e n c e between rhol and r h o 2
.001
-_-
~ l I ~ ~ I S ( P . - c o n f i g u r a t i othan n t h e a c t u a l observed f i n a l c o n f i c u r a t i o n of C""P-vrnorar.
1,. F!
'ale a y e e w i t h Linooes and R o y t h a t t h i s " n r e s e n t p r o n o s a l
has much t o recormend i t i r t e r m o f h e i n q l e s s a r h i t r a r y and more i n f o r m a t i v e i n the r e a l m o f t h e o o s s i b l e " . requirement (see t a b l e 4 ) .
Our example does cnrnnly w i t h t h i s new
I n o t h e r d a t a a n a l y s e d un t i l l now, i t seems
t h a t the previous decision r u l e ( R a t i o t i o n o f Ratio
<
>
3 ) and t h e o r e s e n t r u l e ( o r o p o r -
7 7 ) reach c l o s e r e s u l t s i n terms o f a c c e n t i n q / r e , j e c t i n a t h e
e q u i valence hyDothesi s . We j u s t have g i v e n a v e r v b r i e f o v e r v i e w o f r e c e n t develonments w i t h r e g a r d t o t h e c r u c i a l q u e s t i o n w h e t h e r one can a c c e p t t h e r a d e x - c o n s t r a i n t s
imoosed
I t seems t o us t h a t t h e l a t e s t oronosed W I D E R model rrives a
on the d a t a .
c o n c l u s i v e answer t o t h a t q u e s t i o n , n o t o n l y i n m a t h e m a t i c a l terms, b u t a l s o i n terms of c l o s e a o r e e m n t w i t h what one s h o u l d e x n e c t from v i s u a l i n s n e c L-l,!
c o r r e l a t i o n between IIINISSA
-
CPIDA : r h o 3
.967
: rhoc
,919
n a r t i a l c o r r e l a t i o n o f rho3 degradiition c o e f f i c i e n t K3 =
1-rho32
.254 3.621
r a t i o R = rhoc/ K 3 decision-model f i r s t rule: do b o t h n l o t s have more v a r i a n c e i n common, o r rhoc
>
V-
ves
?
second r u l e : does r h o 3 exceed 3 t i m e s d e q r a d a t i o n K3, o r ratio R > 3 ?
yes
decision : t h e e q u i v a l e n c e of b o t h c o n f i a u r a t i o n s i s
a cceD t e d -__L_
.-
Tab12 3 Lingoes-Borq q u a s i - s t a t i s t i c a l model f o r choosing between two a1 t e r n a t i v e c o n f i g u r a t i o n s . t i o n o f the c o n s t r a i n t c o n f i q u r a t i o n .
46 7
7%e Radex stmcture
P(R
>
w ) = p r c o o r t i o n o f observed r a t i o ' s .044
exceedinn a l l n o s s i b l e r a t i o ' s = g r e a t e r u m e r bound o f
G.U.B.
P(R
>
w), ,045
over a l l oossible d e l t a ' s d e c i s i on-mode 1 f i r s t rule: do h o t h D l o t s have more v a r i a n c e i n cormon, o r rhoc >
C5 ?
yes
second r u l e : t h e GUIDE d i s t r i b u t i o n >
w) < 79r ?
yes
G.U.R.
< 7Y ?
Yes
P(R
decision: t h e e q u i v a l e n c e of b o t h c o n f i g u r a t i o n s i s
acceo t e d
Lingoes-Borg mathematical d e c i s i o n model f o r choosing between two a1 t e r n a t i v e c o n f i g u r a t i o n s .
5 . CONCLUSION When a l a r g e number o f d i f f e r e n t v a r i a b l e s a r e t o be s e l e c t e d i n o r d e r t o f i n d d i s t i n c t ?r?d c r e d i c t i v e o r e d i c t o r s f o r r e g r e s s i o n - a n a l y s i s o r more comD l e x causal models, o u r aooroach, which i s i n i t s essence a method t o s t r u c t u r e t h e o r i g i n a l d a t a a c c o r d i n g t o a radex, can be h e l p f u l . T h i s approach does n o t have t h e disadvantage o f o t h e r " b l i n d " s t r u c t u r i n g techniques l i k e c l u s t e r - o r f a c t o r - a n a l y s i s , which a r e o n l y s e n s i t i v e t o c o r r e l a t i o n s between D r e d i c t o r s , and n e g l e c t s t h e c r i t e r i o n which has t o be predicted.
I n o u r radex-amroach, we reach two o b j e c t i v e s a t once: d i s c r i -
m i n a n t v a l i d i t y , which means t h e g r o w i n g o f i n t e r r e l a t e d v a r i a b l e s , and D r e d i c t i v e v a l i d i t y which grouDs t h e o r e d i c t o r s a c c o r d i n o t o t h e i r r e l a t i o n
w i t h the c r i t e r i o n
.
T h i s aonroach has a l s o some disadvantages.
To f i n d a radex i s n o t so easy,
e s p e c i a l l y when t h e r e a r e more t h a n 2 dimensions t o l o o k a t .
On t h e o t h e r
hand, once a radex i s found and accepted, t h e s o a t i a l r e s u l t i s more
a c c e s s i h l e f o r t h e readers, t o whom t h e i n t e r o l a y between d i s c r i m i n a t i o n and n r e d i c t i o n h e c o w s p o r e v i s i b l e now. v a r i a b l e s t r e a t e d )sfas P3.
1Jr1
t i l l nnw. t h e l a r o e s t number o f
T l i i s number can, o f course, be l a r w r , i f t h e
a l i e n a t i o n does n o t i n c r e a s e t o o m c h .
The h o u n d a r i e s o f o u r annroach do
n o t l i e i n t h e number o f v a r i a b l e s , h u t i n the accuracy and i n t e r n r e t a h i l i t v o f the s o a t i a l r e q r e s e n t a t i n n . Another,
t e c h n i c a l , disadvantane i s t h e l a r n e numher o f r e q i o n a l c o n s t r a i n t s ,
!which i s o f t e n o v e r 100 f o r a r a d e x - s t r u c t u r e (more c o p n l e x than the o r e s e n t examnle) i n 70 o r more i o i n t s .
T+es.
c c r z t r a i n t s , taken tnqether w i t h
b o t h m a t r i c e s o f t h e o r i o i n a l c o r r e l a t i o n s and i n i t i a l c o n f i n u r a t i o n , r e q u i r e a l a r o e number o f coml,uter-memory o r , a l t e r n a t i v e l y , comauter-time
.
A s i m n l e r c o n v e r t i o n o f a radex i n r e n i o n a l c o n s t r a i n t s i s ~ * ~ l c n p r p .
RE FE f'E NCES
B l a l o c k , & f . , Causal models i n t h e s o c i a l s c i e n c e s , Chicaoo, P l e i n e
(1971). Boro, I . , and Lingoes,
J.C., fl model and a l a o r i t h m f o r p u l t i d i m e n s i o n a l
s c a l i n o w i t h e x t e r n a l r e s t r i c t i o n s on t h e d i s t a n c e s , Psvchometrika, 45
(190@), 25-38. Roro, I . , and Linooes, J.T., : ' u l t i d i m e n s i o n a l s c a l i n g w i t h s i d e cons t r a i n t s on t h e d i s t a v c e s . I n J . Lingoes, E . Roskam, and I . 9 o r a (Eds.), Geometric r e n r e s e n t a t i o n s o f r e l a t i o n a l d a t a , Ann A r b o r , ' l a t h e s i s Pres~., ( 1979) , 733-790.
Denisn-..
C.P.,
t ' u l t i d i m e n s i o n a l s c a l i n q and s t r u c t u r a l e q u a t i o n model-
l i n g : a comparison o f m u l t i v a r i a t e t e c h n i q u e s f o r t h e o r y t e s t i n g , h l t i v a r i a t e B e h a v i o r a l Research, 1 7 ( 1 9 8 2 ) , 447-470. Guttman, L . , A ne\v a m r o a c h t o f a c t o r a n a l v s i s : t h e radex, I n
D.F.
La-
zers7elC ( E d . ) , t l a t h e m a t i c a l t h i n k i n g i n t h e s o c i a l s c i e n c e s , New York, Free Press ( 1 9 5 4 ) . Guttman, L . , A n e n e r a l n o n m e t r i c t e c h n i q u e f o r f i n d i n g t h e s m a l l e s t c o o r d i n a t e snace f o r a c o n f i g u r a t i o n o f p o i n t s , Psychometrika, 33 (1968) 469-506. K r u s k a l , J.B., N l t i d i m e n s i o n a l s c a l i n g by o n t i m i z i n g goodness o f f i t t o a n o n m e t r i c h y p o t h e s i s , Psychometrika, 29 (1964a), 1-77. K r u s k a l , J.B.,
Nonmetric m u l t i d i m e n s i o n a l s c a l i n q : a n u m r i c a l method,
Psychometrika, 29 (1964b), 115-129.
469
The Radex structure
Lingoes, J.C.
, The
cuttman-Lingoes n o n m e t r i c nrooram s e r i e s , Mathesis
Press , Ann Arbor, r f i c h i gan (1973) Linqoes, J.C.,
.
T e s t i n q r e o i o n a l hypotheses i n m u l t i d i m e n s i o n a l s c a l i n g ,
I n Borg, I . (Ed.) M u l t i d i m e n s i o n a l d a t a r e p r e s e n t a t i o n s : when and why, Mathesis Press, Pnn A r b o r ( 1 9 0 1 ) . Linqoes, J.C., analysis
-
and Borq, I
., Cr1DA-U:
c o n f i r m a t o r y monotone d i s t a n c e
U n c o n d i t i o n a l , J o u r n a l o f b j a r k e t i n g Research, 15 (1978)
,
610-611. Lingoes, J.C.,
and Born, I., P model f o r choosing between a l t e r n a t i v e
configurations derived f r o r o r d i n a l l y constrained data, U n i v e r s i t y o f p d i c h i qan
, Computer
Linqoes, J.C.,
Centre , t e c h n i c a l r e n o r t , 1 (1981)
and Soro,
I.,
, 1-28.
A c r i t i q u e o f two Drooosals f o r comparinq
PIDS c o n f i g u r a t i o n s , l h i v e r s i t y o f Michiaan, Comouting Centre, 'iechnic a l r e p o r t , 1 (1982) Linqoes, J.C.,
, 1-26.
and Boro, I . , A q u a s i - s t a t i s t i c a l model f o r choosing
between a1 t e r n a t i ve c o n f i g u r a t i o n s d e r i v e d f r o m o r d i n a l l y c o n s t r a i n e d data, B r i t i s h j o u r n a l o f mathematical and s t a t i s t i c a l osycholoqy, 36 (1983a). Lingoes, J.C.,
and Boro, I . , P mathematical d e c i s i o n model f o r t h e
Lingoes-Borg c r i t e r i a i n t e s t i n a t h e e q u i v a l e n c e h y n o t h e s i s .
Univer-
s i t y o f Michigan, C o m u t e r Centre, t e c h n i c a l r e o o r t , 1 (1983b), 1-20. Lingoes, J.C.,
and Roskam, F.,
A mathematical and e m i r i c a l s t u d y o f
two mu1 t i d i m e n s i o n a l s c a l i n n a l g o r i t h m s , Psvchometric llonographs, 38 (1973). Sheoard, R.N.,
The a n a l y s i s o f o r o x i m i t i e s : m u l t i d i m e n s i o n a l s c a l i n o
w i t h an unknown d i s t a n c e f u n c t i o n , I , Psychometrika, 27 (1962a), 125-140. Sheoard, R.N.,
The a n a l y s i s o f w o x i m i t i e s : m u l t i d i m e n s i o n a l s c a l i n g
w i t h an unknown d i s t a n c e f u n c t i o n , 11, Psychometrika,
27 (1962b)
,
219-246. Van Overwalle, F.
, Region-proaram
( F o r t r a n 4) t o comoute o r d i n a l l y con-
s t r a i n t s f o r t e s t i n g r e g i o n s i n t h e Lingoes-Rorg CIIDA-program ( 1 9 8 3 ) .
I would l i k e t o thank P r o f . D r . J . Lingoes f o r h i s a r t i c l e s on r e c e n t devel o o m n t s i n e q u i v a l e n c e t e s t i n g and f o r h i s comments on t h i s paoer. also grateful t o
B. Mergaerts and M. Van d e r P o s t who h e l p e d w i t h t h e
English trans1a t i o n .
I am
470
I;. Van Overwalle
( k e c m r a k e o b j e c t i o n s a q a i n s t t h e r a t h e r unusual l e v e l of 7:' i n the L i n coes-Rorn (1983b) F u i d e r - d e c i s i o n model t o a c c e n t o r r e j e c t t h e e q u i v a l e n c e o f c o n f i r y r a t i o n s v i tt: o r w i t h o u t s t r u c t u r a l c o n s t r a i n t s .
Fortunately,
a n o t h e r h e u r i s t i c mdethod was develoaed b y H u b e r t and Golledge ( H u b e r t , L . J . and Golledge. R.G.,
P h e u r i s t i c method f o r t h e cornoarison o f r e l a t e d s t r u c -
t u r e s , J o u r n a l o f P a t h e m a t i c a l Psychology, 23 ( l q e l ) , 214-276).
Similar i n
i t s o o s s i b l e a o p l i c a t i o n s , t h e i r h e u r i s t i c can a l s o he used t o t e s t t h e e q u i v a l e n c e of c o n f i n u r a t i o n s d e r i v e d f r o m t h e same d a t a . The L i n g o e s - 8 o r q and Hubert-Go1 ledoe anpmaches kzve i n common t h a t t h e y are n o t based on a t h e o r e t i c a l d i s t r i b u t i o n , b u t on a r e f e r e n c e d i s t r i b u t i o n instead.
There a r e however, i m o r t a n t d i f f e r e n c e s between h o t h aoproaches,
namely i n t k e g e r e r a t i o n o f t h e r e f e r e n c e d i s t r i b u t i o n .
Lingoes and Bor?
(iSC3b) generate a r e f e r e n c e d i s t r i b u t i o n o f n a r t i a l c o r r e l a t i o n s , and hence derived r a t i o s .
T h i s r e f e r e n c e d i s t r i b u t i o n i s l i m i t e t k v t h e two raw c o r -
r e l a t i o n s between b o t h c o n f i n u r a t i o n s and t h e d a t a , i n c r e a s i n g f r o m z e r o t n t h e i r actual values.
I f l e s s than 77 p o s s i b l e r a t i o s o f t h i s r e f e r e n c e d i s -
t r i b u t i o n exceeds t h e a c t u a l r a t i o , t h e e q u i v a l e n c e can be accented.
Hubert
and Golledge, on t h e o t h e r hand, nermutate randomly t h e d a t a m a t r i x o f ( d i s ) s i m i l a r i t i e s and comnute f o r each n e r m u t a t i o n a c o r r e l a t i o n c o e f f i c i e n t .
-
I f t h e observed c o r r e l a t i o n does n o t d i f f e r s i o n i f i c a n t l y ( e . ? . a t a l e v e l
o f .01 o r .05) f r o m a l l c o r r e l a t i o n s g e n e r a t e d by these random nermutations, t h e y contend t h a t t h e d i f f e r e n c e between b o t h c o n f i a u r a t i o n s i s n o t s i a n i f i can t .
I n the n e x t t a b l e , we g i v e some n u m e r i c a l examples.
I n o u r examnles ( 6 7
o r m r e v a r i a b l e s ) , a f i r m s i m i l a r i t v between t h e Linaoes-Rorg and H u b e r t Golledge w s u l t s anpears.
One can observe t h a t L i n o o e s - B o r o ' s 7% l i e s some-
what h a l f - w a y between t h e o r o h a b i l i t y l e v e l o f 1% and 57' o f t h e Hubert-Golledoe h e u r i s t i c .
tlorrever, i n L i n g o e s ' examnles, which c o n s i s t o f 8 and 17
p o i n t s , b o t h methods do n o t f i t .
The Lingoes-Bora a u i d e r i s much more d i f -
f i c u l t t o s a t i s f y t h a n the I!ut-ert-Golledge
heuristic.
I n order t o accept
these s m a l l e r c o n f i o u r a t i o n s , Lingoes and B o r q (1983b) r e f e r t o t h e i r n r e vious apnroach ( L i n q o e s and Bora ( 1 9 8 3 a ) ) , which i s even more a r b i t r a r y .
To a v o i d t h i s , one can use t h e H u b e r t and G o l l e d n e (1981) h e u r i s t i c model.
I
I I
HUBERT-GOLLEDCE HEURISTIC
LINGOES-BORC GUIDE
DESGRIPTION O F DATA
hypothesis
variables
ratio
data
proportion of ratio
permutat ions
GLUSHKO
EPUIVALEYCE
17
I ,004
.328
lo00
CUTTUAN
DOUBLE-FACET
8
1.123
.295
800
K
. 129 -.101
z-value
1.219 -.223
INTEGRATIE(5)
FINAL ( I )
75
3.299
.052*
lo00
-.097
- I . 764
INTEGRATE())
WEINER I
75
2.808
.070'
I000
-.092
- I .883
INTEGRATIE(5)
WEINER 2
75
3.834
,040'
I000
-.062
- I .082
BLOK ( 2 )
F I U L (3)
67
3.031
.061'
lo00
-.105
- I . 758
BLOK ( 2 )
WEINER I
67
3.065
.059*
1030
-.079
- I .298
BLOK ( 2 )
WEINER 2
67
3.204
.054*
I000
-. 075
- I . 149
W S (2)
FINAL ( 5 )
81
2.418
.087
lo00
-.1 2 2
-2.521
*
equivalence accepted (proportion < .07)
x*
probability of d i f f e r e n c e
equivalence accepted ( p r o b a b i l i t y o f d i f f e rence > .05)
* equivalence
accepted ( p r o b a b i l i t y of d i f f e rence > .03)
Table 5
Thanks t o Dr. G . L k Soete f o r brinaing the o o s s i b i l i t i e s o f the Hubert and Golledqe h e u r i s t i c t o my a t t e n t i o n .
This Page Intentionally Left Blank
413
AUTHOR INDEX Abdi, H . 6,7 Adam, E .I*/. 286 Andersen, E.B. 16f',257,414,422 Anderson, N.A. 37 Andri cli, D. 299,414,422,429 Anthonisse, J . V . 366 Arabie, P 5,223,379 Arbuckle, J . 240 Arend, L.E. 67 Armstrong, I.1.E. i,~. Arrow, K. 180,365,374 122,124,125,135 Atkinson, R.C. Ayer, !!. 264
.
Baoozzi, R.P. 318 B a i l e y , D.E. 378,379 B a l i n s k i , 1d.L. 366 241-244,246,249,264, Barlow, R.E. 266-261: BarthCl@my, J c P . 183,184 Bartholomew, D.J. 241,242,244,246 B a r t l e t t , F.C. 3,14,17 Bass, F.Y. 387 Bauer, H . 319,341 Beals, R . 194,227,359 Benzecri , J.P. 7 Bero, J . A . 417 Beroe, C. 219 Betson 126 B i r k h o f f , G. 230 Bimbauln, A. 167 B i rnbaum, M.H. 37,303 Blake, R . 60 Bock, R.D. 257 Conacich. P . 219 Boorman,'S . A . 5,223 Borc, I. 360,453,460,463,465-467, 470 Bouchet, A. 274 Bougl@, c. 219 B o u s f i e l d , !*'.A. 7 122,124 Bower, G.H. Bradley 427 Rreimann, L. 319,341 Bremner, J.I.1. 241 B r i e n 125 Broadbent, D.E. 89 Bruner, J . 123 Brunk, H.D. 241 Buehler, J.N. 67 B u f f a r t , H . 42-44,51,56,57
Bunernan, P . 8 Bush, R . 122,124,150 Campbell, D.T. 318 C a r r o l l , J D. 3,8,380-382,384-386, 389.390 Chana, J . J . 3,8,380 Chapanis, A. 127,129 Chipman, J . S . 275 C1 ark, L . h . 380,385,389.390 . . _ Cogan, A . I . 60 Copis, 0. 274 Cohen, M . A . 60,67,79,84 Condorcet 183.185 Conway. J . 61' Cook,-T.D. 318 159,171,283,383,415,438 Coombs, C.H. Cornsweet. J . C . 60 Cornsweet; T.N. 60 Cowan, T.M. 105 280 Cozzens, r7.B. Crothers, E.G. 124 Cunninqham, J.P. 3,8,11,247,380
.
Dawes, R.M. 438 Degenne, A. 7,219.224 de Haan, B . 60 450 Denison, D.R. DeSarbo, b'.S. 378-380,384,385,387, 389,390 De Soete, G . 380,384 de Ideert, C.M.H. 61,74 DObson, A . G . 386 Dobson, J . 8 Doignon, J .-P. 210,216,231,233,234, 274,359 Domhoff, 1.1. 219 Doreian, P. 219 296 Douglas, G . A . Draper, 105 DrBsler, J . 230,359 Ducamp, P . 210,215,216,231,233,234, 274.293.297.300 .~ Duval ,- S 177 Duval, V.H. 177 S.141.
.
Ebbinghaus, H. 121,125 E hrens t e i n 70,72-75 Engelman, L. 379
474
./Lulkor
1@5
Eschcr, I'.C. ESteS, 1.I.K.
122
210,215,216,231, 233,234,256,274,286,2!?3,297,300, 359 F a r r i s , J . 5 . 385 160,296,298,393,398, F i s c h e r , C. H 399,420-422 Fishburn, P . 1 9 3 , 1 ~ 4 , 7 1 0 , ~ 1 1 , ? 1 6 , 272,278,360 Flament, C. 7 Fox, R . 60 Franke, C.H. 285,286 F r i e n d l y , PI. 3,6,27,28 Furnas , G , I d . 380-382,384,385 Falmacne, J.-C.
.
G a l a n t e r , E.H. 171,284 C i n s l e r 319,341 Garner, V . R . 33,56 G a t i , I . 178 Geissler, Y . 4 . ?1,35,?7,42,43,56 ( l e l l a t l y , A.R.H. 61 G e r r i t s , H.J.M. 60.80 G i l l , P.E. 263 Golledge, R . G . 470,471 Goodman, N. 284 Gordon, A . G . 184 Green, D.M. 87,161 Greeno, J.G. 122 Gregory, R.L. 105,106 Grossberg, S . 60-63,67,70,73,75,7€,
79,84 Guilbaud, G.Th. 183 258,259,265,767 Gui 1 f o r d , J .P C u l l i k s e n , H. 398 Gus t a f f s o n , J .E. 399 Guttman, L. 210,231,274,293,297,
.
300,414,415,452,454,455,465 152
Gvozdev, Y . A .
Hakami, S.L. 8 Halmos, P . R . 319 H a m i l t o n , A . 368,369,373 Hanter 127 flarbeck, A . f 1 Harshman, R . 390 H a r t e r 125 H a r t i pan, J . A . 8,244,247,379,380,
385 Hasse
186 ,:E7,229
index
tlayes-Roth, B . 56 Hayes-Roth, F. 56 Hebb, D.O. 60 ' i e i s e . D.P. ?It H e n d r i ckson, A.F 66 Henlev, N. H. 3,6,7,14,25,26 H e r b s t , K . 230 Heron, 14. 60 Hick, 1d.E. 34 H i l d r e t h , E . 62 Hil1,M.O. 7 H i r o t s u , C . 246 Hoffmann, J . 40 Hohle, H. 259 Holman, E.1-I. 350 Hoogstraten, J. 147 Horton, J.C. 66 H o t e l l i n p 453,463,465 Hubel, D.H. 66 Hubert, L . J . 470,471 Puffwan, D.A. 105,113 H u l l , C.L. 122 Hunt, S.P. 66
.
.
Jacquet-1 scrPze , E 280 J a f f r a y , J.Y. 359 James, G . S . 246 Jansen, t4.G.H. 409 Jansen, P.G.N. 420,424 Jas t r o w , J 160,166 ,lerison, P . J . 89,91,92 Johnson, S.C. 6,385 Jones, L.V. 257 Joreskoo, K.C. 394 J u l e s z , B . 61
.
K a n i i s a , C. 61 Kailfman, L . 60 Kempf, W.F. 296 K e n d a l l , M.G. 184 Kennedy, J.H. 61.78 K e r l i n g e r , F.N. 6 K l i n g , G.W. 122 K l i x , F . 37 Kow 126 Kran t z , D.H. 194,197,204,227,239,
256,259,295,348,349,359,360 K r i n c h i k , E.P. 136 K r i s t o f , 14. 393,398 Kristofferson, A.B. 89 K r u s k a l , J.B. 7,12,248,265,452
475
Author index
Lamino, D. 93,99 Lanczos, C. 172 60,77,78 Land, E.H. L a r i w r , J . 240 399 Lawley, D.N. Leewenbera, E .L . J . 42-44,56,57, 61 Lehnann, D.R. 386 Leonard, T. 407 136 Leontyev. A.N. Levey, (1. 61 L e v i t t , P.R.. 223 Lewis 409 L i k e r t , 9 . 414,429 L i n d l e y , D.V. 405,407 415,452,453,455-457, L i nqoes , J C 460,463,465-467,470 L i v i n n s t o n e , P.S. 66 Lockhead, G.R. 67 Loeve, 11. 319,341 Loh, H.K. 160 127,128,155 Lomov, R.F. Lord, F .H. 393,399,405,406,420 L o r r a i n , F. 221,223 Luce, R.D. 88,122,209,239,240,256, 271,272,277,279,284,286,299,301, 348,359,427 Luoncr, X . 10
..
241 Mac Donald, R.R. 160 flac Rae, A.!!!. Ilaqazanik, Y.V. 147 Ilandel, J . 241 Harks , 1.1. 6 1 Narr, D. 62 Masters, <.PI. 414,429 rlausfeld, R. 257 Hac C l e l l a n d , 6 . 240 Hc Cornick, V.T. 379 Mever. F. 101 rli”l1 e r , G .A. 3,6,7,22,23,378 Mingol l a , E 60,62,70,75,78 Fon j a r d e t , B 183,184,210,211,274 Mostel l e r . F. 122.124.150.259 . . . rlurray, (4. 263
.
.
Narens, L. 286,359 Necker, L.A. 106 Ng, Y.-K. 193 378 N i c e l y , P.E. N i s h i i a t o , S. 7 Novick, V.R. 393,398,399,405,406
O‘Brien, V. 59,67,69,71 Orth, B . 359 Owen, J . 405,407 Parks, T.E. 61 Pavlov 121 Pendl , P. 296 Penrose, L.S. 105,106,113 penrose, R. 105,106.113 Pessimier, E . A . 387 Petry, S. 61 Pfanzagl , ,I. 286 P i c k e t t , R.M. E9,91,92 127 Platonov, K.K. Poincar6, H. 284 Popper, K.R. 394 Prenowitz, \
.
Samejima, F. 414,429 S a r l e , W.S. 379 S a r l e r , L.D. 221
416
S a t t a t h , q . 3,8,9,11,380,321 Schaher, C. 394 Scheffe, Y . 248 Scheihlechner, H . 296 S c h e i d e r e i t e r , U . 37 Schiffman, S . 7 Schijnernann, FJ .H. 360 Schwarz, L . t l . 127 Schweitzer, P.J. 379 q c o t , D. 213,257,284 qeidnan, S . R . 219 Sbenarc', Q . N . 379,452 243,245-247 Shorack, C.P. Simnn, H.A. 321 405 Smith, A.F.P. S m i t h , P . T . 241 Sneath, P . F . P Sneider, F. 155 q o k a l , P.0. 8 7olomon-Ilinn 124 Snada, H . 296 qnillman, L . 61,70,73 S t e q m ~ l l e r ,( 1 . 322 tenson, H .ti. 89,91,92 Stern, 37 171 s t e v e n s , 5.7. Steyer, 9 . 317,318,321,323,330, 331,340,341 Stone, V . H . 407 Stumnf, P . 257 S t u t e . 11. 319,341 Suck, R . 359 Cugihara, K . 105 Suppes , P . 210,239,256,284-286, 296,301,322,348 Swets, J.A. 87,89,161 S v i f t 175 121.
Teroan, 5 . 0 . 100 Terouanno , E . 105,106 Terry 427 Thomnsen 205,206 Thorndike, E . L . 121 Thurstone, L . L . 257,258 Timnerrnan, J.G.bq.E.N, 61: l'orcerson, 14 .S . 7,159 Tryon, R . C . 378,379 Tukey, J.'.'. 239-241 Tulvino, E . 7 Tverskv, A . 3,8,9,11,12,177-180, 182,188,193,194,227,239,256, 348,359,360,380,381,438
5 Vallacher, R . R . Van PcCer, D. 104 Van der Ven, P.H.c.c. 393,405,407 van Feden, C . 264 Van Heerden, , l . 517 Van O v e n i a l l e , F . 457,460 van T u i j l , II.F.,1.K. 61,74 Venda, V.F. 130,131,136,137 60,F'O Vendrick, A.J.11. Veroes, P . 7 Vincent, S.R. 133,134 Vincke, Ph. 214,215,772,280
' ! e y e r , D.!1. 5 I h i t e , t 1 . C . 221,223 I J h i t e , T . I 1 . 379 '.liesel, T.N. 66 !-finer, R.J. 243,245,243 l.linsherg, S . 259,262 'alold, H . 366 '.loodroofe 463,465 '.loo &or t h 122,125 ,12 7 I J r i a h t , R . D . 407 "Iu, J:-Y. €6 Uykoff, L . R . 90 Yannakakis, I:. 234 Yarbus, A.L. 60,75 Yan, S.S. 8 vouna, F.I.1. 7 vouno, H . P . 366 Zarakovskv 130,134 Zitnolong, !?. 101 Zinnes, J . 284-286
477
SUBJECT INDEX adantation dynamics 121 adantive 60,61 addi t i ve 25 7 ,263,295,347-35 3,35 7, 358,377,378,380,385,386,389,390 a d d i t i v e d i f f e r e n c e 193-195.197. 202,205 a d d i t i v i t y 239,241,244,247-249, 334 -336 as,vvnmet r y 177- 180,182- 184,186- 183 a t t e n ti on 87,89-93,9C, 100-102 a t t i tude theory 294,304,305 binocular r i v a l r y 59,60,63,64,79, 80 biorder 210-212,714-216,227,232, 235,236,274 h i o r d e r dimension 227,231,233-236 b i o a r t i t i o n 4,5,12.14 b r i g h t n e s s percention 59-62,67 c a t e g o r i e s 33,36,41-46,48,51,52, 54,55 causal model 317-319,321,330, 333-338,340,451,468 choi ce 34,35,56,88,94,101,159,160, 172,255,299 chromatic number 234 c l a s s i f i c a t i o n 33-35,37,38,40-42, 45 -50.5 2-55 c l u s t e r i n g 3,6,7,13,14,378-380, 382,385,387-389,393,394,396,397, 450.452.467 coding- 33;43-45,52,54,55 comnlementary c o l o r induction 59 cow!!ete o r d e r 183,185,187 comosi te measurement 293,300 condorcean a n a l y s i s 180,183-186 Condorce t re1 a t i on 183 ,185 c o n j o i n t measurement 205,239,241, 244,257,347,343,350 contour oerceotion 59 convex 106,110,111,113,116,264, 45 7 Craik-O'Brien e f f e c t 59,67,69,71 d e t e c t a b i l i t y 255 d e t e c t i o n 87-102,161 dichotomization 393,395,413.414, 416-419,422-428 di cho t o m u s 29 7,332 d i s c r i m i n a t i o n 87,88,90,91,93, 95-97,100,101,169-172,255,257,
303
E f f e t Condorcet ecrui orderedness 333,336
183 319,321,324-330,332;
F e r r e r s r e l a t i o n 210.272-275 f i 1 l i n a - i n 35,59-61,64,67,68,75-79, 84.85 form oercention 59-62 fuzzv o r d e r 309 ~~
~
game theorv 433,435,438,439,441 graoh 7,111,113,116,219,220,222-224, 234,283,284,286,219,310,314 guided i n f e r e n c e 32-34,41-43,54-56 Guttman s c a l e 210,293.297 homoneneous favi l y 271,272,275,276, 279,280 hypergraph 219,220,223,224,234 i 11 usory contours 59 i l l u s o r y f i q u r e 61,62,70,78 impossible f i qure 105,106,108 incomoarabili t y 309,312,314 i ndi f f e rence 209, 210,214,309 ,312, 314,351-353,356,358,359 information 33 i n ter n r e t a t i on 105,106 ,111,113,115 , 116,118 i n t e r v a l o r d e r 209-211.216,274, 309-311.313-314 i n t e r v a l s c a l e 347 i n t r a n s i t i v i t , y 193,194 i s o t o n e regression 239,241-244, 246-249
j o i n aeometry
115
27 1-2 73,276- 280 Land r e t i n e x theory 59-61,77-78 l a t e n t t r a i t theory 159,160,163-165, 169,171 l a t t i c e 227-230,236 l a t t i c e dimension 227-230.236 l e a r n i n g 35,36,91,121-127,129-133, 135 -137-156.255 1earni;a 01 a t e a u 124-127,133,137, . . . 145 ,i47,155 lexicographic o r d e r i n g 193,194 1i n e a r o r d e r 180.183.184,264,265, . . 267,272,273,275 , 2 7 j Y 2 7 a
A- r e 1 a t i on
418
Subject index
n a j o r i t y r u l e 177,1C0,185 llarkov chain 438 Yarkov orocess 122,123,138 matchino 59,63,64,66,79 ,C0,65,
Kidex s t r u c t u r e 449,454457,459, 'J',G, :53, :66-.
283,284,286-288,319,340 227,2E3-287,303-305, 333,435 me as u re men t 5 6 ,10 1,177,25 7 ,25 9 ,
r e a c t i on-ti me 34,37-40,43,45,48 reseau 219-221,223-724 reseau soci a1 219,220
meaninoful
414,419,420,423,424,426,427,429
271,283,284,286,2~3,295,296, 303-305,349,353,330,381 medi an order 177 ,180 ,183-188 mu1 t i dimensional s c a l i n g 7,227, 348,359,360,383,390,449,451, 45 2
semi-order 13,193,209-212,271,272, 275-277,280,233-2E6,288,309 s i mi 1 a r i t y 6-8 ,10,11,39,177- 179,
neon color spreading
s i m p l e ~ o r d e r 211,212,246,288-290,
1G1-183,185,186,188,190,193,220,
223,227,271,272,283,360,381,~96, 37,452-454.470
59,61,70,
73
NP-comolete
298,351,356
234
soecific objectivitv
293-296,303-
305
observation 87,98 order( i ng) 56 ,108,110,115,116, 118,119,169 ,171,172,239-245, 263-267,279,283,284,286,287, 289,293,296-300,302-304 ,357, 359
Dartial i n t e r v a l order
309,312-
314
p a r t i a l order
s t a h i 1 ized images 59-61,75,79,80 s t r a t e g y 34,40,121,124,133,136-143, 145-147,149-152,155,156 325
s t r i c t order
threshold 7,86,88,89,92 ,a6 ,98 ,209212,214-216,257,272,278,280,284 t r a n s i t i v i t y 115 ,116,183,185,195, 197,215,220,222,256,272,273,283,
184,187,228,246,
264,255
DerceDtion
59-62,67,79,89,91,93, 101,102,105 ,136,139 ,149,159, 171,353 oerceotual 37,42,160,161 ,166 ,167 nerceDtua1 decision 87-90,97 ,10 1 polychotomization 413,414,416, 418-420,423,427,429 polyhedral 101,110,111,113,115 , 116 ,118, 119 nreference 160,193,194,203,206 , 209-212,214,215,255,260,268, 284,309,312-314,377 oreorder 179,309,310,312-314 preorderedness 317,319,321-326 , 328-336.338-340 prisoner ' s - d i 1emma 433,435,438, 439.442 nrobabf 1 i s t i c consistency 271,272 oroximi t.v ( d i s t a n c e ) matri x 3,5,7,
8,12-17,19-21,25,27,348,377-381, 383,384,389,390 309
pseudo-order
quasi order 273,274 ,277,349,358 questionnaire 305,414,415,417,427
284,297,298~300,310,313,314,325327,329,330,349,354,357
t r e e 3,7,8,10-12,15 ,17 ,19,24,26,28, 37-40,47,49,50,52,53,55,377,378, 380-383,385-387,389,390
unfoldinq 171,360-383 u t i l i t y 209-211,256,257,271 visual cortex 59 weak causal model 317-319,338-341 weak order 195.197.198.205.231. ?72-277,279,328-j3OY347-j50 356,358,359,423
,j53
,
