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0; (~1' ad¥:(~2' a2) and
will provide such an insight into several features of a distribution. Indeed one of us (NLJ) has some sympathy with the .
'\", .1) I' lill 'II
cp(x;~,a) = (j21t
quite drastically.
137
Effects of non-normality
.,,1
Iii,
i
II.
11
the indices Cp, Cph Cpu, Cpmand Cpk adds any knowledge or understanding beyond that contained in the equivalent basic parameters J1,a, target value and the specification limits'.. McCoy (1991, p. 50) also notes: 'In fact, the number of reallife production runs that are normal enough to provide truly accurate estimates of population distribution are more likely than the exceptions', although (p. 51) '... a well-executed study on a stable process will probably result in a normal distribution'.
Ilc
I I 'I
4.2 EFFECTS OF NON-NORMALITY
iln (JO.
ny
1111
.
The discussion of non-normality falls into two main parts. The first, and easier of the two, is investigation of the properties of PCls and their estimators when the distribution of X has specific non-normal forms. The second, and more difficult, is development of methods of allowing for non-normality and consideration of use of new PCls specially designed to be robust (i.e. not too sensitive) to non-normality. There is a point of view (with which we disagree, in so far as we can understand it) which would regard our discussion in the next three sections as being of little importance. McCoy (1991) regards the manner in which PCls are used, rather than the effect of non-normality upon them, as of primary importance, and writes: 'All that is necessary are statistically generated control limits to separate residual noise from a statistical signal indicating something unexpected or undesirable is occurring'. He also says that PCls (especially Cpk)are dimensionless and thus'... become(s) a handy tool for simultaneously looking at all characteristics of conclusions relative to each other'. We are not clear what this means, but it is not likely that a single index
ntl
I
viewpointof M. Johnson(1992),whoasserts that '... noneof
ull tll nos
IlJ .IsIIS 2 Itnd ,
3 ,'"SI
nfl'lce, led
,
1. a skew distribution with finite lower boundary
A ,at ue
~
dex
- namely
a
xL distribution (Chi-squarewith 4.5 degreesof freedom) 2. a heavy-tailed([32>3)distribution- namelya t8distribution 3. a uniform distribution. In each case, the distribution is standardized (shifted and scaled to produce common values (~=0, 0' = 1)formeanand standard deviation). The proportions (ppm) of NC items outside limits:!: 3a are:(approximately)
lI'nl ~Ol~~
1-
Gunter (1989) has studied the interpretation of Cpk under three different non-normal distributions. These distributions all have the same mean (~) and standard deviation (a), they therefore all have the same values for Cpk, as well as CpoThe three distributions are:
.
For (1.) 14000 (all above 3a). . For (2.)4000 (half above 3a~half below
.
For (3.) zero!
- 3a).
Arid for normal, 2700 (half above 3a, half below Figure 4.1 shows the four distribution curves.
- 3a).
138,
Process capahility indices under non-normality
Table 4.1 Values of Pr[CpcJ
0.48 t-
'Ill
, I
\ \ ..
I'
\\1
0.00t-
.../' I I
I
-2.5
0.0
---
'
. 7.5
I
I
A number of investigationsinto effectsof non-normality on estimators of PCls have been published. English and Taylor (1990)carried out extensiveMonte Carlo (simulation)studies of the distribution
I I
of Cp (with Cp= 1) for normal, symmetrical'
alia,
10
0.50 0.75 1.00 1.25 1.50 2.00
2.50
1.000 0.933 0.568 0.243 0.091 0.013 0.002
1.000 0.959 0.529 0.189 0.064 0.011 0.003
1.000 0.989 0.501 0.127 0.035 0.005 0.001
0.985 0.865 0.644 0.423 0.259 0.097 0.039
1.000 0.979 0.548 0.129 0.017
1.000 0.994 0.523 0.087 0.011
1.000
0.996
I
0.50 0.75 1.00 1.25 1.50
1.0pO 0:508 0.041 0.002
0.900 0.617 0.315 0.135
I
') I\(\
fI flfI1
fI flfIfI
fI flAA
A A,.".,
J
I
Ille,
i
I
"IIInl
--~~n
I
AI., ~J).
that for n less than 20, there can be substantial,
departures from the true Cp value (as well as differencesin interpretations of these values for non-normal distributions). .. All four distributions have the same expected value (~=O), and standard deviations (0'= 1).The specification range was LSL= -J.to USL=3, so the value of Cp was t for all four distributions.
I ,
~,r
j
Shll
~
dill('II he ru
,
I
-
50
L~N 1'1
'Iill II ..h II .Id
I
we note,
20
. ell ",v
I I
sample sizes n = 5,10 (10)30, 50. The results are summarized in part in Table 4.1. From these values of Pr[Cpc]
..11
i
triangular, uniform and exponential distribution"',with inter
0.969 0.846 0.683 0.527 0.404 0.240 0.150
I'0),
I
Exponential
1.000 0.924 0.522 0.267 0.152 0.058 0.025
I 20 7.5
Fig.4.1 Four different shaped process distributions with the same c; and 0', and hence the same Cpk'(Redrawn from Gunter, RH. (1989), Quality Progress, 22, 108-109.)
Un!form
1.000 0.881 0.561 0.320 0.193 0.079 0.039
--r
I
Triangular
0.996 0.873 0.600 0.373 0.223 0.089 0.041
Ii)
5.0
Normal
0.50 0.75 1.00 1.25 1.50 2.00 2.50
(3)
2.5
from simulation (Cp= 1)
5
(iulIssiulI (normul)
0.16t-
-5.0
Distrihution of X 11= c=
1/ 'I 'VI .t \
(2) 0.32 J-
139
Effects of non-normality
"
i
'I
I
.
Also, the values for the (skew) exponential distributions differ sharply from the values for the three other (symmetrical) distributions especially for larger values of c. On the more theoretical side, Kocherlakota et af. (1992)
140
Process capability indices under non-normality
'have established the distribution of (;p = idj8 in two cases: when the process distribution is (i) contaminated normal with' (11
= (12
= (1, see (4.1); and (ii) Edgeworth
fx(x)
= [1-iA3D3
t'r
The essential clue, in deriving expressions for the moments of (;" for this model. is the following simple observation.
11',lh
series
+~A4D4 +-h-A~D6Jcp(x; 0,1)
where Dj signifies jth derivative with respect to X,
(4.2)
1)
(4.3a)
~a)
A random sample of size n from a population with PDF (4.4) can be regarded as a mixture of random samples of sizes N 1>"', N k from k populations with PDFs CP(X;~1,(1),...,cp(x;~k>(1),where N=(N1)...,Nk) have a joint multinomial distribution with parameters (n; P1,..., pd so that
.
aJ:1,d
A3
=
Ji3(X)
.
{Ji2(X)}i=JP;
rr[N=nJ=r{jOl
and
(Nj=nj)]
nk P
-~ nl
-
A4=
Ji4(X) {Ji2(X))2 -3=/32-3
(4.3 b)
j
k
Lnj=n
,I.
j=1
~I
LPj=l
j=1
n=(n1",.,nk) (4.5)
iI
I I
The conditional distribution of the statistic
4.2.1 Contaminatednorm~l
"
We will refer in more detail, later, to the results of Kochlerlakota et ai. (1992). First, we will describe an approach used by ourselves (Kotz and Johnson (1993)) to derive moments of Cp for a more. general contamination model, with k components (but with each component having the same variance, (12), with PDF
L (XJ-~O)2
,.
j= t
"
(wl.r (71., nll.,
(4.6)
(where ~o is an arbitrary, fixed number), given N = n, is that of (12x [noncentral chi-squared with n degrees of freedom and
noncentrality
. k
k
L
"J
k
Onj!j=t j= 1 k
10)
are standardized measures of skewness and kurtosis respectively. (See section 1.1 for definitions of the central moments Jir(X) and the shape factors A3,A4,$, /32')
J=1
141
Effects of non-normality
PjCP(x;~j, (1)
I~
(4.4)
fl
parameter
A~
=
L ni~j-'o)2]-symbolically, j= 1
1-
","-'-"." . . ::, . '.\,.:',112 (",", - '. ' '~~> ;, . ;' . '," ..."'"'. ~.,. ~,., . ,/'
(12X;,\.t~) (4.7)
142
1\
Process capahility indices under non-normality
,
The conditional distribution of
8~
.
and in particular
n ,- 1
.
-
1
I
E[C1,IN= n] = C,,(n--1)2 exp( -'2)'n)
L (Xj-X)2 j= 1
(n-1)82=
143
Effects of non-normality
is that of
1
CfJ
(4.8)
0"2X~2-1 (An)
X
,
,~o
r
i
b::()
n
(
2
)
L'
'
1-/-
J2r(~;l
J.8)
(4.11a)
+)
with
E[C~IN =nJ = C~(n-l) exp(-.tAn)
\~
k
An=
1
L nj((j-~1)2
where
~I=-
j~ 1
n j
k
L nj(j
x CfJ(Hn)l
1=0---:-;l. n- 3 + 2.1
Expressions for moments of noncentral chi-squared distributions were presented in section 1.5 and have been used .in Chapters 2 and 3. We will now use these expressions to derive formulas for the moments of
A';~-
.
I!I/' ,
E[C~J=L n
, l~
Conditional on N = n, Cp is distributed as
(4.qI b)
~ve
8
d CP=38
.
Averaging over the distribution of N, we obtain
i 10
-
~ TInj!
( j= 1
)E[C~lnJ
1'=1,2,...
.iI P?
)(
J=1
(4.12)
The summation is over all nj~O, constrained by
I
k
!. 3 d(n -1)% !x~0"
1(An) = Cp(n -1)i /x~ - 1(An)
(4.9)
L nj=n.
t(»)
Recalling the formula for moments of noncentral chi-squared from Chapter 1, we have
~~rcd I'd
E[C~IN =nJ = C~(n-1)ir exp(-lAn)
j~ 1
In the special symmetric case of k=3, with ~dO"=-'a, ~2= 0, ~3/0"= a, (a> 0) and PI = P3= P, say, the noncentrality parameter is
'
x
1
L
I
rC; I {(t~n)i z! }
, i= 0
)
J+i-~ n-1 2 2r/2 r( -2+i
1 An=
(4.10)
j
'I"~ h I
I
LIO) -. 14I
{ nl +n3--~(n3-nd
2
}a
2
(4.13)
Some values of E[CpJ/Cp and S.D.(C\)/Cp, calculated from (4.11a) and (4.11 b), are shown in Table 4.2 (based on Kotz
Effects of non-normality .cr.)
q
r "
\0 r-"
0
000,000
000000
V)~;:,"
O\\Ooo "
"
"
Iu.J 0
II", 0 -Ivi u I .
u.J
t:: <1) ..c
(I
II,
900000
N-OO',...\o
--O\N"
OO\oM""""""o\ 000000\ "":"":"":"":"":0
\OMO\r-Ooo 0\0\ r-r-"
\0 V) 0\ O\O\r-r-"
,
Iq
I 0'1
0
0\
1. for given n, the bias and S.D. both decrease as a increases; 2. for given a, the bias and S.D. both decrease as n increases; 3. for given a and n, the bias and S.D. do not vary greatly with Pt and P2 (when k=3) or with P (when k=2). The bias is numerically greater when the 'contaminating' par. ameters are smaller; 4. on the other hand, the greater the value of a, the more marked is the variation (such as it is) with respect to Pt and P3 (or p).
000000
000000
V)
0
"
NN----
000000
'-
and Johnson (1992)) for selected values of a and p. (Hung and Hagen (1992) have constructed a computer program using GAUSS for calculation of these quantities.) , The table is subdivided into two parts. In both parts Cp(=1d/0')=1. In Table 4.2a the cases k=3, n= nt +n2 +n3= 10,20, and 30; Pt =P3 =0.05, and 0.25; a = 0.005, 0.01, 0.125, 0.25, 0.5, and 1.0 are presented. Table 4.2 b covers situations when k = 2 for the same values of n and a, with p=O.l, and 0.5.. From the tables we note that
" II
<
N
0
0'
Noooo\O"
1u.J
cr.) '-' ;::: .8 ..... e<j ';;
qqqqqq
"""\OO\"
:
:
:
:
:
:
r-r-OO\OO 0\0\00r-"
r-\OoO\oo 0\0\00r-"
V)I N
<1) "0 "0
000000
000000
~
......
MO"""oo\OM O\O'I"
"0
;;
:
..... CIJ
:
:
:
:
:
:
r-r-oooo
"0
G3' 0 '-' <1) 0 ;:j
0
<1)
I
~
~I-<
0
;:j .....
>< '8 ~
:
t
:
nll~
Observe, also, that the bias is positive when a is small, and decreases as the proportion of contamination increases. HQwever, the bias is negative for larger values of a and p, and it becomes more pronounced as n increases. This is
w4
contrary
(I' ~-t
N
~
"
~~~~~~
1u.J
13 t)
V)
:
II,
"
c..
:
- - -'...... 000000 N
000000
;>
~
:
r-r-OOOO 0\0\ 00 oo"
0\0\0000"
10 ...... cr.)
I
MN"""O\O1r\ O\O\"
000000
;;
.
:
:
:
:
~~~~~~ :
:
r-r-oooo
~ II
~ I 'V.....
:
"
"
000000
:
:
C'!f'j~~~~
000000
:
:
:
:
.I
r-r-OOOO .0'I0\0000"
000000
:
:
W
0'I0\0000"
.
:
w
~
qqqqqq-......-.
:
b ~
,.,
"J'
/I
,
th'l"
1'1,,' (I II
II
~
Z M
~ .!! .c= Eo-<
II I
145
~
...
II
V)
Ir\ Ir\ Ir\ V). or,
>lJ>,.,ONONON 11::>..000000 r'1"
."J>
... I::>..
N
II
~ ,.
II
....
II
>lJ>
I~
>lJ>1::>.._V)"""V)-1r\ I 000000
0
= :.:-
0 N
0 M
II ~ .. 0 .c:.:-
0 N
0 M
mill
I ,1
to the situation
when p = 0 (no contamination)
in
which the bias of Cp is always positive, though decreasing as n increases. Gunter (1989) also observed a negative bias, when the contaminating variable has a larger vflriance than' the main one. . Kocherlakola et al. (1992) provide more detailed tables for the case k = 2. They also derived the moments of Cpu= (USL - X)/(38). The distribution of Cpu is essentially a mixture of doubly noncentral t distributions; see section 1.7.
146
Process capahility indices under non-normality
4.2.2 Edgeworth series distributions
III!
The use of Edgeworth expansions to represent mild deviations from normality has been quite I~lshionable in recent years. There is a voluminous literature on this topic, including Subrahmaniam* (1966, 1968a, b) who has been a pioneer in this field. . It has to be kept in mind that there are quite severe limitations on the allowable values of .j7f;and fJ2to ensure that the PDF (4.2) remains positive for all x; see, e.g. Johnson and Kotz (1970, p.l8), and for a more detailed analysis, Balitskaya and Zolotuhina (1988)). Kocherlakota et al. (t 992) show that for the process dislribution (4.2) the expt:<.;(t:dva/lit:
.
KocherIakota et al. (1992) also show that the variance of Cp isi
I 2 1 1 { { [ 1+~ 1+g"
_I.
n -1 2 n(n'-3)Cp
lint
~
(4.14c)
~I.
Table 4.3 is based on Table 8 of Kocherlakota et al. (1992).
IfQI'
As is to be expected from section 1.1, for given fJz the values~of E[CpJjCp do not vary with j7f;, and the values of S.D.(Cp)jCp vary only slig~htly with ft. Also, for
given J7I!, the values of S.D.(Cp)jCp increase with fJ2'
Kocherlakota et at. (1992) carried out a similar investigation for the distribution of the natural estimator of Cpu (=!(lISL-~)j(j),
of Cp is
iI! ~
E[C.]
(n-1)Jr(~(n-2))
Cpu=
C,,{t +~y"(fJ2-3)--~h,,fil}
fir(i(n-I)j
(4.14")
~(/)
where
,
I
.
n-1 g,,= n(n+1)
n=
;
10
0.0
:1:0.4
(n-2)
"-: n(n + l)(n + 3)
30
The premultiplier of Cp will be recognized as the bias correlation factor h"- I, defined in (2.16), so (4.t 4 a) can be written
3 J I E[C,,] = E[Cplnormal] {J + /J?!,J/J2 -- 3)-, Hh,,/J1i (4./4h) K. Subrahmaniam was the (pen) name used by K. Kocherlakota in the 1960s.
0.0
Ia-
:1:0.4 I '1"1
I Ib) II
...
1"'f1he
USL.-X 3S
(4.15)
Table 4.3 Expeltcd vailicand standard deviation of ,Cp/Cpfor Edgeworth process distributions
and
h-
147
Effects of non-normality
/32
3.0 4.0 5.0 3.0
4.0 5.0 3.0 4.0 5.0 3.0 4.0 5.0
E[Cp]/Cp
1.094 1.128 1.161 1.094 1.128 1.161 1.027 1.039 1.051 1.027 1.039 1.051
S.D. (Cp)/Cp
0.297 0.347 O.3R9 0.300 0.349 0.392 0.140 0.169 0.193 0.141 0.170 0.194
If the process is symmetrical (so that fJl =0), E[CpuJ is proportional to Cpu' Table 4.4 (based on Kocherlakota et al.
148
Table 4.4 Expected value and standard deviation of Cpu for Edgeworth process distributions E[Cpu]
n
Jpl
10
- 0.4 3.0
30
4.0 5.0 3.0 4.0 5.0 3.0 4.0 5.0 3.0 4.0 5.0 3.0 4.0 5.0 3.0 4.0 5.0
0.0
0.4
-0.4
0.0
0.4
P2
Cpu= 1.0 1.086 1.119 1.153 1.094 1.128 1.161 1.100 1.134 1.107 1.024 1.036 1.048 1.027 1.039 1.051 1.029 1.041 1.053
Cpu= 1.5 1.632 1.682 1.733 1.641 1.592 1.742 1.647 1.697 1.748 1.538 1.556 1.574 1.540 1.55H 1.576 1.542 1.560 1.578
149
Effects of non-normality
Process capability indices under nOI1-normalit y
S.D.(Cpu) Cpu= 1.0 Cpu= 1.5 0.287 0.426 0.337 0.502 0:381 0.569 0.320 0.433 0.366 0.533 0.407 0.596 0.345 0.486 0.388 0.554 0.427 0.615 0.136 0.201 0.166 0.246 0.190 0.283 0.154 0.220 O.IHO 0.261 0.203 0.297 0.168 0.235 0.193 0.274 0.214 0.308
II. for,
I"~ . I
r' II 1\ II II, III; '11
I'll 111
WI
(1992)) shows that even when /31is not zero, E[CpuJ is very nearly proportional to CpuoOf course, if /31is large, it wil1not be possible to represent the distribution in the Edgeworth form (4.2). Table 4.3 exhibits reassuring evidence of robustness of Cp to mild skewness.
(II h ~ry
Distribution number and name [I] Normal (50,1) [2J Uniform (48.268,51.732) [3J 10 x Beta (4.4375,13.3125)+47.5 [4J 10 x Beta (13.3125,4.4375)+42.5 [5J Gamma (9,3)+47 [6] Gamma (4,2)+48 [7J (xl. 5) Gamma (2.25, 1.5)+ 48.5 [8J (Exponential) Gamma (1,1)+49 [9] Gamma (0.75,0.867)+49.1340 [10] Gamma (0.5,0.707)+49.2929 [l1J Gamma (0.4,0.6325) + 49.3675 [12J Gamma (0.3,0.5477) + 49.4523 [13J Gamma (0.25,0.5)+49.5
Skewness (J f31 ) 0 0 0.506 -0.506 0.667 1.000 1.333 2.006 2.309 2.828 3.163 3.651 4.000
In all cases; the expected value of the process distribution is 50, and the standard deviation is 1. The values shown in the tables below are based on computer output kindly provided by Drs Barbara and Kelly Price of Wayne State University in 1992. The specification limits are shown in these tables, together with the appropriate values of Cp and Cpk' The symbols (M), (L) and (R) indicate that T =--=, >, < 1(LSL+ USL) respectively. In all cases, T = 50.
~Ol
itlh I'llIt
,
I~
(M) (L) (R) (M).
LSL 48.5 45.5 48.5 47.0
USL 51.5 51.5 54.5 53.0
Cp 0.5 0.5 0.5 1.0
Cpk 0.5 1.0 1.0 1.0
(L) (R) (M) (L) (R)
LSL 44.0 47.0 44.0 41.0 44.0
USL 53.0 56.0 56.0 56.0 59.0
Cp 1.0 1.0 2.0 2.0 2.0
Cpk 1.5 1.5 2.0 2.5 2.5
4.2.3 Misccllancous Price and Price (1992) present values, estimated by simulation, of the expected values of Cp and CPk>from the foHowing process distributions, numbered [lJ to [13J
n
~
-
II, g .
Ihl
The value of E[CpjCp] does not depend on ~, so we do 11Ot need to distinguish (M), (L) and (R), nor does it depend on Cpo Hence the simple Table 4.5 suffices.In this table, the distributions are arranged in increasing order of I~I.
150
Construction of robust PCls
Process capahility indices under non-normality O'
Table 4.5 Simulated values of E[Cp/CpJ Distribution [lJ [2J [3J [4J [5J [6J [7J [8J [9J [10] [11J [12J [13]
(normal) (Uniform)
'
(Beta) (Beta)} (Gamma) (Gamma) (Gamma) (Gamma) (Gamma) (Gamma) (Gamma) (Gamma) (Gamma)
n= 10
n=30
n= 100
1.1183 1.0420 1.1171
1.0318 1.0070 1.0377
1.0128 1.0017 1.0137
1.1044 1.1155 1.1527 1.2714 1.3478 1.5795 1.6220 1.8792 2.2152
1.027 ' 1.0371 1.04"/4 1.0801 1.1155 1.1715 1.2051 1.2595 1.2869
1.0091 1.0143 1.0146 1.0242 1.0449 1.0449 1.0664 1.0850 1.0966
Hn
-
j
HI'
al~ HIiJ a~. fn 0
8h till PI', III' 111
These values do not depend on '/: The values for the two beta distributions have been combined, as they should be the same.
10 IOns 11.1
I.hl
Comparing the estimates for the normal distribution ([lJ) with the correct values (1.0942, 1.0268, 1.0077 for 11= 10,30, 100 respectively) we see that the estimates are in excess
by about 2% for n = 10,t% for n= 30or 100.
0
00
0
Sampling variation is also evident in the progression of values for the nine gamma distributions, especially in the n = 100 values for distributions [6J and [7J, and [9J and [10]. As skewness increases, so does E[CpjCp], reaching quite remarkable values for high values of y1J;. These howeyer, correspond to quite remarkable (and, we hope and surmise, rare) forms of process distributions, having even higher values of y/f3; than does the exponeI2:tialdistribution. For moderate skewness, the values of E[CpjCpJ are quite close to those of the normal distribution (d. Table 4.3). Table 4.6 presents values of E[CpkJ estimated from simulation for a number of cases, selected from those presented by Price
and Price (1992).
0
We again note the extrem€ly high positive bias - this time
st~11J) (/J ~== llilficss Ya, 0111.of
Illhe MlllfOJ. III,lIe Ih,~w1I\\.,nd hll~cn ton. fllte
of Cpkas an estimator of Cpk- fordistribution[13] whenn= 10, and only relatively smaller biases when n=30 and n= 100. For the exponential distribution [8J there is a quite substantial positive bias in Cpk' The bias is larger when ( is greater. than l(LSL+ USL) - case (L) - than when it is smaller"-~ case oCR).The greater among these biases for exponential, in Table 4.6, are of order 25-35% (when n= 10), falling to 21-5% when n= 100. As for Cp, the results for the extreme distribution [13J are sharply discrepant from those for normal, and mildly skew distributions. Of course, [13J is included only to exhibit the possibility of such remarkable biases, not to imply that they are al1ything like everyday occurrences. Coming to the variability of the estimators Cp and Cpk>we note that the standard deviation of Cp might be expected to be approximately proportional to J({32-1) where f32(= J14/(J4)is the kurtosis shape factor (see section 1.1) for the process distribution. We would therefore expect lower standard deviations for uniform process distributions (f32 = 1.8) than for normal (f32 = 3) and higher standard deviations when f32> 3 (e.g. for the exponential [8J with f32= 9). Values of J(f32 -1) are included in Table 4.7, and the estimated standard deviations support these conjectures. It should be realized that Tables 4.5-7, being based on simulations, can give only a broad global picture of variation in bias and standard de:viation of Cp and Cpkwith changes in the shape of the process distribution. More precise values await completion of relevant mathematical analyses which we hope interested
readers might undertake. 0'
4.3 CONSTRUCTION
UaThtfby ll'llt nl1l4ll11e
151
OF ROBUST PCls
The PCls described in this section are not completely distribution-free, but are intended to reduce the effects of nonnormality.
I
-~-
-~~~-
~
---
,--_.
Table 4.6 Values of E[Cpk] from simulation (Price and Price (1992» Cpk
min(-LSL,
USL-'
1
0.5
[1] [2] [3][4] [7] [8] [13] [l]LR [2]LR [3]L[ 4]R/[3]R[ 4]L [7] L/[7] R [8]L/[8]R [13]L/[13]R [1] [2] [3] [4] [7] [8]
1/2
1
1.0
0.468 0.453 0.471 0.474 0.487 0.582 0.516 0.504 0.525/0.509 0.528/0.519 0.548/0.532 0.670/0.617 0.985 0.957 0.989 0.998 1.057 . 1.028 '1.163
0.464 0.424 0.464 0.480 0.527 0.904 0.559 0.521 . 0.569/0.548 0.597/0.555 0.671/0.600 1.259/0.956 1.023 0.945 1.022 -
2.012
[13] "iIiio
-..;
- ""'1"
;:'
-
[13]
--
---
2/3 -
2.0
[l]LR
'!a... ~-~~
1
z~
-
H18 1.042 1.129/1.111 1.174/1.132 1.307/1236 2.337/2.064 2.141 1.987 2139 2209 2.435 4.227 2.237 2.084 2.245/2.233 2.326/2.284 2.578/2.507 4.582/4279
~
..
. 0.997-
1.067
~.
1.226
-- -
0.480 0.474 0.481 0.480 0.485. 0.519 0.506 0.501 0.513/0.504 0.510/0.504 0.515/0.509 0.557/0.540 0.986 0.975 0.988 0.987
1.226
"""'"
"" ,j
---.
[2]LR [3]L[ 4]R/[3]R[ 4]L [7]L/[7]R [8]L/[8]R [13]L/[13]R [1] [2] [3][4] [7J [8J [13J [lJLR [2JLR [3JL[ 4JR/[3JR[ 4JL [7] L/[7J R [8JL/[8JR [13JL/[13JR
4/5
*Notes:
- ---
.::c.-""
n= 100
n=30
n=lO
Process distribution*
d
-
1.032 1.007 1.043/1.032 1.052/1.043 1.088/1.072 1.J13/1.261 2.016 1.964 2.027 2.045 2.108 2.513 2.064 2.014 2.081/2.070 2.099/2.090 2.168/2.153 2.600/2.548
~,,"~;. ~-
-"'-'
"" JijJ
i I ! i
1.007
-,
.--
. _.
..
--="
'-' .J
1.013 1.002 1.017/1.011 1.017/1.012 1.027/1.021 1.105/1.088 1.999 1.976 1.998 2.002 2.021 2.164 2.026 2.003 2.030/2.024 2.032/2.026 2.052/2.046 2.201/2.185
-
(a) When min(~-LSL, USL-Q/d=l, we have ~=1(LSL+USL1 so only (M) applies. (b) Since [3] and [4] are mirror images, results can be merged as shown. (c) For symmetrical-distributions (normal [1] and uniform [2]), the L and R values should be the same and so they have been.averaged.
i
----
Table 4.7 Estimates of S.D. (Cp) and S.D. (Cpd from simulation (Price and Price (1992» Distribution
(Jh-1)!
sgng-1(LSL+USL)}
Normal [1J
. 1.414
O(M) 1(L), -l(R) }
0.894
O(M) l(L), -l(R) }
Uniform [2J
n=30 S.D.(Cpk) S.D.(CpfCp) n=l00 S.D.(CpJ S.D.(CpfCp) 0.077 0.148 0075 0148 } { { 0.160} . . 0.082 0.047 0045 0089 SO.088 } { } . lO.110 .' 0.056 0.109 0.192
.
xi.s [7J
2.160
O(M) l(L)
0,199
}
-l(R) Exponential [8J
2.828
Values for S.D.(Cpi;) correspond
-I sgn(h)=
--.
l
-
0.142
{ 0.169} 0.119
with Cpk = 1.
}
for h> 0
4.8 0.135%
-
0.237
{ 0.255'}
0 for h=O
--
. Table
0.273
{ 0.137} 0.092 0.139
0.258
}
to process distributions
0.111
for h
{ ,... I
,
O(M) l(L) l(R)
} { 0.245 0.~66
-
and
- ~'~F
-
99..8.65D/. poiD1s ol:
~'-
---
~
p~
~
~
~~
~
-
--;- ,>'
Table 4.8 0.135% and 99.865% points of standardized Pearson curves with positive skewness (vip v7f;. <0, interchange 0.135% and 99.865% points and reverse signs. {Clements (1989»
Ji;
1.0
1.2
1.4
u
---
1
> 0). If
0.0
0.2
0.4
0.6
0.8
1.8
-1.727 1.727
-1.496 1.871
-1.230 1.896
-0.975 1.803
-0.747 1.636
2.2
-2.210 2.210
- 1.912 2.400
-1.555 2.454
-1.212 2.349
-0.927 2.108
- 0.692 1.822
2.6
- 3.000 3.000
- 2.535 2.869
-1.930 2.969
-1.496 2.926
- 1.125 - 2.699
-0.841 2.314
-0.616 1.928
3.0
-3.000 3.000
- 2689 3.224
- 2.289 3.358
-1.817 3.385
-1.356 3.259
- 1.000 2.914
-0.739 2.405
-0.531 1.960
3.4
- 3.261 3.261
- 2.952 3.484
- 2.589 3.639
- 2.127 3.175
-1.619 3.681
- 1.178 3.468
-0.865 2.993
-0.634 2.398
3.8
- 3.458 3.458
-3.118 3.678
-2.821 3.844
- 2.396 3.951
- 1.887 3.981
-1.381 3.883
-1.000 3.861
-0.736 2.945
~0.533 '2.322
4.2
- 3.611 3.611
-3.218 3.724
-2.983 3.997
-2.616 4.124
- 2.132 4.194
-1.602 4.177
-1.149 3.496
-0.840 3.529
-0.617 2.798
4.6
-3.731 3.731
- 3.282 3.942
- 3.092 4.115
-2.787 4.253
- 2.345 4.351
-1.821 4.386
-1.316 4.311
-0.950 4.015
- 0.701 3.364
-0.510 2.609
5.0
- 3.828 3.828
- 3.325 4.034
-3.167 4.208
-2.91§ 4.354
-02.524 4.468
- 2.023 4.539
- 1.494 4.532
- 1.068 4.372
-0.785 3.907
- 0.580 3.095
/32
-
1.6
For each (J Ph P2) combination, the upper rmv.contains-9:-!-3-5%,-puiub(O;T1fmillierower, 99:8'65°)'0' points (0.).
1.8
Construction of robust PCl s
Moment esti'mators for (4.17)and (4.18) are obtained hy
-~
,v
.
§ 88~:2~
'-'g
~ ..0
1
d<""i<""i..f'l-;
.~ R ::;j l
'"
1-< c.8
MO-'
f6
I
0...
II
~~~~~ OMM'
'b' °
<:I:>I M ~I'"
~"
+
OMM'
OOO\Mv)
HJ'
'
VI IOI N ~ VI b
1.0
d<""i<""i'
V)
M
V)
~ - d<""i<""i'
I
1
I
I
...... cod ...c::
-
0 1
...c::
u ::;j '"
<:1:>100 ...... 0
~
:::I 1 \0
~ > oi:i 11)
o
'
i,
V)
OOV)NV) ooNoo\Od..f<""i..fV'i 00'
I ,
4
I-<
g .0'
'r .::: .U ~
'
~
~ ~
.g i!) 1:1
i!)
~
~ ~ I-<
~
Q
159
V) 00
0\
0\0\0\ d d d
II II II 0...
0...
c...
I-<
I-<
I-<
~ ",c.8c.8c.8 <::0.<::0.<:1:><:1:><:1:>
I I
.
I I
II I IL
d "I
",
replacing (Jby an appropriate estimator.
At this point we reiterate the difference between Clements' method and the Johnson-Kotz-Pearn method. In the first method there is an attempt to make a direct allowance for the values of the skewness and kurtosis coefficients, while the second method aims at getting limits which are insensitive to these values. In the second method we no longer have guaranteed equal tail probabilities, but we do not have to estimate .jlf; and {32which it may be difficult to achieve with accuracy, because of bse of third and fourth sample moments, which are subject to large fluctuations. Both methods rely on the assumption that the population distribution has a unimodal shape close to a Pearson distribution for Clements' method,and more restrictively, close to a gamma distribution for Johnson-;Kotz-Pearn method. Another approach, also aimed at enhancing connection between PCI values and expected proportions NC, tries to 'correct' the PCI, so that the corrected value corresponds (at least approximately) to what would be the value for a normal process distribution with the same expected proportion NC. Munechika (1986, 1992) utilized an approximate relation het\vccncorresponding percentiles of standardized normal and Cham-Charlier (Edgeworth) distribution (see Johnson and,Kotz (1970, p. 34)) to obtain an approximate relationship between the process PCI and the corrected (equiv~Jent normal) pel values. He appHed this only to the case where there is only an upper specification limit (110LSL), with the CpkU index (see equation (2.25b)).The approximation is quite good for gamma process distributions which are not too skew. From the relationship (in an obvious notation) XGt~ Ua+ i(U; -1)J7f; he obtained j(process index) ~ 3(corrected index) + i {(corrected index)2-1 } leading to (corrected index~(1/3..JP~)[{j31 + 18(process index)~ + 9}1/2- 3J. The inverse (also approximate) relationship UGt~XGti(x;: 1)J7f; would give the somewhat simpler formula ~
160
-,I
Process capahility indices under non-normality
(corrected index)~ (processindex)-
Construction of robust PCI s
The bases for their choices are that for a normal distribution
fa{9(process index)2~ 1~
.
(This formula was not used in Munechika (1986, 1992).) Of course, use of this correction requires a value for -Jf31' Unless this is well established, it is necessary to estimate it Such estimation may well be subject to quite large sampling
variability.
.
.
4.3.3 'Di,stribution-free' PCls
with smaller values of f3(but still with IJ(= 0.05) requires smaller
samples (lessthan 300).They recommend taking
place of 68.
the interval (~- (1,~ + (1) of length 2(1 contains 68.26% of values (and, of course, 6(1=1 x 4(1= 3 x 2(1).
~
4.3.4 Bootstrap methods Franklin and Wasserman (t 991), together with Price and Price (1992) should be regarded as the pioneers of application of bootstrap methodology in estimation of Cpk' The bootstrap method was introduced some twelve years ago (see Efron, 1982) and has achieved remarkably rapid acceptance among statistical practitioners since then. (Over 600 papers on the bootstrap method were published in the period 1979, 90!) It is not until very recently, however, that its application in the field of PCls has been developed. . The bootstrap method is a technique whereby an estimate of
I
I t
. \
I. I, ~
,. i
~ . .
.
the.distribution functionofa statisticbased on a samplesizen, say, is obtained from data in a random sample,of sizem (~n) say, by 're-sampling' samples of size n - with replacement from these m values and calculatingthe correspondingvalues of the statistic in question. Usually m= n,but thisneednot be the case. Here is a brief formal description of the method. Given a sample of size m with sample values XI,X2,"',X", we choose (with replacement) a random sample ([1], say) .x[IJ1>"" X[IJII of size n, and calculate
I
. f3= 0.9546 and place of 68, or . f3= 0.6826 and
the interval (~--2(1, ~+2(1) oflength 4(1contains 95.46% of ..values, and
It is here that we must part company with them, as it seemsunreasonable to use relationships based on normal distributions to estimate values which are supposed to be distribution-free!
I 1
Chan et at. (19~H~) proposed the following method or Qbtaining 'distribution-free' PCI s. They note that the estimator, 8, in the denominator of Cp is used solely to cstimtltc the length (6(1)of the interval covering 99.73% of the values of X (on the twin assumptions of normality and perfect centring, at ~=~(LSL + USL)). They propose using distribution-free tolerance intervals, as defined, for example, in Guenther (1985) (not to be confused with Gunter (1989)1) to estimate this length. These tolerance intervals (widely used in statistical methodology for the last 50 years) are. designed' to include at least 100f3% of a distribution with preassigned probability 100(1-1J()%, for given f3 (usually close to 1) and IJ((usually close to zero). In the PCI framework, a natural choice for f3 would be f3= 0.9973,with, perhaps, 1J(=0.05. Unfortunately construction of such intervals would require prohibitively large samples (of size 1000 or more). Chan et at. (1988) suggest that this difficulty can be overcome in the following way. Construction of tolerance intervals
161
C(1JPk
from this new
'sample'. This is repeated many (g) times and we obtain a set
using! x (length of tolerance interval) in I
using 3 x (length of tolerance interval) in ~ 1
of values C[I]Pk,C[2]pko""C[9Jpk'which we regard as approximating the distribution of Cpk in samples of size n - this estimate is the bootstrap distribution. (The theoretical basis
162
Process capahility indices under non-normality
Flexible PCls
of this method is that we use the empirical cumulative distribution from the first sample - assigning probability m - 1 to each va:lue - as an approximation to the true CDF of X.) Practice has indicated that a minimum of 1000 bootstrap samples are needed for a reliable cakulation of bootstrap confidence intervals for Cpk'
4.3.4c The bias-corrected percemile confidence interval
This is intended to produce a shorter confidence interval by allowing for the skewness of the distribution of Cpk' Guenther (1985) pointed out the possibility of doing this in the general case, and Efron (1982) developed a method applicable in bootstrapping situations. The first step is to locate the observed Cpkin the bootstrap' order statistics Cpk(l)~... ~ Cpk(lOOO).For example, if we
itIJ
According to Hall (1992) a leading expert on bootstrap ~
methodology - difficulties 'in applying the bootstrap to process capability indices is that these indices are ratios of random variables, with a significant amount of variability in the variable in the denominator'. (Similar situations exist in regard to estimation of a correlation coefficient, and of the ratio of two expected values. The bootstrap performs quite poorly in these two (better-known) contexts.) Franklin and Wasserman (1991) distinguish three types of bootstrap confidence intervals for Cpk, discussed below.
1
have
Cpk = 1.42 from
the original data, and among the,
bootstrapped values we find
Cpk(365)= 1.41 and \
I!
I
t'
4.3.4a The 'standard' confidence interval
(C~k - ZI -a/2S*(Cpk)' C~k + Zl-a/2S *(Cpk))
(4.19)
"
365
A
""
r u'
-Ii
,
I'
;
II
,i
1,000= 0.365 = Po, say
We then calculate <1>-I(po)=zo (i.e. <1>(zo)=Po)'In our Zo= <1> - 1 (0.365) = - 0.345. We next calculate
example,
PI =<1>(2z0-Z1-a/2)
where <1>(ZI-a/2)=1-a/2.
and
Pu = <1>(2zo
+ZI-a/2)
(in our case, with a=0.05; ZI-a/2 =<1>-1(0.975)=1.960, PI= 11>( - 2.650)= 0.004; Pu= <1>( 1.270)==0.898, and form the confidence interval (Cpd1000PI)' Cpk(1000pu))'
It, 4.3.4b The percentile confidence interval Ii
The 1000 CpkS are ordered as Cpk(1)~Cpd2)~...~ Cpk(1000). The confidence interval is then (Cpk«500a»), Cpk«500(1-cx))), where < ) denotes 'nearcst integer to'.
:i
Cpk(366)= 1.43
then we estimate
- Pr[C11k~ 1.41] as
One thousand bbotstrap samples are obtained by re-sampling from the observed values XI, X 2"", X n (with replacement). The arithmetic mean C~k and standard deviation S*(Cpk) of the CpkSare calculated. The 100(1-a)% confidence interval ~~~~ -
163
I
1,,1 MI-I
II
In our example, this is (Cpk(4),Cpk(898)). The rationale of this method is that since the observed Cpk is not at the median of the bootstrap distribution (in our case, below ~he median) the confidence limits should be adjusted approximately (in our case, lowered, because Zo<0). Evidently, the method can be applied to other PCls.
164
Flexible PCl s
Process capability indices under non-normality
165
(J2 and expected value l' we would have
Schenkler (1985) has presented results casting doubt on the efficiency of the 'percentile" and 'bias-percentile' methods above. Hall (1992) suggests that the method of bootstrap iteration described by Hall and Martin (1988) might be more useful in this case. Hall, et al. (1989) describe an application of this method to estimation of correlation coefficients, but, to the best of our knowledge, it has not as y~t been tested. for estimation of PCls.
EX>T[(X - 1')2J =EX
~
~'
Finally they define
1
.
(
USL-
1'-LSL
i.
i
Johnson et ai. (1992) have introduced a 'flexible' pct, taking into account possible differences in variability above and below the target value, 1'. They define one-sided PCls
I
-
{EX
-
1')2]}!
Note that if we have 1'=!(USL+LSL)=m, 1'-LSL=d, then
)
(4.23)
so that USL- 1'=
d
USL - l'
CUjkp= 3)2 {EX>T[(X -1')2]}!
l'
Cjkp=min (CUjkp, CLjkp)= 3)2 mID {EX>T[(X -1')2]}!'
4.4 FLEXIBLE PCls
1
(4.22)
i
(4.20a)
Cjkp= 3)2max([Ex>T[(X
-1')2], EX
I
and
t. 1 CLjkp
l'
- LSL
= 3)2 {Ex
4.4.1 Estimation of C]kp
(4.20b)
1')2]}!
I
A natural estimator
where (as in section 3.5)
~
EX>T[(X - 1')2] =E[(X
-
1')2IX> 1']Pr[X> 1'] (4.21a)
f
and
of Cjkp is 1.
Cjkp
= 3)2
mID
USL- l' 1'-LSL (8 + In)! ' (8- In)!
(
)
(4.25a)
where Ex< T[(X - 1')2] = E[(X - 1')21X< 1']Pr[X < 1'] (4.21b)
8+= I
(It is assumed that Pr[X
==
T] =0). Note that the radon;
Pr[X> 1'], Pr[X < 1') make allowance for how often positive 'and negative deviations from l' occur. The multiplier 1/(3)2), while the earlier PCls use 1. aris~s from the fact that for a symmetricaldistribution with variance
L
Xi>T
(X/-1'f
and 8_=
L
X,
(X1-1')2
Note that
'.
.
E[S +]
= nE[(X .:..7')21X > 1']Pr[X> 1']
so that S+ In (notS+ I[number of XIs greater than T]) is a.n
166
Flexible PCls
capability indices under non-normality
Process
I,
distribution, consideration of this case can provide an initial point of reference. Later we will indicate ways in which the results can be extended to somewhat broader situations though not as broad as we would wish. Table 4.10 presents :71Umerical values of
unbiased estimator of (4.21a) (and S- jn) is an unbiased estimator of (4.21b). The es.timator CjkP can be calculated quite straightforwardly - the only special procedure needed is separate calculation of sums of squares for Xi> T and Xi< T, as for Boyles' modified Cpm index, described in section 3.5. Our analysis will be based on the very reasonable condition LSL< T
,
Cjkp=
d~
;;::;
USL-T
. mill
(3y 2)0'
(
d ~
(S(fz
)
!'
T-LSL "
(
(fz
E[CjkPjCjkP] and S.D. (CjkPjCjkp)
(4.25 b)
1
))
d~-='
II
(assuming normal process distribution) for several values of (USL-T)jd, and n=1O(5)50. Note that if (USL-T)jd=1, then, T = (LSL + USL)j2. It is instructive to compare these values with similar quantities for CpkjCpk and CpmjCpnu in Kotz and Johnson (1992) and Pearn et al. (1992) respectively (see also Tables 2.4 and 3J). In general the estimator CjkPis biased. The bias is negative when T=1:(USL+ LSL) but increases as (USL- T)jd decrea.ses.It is quite substantial when (USL- T)jd is as small as 0.4. As might be expected the variance of CjkPdecreases as ,t increases .- it increasesas (USL- T)jd decreases,as the target value gets nearer to th6 upper specification limit.,The bias, alsb, decreases asrdncreases; fliiseffeci'Is particulariy noticeable for snulller values of (USL - T)jd. For n ~ 25, and (USL - T)jd ~ 0.60 thc stability of both E and S.D. is noteworthy. Since the index Cjkp is intended to make allowance for' asymmetry in the distribution of the process characteristic X it is of interest to learn something of the distribution of the
1.1
estimator CjkP undersuchconditions.The analysisin Appen-
14'
dix 4A can be extended to certain kinds of asymmetry of distribution of X. We note two ways in which this may be done.
I I
I
I
where 0' is an arbitrary constant. To study the distribution of CjkPit will be convenient to consider the statistic
t ..
:Po I t,
'J "
'I
,I
n d z -Z -Z ' -Z D= 18;;: Cjkp=max(alS+(f ,azS-O' )
()
A
A
(4.26)
:,
"
1
4
where I
USL~ T
(
al= -- d
-z
)
T -LSL and a2= d
(
-Z
)
I' I I.
I
I
'11
Note that ai"!+ai"!=2, and I
i
'1!
III
E[CjkP]= Ln8 (~Yrr E[J5-!r] 1
The distribution of jj will be, in general, quite complicated. In Appendix 4A we discuss a special case in which the distribution of X is, indeed, normal with expected value T and variance 0'2.Although this is not, in fact, an asymmetrical
167
t
1. If the population density function of X is I
l'
Hg(x; T, O'd+g( -x: - 'T,0'2)]
(4.27)
_..~ -.~
-~-
-
c
-
0\
'"
>. ..0 "0
.8 ......
;:j ..0 .~ ......
CIj
en :.a
~ 8 H
">
.......
vQ.,.
<:,,)
:z
.~ ~
k:
~
..c:: II) ..c:: ...... ..... 0
~ 0 p..
~
H
H
.8
0 s::
<.!..
h V
h 1\\ ~
.8
~ I
b
"--'"
~
~ 0 ~ 0 ..c:: . .~
II b h" >f ~
'08"0"0
~
8CIj.
0 0
s:: d'
+->.8
o~
..0
0
:;::
oS
~
8
~ --.E s::
CIj <1>-
':i--. N b
--
:.a
I
en
en 0
b
0
8 0..
-I~ --I
0 ..c:: .....
NN
b
+
Nb
-IN
..0 'i: ~..c:: .-; +-> ~ II.~ °- O""'N "O':
N
II
b
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l
170
Proccss cafJahility indices /llIdl'/' nOll-normality
Asymptotic
4.5 ASYMPTOTIC PROPERTIES.
Large-scaleapproximations to the distributions of Cp,
2 .1 .l 2 0'pk = 9 + 2 C pk
- HI U1/+!Cpk(,82 -1)!U 2}
(note that Cp= Cpk in this case) where Uland U2 are standardized bivariate normal variables, with correlation coefficient .j7J~/(,82-1)t. If the process distribution is normal the distribution is that of
Cpk
-!
C~ = (~)8 ~
3{
the asymptotic distribution of Cp can easily be approached by considering the asymptotic distribution of 8. Chan et ai. (1990) provided a rigorous proof that ~(Cp - Cp) has an asymptotic normal distribution with jmean zero and variance
j ~
II
of vi n(Cpk - Cpk) is normal
kU2
J2
P
l
(4.33)
J
U2 are independent standardized
normal
For Cpm, we have ..j';;(Cpm - Cpm)asymptotically normally distributed with expected value zero and variance
I
-
~-m
(~-m) +jp, (7 )+:[' + (~ )
1(/3 -1)
0'
2
with expected
value zero and variance
IU11+ ~C
2
If /32=3 (as for normal process distribution), O'~=~C~. For Cpk, Chan et ai.jI990) found that if ~¥:m the asymptotic distribution
where Uland variables.
(4.29)
t(/32 -1) C~
(4.32) I
I I
Since
O'~ =
(4.31)
In the very special case when ~= m, the limiting distribution of ~(Cpk - Cpd is that of
have been studied by Chan et ai. (1990).
Cpm
171
/32= 3, so that
There are some large-sample properties of PCls which apply to a wide range of process distributions and so contribute to our knowledge of behaviour of PCls under non-normal conditions.' Utilization of some of these properties calls for knowledge of the shape factors A.l and ..14,and the need to estimate the values of these parameters can introduce substantialerrors, as we have already noted. Nevertheless, asymptotic properties, used with clear understanding of their limitations, can provide valuable insight into the nature of indices. and
properties
6;m=
{1
}
6m
C;m (4.34)
2
.
.~ 2
1
O'pk= '9
where
ro
+ 2Y PI Cpk+
1
2
'4(fJ2-1)Cpk
(4 .10)
0'2pm
If the process distribution is normal, then
If the process distribution is normal, .j7f; = 0 and /32= 3, hence
~-m
= 1 if ~> m, = ,-I if ~< m.
'f
.jji;=.0
and
---
2
(-a ) +!
(4.35)
')
{I+ e:m)';'
C;m
172
Process capability indices under non-normality
Appendix 4.A
.'III
and if, also, ~= m ".2
_1Z C pm
" pm -
2
,
~
I
,
From (4.26)
(4.36) I
ale
~
for 8- + 81
'-2
S
I
III "
173
+(f
D=
(In this case, also, Cp=Cpm)'
az8-
{ APPENDIX 4.A
So 15 is distribu,ted as
As noted in the text, it is assumed that the distribution of X is normal, with expected value T and standard devia-
Q1HXnZ
tion (f.
al
for -8+ < -az
(f-Z
8-
t'10r
> az -
H (1 - H»
az . -l.e. al
(4.38)
al
H>- az al+aZ
Z t' az ( { az 1-H)x1l lor H<- al + az
With the stated assumptions, we know that: I
1. the distribution of (Xi- T)2 is that of XT(f2; 2. this is also the conditional distribution of (Xc- T)z, whether Xi>T or Xi
The overall distribution of 15can be represented as I'
.
3. the number, K, of Xs which exceed T has a binomial distribution with parameters n, 1-- denoted Bin(n, hand hence 4. given K, the conditional distributions of 8 + (f - Z and 8 - (j-z are those of xi:, X;'-K respectively, and 8+ (f-z and 8 - (f- Z are mutually independent; and also 5. the distribution of H = 8 + /(8 + + 8-) is that of xV(xi+ X;-K) which is Beta(1-K,l(n-K)) so that the density function of H is
fJ "' alH
{ az(1-II)
I
1\ BetaO:K,1-(n-K» 1\ Bin(n,1) Ii K [I
(4.39)
I \I
t' ~
,-
where the symbol /\ y means 'mixed with respect to Y' having the distribution that follows, and h=az/(al +a2). Without loss of generality, we assume that
USL- T:::;T - LSL. Then, for a symmetrical distribution with mean T and variance (fz,
O
C
(4.37)
jkp
I
!. , for K=I,2,...,n-l, and 11 and 8++S- arc mutually I independent; and further 6. the conditional distributions of 8 + (f - Z and 8 - (f - z, given K and H, are those of H X; and (1- H)X;, respectively.
z for H>h for H < b} XII
I
III
, fH(h)={B(lK,l(n-K))}-lhlK-l(1-h)l(n-KJ-l
.
.11
-~ USL-T=~~ - 3(f d 3(f r;; y al
.and
I
1 C+
Val
1 ;:--=2
V Qz
(4.40)
I
174
where
Hence (from (4.26) and 4.40)) CjkP
=
na1
( )
! 15-!
Bv(Ul,U2)=JoyUI-I(1--y)"2-ldy
,
and
and B(UloUZ)=B1(1l1,llz) (see section 1.6). In particular
i"
E[(~~::YJ ~ (n~1yr E[15~!r]
(4.42)
Now, from (4.37), (4.38) and (4.39), noting that
I
II
E
CjkP
[,Cjkp ]
= rO:(n.-1»~ 2"+1r(-!n)
at
[( ) + az
\
1
\
I
I
= {2!rr(tn)}-1r(t(n - r))
E[(X;)-!r]
I"
I. i,
(4.41)
2
Cjkp
175
Bibliography
Process capability indices under non-normality
,,-1
(~)
+ k~l B(tk, t(n-k))
{ B1-b(t(n-k),1(k-1))
I
~-!r - r(t(n-r))
E[D
]-
~
[
-!r
2\rnin) 2" az
(~)
,,-1
I
x {ai\rB1-b(!(n-k),
+
I
+ k~l 8(1k, l(n-k))
(::yr Bb(1k,1(n-k-r))}]
I t(k-r))
BIBLIOGRAPHY
1
1
+ a2!rBb(tk,!(n-k-r))}
+al!rJ
Balitskaya, B.O. and Zolotuhina, L.A. (1988) On the representation of a density by an Edgeworth series, Biometrika, 75, 185187. B,\rnard, G.A. (1989) Sophisticated theory and practice in quality improvement, Phil. Trans. R. Soc., London A327, 581-9. Boyles, R.A. (1991). The Taguchi capability index, J. Qual. Technol.,
I
I
whence r
[( Cjkp)
E
CjkP
]
=
n!r:l~(n;r)) 2
r(zn)
,,-I
a1
[( az )] (~)
23, 17-26.
!r+ 1
r
II.
+ k~l B(tk..Hn-k» { 81-1,(z(n-k)02(k-r))
.
+ (::)!rBb(1k,!(n-k-r))}]
(4.43)
I I
.1.
.
Chan, L.K., Cheng, S.W. and Spiring, FA (1988).The robustness of process capability index Cp to departures from normality. In Statistical Theory and Data Analysis, II (K. Matusita, ed.), NorthHolland, Amsterdam, 223-9. Chan, L.K., Xiong, Z. and Zhang, D. (1990) On the asymptotic distributions of some process capability indices, Commun. Statist. - Theor. Meth., 19, 1-18. . Clements, J.A. (1989). Process capability calculations for non-
normal calculations,Qual.Progress,22(2),49-55.
176
177
Process capahility indices under non-norl'l'!ality
Bibliography
Efron,B. (1982)The Jackknife, the Bootstrap and Other Re-sampling
Kocherlakota, S., Kocherlakota, K. and Kirmani, S.N.U.A. (1992) Process capability indices under non-normality. Internat. J. Math. Statist. 1. Kotz, S. and Johnson, N.L. (1993) Process capability indices for non-normal populations, Internat. J. Math. Statist. 2. Marcucci, M.a. and Beazley, C.e. (1988) Capability indices: Process .performance measures, Trans. ASQC Tech. Conf., Dallas, . Texas, 516-22. McCby, P.F. (1991) Using performance indexes to monitor production processes,Qual.Progress,24(2),49-55. Munechika, M. (1986) Evaluation of process capability for skew distributions, 30th EOQC Conf. Stockholm. Sweden. Munechika, M. (1992) Studies on process capability in matching processes, Mem. School Set. Eng., Waseda Univ. (Japan), 56, 109~124. Pearn, W.L. and Kotz, S. (1992) Application of Clements' method for calculation second and third generation PCls from nonnormal Pearsonian populations. (Submitted for publication). Pearn, W.L., Kotz, S. and Johnson, N.L. (1992) Distributional and inferentialproperties of process capability indices,J. Qual. Tech1'101. 24, 216-31. Pearson, E.S. and Tukey, J.W. (1965) Approximate means and standard deviations based on differences between percentage points of frequency curves, Biometrika, 52, 533-46. Price, B. and Price, K. (1992) Sampling variability in capability indices, Tech. Rep. Wayne State University, Detroit, Michigan. Rudo1t\~E. and Holfman, L. (1990) Bicomponare Verteilung - eine erweiterte asymmetrische form der Gaul3schen Normalverteilung,
Plans, SIAM, CBMS-NSF Monograph, 38, SIAM: Philadelphia, Pennsylvania. '.
.'
English,J.R. and Taylor, G.D. (1990)ProcessCapability AnalysisA Robustness Study, MS, Dept. Industr. Eng., University of
Arkansas,Fayetteville.
.
Fechner, G.T. (1897) Kollektivmasslehre, Engelmann, Leipzig. Franklin, L.A. and Wasserman, G. (1992). BQotstrap confidence interval estimates of Cpk: An Introduction, Commun. Statist.Simul. Comp.,20, 231-44' Gruska, G.F., Mirkhani, K.' and Lamberson, L.R. (1989) Nonnormal Data Analysis, Applied Computer Solutions, Inc.; St. Clare Shores, Michigan. Guenther, W.H. (1985). Two-sided distribution-free tolerance intervals and accompanying sample size problems, J. Qual. Teclmol., 17, 40-3. . Gunst, R.F. and Webster, J.T. (1978). Density functions of the bivariate chi-squared distribution, J. Statist. Compo Simul., 2, 275"';88. Gunter, B.H. (1989) The use and abuse of Cpk' 2/3, Qual. Progress, 22(3), 108-109; (5), 79-80. . Hall, P. (1992) Private communication, Hall, P. and Martin, M.A. (1988) On bootstrap resampling and iteration, Biometrika, 75, 661-71. . Hall, P., Martin, M.A. and Schucany, W.R. (1989) Better nonparametric bootstrap confidence intervals for the correlation
coefficient,J. Statist. CompoSimul,33, 161-72.
I
I
.
.
Hsiang, T.e. and Taguchi, G. (1985) A Tutorial on Quality Control and Assurance - The Taguchi Methods, ASA Annual Meeting, ,
Las Vegas,Nevada.
Hung, K. and Hagen D. (1992) Statistical Computation Using GAUSS: Examples in Process Capability Research, Tech. Rep. , Western Washington University, Bellingham. Johnson, M. (1992)Statisticssimplified,Qual.Progress,25(1),10-11.
I
Textiltechnik,40, 49-500.
I.
Johnson. N.L., and Korz, S. (1970) Distributions in Statistics: Continuous Univariate Distributions, John Wiley, New York.
Johnson, N.L., Kotz, S. and Pearn, W.L. (1992)Flexible process capability indices(Submittedfor publication). Kane, V.E. (1986)Process capability indices,J. Qual. Technol.,18, 41-52.
..
Schenker, N. (1985) Qualms about bootstrap confidence intervals, J. Amer. Statist. Assoc., 80, 360-1. .' Subrahmaniam, K. (1966, 1968a, 1968b). Some contributions to the theqry of non-normality, I; II; III. Sankhya, 28A, 389-406; 30A, 411...:32;30B, 383-408.
\
l .1
5
Multivariate process capability indices
5.1 INTRODUCTION
I ! I
;1
" I~ II II ill "
,. Ii
L
Fr~quently - indeed usually - manufactured items need values of several different characteristics for adequate description of their quality. Each of a number of these characteristics must satisfy certain specifications. The assessed quality of the product depends, inter alia, on the combined effect of these characteristics, rather than on their individual values. With modern monitoring devices, simultaneous recording of several characteristics is becoming more feasible, and the utilizationof such measurements is an important issue. Multivariate control methods (see All (1985) for a comprehensive review), although originating with the work of Hotelling (1947), have only recently become an active and fruitful field or research. The use of PCls ih connection with multivariate measurements is hedged arohnd with even more cautions and drawbacks than is the case for univariate measurements. IIi particular, the intrinsic difficulties arising from use of a single index as a quality measure are increased when the single index has to summarize measurements on several character~ istics rather than just one. In fact, most of the multivariate PCls which have been proposed, and will be discussed in this chapter, can be best understood as being univariate PCls,
\ II I
180
Multivariate
process capability indices
1
I'
based on a particular function of the variable.s representing the characteristics. Further, so far as we are aware, multivariate PCls are, as yet, used' very rarely (if at all) in many
industries.
I
i
'
The contents of this chapter should, therefore, be regarded mainly as theoretical background, indicating some interesting possibilities, and only an initial guide to practice. Nevertheless, we believe that study of this chapter will not prove to be a barren exercise. The reader should gain insight into the nature and consequences 'of multiple variation, and also to understand pitfalls in the simultanepus use of separate PCls, one for each measured variate.
5.2 CONSTRUCTION
;1 I
I
JI ill
']
I
I
I
I
OF MULTIVARIATE PCls
l'
Just as in the univariate case, we ne6d to consider the inherent structure of variation of the measured characteristics - but, this time, not only variances but also correlations need to be taken into account. Deviations from target vector (T) also need to be considered. The structure of variation has to be related to the specification region (R) for the measured variates Xl"", X v' In principle this region might be of any form - later we will discuss a favourite theoretical form though in practice it is usually in the form of a rectangular paralellopiped(a vdimensional 'box').defined by
'.1 1m
I
I
.
I
LSL/~X/~USL/ i=1,...,v
(5.1)
If the specification region is of this form, Chan et al. (1990) suggest using the product of the v univariate Cpmvalues as a multivariate PCI. A moments' reflection however, shows that apart from the serious defects of the univariate Cpm,described in Chapter 3, this could lead to absurd situations, even if the v measured variates are mutually independent. The reason for
this is that a very bad (i.e. small) Cpmvalue for one variate can be compensated by sufficiently large Cpmvalues for the other values, giving an unduly favourable value for the multivariate PCI. (If one component is only rarely within specification limits, it is small consolation if all the others are never nonconforming!) A more promising line of attack may be possible using results of Kocherlakota and Kocherlakota (1991), .who have derived the joint distribution of Cpms calculated for two characteristics, with a joint bivariate normal distribution. We will return to this work in section 5.5. As with univariate PCls, there are two possibe main lines of approach - one based on expected proportion of NC items, the otper based primarily on loss functions. Both are based on assumptions: about distributional form in the first approach; and about fo~'mof loss function in the second. These more or less arbitrary assumptions are, of necessity, more extensiv,ein the multivariate case, just because there are more variates involved. Note that the approaches of Lam and' LittIg (1992) and Wierda (1992a) (see chapter 2, section 2) can
be extended straightforwardlyto multivariate situations.
It
Corresponding to the assumption of normality in the univariate case, we have, in the multivariate case, the assumption of multinormality, with
11
-
.
181
Construction of multivariate PCl s
"
E[XiJ = ~/ var(Xd= at corr(Xj,Xr)=p// i, if= 1,..., v
i I'
1I '
i.e. expected value vector ~=(~1'~2""'~v)
I
"1
:~
i
I
I
.
I
j
covariance
matrix
V0 =(p/i'Q"/al)
with
PII
and varianceall i. This
= 1 for
means that the joint PDF of,X = (X b ... , X v) is
fx(x)=(2n)-v/2IV 0I-~ exp{ -!(x -~)'Vo l(X-~)} (see section 1.8.)
I~ I
(5.2)
I
182
Multivariate
process capability indices Multivariate
If a loss-,function approach is employed, a natural generalization of the univariate ~oss function k(X - T)2 is the quadratic form L(X)=(X-T)'A
-l(X-T)
I I
(5.3)
183
control charts. Nevertheless, a vector-valued multivariate PCI has been proposed. This will be discussed, briefly, in section 5.6.
I
I
with a specification region L(X)::::; C2. (A- 1 is called.. the generating matrix of this quadratic form). T. Johnson (1992) interprets C2 as a maximum 'worth', attained when X = T i.e. all characteristics attain their respective target values and L(X) is a loss of 'worth', so that .zero worth is reached at the boundary of R. Note that A does not have any necessary relationship with the vari~nce covariance matrix Vo. It is, indeed, very unlikely
5.3 MULTIVARIATE ANALOGS OF Cp
If we accept the assumption of multinormality (5.2), it would be natural to use, as a basis for construction of PCls, the quadratic form
w =(X-~)Vo l(X-~)
(5.4)
,
that. A would be identical with V0, (though tempting for theoreticians to explore what would be the consequences if this were the case, as we will see later). In the following se<:lions we describe ~I few proposed multivariate PCls and provide some critical assessment of their properties. , Before conclu'ding this section however, we stress that any inClexof process capability, based on multivariate characteristics, that is a single number, has an even higher risk of being misused and misinterpreted than is the case for univariate PCls. The time-honoured measures and tech-' niques of classical statistical multivariate analysis - e.g. Ho,teiling's T2, Mahalanobis' distance, principal component analysis - provide ways of obtaining more detailed assessment of process variation. It would be inefficient, not to utilize appropriate forms of these well-established measures and techniques. Reduction to a single statistic, however ingeniously constructed, is equivalent to replacing a multivariate problem by a univariate one, with attendant loss of information. However, if a process is stable, it is much easier to monitor a single index than, for example to use many mean and range
analogs of Cp
The statistic W would have a (central) chi-squared distribution with v degrees of freedom. If, also (see remarks I near the end of section 5.2), the specification region R were of I form
:11 III I
'
l
"
III I
,II
/1'
I 11
:;1
111
,
Ii;
v III
.'
tll 11'1
i
"'1
I
j
2 2 W::::;XV,0.9973=Cv
the expected proportion of NC product would be 0.27%. We also note that it has been suggested that an estimate (p, say) of the expected proportion of NC items. based on obsetved data, and exploiting the assumed form of process distribution (usually multinormality), might be, itself, used as a PCI. We have already noted this suggestion for the univariate case, in section 2.1 It is important to realize that the dependence on correct form of process distribution is even heavier in the multivariate than in the univariate case. Littig et al. (1992) suggest using $-1(1(p+l)) as a PCI (as in the univariate case). See also end of section 5.5. Recall that the univariate index Cp was defined as length of specification interval Cp = 6 x (standard deviation of X)
(5.5)
I-' 184
Multivariate
I
process capability indices
I I
the factor 6 being used because, if C p = 1 and variation is
I
Multivariate
normal, it isjust possible- by making ~:;::t(USL + LSL)- to
not satisfied if the 'box' region LSLi~Xi~USLi (i= 1,..., v)is used. The volume of the rectangular paralellopiped is
I I I I
have the expected proportion of NC product as small as 0.27%. Taam et al. (1991), regarding the denominator in (5.5) as 'length of central interval containing 99.73% of values of X', propose the natural generalization
185
analogs of Cp
I
I
v
n (USLi-LSLi)
I
I
I
r
(5.10)
i= 1
j
C P
=
vol.ume of s~e~ification region
but Pr[LSLi ~ Xi ~ USLi for all i = 1,... , v] is not necessarily equal to 99.73%, even if
.
volume of regIOncontaInIng 99.731Yo of values of X.
(5.6)
v
.n (USLi-LSLi)=
The denominator is the volume of the ellipsoid (x - ~)'VO'1 (x -~) ~
X~. 0.9973
.
1=1 (5.7)
(5.11)
(nx~.9973)hIVol ntv+l)
The property would hold if R were of form
which is .
(X-~)'VO' 1(X-~)~K2
'v
(nX~.0.9973)2 IVoil
ntv+ 1)
(5.8)
but, as we noted in section 5.2, this is unlikely to occur. Even if an ellipsoidal R, of form
~
.
[Of course, this would be the volume of the ellipsoid (5.7), whatever the value of ~. The ellipsoidwould contain 99.73% of values of X however, only with ~ equal to the expected value vector.] This leads us to the d'efinition
-
c P
I
-
(5.12a)
(X-~)'A -l(X -~)~K2
, (5.12b)
were specified, it is unlikely that A would equal Vo. Taam et al. (1991) propose to avoid this difficulty by using
~
.
a 'modified specification region' -. R*, say - defined as the volume of R Qv
IVoll
greatest volume ellipsoid with generating matrix VO'l, contained within the actual specification region, R. If this ellipsoid is given by (5.12a), then
~
(5.9)
~
where a:=(nx~.0.9973)!v/r(tV+ 1).
III
However, some modification of this definition is needed in order to provicie a genuine generalization of the coverage property of Cp(v= 1),which would be that, if ~=T and T is the centre of the specification region, then Cp= 1 should ensure that 99.73% of values of X fall within R. This property is clearly
.
I'
C~= volume
I
volume of R* of(X-~)'VO1(X-~)~X;.0.9973
I /1 I~
(
K2
V/2
)
= X;.0.997;
(5.13) /
.
for any ~ and any v.
The most one might reasonably hope for, in practice, is that the region R is defined by (5.12b), and it is known that
.I
J
186
Multivariate
process capability indices
Multivariate
v 0 is proportional to A, i.e. V0= 02A, though the value of the multiplier, 02, is not known.. Total rejection of this possibility (an opinion held in some circles) seems to be an unduly pessimistic outlook, assuming that practitioners have so little knowledge of the process as to be unwilling to be somewhat flexible in adjusting engineering specifications to the actual behaviour of the process in order to benefit from available theory. We consider such an attitude to be unhealthy, and encourage practitioners to, take advantage of knowledge of likely magnitude of correlations among characteristics in
setting specification,if possible. If V 0
= ()2A
is distributed as e2X~-l)v, and S/{(n-1)v} estimator of (J2. A natural estimator of vCp is ~
K X",O.9973
is an unbiased
(n-1 )v 1 S
( )
,
(5.16a)
which is distributed as K
-
.
((n-1)v)!
Xv,O.9973 °X(n-l)v
then
Cp
-
vC p-
187
analogs of Cpm
=
((n-1)v)j
"Cp
X(n-l)v
and a 100(1- iX)%confidence interval for vCp is
volume of x'A - lX ~ K2
= voumeo --- 2 I f x v 0- 1X"'::Xv.O.9973 I
I
IX~(K/()f = volume of x'V() ' - 1 --- 2 vo Iume 0f x V0 X"'::Xv.O.9973
~
=
(
K2 2
0
2 Xv,O.9973
(
I
{(n-1)v}!
p,
X(n-l)v,l
~
t
{(n-1)v}7
v
)
(5.16 b)
p
(cf (2.9 c)).
)
(5.14) ,
I j tI
5.4 MULTIVARIATE ANALOGS OF Cpm
Taam et al. (1991) proceed to define a mutivariate analog of Cpmby the formula C* pm
K
°Xv,O.9973
= C~/v
I I~
(5.15)
Estimation of Cp (or vCp) is equivalent to estimation of O.If Vo={)2A, and valucs X/=(xlj, ...,x"j) arc availahle for 11 individuals (j= 1, .", n), the statistic, 11
s=
v
V/2
Pearn et at. (1992) reach a similar PCl hy regarding X;,O.9973 as a generalized 'length' corresponding to a proportion 0.0027 of NC items, and K2/h2 as a generalized 'allowable length'. They define
vCp=
C
X(n-l)V,iXI2
I
I
L (Xj-X)'A -l(Xj-X) j= 1
=
volume of R*
.mm
...
..
volume of ellipsoid x'Vi lX
(5.17)
(cf. (5.13)) where
VT= Vo+(~-T)(~-T)'
~,l..
I I
I
Since IVTI= IVol{1 +(~- T)'VC;l(~- T)}j C:m = Ct{ 1 + (~- T)'VC;l(~ - T)} -j
,j
,J_1
(cf. (3.8))(5.18)
188
Multivariate
(Taam et at. (1991)usethe notationMCPfor C:m). Thisindex has the property that when the process mean vector ~ equals the target vector T, and the index has the value 1, .then 99.73% of the process values lie within the modified specification region R *. This is analogous to the property of the univariate Cpmindex. The matrix VT can be estimated unbiasedly as
I
Cpm as
' I
Cpm
1
n
,
.
L
n j= 1
(Xj-T)(Xj-T)
(5.20)
If V0 is known, an unbiased estimator of the denominator of 2 . C pm is
t
(5.19a)
!n j=1 (Xj-T)'V(j1(X-T)
.
and V0 by
and a natural estimator of Cpmis ~
V= -
1
n-1
~ L. (Xj-X)(Xj-X)'
(5.19b)
Cpm =
j=1
where 1
I I
n
L Xj n j= 1
X=-
I I' 0
Note that n--1 - n V +(X-T)(X-T)'
.j A 'I
~
~T= ,
=[E[(X- T)';(j l(X- T)]T = {I + V-1( - T)V(j 1( - T)} -i
I
.
\\= -
189
Multivariate analogs of Cpm
process capability indices
(5.19c)
~
Chan et at. (1990) assume that the specification region R is
of form
'1"
'
nv
n
[ j~1
2
(5.21)
(Xj-T)'V()1(Xj-T)
J
At this point we note that Cpm is subject to the same drawback as Cpm,noted in section 3.2. Identical values of Cpm can correspond to dramatically different expected proportions of NC items, if the specification region is not centered at T. Also, calculation of C"", requires knowledge of the correct Vo. If all arbitrary matrix is used in place of V0 ill defining Cpm, we can encounter a similar effect of ambiguity in meaning of the value of Cpm' (See Chen (1992) and also Appendix 5.A). The numerator v is introduced because if ~= T, then
.
E[(X - T)'V (j 1(X- T)] = v
(X-T)'V() I(X-T)~
K
i.e. the generating matrix of the specification region. R. is
proportional to V0, the variance-covariance matrix of the measured variates. They define a multivariate analog of
, lit,
If ~= T, then C;.; is distributed as (nv)-1 X;v. Hence, in thes~ special circumstances E[ Cpm] =
nv ! r( t(nv - 1))
() 2
l( tnv)
(5.22a)
190
Multivariate process capability indices Multivariate
ap.d
r(!nV)r(tnv-t)-
var(Cpm) =Jnv -
[
{r(!(nv-t)Y
{r(tnv)}2' -
] (5.22 b)
(Chan et ai. (1990)). Percentage points for Cpm(on the assumption that are easily obtained since
analogs
191
of Cpm
when nv= 100, we find Go.o2ss:f0.8802(correct value 0.8785) and Go.os~0.8978 (correct value 0.8968):
!
and when nv = 200, we find GO.O25 ~0.9118 (correct value 0.9109)
-I.
-
and Go.os~0.9251 (correct value 0.9245)].
Peam et al. (1992) felt that Cpmis not a true generalization of the univariate Cpm,and proposed the PCI
~= T)
Pr[Cpm~ GaJ = Pr[C;'; ~ G;2J = Pr[(nv) - Ix;v ~ G; 2J
-i
1 vCpm=vCp{ 1+ ;(~-T)'VOl(~-T)
I
I
} (5.25)
=vCp X Cpm
=Pr[X~v~nvG;2]
II
I
I
(see (5.15) and (5.20)). Hence, to make Pr[Cpm ~ GaJ = Ctwe take
\ I
Note that the univariate
indices Cpm and Cp a!e related by
I
Ga-- (nv)! Xnv.l-IX
(5.23) Cpm=Cp{l+(~~TY}
\
t
Except when nv is small the approximation
-1'
I
where ~ = E[XJ, (J2 = var(X) and T is the target value for X. A
natural estimator of vCpmis GIX~
-
(5.24 a)
(2nv)! zl-IX+(2nv-1)1
I K(nv) "Cpm= 0--'
Xv,O.9973
where
(z1 - IX) = 1- Ct,or even
, , Zl-IX +1-GIX= C2nV)!
1
-,
- 1 if
(A--T)f
1 -!
(5.26)
In all the previous work, distribution theory has leaned heavily on the assumption that:
-1
)
f
x (S+n(X-l)Vo
(5.24b)
1. R is of form (X t)' A-I (X- T)~ K2 and.
4nv-
~
I~
give values which are sufficientlyaccurate for practical purposes. [For example,using (5.24a) when nv=40 we find Go.o25s:f0.8245(correct value 0.8210) and Go.os~0.8492 (correct value 0.8470):
2. 1\,= V0, or, at least, A is proportional as in (5.14).
to V0, i.e. (12A = V0,
In this second case, even with ~#T, it is possible to evaluate, the distribution of n ()2
L (Xj-T)'Vo
j= 1
l(Xj-T)
192
Multivariate
Multivariate
process capability indices
analogs of Cpm
193
regIOn IS
as that of
h( x -- T) < r 0
02 X(noncentral X2 with nv degrees of freedom and noncentrality parameter n(~ - T)'Vo l(~ - T))
Table 5.1 Scalar indices for multivariate data
However,if A is not proportional to V0, the distribution of (X-T)'A -l(X-T), although known, is complicated (Johnson and Kotz (1968)). (Appendix SA outlines some details.) Other approaches to measuring 'capability' from multivari-
ate data include:
.
Symbol
Taam et al. (1991)
Cp
1. Use of the PCI
Definition vol. speen. region vol. of (x-~)'VOI(x-~)~X;.O,9973 As Cp, with modified speen. region Cl/v p , As C~, but denominator =(vol. of
Taam et al. (1991) C: Pearn.et al. (1992) vCp Taam et al. (1991) qm
.
vCR= IVoA-II +(~- T)'A -l(~-T)
Source
(x'Vi 1X
(5.27)
. The motivation for this PCI is that it is expressed in terms of the matrix A used in the specification, rather than the process variance-covariance matrix - on the lines of Taam et al. (1991). If v=l and A=(1d)2 we have ICR=C;n~' Note that small values of vCRare 'good', and large values are 'bad'. 2. Proceeding along the lines of Hsiang and Taguchi (1985) and Johnson (1992) (see section 3.2) we may introdu.ce
, I
:
I
q
Chan et al. (1990)
Cpm
Pearn et al. (1992) Proposall. Proposal 2.
vCpm vCp X Cpm iVoA-ll+(~-T)'A -l(~-T) vCn E[L(X)] tr(A-1Vo)+(~-T)'A -l(~-T)
v 0 is the variance-covariance characteristics,
and
.
As but denominator E[(X-T)'Vo l(X- T)]
matrix of the joint distribution of the process
V 1'= V 0 +(~
--T)(~ -T)',
where h(') is a nonnegative scalar function, satisfying the
condition h(tx)= th(x) for all t > O. Then a multivariate' PCI can be defined as
L(x)=(x-T)'A
-l(x-T)
(5.28a) MCp=r/ro
as a loss function (generalizing L(x) = k(x- T)2).We.have E[L(X)] =tr(A -IV o)+(~- T)'A'-l(~-T)
.
where
(5.28 b)
Pr[h(X-T»r]
where tr (M) denotes 'trace of M', which is the sum of diagonal elements of M. Large values of E[L(X)] are, of course, 'bad', and small values, 'good'. For readers' convenience we summ<Jrize,in Table 5.1,
the scalar indicesintroduced in this chapter.
I
.
3. Chen (1992) has noted that a broad class of PCls can be defined in the following way. Suppose the specification
This ensures that if MCp= 1 the ~xpected proportion of NC items is a. This definition includes, for example, the rectangular specification region IXj-Tjl
-
=a
(j=l,...,v)
194
Multivariate
process capability indices
by taking
J4
h(X-T)=
n"iaxrj-lIXj-
111
Tjl
1 t;it; v
I
and 1'0= 1. In this case, and several others, actual estimation of MC" (Le. of r) can entail quite heavy compulation. Some details of possible methods of calculation are provided by Chen (1992)~ 4.' Wierda's (1992a, b) suggestion that -1<1>-1 (expected proportion (p) ofNC items) be used as.a PCI, is applicable in ,multivariate, as well as 'univariate situations. It can then 'be regarded as a multivariate generalization of Cpk' The same comments - that one might as well use p, itself, and that p is estimated on the basis of an assumed form of process distribution, accuracy will depend on correctness , of this assumption - apply, as mentioned in section 2.
195
Vector-valued PCls
I
d
joint distribution of Cps for v = 2, under assumed multinormality. A different type of vector PCI has been proposed by Hubele et al. (1991),in which they suggest using a vector with I three components. 1. The ratio ar~a of specification rectangular paralellopiped + area of smallest rectangle containing the 99.73% ellipsoid, (x- ;)'V 0 l(X-;) ~ X~,0,9973 ~ ]
,
,I
= 'specification rectangte'
[
'process rectangle'
+
(5.29a)
]
The numerator is (see (5.10») 5.5 VECTOR-VALUED PCIs
As we have noted (several times) it is a risky undertaking to represent variation of even a univariate characteristic by A single index. The possibility of hiding important information ,is much greater when multivariate characteris.tics are I
under consideration, and the desirability of using vector
i
valued PCls arises quite naturally.
One such vector would
I consist of the v PCls, one for each of the v variables. These I might be Cps, CpkSor Cpms,and would probably be related I to .the concept of rectangular parallelopiped specification " regIOns. I
[/~ (USL/- LSL/)J+ Hubele et al. (1991) show that the denominator is
I I
I I 2, 1
2Xv.O.9973(
tl1lVOill}+ Ilvo
11)!'
(5.29b)
where, Voil is obtained from Vo 1 by deleting the ith row and the ith column. 2. The P-vatue of the Hotelling T2 statistic
'
We have already mentioned (in section 5.1) a suggestion to
I
use thc product of Cpms, and havc pointcd out thc dcli~icncics
i
proposal. Interpretation of the set of estimated PCls would be assisted by knowledge of the joint distribution. The
,
work of Kocherlakota and Kocherlakota (1991)provides the,
w2 =n(X- T)'~-l(X- T)
I of this
j
(5.30)
where ~ is the s@!p~an~~:£QYariance matrix. This component includes information about the relativelocation of the process and specificationvalues.If ~'=T - i.e.
196 ,
Multivariate
process capability indices
thc proccss is accuratcly ccntrcd distribution
!
thcn WZ has the
of I
v(n-l) n-v {Fv.n-v variable}
,
When X is close to T the P-value of WZ is nearly I;thc further X is from T, the nearer is the P-value to zero. 3. The third compone:nt measures location and length of sides of the 'process rectangle' relative to those of the 'specification rectangle'. For the case v= 2, and using the symbols UPRi, LPRi to denote upper and lower limits for variable Xi in the process rectangle, the value of the component is
197
evant distribution theory. Nevertheless, (1.), (2.) and (3.), comhincd will give a useful idca of thc process capability especiaIIy indicating in what respects it is, and is not, likely to be satisfactory.
J I ;
APPENDIX 5.A
"; I
We investigate the distribution of ~
II
max l. IUPR1-LSLd, ILPR1- USLll, IUPRz-LSLzl,
[
5.A
Appendix
I
USLt - LSL1 USL1- LSL1 USLz- LSLz
I
1
W=(X-T)'A
-l(X-T)
where X has a v-dimensionfll multinormal distribution with expected value vector ~ and variance-covariance matrix V0, and A is a positive definite v x v matrix. From Anderson (1984, p. 589, Theorem A2.2) we find that there exists a 110nsingular v x v matrix F such that
ILPRi - USLzl FVoF'=I
USLz - LSLz J
I A value greater than 1 indicates that part or whole, of the process rectangle falls outside the specification rectangle. Taam et al. (1991) point out that components (1.) and (2.) of : this triplet are similar to the numerator (Cp) and the estimator of the denominator, I
[1 +(X- T)'V-1(X- T)]i = [1+n-l WZJi of C~III(sec (5.1R», rcspectively. Although these three components do measure dill'erent aspects of process capability, they are by no mearis exhaus, Hve. Also, when v> 2, calculation of component (3.) is cumbersome, and more importantly, there is no practically rel-
I
i
Ii'
r
i \
I I
(5.31a)
and
iI
FAF' = diag (),)
(5.31b)
where diag (),) is a IIx IIdiagonal matrix in which the diagonal elements A1, ..., Av are roots of the determinantal equation
IA-AVO/=O
(5.32)
and 1= diag (1) is a v x v identity matrix. Note that (5.31a) can be rewritten F'F=VOl
I
(5.31 c)
and (531 b) as (F- l)'A -1 F-1 = diag(1/),)
(5.31d)
198
Multivariate
If Y = F(X - ;) then Y has a v-dimensionalmultinormal distribution with expected value 0 and variance-covariance matrix
(W is the weighted sum of n independent
The general distribution of W in this case has been studied by centage points of the distribution for v= 4 and v= 5 for various combinations of values of the ASsubject to the condition
from (5.31a). Then since X-;=F-Iy,
v
These lOO(l-IX)% percentage points are values of c~ such that Pre W < c~] = 1 -IX. From the tables one can see how c~ varies with variation in the values of the AS. Some values for v = 4 are shown in Table 5.2.
v
Aj-1(Yj+<5Y
1
L -=v j= 1 Aj
W=(F-Iy + ;-T)'A -1(F-Iy +;-T) "';(Y'+(~-T)'F'}(F:-1)' A-1F-1(Y +F(~-T)) =(Y +F(;-T))' diag(1/l)(Y+F(;-T))
I
xi variables.)
Johnson and Kotz (1968)inter alia. They provide tables of per-
E[YY'] = FE[(X- ;)(X - ;)']F' = FVoF'= I
=
199
Appendix 5.A
proce§s capability indices
(5.33).
j= 1
Table 5.2 Percentage points of W
where 0' =«51>"" <5v)=(;- T)'F'. We note that in the particular case when A = ()2V 0 we have from (5.32), A1=A2="'=Av=82 so v
W=O--2
I
(Yj+<5j)2
rl3
Ail
97.5%
99%
99.5%
99.73%
2.5 2.0 1.5 1.5 1.2 *1.0
0.7 1.0 1.5 1.5 1.2 1.0
0.4 0.7
0.4 0.3 0.2 0.5 0.4 1.0
14.3 12.8 12.4 12.3 11.7 11.1
18.3 16.0 15.2 15.0 14.1 13.3
21.4 18.4 17.3 17.1 15.9 14.9
24.1 20.7 19.2 19.0 17.5 16.25
OJ
0.5 1.2 1.0
In all cases !:j=IAjl=4. *Thc values in this row are percentage
and is distributed as
points of X
-
chisquared
with four degrees of freedom.
8-2 x (noncentral chi-squared with v degrees of freedom and noncentrality parameter /I
I
..1.-1 2
(5.34)
J= I
j= 1
A-I I.
t c
<5}=o'o=(;-T)'F'F(~-T)=(~-T)'Vo
If we have T =;, i.e. the process is value, ~hen ,)= 0 and W reduces to n
w= I
1(~-T)).
centred at the target
Y2
-L
j= I Aj
II
~. II II
The more discrepant the values of the AS, the greater the correct value of c~. This will, of course, decrease the values of the PCls defined in sections 5.3 and 5.4, implying a greater proportion of NC items than would be the case if all ASwere equal to 1. Of course, one must keep in mind that this conclusion is valid with regard to situations restricted by the requirement of a fixed value of th~ sum n
(5.~5)
I-
1
j= 1AJ
200
M /III iIJi/rii/1 t'
l'nJ('('SS
(,ill'ilhi fil.l' iI/ii jet's
Bibliography
If all the AjS are lar'ger in the same proportion, one would obtain proportionutely smaller values of c;.
Peam, W.L., Katz, S. and Johnson, N.L. (1992) Distributional and inferential properties of process capability indices, old and new, J. Qttal. Technol. 24, 215-31.
BIBLIOGRAPHY
Taam, W., Subbaiah, P. and Liddy, J.W. (1991)A note on multivariate capability indices. Working paper, Dept. Mathematical Sciences, Oakland University, Rochester MI.
II Alt, F.B. (1985) Multivariate qliality control. In Ellcyclopedia (!I' Statistical Sciences, (S. Kotz, N.L. Johnson and.e,B. Read, eds.) 6, Wiley: New York, 110-22. . Anderson, T.W. (1984) An Introduction to Multivariate Statistical Analysis (2nd edn) Wiley, New York. . Chan, L.K. (1992), Cheng, S.W. and Spiring, FA (1990) A .multivariate measure of process capability, J. Modeling Simul., 11,1--6. [Abridged version in Advances in Reliability and Quality Control (1988) (M. Hamza, ed.) ACTA Press, Anaheim, CA; 195-9]. Chen, H.F. A multivariate process index over a rectangular paralellopiped, Tech. Rep. Math. Statist. Dept, Bowling Green State
University,BowlingGreen, Ohio.
I
Wierda, S.l. (1992a) Multivariate quality control: estimation of the percentage good products. Research memo, No. 482, Institute of Economic Netherlands. Research, Faculty of Economics, University of Groningen,
.~
r
'
.
.
Hotelling, H. (1947)in Techniques of Statistical Analysis (e. Eisenhart, H. Hastay and W.A. Wallis, eds.) McGraw-Hill: New York, 111-84. Hsiang, T.e. and Taguchi, G. (1985) A Tutorial on Quality Control and Assurance - The Tayuchi Me/hods, Amer. Statist. Assoc. Meeting Las Vegas, Nevada (188pp.). 'Hubele, N.F., Shahriari, H. and Cheng, e.-S. (1991) A bivariate process capability vector, in Statistical Process Control in Manu, facturing (lB. Keats and D.e. Montgomery, eds.) M. Dekker: New York, 299-310. . Johnson, N.L. and Kotz, S. (1968) Tables of the distribution of quadratic forms in central normal variables, Sankhyii Ser. B, 30, 303-314. Johnson, T. (1992) The relationship of Cpmto squared error loss, J. Qual. Technol., 24, 211-15. Kocherlakota, S. and Kocherlakota, K. (1991) Process capability indices: Bivariate normal distribution, Commull. Statist. - Theor. Meth., 20, 2529-47. Littig, J.L., Lam, C.T. and Pollock, S.M. (1992) Process capability . measurements for a bivariate characteristic over an elliptical tolerance zone, Tech. Rep. 92-42, Dcpt. Industrial and Operations Engineering, University of Michigan, Ann Arbor.
201
I I
I
I I
r '/ , I
Wierda; S.l (1992b) A multivariate process capability index, P"oc. 9th Int(!l'Ilat.Conf. Israel. Soc. Qual. Assur. Jerusalem, Israel.
Note on computer progralns
I
The following SAS programs, written by Dr Robert N. Rodriguez of SAS Institute Inc., SAS Campus Drive, Cary NC 27513-8000, USA, will be available in the SAS Sample Library provided with SAS/QC software (Release 6.08, expected in early 1993). A directory of these programs is given in the Sample Library program named CAPDIST.
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1
Chapter 2 Expected value and standard deviation of Cp Surface plot for bias of Cp . ' Surface plot for mean square error of Cp Expected value and standard deviation of Cpk Surfm:e plot for bias of Cpk Surface plot for mean square error of Cl'k Plot of probahility dcnsity function of Cp
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Plot of probability density function of Cpk I
.Confidence
intervals for Cp Approximate lower confidence bound for Cpk using the approach of Chou et al. (1990)(reproducestheir Table 5) Approximate lower confidence bound for Cpk using the approach cf Chou et al. (1990) and the numerical method of Guirguis and Rodngucz (1992) (generalizes Table 5 of Chou
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et al.) .
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Approximate confidence intervals for Cpk using the methods' of Bissell (1990) and Zhang et al. (1990) 203
204
,
Note 011computer programs
Chapter 3 Expected value and standard deviation of Cpm Expected value and standard deviation of Cpmk Surface plot for bias of Cpm Surface plot for mean square errr of Cpm Approximate confidence limits for Cpmusing the method of Boyles (1991) Chapter 4 Expected value and standard deviation of CjkP/Cjlil~ Surface plot for bias of CjkP Surface plot for mean square error of (\.\ ,
Postscript
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Our journey down the bumpy road of process capability indices has come to an end, for the moment. We hope that those who have patiently studied this b90k, or even those who have browsed through it in less detail, have been motivated to seek for answers to the following questions. I. Are the resistance and objections to the use of PCls in 'practice justified? 2. is there truth in the statement that 'a manager would rather use a wrong tool which (s)he understands than a correct one which (s)he does no1'? 3. Are PCls too advanced concepts for average technical workers on the shop floor to comprehend? Let us state immediately that our answer to the last question should be an emphatic 'No'. In our opinion it is definitely not too much to expect from workers a basic understanding of elementary statistical and, prohahilistic concepts such '~s me,an and standard deviation (and target value and variab,ility in general). We also sincerely hope that the answer to, the second question is an equally emphatic 'No'. In spite of rather disturbing catch phrases and slogans, such as 'KISS (keep it simple, statisticians)' and even 'Statisticians keep out' (ascribed to Taguchi advocates at a meeting in London), we strongly believe that a great majority of managers and engineers are sufficiently enlightened and motivated to explore new avenues, and are not lazy, or afraid o~ the 'unknown'. Deming's philosophy and a shrinking world (from development of 'instant' communications) are factors
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206
Postscript
Postscript
contributing to an atmosphere favourable to development of innovative techniques C!n the shop floor, and improvements in educational methods may be expected to produce many more open-minded individuals. Having explained our reactions - we hope, satisfactorily to the second and third questions we now attempt to reply to
the first, and most difficult,one. We rephrase the question -
.
should the use of PCls be encouraged? As we indicated in the introduction to Chapter 1, PCls can be used with advantage provided we understand their limitations - in particular, that a PCI is only a single value and should not usually be expected to provide an adequate on its own, and PCls are subject to sampling I measure .
, variation,
as are other statistics. In the case of sample mean
(X), for example, the sampling distribution of X is usually approximately normal even for moderate sample sizes (n> 6, say), even if the process distribution is not normal. We therefore use a normal distribution for X with some confidence. On t4e other hand none of the sampling distributions of PCls in this book is normal, even when the process . distribution is perfectly normal. , Tables of the kind included in this book provide adequate
~ackground . for interpreting estimates of PCls when the
process distribution is normal. In our opinion, objections, ioic~d by some opponents of PCls, that are based on noting that the distribution of estimates of PCls when the process distribution is non-normal is unknown and may hide unpleasant surprises are somewhat exaggerated. The results of studies, like those of Price and Price (1992) and Franklin and Wasserman (1992) summarized in Chapter 4 provide some reassurance, at the same time indicating where real danger maylurk.In particularthe effectsof kurtosis(fJ2< 3, or more
importantly {J2» 3) are notable.
.
We appreciate greatly the efforts of those involved in these enquiries, although we have to remark - with all due respectthat their methods (of simulation and bootstrapping) do not, ,
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207
as yet; provide a sufficiently clearcut picture for theoreticians to guess the exact form of the sampling distribution, on the lines of W.S. Gosset ('Student')'s classical work on the distribution of XIS in 1908 wherein hand calculation coupled with perceptive intuition led to a correct conjecture, deriving the t-distribution.
Index
Approximations 10, 12, 46, 109 Asymmetry 113 Asymptotic distributions 170 Attributes 78
1
Bayesian methods 112, 151 Beta distributions 22, 118, '121, 128, 148, 172 functions 22 incomplete 23 ratio, incomplete 23, 80, 129 Bias 33, 57, 98 Binomial distributions 15, 114, 172 Bivariate normal distributions 179 .
Blackwell-Rao theorem 33, . 80, 133 Bootstrap 161
Capability indices, process, see Process capability indices Chi-squared distributions 18,45,75, 137, 172, 181
nO~1central20, 99, 111, 119, 141, 198 Clements' method 156 Coefficient of variation 7 Computer program 203 Confidence interval 45, 70, 107
for Cp 45, 47 . for Cpk 69 for Cpm 107 for Cpm 187 Contaminated distribution 136, 140 Correlated observations 76, 106, 179 Correlation 7, 29, 78, 82, 106, 181 Cumulative distribution function 4 Delta method, see Statistical differential method Density function, see Probability density function Distribution function cumulative 4 Oistribution-free PCls 160
210
Index
Edgeworth series 140, 146 Ellipsoid of concentration 18,4 Estimation 43, 98 Estimator 32 maximum likelihood 32 moment 32 unbiased 79 unbiased minimum variance (MVUE) 79, 127 Expected proportion NC 40, 53, 92, 183 . value 6
Exponential distribution 19, 138, 149 Folded distributions 26 beta 27, 174 norl1lul26, 56, 17] Gamma distributions 18, 149 function 18 incomplete 19 ratio 19 incomplete 20 Gram-Charlier series, see Edgeworth series History 1, 54, 89 Hotelling's T2 195 Hypothesis .35 Independence 7 Interpercentile distance 164 Johnson-Kotz-Pearn method 156
Index
Kurtosis 8
Parallelepiped 178, 194 Pearson- Tukey (stabilizing) results 157 Percentile (percentage) point . 18, 190, 199
Likelihood 32 maximum 32 ratio test 35, 75 Loss function 91, 181, 192
Performance indices, process, see Process performance indices
Maximum likelihood 32 Mean deviation 47 Mean square error 92, 101 Mid-point 39 Mixture distributions 27, 99, 121, 169 Moments 7,59, 102, 122, 168 Multinomial distributions 16, . 141 Multinormal distributions 29, lXI, 19X Noncentral chi-squared distributions, see Chisquared distributions,. noncentral t-distributions, see [distributions, noncentral Nonconforming (NC) product 38 Non-normality 135 Normal distributions 16, 53, 161 bivariate, see Bivariate normal distributions folded, see Folded distributions, normal Odds ratio 78
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211
MCp 193 MCP 188 comparison of 74, 95 tlexible 164 . for attributes 78 for multivariate data 177 vector-valued 194 Process performance indices (PPls) Pp 41 Ppk 55
Poisson distributions 15, 21, 99, 107 Probability density function 5 Process capahility indices Quadratic form 30 (PC Is) Cjkp 165 Ratio 47 Cp 38 Robustness 151 C p 186 Cpg 90 Semivariance 116, 153 C pk 51 Shape factor 8, 67, 151 Cpk 67 kurtosis 8 Cpk 66 skewness 8, 149 Cpkl 55 Significance test 35 Cpku 55 Simulation 75, 138 C pk(8)157 Skewness 8, 149 Cpm 90, 187 Specification limit 38 <;in 112 lower (LSL) 38 Cpm 187 upper (USL) 38 Cpmk 117 region 178 Cpm(a) 121 Stabilization 157 Cpp 40 Standard deviation 7 TCpkl 55 Statistical differential methOd 'TCpku 55 10, 107 vC p 187 Sufficient statistic 32 vC R 192 Systems of di~tributions 30 CLjkp 164 Edgeworth 31 Gram-Charlier 31 CUjkp 164 Le 90 Pearson 31, 156 '
212
Index
t-distributions
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Variance 7 Vector-valued PCls 194
24; 137
noncentral 25, 63 Total variance (Otolal) 41 Triangular distribution 138
Worth 9.4
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Uniform distribution 22, 136, 149
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