R
Methods for experimental design
DATA HANDLING IN SCIENCE AND TECHNOLOGY Advisory Editors: B.G.M. Vandeginste and S.C. Rutan
Other volumes in this series: Microprocessor Programming and Applications for Scientists and Engineers by R.R. Srnardzewski Volume 2 Chemometrics: A Textbook by D.L. Massart, B.G.M. Vandeginste, S.N. Deming, Y. Michotte and L. Kaufman Volume 3 Experimental Design: A Chemometric Approach by S.N. Derning and S.L. Morgan Volume 4 Advanced Scientific Computing in BASIC with Applications in Chemistry, Biology and Pharmacology by P. Valko and S. Vajda Volume 5 PCs for Chemists, edited by J. Zupan Volume 6 Scientific Computing and Automation (Europe) 1990, Proceedings of the Scientific Computing and Automation (Europe) Conference, 12-15 June, 1990, Maastricht, The Netherlands. edited by E.J. Karjalainen Volume 7 Receptor Modeling for Air Quality Management, edited by P.K. Hopke Volume 8 Design and Optimization in Organic Synthesis by R. Carlson Volume 9 Multivariate Pattern Recognition in Chemometrics, illustrated by case studies, edited by R.G. Brereton Volume 10 Sampling of Heterogeneous and Dynamic Material Systems: theories of heterogeneity, sampling and homogenizing by P.M. Gy Volume 11 Experimental Design: A Chemornetric Approach (Second, Revised and Expanded Edition) by S.N. Derning and S.L. Morgan Volume 12 Methods for Experimental Design: principles and applications for physicists and chemists by J.L. Goupy
Volume 1
DATA HANDLING IN SCIENCE AND TECHNOLOGY -VOLUME
12
Advisory Editors: B.G.M. Vandeginste and S.C. Rutan
Methods for experimental design principles and applications for physicists and chemists
JACQUES L. GOUPY 7, Rue Mignet, 75016 Paris, France
ELSEVIER Amsterdam - London - N e w York -Tokyo
1993
ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 211,1000 AE Amsterdam, The Netherlands
Translation and revised edition of: La Methode des Plans d’Experiences. Optimisation du Choix des Essais et de I’lnterpretation des Resultats 0 Bordas, 1988 0 Dunod for updatings Translated by: C.O. Parkes
ISBN
0-444-89529-9
0 1993 Elsevier Science Publishers B.V. All rights reserved.
No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V., Copyright & Permissions Department, P.O. Box 521,1000 A M Amsterdam, The Netherlands. Special regulations for readers in the USA - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the USA. All other copyright questions, including photocopying outside of the USA, should be referred to the publisher. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper. Printed in The Netherlands
To my wife Nicole
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PREFACE This book is devoted to researchers who, because of limited time and resources, must use a minimal number of experiments to solve their problems. It was written with the aim of avoiding theoretical statistics or mathematics. It is not intended to replace the texts on analysis of variance, regression analysis or more advanced statistical treatments. It was written for experimenters by an experimenter. It is an introduction to the philosophy of scientific investigation. This book has grown out of my consulting practice and a series of short courses given to industrial researchers. These experiences taught me that a good method and solid concepts are more useful than complex theoretical knowledge. Therefore, I have attempted to preserve the balance between the practice necessary to carry out a study and the theory needed to understand it. I have tried to write a book that is usehl and clear. While mystudentsmay have learned something from me, I have certainly learned from them. As a result this new English edition contains considerable additional material not included in the original French book. The presentation makes extensive use of examples and the approach and methods are graphical rather than numerical. All the calculations can be performed on a personal computer. Conclusions are easily drawn from a well designed experiment, even when rather elementary methods of analysis are employed. Conversely even the most sophisticated statistical analysis cannot salvage a badly designed experiment. Readers are assumed to have no previous knowledge of the subject. The presentation is such that the beginner may acquire a thorough understanding of the basic concepts. There is also sufficient material to challenge the advanced student. The book is therefore suitable for an introductory or an advanced course. The many examples can also be used for self-tuition or as a reference.
ACKNOWLEDGEMENTS I am gratehl to the many researchers whose work provided the examples cited in this book and who have asked me so many questions on how to use experimental designs efficiently. I am also grateful to Owen Parkes who translated this book and was a continual source of advice. I wish to thank the staff at Dunod, particularly Maryvonne Vitry and Jean-Luc Sensi, and the staff at Elsevier, for their encouragement and support. A special thanks to my wife who sustained me with love and made this work possible.
Paris February 1993 Jacques GOUPY
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ix
CONTENTS Preface
vii
Acknowledgements
vii
Chapter 1
Research strategy : Definition and objectives
1. Introduction 2. The process of knowledge acquisition 2.1. Gradual acquisition of results, 4 2.2. Selection of the best experimental strategy, 4 2.3. Interpretation of results, 4 3. Studying a phenomenon 3.1. The classical method, 5 3.2. Experimental design methodology, 6 4. Historical background Chapter 2
Two-level complete factorial designs:2*
1. Introduction 2. Two-factor complete designs: 22 2.1. Example: The yield of a chemical reaction, 10 3. General formula of effects 4. Reduced centred variables 5 . Graphical representation of mean and effects 6. The concept of interaction 6.1. Example: The yield of a catalysed chemical reaction, 21 7. General formula for interaction Chapter 3
Two-level complete factorial designs: 2k
1. Introduction 2. Complete three factor design: 23 2.1. Example: The stability of a bitumen emulsion, 29 3. The Box notation 4. Reconstructing two 22 designs from a Z3 design 5 . The relationship between matrix and graphical representations of experimental design 6. Construction of complete factorial designs 7. Labelling of trials in complete factorial designs 8. Complete five factor designs: 25
1
1
2
4
7 9
9 10 15 16 19 21 24 29 29 29 34 35 36 37 38 38
X
8.1, Example: Penicillium chrysogenum growth medium, 38 9. Complete designs with k factors: 2k 10. The effects matrix and mathematical matrix 10.1. Matrix transposition, 45 10.2. Matrix multiplication, 45 10.3. Inverse of X, 46 10.4. Calculation of X'X, 47 10.5. Measurement units, 47
Chapter 4
Estimating error and significant effects
43 44
49
1. Introduction 49 50 2. Definition and calculation of errors 2.1. Arithmetic mean, 5 1 2.2. Dispersion, 51 53 3. Origin of the total error 56 4. Estimating the random error of an effect 4.1. The investigator knows the experimental error of the response, 56 4.2 The experimental error of the response is unknown, 59 Several measures on the same experimental point, 59 Repeat the whole experimental design, 60 4.3. The experimental error of the response is unknown, and the experimenter does not want to perform any supplementary experiments, 62 5. Presentation of results 63 5.1. Numerical results, 63 5.2. Illustration of results, 63
Chapter 5
The concept of optimal design
1. Introduction 2. Weighing and experimental design 2.1. Standard method, 70 2.2. Hotelling method, 70 2.3. Strategy for weighing four objects, 71 3. Optimality criteria 3.1. Unit matrix criterion, 76 3.2. Maximum determinant criterion, 77 3.3. Minimum trace criterion, 79 3.4. "The largest must be as small as possible" criterion, 80 4. Positioning experimental points 4.1. Positioning experimental points for one factor, 8 1 5. Measurement of an electrical resistance Example: Measuring an electrical resistance, 83 6. Positioning experimental points for two factors 7. Positioning the experimental points for k factors
67
67 69
76
80 83 85 89
xi
Chapter 6
Two-level fractional factorial designs: 2k-P The Alias theory.
1. Introduction 2. First fractional design: Z3-' 2.1 Example: Bitumen emulsion stability (continued from Chapter 3), 3. Interpretation of fractional designs 4. Calculation of contrasts 5. Algebra of columns of signs Alias generators, 99 6. Construction of fractional designs (one extra factor) 7. Notation of fractional designs 8. Construction of fractional designs (two extra factors) 9. Construction of fractional designs (p extra factors) 10. Practical rules 10.1 Going from AGS to contrasts, 110 10.2 Going from contrasts to the AGS, 110 11. Choosing the basic design 1 1.1. Total number of factors to be studied, 1 11 1 1.2. Number of trials to be performed, 112
Chapter 7
Two-level fractional factorial designs: 2k-P Examples
91
91 92 92 94 95 98 100 104 104 108
110 111
115
1. Introduction
115
2. 2*-' fractional design 2.1. Example: Minimizing the colour of a product, 116 2.2. Techniques for dealiasing main effects from interactions, 119 2.3. Construction of the complementary design, 121 2.4. Contrast calculation, 122 2.5. Interpretation, 123
116
3. 274 fractional designs 3.1. Example: Settings of a spectrofluorimeter, 127 3.2. Calculation of contrasts, 130 3.3. Interpretation of the initial design, 132 3.4. Construction of the complementary design, 134 3.5. Interpretation of the initial and complementary designs, 140 4. Studying more than seven factors 5. The concept of resolution 5.1. Definition of resolution, 142
127
5.2. An example of a 2:
design: Plastic drum fabrication, 144
142 142
xii
Chapter 8
Types of matrices
1. Introduction 2. The experimental matrix 3. The effects matrix 4. The basic design matrix for constructing fractional designs Chapter 9
Trial sequences: Randomization and anti-drift designs
151
151 151
153 154 159
1. Introduction 1.1. Drift errors, 161 1.2. Block errors, 161 2. Small uncontrollable systematic variations 3. Systematic variations: Linear drift 4. #en should trials be randomized? Example: The powder mill, 166 4.1. Powder mill: First investigator's strategy, 167 4.2. Powder mill: Second investigator's strategy, 168 4.3. Powder mill: Third investigator's strategy, 171 5 . Randomization and drift
159
Chapter 10
179
Trial sequences: Blocking
162 162 166
175
1. Introduction 179 2. Block variations 180 3. Blocking 180 Example: Preparation of a mixture, 180 185 4. Blocking on one variable Example: Penicillium chrysogenum growth medium (continued), 185 190 5 , Blocking on two variables 5.1. Example: Yates' bean experiment, 190 5.2. Interpretation of experimental results, 194 20 1 6.Blocking of a complete design
Chapter 11
Mathematical modelling of factorial 2k designs
1. Introduction 2. Mathematical modelling of factorial designs 3. Formation of the effects matrix 4. Evaluation of responses throughout the experimental domain 4.1. Example: Study of paste hardening, 210 4.2. Interpretation, 21 1 5. Test of the model adopted 6. Selection of a research direction 6.1. Mathematical model, 21 5 6.2. Isoresponse curves, 216
203
203 204 208 209 213 213
...
Xlll
6.3. Steepest ascent vector, 2 17 7. Choice of complementary trials 8. Analysis of variance and factorial designs Example: Sugar production, 220 8.1. Analysis of the problem by factorial design (one response per trial), 220 8.2 Analysis of the problem by analysis of variance (one response per trial), 223 8.3 Analysis of the problem by factorial design (two responses per trial), 226 8.4 Analysis of the problem by analysis of variance (two responses per trial), 227 9. Introduction to residual analysis 10. Error distribution
230 234
Chapter 12
239
Choosing complementary trials
220 220
1. Introduction 2. A single extra trial Example: Clouding of a solution, 240 3. Two extra trials Example: Clouding of a solution ( block effect), 244 4. Three extra trials 5. Four extra trials 5.1, Reconstruction of the experimental design, 252 5.2. Presentation of results, 252
239 240
Chapter 13
257
Beyond influencing factors
1. Introduction 1.1. IdentifLing the domain of interest, 257 1.2. Looking for an optimum, 258 1.3. Finding the minimum response sensitivity to external factors, 258 2. Identifying the domain of interest 2.1. Example: Two-layer photolithography,258 2.2. Examination of the results for response Lz, 263 2.3. Examination of the results for response L,, 265 3. Finding an Optimum 3.1. Example: Cutting oil stability, 268 3.2. Interpretation, 270 4. Finding a stable response 4.1. Example: thickness of epitaxial deposits, 271 4.2. Interpretation, 275
244 246 250
257
258
268 27 1
xiv
Chapter 14
Practical method of calculation using a quality example
1. Introduction 2. A quality improvement example Example: Study of truck suspension springs, 284 3. Interpretation, step 1 3.1. Calculation of responses, 287 3.2. Analysis of results (interpretation, step 1) , 289 4. What is a good response for dispersion? 4.1. Variance, 292 4.2. Logarithm of variance, 293 4.3. Comparison of variance and logarithm of variance, 293 4.4. The signal-to-noise ratio, 295 5. Interpretation, step 2 5.1. Calculation of responses, 295 5.2. Analysis of results (second step of interpretation), 296 6. Optimization
283 283 284 287 292
295
302
Chapter 14 (continued) Detailed calculations for the truck suspension springs example
309
1. Calculation for the first interpretation 2. Calculation for the second interpretation 3. Calculation for optimization
309 316 326
Chapter 15
Experimental designs and computer simulations
333
1. Introduction 2. Example 1, Propane remover optimizing 2.1. The problem, 335 2.2. Simulation, 337 2.3. Interpretation, 337 3. Example 2: Optimization of a hydroelastic motor suspension 3.1. The problem, 340 3.2. Calculations, 343 3.3. Interpretation, 345 3.4. Conclusion, 346
333 335
4. Example 3: Natural gas plant optimization 4.1, The gas production system and the problem to be solved, 347 4.2. Choice of responses, 349 4.3. Choice of calculation design, 351 4.4. Calculations, 352 4.5. Interpretation, 353 4.6. Optimization, 361 4.7. Conclusion, 363
347
339
xv
Chapter 16
Practical experimental designs
365
1. Introduction 2. Calculation of effects and interactions when an experimental point is misplaced 3. Calculation of effects and interactions when all the experimental points are misplaced 4. Error transmission 5. Experimental quality
365
Chapter 17
391
Overview and suggestions
1 . Introduction 2. Selection of the best experimental strategy 2.1. Defining the problem, 392 2.2. Preliminary questions, 394 2.3. Choice of design, 397 3. Running the experiment 4. Interpretation of results 4.1. Critical examination of the results, 398 4.2. Follow up, 400 5. Gradual acquisition of knowledge 6. What experimentology will not do
Appendix 1
Matrices and matrix calculations
1. Introduction 2. Definitions 2.1. General, 403 2.2. Definitions for square matrices, 405 3. Matrix operations 3.1. Operation on array, 407 3.2. Operations between arrays, 408 3.3. Calculation of an inverse matrix, 41 I 4. Matrix algebra 5 . Special matrices
Appendix 2
Statistics useful in experimental designs
1 . Normal distribution Population, 4 18 Sample, 418 Variance, 419 2. Variance Theorem One random variable, 419 Error of the mean, 420
366 372 377 388
391 392
398 398 40 1 402
403 403 403 407
413 414
417 417
419
xvi
Appendix 3 Order of trials that leaves the effects of the main factors uninfluenced by linear drift. Application to a Z3 design
421
Bibliography Author index Example index Subject index
43 1 440 443 447
CHAPTER I
RESEARCH
STRATEGY:
DEFINITION AND OBJECTIVES
1.
INTRODUCTION
Experimental scientists and technicians employed in laboratories, industry, medicine or agriculture throughout the world run experiments. The classical experimental approach is to study each experimental variable separately. This one-variable-at-a-time strategy is easy to handle and widely employed. But is it the most efficient way to approach an experimental problem? The first people to ask this question were English agronomists and statisticians working at the beginning of the century. Agronomy is somewhat different from most experimental sciences in that there are almost always a large number of variables and each experiment lasts a long time. As they could not run large numbers of trials, they worked to develop the best research strategy. They found that the classical method was not appropriate and developed a revolutionary approach which guaranteed experimenters an optimal research strategy.
2
Since then, many investigators have contributed with such topics as: optimal experimental designs, study of residuals, composite designs, Latin squares, fitting equations to data, multivariate calibration, empirical model building, response surface methodology, etc. All these techniques can be thought as components of a new discipline which, strangely, has no name. The name suggested for the field is Experimentics or Experimentology [ 11. But the scientific community has yet to decide. The factorials designs which are the subject of this book form just one part of Experimentics. This first chapter outlines the areas in which experimental designs can be applied, defines objectives and raises the general problem of how to study a phenomenon. The main points covered will be: I . The general process by which experimental knowledge is acquired. 2 . The three essential aspects of knowledge acquisition using the methodology of Experimental Designs:
Gradual acquisition of results. Selection of the best experimental strategy Interpretation of results. 3 . A comparison of classical approach and experimental design to study a phenomenon 4. A brief historical background.
2.
THE PROCESS OF KNOWLEDGE ACQUISITION
Any search for new information begins by the investigator asking a number of questions (Figure 1.1). For example, if we want to know the influence of a fertiliser on the wheat yield of a plot of land, we could ask several questions, such as : How much fertiliser is needed to increase the yield by 10% ? How does rainfall affect fertiliser efficiency ? Is the wheat quality influenced by the fertiliser ? These questions define the problem and determine the work to be carried out to solve it. It is therefore important to ask the right questions: those that can help us to resolve the problem. This is not quite as simple as it may appear. Before actually beginning any experiments, it is always wise to check that the information required does not already exist. The experimenter should first prepare an inventory of the available information, by compiling a bibliography, consulting experts, theoretical calculations, or any other method which provides himher with answers to the questions asked without actually carrying out any experiments. This preliminary survey may answer all the questions, resolving the problem. If it does not, some questions may remain to be answered, or they may be modified in the light of the information obtained. It will then be necessary to carry out experiments to obtain all the answers required. This preliminary study is a routine part of all experimental work and we shall not discuss it further. Our concern is not with this initial phase, but with those that follow. These are the steps in which the experimenter thinks about the experiments to be
3
performed, and our problem is how to select the experiments that must be done and those which need not be done. Is there a single ideal strategy? Such an ideal strategy should: give the desired results as quickly as possible. avoid carrying out unnecessary experiments. ensure that the results are as precise as possible. enable the experiments to progress without setbacks. provide a model and optimisation of the phenomena studied. There is such an ideal strategy, and it is effective because it simultaneously takes into account three essential aspects of knowledge acquisition.
0
gradual acquisition of results. selection of the best experimental strategy. interpretation of results.
SYSTEM TO STUDY
QUESTIONS Q1, Q2 ...Qn
INFORMATION INVENTORY
J OlCE OF AN EXPERlMENTAL STRATEGY I I GRADUAL ACQUISITION OF RESULTS
1 1
EXPERIMENTATION
INTERPRETATION OF THE RESULTS
5 KNOWLEDGE OF THE SYSTEM STUDIED
Figure 1.1 : The boxed steps define the areas of Experimentics.
4
The experiments should be organised to facilitate the application of the results. They should also be organised to allow the gradual acquisition of relevant results.
2.1. Gradual acquisition of results The experimenter clearly does not know the results when the study begins. It is therefore wise to work progressively and to be able to reorientate the study in the light of the early trial results. A preliminary rough outline can be done and then used to select any change in research direction that may better identi@ the most important points of the study and those avenues that should be abandoned to avoid any waste of time. This is why we recommend working progressively. An initial series of trials can provide provisional conclusions. A new series of trials can be done based on these provisional conclusions. The results of both these series should then be used to obtain a better picture of the results. Then, a third series of trials can be run if necessary. In this way the experimenter accumulates only those results that he requires, and the study stops when the original questions have been answered.
2.2. Selection of the best experimental strategy This strategy should facilitate the organisation of gradual acquisition of results. It should also minimise the number of trials, but it must not compromise the quality of the experimentation. On the contrary, it should ensure that the results are the most precise possible. Experimental designs, response surface methodology, and other approaches, such as steepest ascent and Simplex, are perfect for our requirements : 0 0 0
progressive acquisition of knowledge. only the required number of experiments the most precise results.
We will see that they provide the maximum of usefid information for the minimum number of experiments.
2.3. Interpretation of results The initial choice of experiments should facilitate interpretation of the results. Results should be readily interpreted and easily understood by both specialists in the field and those that are not. The methods recommended above can help us attain both these objectives. Microcomputers have made what used to be a long and tedious process of calculating results much more accessible. Not only are the calculations done quickly and accurately, but graphical outputs are a spectacular means of displaying results.
3.
STUDYING A PHENOMENON
The study of a phenomenon can be outlined as follows: the scientist may want to know, for example, the yield of wheat from a plot of land, the profit made on a chemical product or
5
the wear on a car motor component. This yield, price, or wear depends on many variables. The grain yield will vary with the nature of the soil, the amount and type of fertiliser, the exposure to the sun, the climate, the variety of wheat seed sown, etc. The profit from sale of a chemical may depend on the quality of the feedstock, industrial production yields, product specifications, plant conditions, etc. A similar set of variables will influence the wear of the car motor component. We can assess this as the response, y. This quantity is a finction of several independent variables, xi.which we shall call factors. It is possible to link mathematically the response y to the factors, Xi, as follows :
y =f(xl,
x2>
x37.,.>
xn..,.)
The study of a phenomenon thus requires measuring the response y for different sets of factor values. Let us, first, examine briefly the "classical" method of establishing the fhction.
3.1. The classical method The levels of all the variables except one are held constant. The response y is then measured as a hnction of several values of this unfixed variable xI.
A
B
C
D
E
X
Figure 1.2 :Only the levels of the variable x, are modified, the 8 other variables are held constant.
6
At the end ofthe experiment on this first variable, a curve is drawn of y = f (x,) (Figure 1.2). If the experimenter wishes to study all the variables, the whole experiment must be repeated for each one. Using this method, if he wanted to study just seven factors, with only five points per variable, he would have to carry out = 78,125 experiments or trials. This represents an enormous amount of work, and is clearly not feasible. The experimenter must therefore find a way of reducing the number of tnals. There are only two ways of doing this: reduce the number of experimental points per variables or reduce the number of variables. Reduce the number of experimental points If he elects to examine only three points per variable instead of five, he would have to carry out 37 = 2 187 trials. Two points per variable would require 27 = 128 trials. This is still a lot of work, and is often too much for either the budget or the time available. As there must be at least two experimental points per variable, the experimenter has no option but to:
Reduce the number of variables But even a system with four variables, testing each of them at three values, requires 34, or 81 trials. This way of working is both tedious and unsatisfactory. If some variables are ignored, people could be dubious about the results, and the investigator will be obliged to apologise for presenting incomplete conclusions. The inconvenience of this approach is particularly evident when safety or large sums of money are involved. This is precisely why we shall now proceed to examine the method of experimental design.
3.2. Experimental design methodology The essential difference between the classical one-variable-at-a-time method described above and the experimental design is that, in the latter, the values of all the factors are varied in each experiment. The way in which they are varied is programmed and rational. While this may appear somewhat disturbing at first sight, this approach of multiple simultaneous variable settings, far from causing difficulties, offers several advantages. Some of these are :
0
fewer trials. large number of factors studied. detection of interaction between factors detection of optima. best result precision. optimisation of results. model-building from the results.
Experimental designs can be used to study a great number of factors while keeping the total number of trials within reason. This is why one of its major applications is the search for influencing factors.
7
Instead of limiting the number of factors studied, the experimenter initially reduces the number of experimental points per factor. The term factor will be used rather than variable because it can include both continuous and discrete variables. The search for influencing factors consists of setting only two values for each factor, these values are called the levels. studying as many factors as possible, even those that may appear, at first sight, to have little influence. Many of the factors studied will probably have no influence, only a few will act upon the response. The results can then be used to choose new experimental points to define one or more specific aspects of the study. Thus, all the influencing factors will have been detected and studied, while keeping the number of trials to a minimum. Hence, the study can be completed without waste of either time or money.
4.
HISTORICAL BACKGROUND
Agronomists were the first scientists to confront the problem of organising their experiments to reduce the number of trials. Their studies invariably include a large number of parameters, such as soil composition, effect of fertilisers, sunlight, temperature, wind exposure, rainfall, species studied, etc., and each experiment tends to last a long time. At the beginning of the century Fisher [ 2 , 31 first proposed methods for organising trials so that a combination of factors could be studied at the same time. These were the Latin square, greco-Latin square, analysis of variance, etc. The ideas ofFisher were taken up by agronomists such as Yates and Cochran, and by statisticians such as Plackett and Burman [4], Hotelling [S], Youden [ 6 ] ,and Scheffe [7], and used to develop powerful methods. However, their studies were often highly theoretical, and involved difficult calculations. These difficulties, plus the revolutionary concepts developed by these pioneers, undoubtedly hindered the rapid spread of the new methods into the worlds of industry and universities. During the World War 11, major industrial companies realised that these techniques could greatly speed up and improve their research activities. Du Pont de Nemours adapted the techniques employed in agronomy to chemical problems, some years later ICI in England and TOTAL in France began using experimental designs in their laboratories. Other major companies, such as Union Carbide Chemicals, Proctor and Gamble, Kellogs, General Foods, have also adopted this approach. But the applications of these methods have never become generally known and for the most part have remained restricted to their original discipline of agronomy. They feature in few courses and despite the efforts of certain teachers, few students have learned them. The outstanding teachers in this field include Professors Box [8], Hunter and Draper [9], Benken in the USA, Phan Tan Luu in France, and Taguchi [ 101 in Japan. Thus, although they have been known and applied in certain areas for over half a century, the techniques are poorly understood and not generally used. The calculations are no longer a problem, thanks to the widespread availability of microcomputers. The challenge now is to overcome the reticence of users by clearly demonstrating the advantages afforded by
8
experimental design. The method described in the foilowing chapters, together with the examples which are given, will, it is hoped, make experimental designs accessible to researchers in both the industrial and academic worlds.
CHAPTER 2
TWO-LEVEL COMPLETE FACTORIAL DESIGNS: 22
1.
INTRODUCTION
Two-level factorial designs are the simplest, but are widely used because they can be applied to many situations as either complete or fractional designs. This chapter deals with complete designs. We will first examine a simplified example using only two factors. We will use it to introduce several important basic concepts which will be used in later chapters: experimental matrix, effect of a factor, iInteraction between factors, reduced centred variables, etc. This chapter also indicates how to calculate the effects of each factor and the interactions between factors. The reader will find usefkl tools to facilitate the interpretation of results and the presentation of conclusions.
10
2.
TWO-FACTOR COMPLETE DESIGNS: 22 The important concept of effect is best understood with the help of an example.
2.1. Example: The yield of a chemical reaction The Problem: The yield from this reaction depends on two factors: temperature and pressure. The chemist carrying out the study needs to know if the yield increases or decreases with increasing temperature. He also wants to $2 know the effect of pressure changes on the yield. The experirnental setup allows the reaction temperature to be varied from 60°C to 80°C,and the pressure from 1 to 2 bar The experimental domain (Figure 2.1) is I thus defined by the four points I '
~
A
1
60" C
lbar
{
80" C
60" C
lhar
12har
{
80"
c
2bar
The experimenter must adopt a specific research strategy in order to obtain the responses he requires. He could, €or example, fix the temperature at 70°C and carry out 3 experiments at 1, 1.5 and 2 bar as shown in Figure 2.2. The yield increases with pressure from 70°C to 75OC and 80°C. Pressure
2 bar
1 bar
60 OC
8o
Temperature
oc
Figure 2.1: Definition of the experimental domain
.
11
The effect of temperature can then be studied by keeping pressure constant at 1.5 bar and carrying out 3 experiments at 60"C,70°C and 80°C.The yield increases with temperature, 65%, 75% and 85%, indicating that both temperature and pressure must be increased to obtain the best yield. A final experiment at 80°C and 2 bar confirms these assumptions and the study is complete. But it has taken six experiments to obtain this result. Pressure
Figure 2.2: Example of a research strategy.
The experimenter could have selected another strategy by using other experimental points. These points could be evenly distributed throughout the experimental domain, or they could be selected randomly. But is there a best strategy? Clearly it is one that minimises the number of experiments without sacrificing precision, so that the same conclusions are reached. This best strategy exists; it consists of using the points A, B, C and D, the extremities of the experimental domain (Figure 2.3). This is the strategy adopted for two-factor experimental designs. While it provides the same results as the above experiment, it requires only four experiments. Let us now see how this approach can be used to provide a hller analysis of the results. We shall use the convention of -1 for the low level of each factor and +I for the high level. We can then place all the experimental information in a table, called the Experimental Matrix or Trial Matrix (Table 2.1). Each experiment is defined in this matrix. For example, in trial no 3, factor 1 (temperature) will be held at 60°C and factor 2 (pressure) at 2 bar. The trial is run under these conditions and the yield is measured. The other three trials shown in the matrix are carried out in a similar fashion and the results entered into a specific column in the experimental matrix and on the graph representing the experimental domain (Figure 2.4).
Pressure
D 2 bar
+I
1 bar
-1
B
A t -1
+I
Temperature
80 "C
60 "C
Figure 2.3: Location of experimental points t o obtain an optimal research strategy.
TABLE2.1 EXPERIMENTAL MA= THE YIELD OF A CHEMICAL REACTION
Pressure
-1 +1
4
1
I
Level (+)
8OoC
2 bar
13
Pressure
4
2bar
+I
1 bar
-1
-1
60 OC
+I
Temperature
80 OC
Figure 2.4: Results are entered in the experimental domain.
Results: Four trials are sufficient, and the experimenter can conclude that the greatest yield is obtained by working at 80°C and with a pressure of 2 bar.
This experiment introduces the important concept of the effect of a factor. When the temperature is increased from 60°C (level -1) to 80°C (level +1), the yield increases by 10 units (Figure 2.5), regardless of the pressure. Thus the overall effect of temperature on the yield is + 10 units. The main effect or effect of temperature is by definition, hay of this value, or +5 yield units.
14
pressure A
2 bar
+I
1 bar
-1 I Temperature
+I
-1 60 OC
80 OC
Figure 2.5: Main effect of temperature: 5 YOyield units.
When pressure is increased from 1 bar (level -1) to 2 bar (level +I), the yield increases by 20 units (Figure 2.6), regardless of the temperature. Thus the overall effect is +20 units and the main effect or effect of pressure is + I 0 yield units. Pressure
1 bar
4
' 70%
-1
I
-1 60 OC
+1 80 OC
Figure 2.6: Main effect of pressure: 10 YOyield units.
Temperature
3.
GENERAL FORMULA OF EFFECTS
The above results can be generalised by using literal values. We shall call y , the response of experiment 1, and y , the response of experiment 2, etc. The global effect of temperature is defined as the difference between the average of the responses at the high temperature and the average of the responses at the low temperature (Figure 2.7).
-1
-1
+I
Xl
Figure 2.7: y+ is the average response at the high temperature level. y - is the average response at the low temperature level. There are two responses at the high temperature level, y , and y, . The average response at the high temperature level, y+, is therefore given by:
The average response at the low temperature level, y-, is:
y-
= -[Yl 1
+Y,l
2
The effect ,?I averages,
of temperature is, by definition, half the difference between these two
16
or
Similarly, the effect Ep of pressure, is given by the expression: E =-[-yl 1 -Y, +y3 P 4
+Y~I
Inserting the numerical values of the responses ( Table 2. l), we get: 1 E - -[-60+
70-80+ 901 = 5%
‘-4
E,= -[-60-70+80+90] 1
=10 %
4 The average, I, of all the responses is given by
I = - [1 + 6 0 + 7 0 + 8 0 + 9 0 ] = 7 5 % 4
The formulae for calculating the values of the effects are easily remembered: the responses occur in the order of the trials and are preceded by the signs + or - that appear in the column of the corresponding factor in the experimental matrix. Thus the sequence of signs for the temperature column is:
-
+
-
+
This method is general and we shall use it to calculate the effects in all two-factor factorial designs, whatever the number of factors. We assume that, in the above calculations, all the phenomena studied vary linearly between the experimental points. This assumption is justified by its simplicity and by all the consequencesthat can be deduced fiom it. This is a very usefbl assumption at this stage, and we shall see that it is a first step towards more complex concepts later. So that we do not forget that there are both measured responses and calculated responses, we shall indicate measured responses as filled circles and calculated responses as open circles in all fbture diagrams or figures.
4.
REDUCED CENTRED VARIABLES
The logic behind assigning the value -1 to the low level and +I to the high level merits closer examination, as it leads to two major changes. The first is a change in the unit of measurement and the second is a change in origin.
17
. . *
80 "C
60 ' C
Normal Variables (Temperature)
1 - 1
20 "C
CRV
Centred Reduced Variables
+
-1
+1
Figure 2.8: Comparison of normal units and reduced units.
Change in units of measurement The temperature increases fkom a low level (-1) of 60°C to a high level (+1) of 80°C. There are thus 20 normal temperature units between the extremes of this experimental domain. But if we use -1 and +1 there are only two temperature units between the same two extremes. The new unit introduced by the notation -1 and +1 thus has a value of 1O"C, or ten normal temperature units. This is therefore a reduced variable, and the value of the new unit in terms of normal units is a step. In this case the temperature step is 10°C. Similarly, the pressure step in the above example is 0.5 bar. Change in origin The mid point of the [-1, +1] segment is zero, and this is the origin of the measurements in the new units. In normal units, the origin is not in the middle of the [-1, +1] segment; it lies outside the 60-80°C interval. The origin of the pressure values is similarly changed. The new variables are said to be centred.
0 "C Normal Variables 1 (Temperature)
60 "C
80 "C
1
Centred Reduced Variables
c--c--I -1
Figure 2.9: Normal origin and centred origin.
0
+1
18
The value of the new origin expressed in normal units can be obtained by taking the centre of the experimental domain, which is 70°C for the temperature and 1.5 bar for pressure. Then, if A- is the low level of a variable expressed in normal units, and A+ is the high level of a variable expressed in normal units. so that A,, is the midpoint ofthe [-1,+1] segment, or zero level of the variable expressed in normal units: A +A, A,=
..
2
For temperature, this gives A"=
60+80 ~
2
=70"C
and for pressure
A,=
~
1 +2 2
=
1.5bar
Thus, assigning the value -1 to the low level value of a factor and +1 to its high level value leads to a change of units, and a change of origin. These new variables are therefore named reduced and centred variables, or coded variables.
Normal Variables (Temperature) Centred Reduced Variables
Normal Variables (Pressure)
60°C
70%
I
I
-1
0
I
I
1 bar
1.5 bar
80°C
I +I
I 2 bar
Figure 2.10: All normal variables can be transformed into centred reduced variables.
19
The use of centred reduced variables greatly simplifies the presentation of the theories underlying two-level factorial designs. These centred reduced variables will be used in all subsequent discussions. Normal variables can be converted to centred reduced variables using the formula: x=-
A-A, step
where,
.
-
is the centred reduced variable measure in units of step, A is the variable in normal units (e. g., degrees Celsius or bar), A, is the value (in normal units) of the variable at the mid-point, i.e., the point chosen as the origin for the centred reduced variable.
x
In this case A, is 70°C for temperature and the step is 10°C Applying the formula to temperature, we get: x=-
A-70 10
Substituting A for the temperatures of 60°C , 70°C and 80°C gives the values of -1, 0, and +1 for x.
5.
GRAPHICAL REPRESENTATION OF MEAN AND EFFECTS The mean of all responses, which can be denoted as I or yo,is given by the expression: I = Y o = q1[ + Y , + Y , + Y , + Y 4 ]
or
or, using the low and high temperature means: I = y = -1[ y + f Y - ] 0
2
As the response is assumed to vary linearly, the point represented by y+ is at the centre of the segment [ y,, y , 1, i.e. at the zero pressure level (Figure 2.11). The same is true for y- , the centre of the segment [ y l ,y3 1. Using the same reasoning, the mean y o ofy+ and y - is at the centre of the segment [ y+ , y - 1. The mean of the responses is thus the value of the response at the centre of the experimental domain: level zero for temperature and level zero for pressure.
20
Y
Y
+
-1
0
+I
Xl
Figure 2.11: A 22 experimental design and t h e response surface.
If we now consider the plane passing through the zero pressure level and including the three responses y+, yo and y-, we observe that the straight line joining the responses y+ and y(Figure 2.11) represents the variation in the response on going from the low to the high temperature levels. It therefore illustrates the overall effect of temperature. The mean temperature effect, or more simply, the temperature effect, is half the overall effect. The effect of temperature is shown by the change in the response on going from the zero temperature level to the high temperature level. We will use this concept frequently in the coming pages and we will use diagrams like Figure 2.12 to represent factor effect.
21
RESPONSE
EFFECT OF FACTOR 1
I
0
-1
+1
FACTOR 1
Figure 2.12: Classical diagram to represent factor effect.
6.
THE CONCEPT OF INTERACTION The following example introduces the concept of intera d o n between factors.
6.1. Examp1e:The yield of a catalysed chemical reaction The problem:
iif The same chemist studied the same reaction under the same d conditions But this time a catalyst was added in order to improve the I yield The question now is how pressure and temperature should be f+
regulated. The experimental conditions and results are summarised in J the experimental matrix (Table 2 2). 9
In the previous example, the mean effect of a factor was defined at the zero level of the other factor. But we can also define the effect of a factor for any level of the other factor, for example the low or high level. We shall now do this to examine the concept of interaction. In Figure 2.12, the effect of pressure at the low temperature level is: E
~
P(t
1 =-[SO% 1 2
- 60%] =10 %
while the effect of pressure at the high temperature level is:
22
1
Ep(t+)= -[95% 2
- 7O%] =12.5 %
Thus, the effect of a factor is not the same at the high and low levels of the other factor. It is said that there is interaction between the two factors. The interaction between temperature and pressure is defined as half the difference between the effects of pressure at the high and low temperature levels: E
Pt
1 = -[12.5% -lo%] ~ 1 . 2% 5
2
TABLE2.2
EXPERIMENTAL MATiUX THE YIELD OF A CATALYSED CHEMICAL REACTION
Trial no
Temperature
Pressure
1 2 3 4
-1 +1 -1 +1
-1 -1 +1 +1
60% 70% 80% 95%
We can calculate the interaction for the temperature in the same way. At the low pressure level it is E
1
=-
t(P-)
2
[ 70% -6O%] =5 %
While at the high pressure level it is:
E
1
+
t(P
= -[95%
1
2
-SO%] =7.5 %
The value of the interaction is thus: E
tP
1 2
= -[7.5%
-5%] ~ 1 . 2% 5
23
which is the same as we calculated earlier. This result applies whether we refer to the pressurehemperature interaction or to the temperature/pressure interaction.
Pressure
+l
-1
60 OC
Temperature
80 OC
Figure 2.13: The effect of temperature is not the same a t the low and high pressure levels: there is interaction.
The main effect of temperature is defined and calculated as in the first example - the study of the yield of a chemical reaction, i.e., which is calculated with respect to the zero pressure level. The average response at the high temperature level is: 1 y + = --[95%+70%] =82.5 % 2
and the average response at the low temperature level is: y
-
1 2
= -[60%+80%]
=70 %
The effect of temperature, E, is thus: 1
E t = -[82.5% -7O%] 2
= 6.25%
24
andthe effect of pressure is:
E
1 2
= -[87.5%
P
-65%] = 11.25%
With these results the experimenter can conclude his study.
Results:
The four trials
indicate that the best yield is obtained with a
I temperature of 80°C and pressure of 2 bar. The catalyst has no effect at 60°C or 1 bar. Increasing the temperature alone does not reveal the effects of the catalyst. Neither does increasing pressure alone. Both @ temperature and pressure must be increased for the catalyst to 8 operate
7. GENERAL FORMULA FOR INTERACTION We can now develop the general formula for calculating interaction using the responses measured at the experimental points. The definition of effects remains the same, whether or not there is interaction. The formulae for the response mean, temperature and pressure effects are thus unchanged:
I = -[+y, 1 +Y, +Y3 + Y s ] 4
E
P
= -1[ - y 4 '
-
The interaction between temperature and pressure is indicated by Etp. At the high pressure level, the effect of temperature E (p+ ) is:
while at the low pressure level, the effect of temperatureE 1 Et@- )= TEY2 - Y1
I
t(P- )
is:
25
The interaction Etp is defined as half the difference of these two effects:
This can be simplified to:
This formula looks very like the one used to calculate the mean and effects. It can be obtained by constructing a list of the +1 and -1 having the same sequence as in the formula. This is easy as long as we note that the products of the pairs of elements in the temperature and pressure factor columns give a 12 column in which the signs are in the same order as those of the interaction (Figure 2.13). We can therefore construct an effects matrix (Table 2.3) fiom which we can obtain: the mean: using a column of four + signs. the effects of factors: using the sequence of signs in columns of the experimental design (experimental matrix). interaction between factors: each sign is calculated by applying the sign rule to the corresponding factors. e.g. in trial number 1, factor one is - and factor 2 is -; thus the interaction 12 has the sign (- ) x (- ) = (+) (Figure 2.14).
Factor 1 I
+
+
Factor 2 I
Interaction
- Multiplication
I
+ I
sign
+ +
sign
I
+
Figure 2.14: Calculating the interaction column using the sign rule.
26
The effects matrix therefore has four main columns: one for calculating each effect, one for the interaction, and one for the mean. Columns for the trial number and for the responses are normally included in the table. The divisor and the calculated results are placed beneath the mean, factors and interaction columns. The arrangement is shown in Table 2.3 TABLE2.3
EFFECT MATRIX THE YIELD OF A CATALYSED CHEMICAL REACTION
Interaction
Response 60%
70% 80% 95%
Effects
76.25
6.25
11.25
1.25
27
RECAPITULATION Our analysis of the yield of a chemical reaction has shown: The strategy used in a two-level experimental design. Using experimental points that are the extremies of the experimental domain for each factor gives the best estimate of the effect of each factor. The notion of effect and the calculation of effects. The tools used: -experimental matrix, -graphical representation of the experimental domain on which are placed the experimental results. -graphing the effects in a plane passing through the centre of the experimental domain. The definition of reduced centred variables. The example of the yield of a catalysed chemical reaction: Introduces the concept of interaction. Gives the general formula for calculating interaction Shows how to construct an effects matrix.
CHAPTER 3
TWO-LEVEL COMPLETE FACTORIAL DESIGNS: 2k
1.
INTRODUCTION
Two-level factorial designs are the simplest, but are widely used because they can be applied to many situation as either complete or fiactional designs. This chapter deals with complete designs. We will first examine a simplified example using only two factors. It will allow us to introduce several important basic concepts which will be used in later chapters. We will analyse a three-factor design and extrapolate the ideas acquired in this first example to an actual experimental design having five factors. Lastly, we will use the matrix approach to interpret two-factor complete factorial designs.
2.
COMPLETE THREE FACTOR DESIGN: 23
2.1.Example: The stability of a bitumen emulsion The Problem:
A manufacturer of bitumen emulsion wants to develop a new ak formulation. He has two bitumens, A and B. He wants to know the I
30
9
2
effects of a surfactant (fatty aad) and hydrochloric acid on the stability of the emulsion
As there are three factors, he decides to use a 23 design with the following factors and response Factors = Factor 1 high and low fatty acid concentrations.
..
Factor 2 diluted and concentrated HCI. Factor 3 bitumen A and B.
Response Emulsion stability index, measured in stability points .The scientist knows that the experimental error of the response is plus or minus two stability points. He wishes to find the most stable emulsion: the one with the lowest stability index. Domain The two levels of each factor are indicated by +1 and -1 as reduced centred (or coded ) variables. The experimental domain is a cube (Figure 3 . 1 ) and the eight experimental points chosen are at the corners ofthe cube.
8
7
6
4
Figure 3.1: Distribution of experimental points within the experimental domain.of a Z3 design.
31
The experimental matrix (Table 3.1) is constructed in the same way as for the 22 design, but contains eight and not four experiments. To simpli@ table 3.1 we have used the signs + and - without the figure 1. The factors studied are not necessarily continuous variables, and two level factorial designs may include both continuous and non-continuous or discrete variables.
Trial no
Factor 1 (fatty acid)
Factor 2 (HCl)
Factor 3 (Bitumen) -
1
-
-
2
+
-
-
3
-
-
4 5
+
+ +
-
-
6 7
+
-
8
+
+ +
+ + + +
Level (-)
low conc.
diluted
A
Level (+)
high conc.
concentrated
B
-
Response
~~
The effects of each factor and the interaction values are calculated fi-om the effects matrix (Table 3.2) as they were for the 23 design, i.e. by taking the experimental matrix signs for the main factors and using the sign rule for the interactions. The effects and interactions are obtained by a three-step calculation: The response is multiplied by the corresponding sign in the factor (or interaction) column, The products obtained are added, The sum so obtained is divided by a coefficient equal to the number of experiments For example, the effect of factor 3 is obtained fiom the formula: 0
1
E, = -[-388
37-26-24
+30+ 28+ 19+ 161
= -4
similarly, the third order interaction, 123, is obtained from: 1 E 123= -[-3 8
8+37+26-24+30-28-19+
161 = 0
32
r' +
6 7 8
+ + +
Effects 27.25
TABLE3.2
~1
EFFECTS MATRIX STABILITY OF A BITUMEN EMULSION
Inter. 23
Response
+ + +
-
+
-
+
-
+
+ +
-1
-6
+ + -4
-0.25
-0.25
0.25
0
The experimenter then analyses the results by drawing up a table of effects indicating, whenever possible, the experimental error estimated by the standard deviation (Table 3.3).
TABLE3.3 TABLE OF EFFECTS STABILITY OF A BITUMEN EMULSION
Mean
27.25
k 0.7 points k 0.7 points +_ 0.7 points 0.7 points
1
-1.00
2 3
-6.00 -4.00
12 13 23
-0.25
123
*
0.25
k 0.7points k 0.7 points k 0.7 points
0.00
k 0.7 points
-0.25
33
We can now begin to interpret these results. All the interactions are smaller than the standard deviation. These can therefore be considered to be zero and neglected. Factors 2 and 3 are much greater than the standard deviation, and thus have an influence, while factor 1 is just a little larger than one standard deviation and much smaller than two standard deviations. It is thus unlikely to have any influence STABILITY
A
33.25 27.25 21.25
-1 DILUTE
o
+ CONCENTRATED
HCI CONCENTRATION
Figure 3.2: Effect of hydrochloric acid (factor 2) on bitumen emulsion stability.
STABILITY
31.25
0
27.25 23.25
-1
+l
A
B
BITUMEN
Figure 3.3: Effect of bitumen type (factor 3) on bitumen emulsion stability.
34
Therefore, the concentration of fatty acid (factor 1 ) probably has no influence on emulsion stability over the range of concentrations tested. The plane passing through the centre of the experimental domain and parallel to factor 2 shows the effect of hydrochloric acid. The plane passing through the centre of the experimental domain and parallel to factor 3 reveals the effect of bitumen. We can now state the results of the experiment: Results:
*
The fatty acid concentration has little or no influence on the emulsion stability. The hydrochloric acid concentration has a large effect The ; I type of bitumen used is also important, the best stability (lowest Q i response) will be obtained with type B and dilute HCI There IS no xx s significant interaction. B
g
Note:
A negative effect is not necessarily an undesirable one. An effect is negative when the response falls as the factor increases from -1 to + I . Conversely, a positive effect occurs when the response increases as the corresponding factor goes from -I to + I
3. THE BOX NOTATION We could also use the Box notation [8] to indicate the effects and interactions. With this notation El is represented by a bold figure 1 (I), and E, = 2, E, = 3, etc. The mean is represented by the letter I The general formulae for the effects and interactions of a 2 3 design are’ Mean
=
1 1 = -[+Y, + Y , + y 7 +Y, + Y ~+ y 6 + y 7 +y81 8
35
4.
RECONSTRUCTINGTWO 22 DESIGNS FROM A 23 DESIGN
Examining the results in a little more detail, we see that, as factor 1 is without influence, the experimental domain is reduced to a design in which only factors 2 and 3 have any influence. This also indicates that the response does not depend on the level of factor 1, but only on the levels of factors 2 and 3 . The responses can therefore be rearranged in pairs ignoring the factor 1 level, as shown in the following table (Table 3.4). TABLE3.4 EXPERIMENTAL MATRIX REARRANGED STABILITY OF A BITUMEN EMULSION.
Trial no
7
8
Response
+
37 24 28 16
38 26 30 19
+
Average 37.5 25.0 29.0 17.5
These results can also be displayed graphically, as in Figure 3.4 29 CONCENTRATED +1
[fx
16 19]i7.5
HCI
A -1
Bitumen
B
+1
Figure 3.4: The bitumen emulsion is most stable when the hydrochloric acid is dilute and bitumen B is employed.
36
5. THE RELATIONSHIP BETWEEN MATRIX AND GRAPHICAL
REPRESENTATIONS OF EXPERIMENTAL DESIGN This relationship is easy to understand for a 22 experimental design. An experimental point A can be defined: 1 . by its coordinates in a Cartesian two dimensional space: a on the Ox, axis (horizontal) and b on the Ox, axis (vertical) as show in Figure 3.5. This is the graphical representation. The coordinates of a and b can be expressed in centred reduced (or coded) units or in classical units.
"T
/A
Figure 3.5: Geometric representation of experimental points 2. by the level of the two factors studied, trial A is defined by level a of factor x, and level b of factor x2. The coordinates of experimental points are the levels indicated in the experimental matrix X2
P
b
TRIAL NAME
a'
P
P'
7b'
-
Figure 3.6: The matrix diagram of experimental points is equivalent to the geometric representation
37
A set of experiments is defined by several points with geometrical representation and by several trials with matrix representation. Figure 3.6 illustrated these two ways of representating two experimental points and the two corresponding trials. While it is also possible to produce a graphical representation of a three factor experiment in a three dimension space it is clearly impossible to do so for four and more factors. It is therefore necessary to find a way of representing experimental points in these hyper-spaces which is both convenient and applicable to any number of dimensions. The most common solution is to use matrix representation, which works for any numbers of factors. Table 3.5 shows four trials defined by the level of seven factors. TABLE3.5
The geometrical counterpart of Table 3.5 is a set of four points defined by their seven coordinates. Hence the experimental matrix gives the location of experimental points in the experimental space, Anyone producing experimental designs must learn to think in ndimensional space without graphical representation. It is easy to pass from geometrical to matrix representation for two or three factors and experimenters must become accustomed to switching from n factor matrices to n dimensional space and vice versa.
6.
CONSTRUCTION OF COMPLETE FACTORIAL DESIGNS
All factorial designs are constructed in the same way as those shown in Tables 2.2, 2.4 and 3.7. The sequence of the signs for factor 1 is: -
+
-
+
-
+
-
+
,etc.
They alternate, commencing with a negative (-), The sequence of the signs for factor 2 is a series oftwo -, followed by two +: _ _
+ +
- -
+ + ,etc.
38
The sequence for factor 3 is four negatives (-), followed by four positives (+). Any hrther factors have 8, 16, 32, - signs followed by 8, 16, 32 + signs. There is always the same number of + and - signs in the column for each factor.
7.
LABELLING OF TRIALS IN COMPLETE FACTORIAL DESIGNS
When the + and - signs for each factor are laid out as shown above, the trials are numbered sequentially using whole numbers.(see Tables 2.2, 2.4 and 2.7). This is Standard numbering. As we will see later, the order of the trials can be changed, for randomisation, drift or blocking designs. But the number of each trial will be retained, regardless of its position in the layout. For example, trial number 23 of a complete 25 design (Table 3.7) always has the sequence of levels taken by factors 1 , 2 , 3 , 4 and 5:
-++-+ There are other ways of labelling trials, but we shall not discuss them here
8.
COMPLETE FIVE FACTOR DESIGNS: 25
8.1. Example:
Penicillium chrysogenum growth medium
The Problem: This design was used in a study to increase the yield of a penicillin production plant It was reported by Owen L Davies [Illin his book "The design and analysis of industrial experiments" Penicillium chrysogenum is grown in a complex medium, and the experimenter wanted to know the influence of five factors
8 *3. i
y.
f %-
F
-f
5
1 concentration of corn liquor 2 concentration of lactose
3 concentration of precursor 4 concentration of sodium nitrate 5 concentration of glucose
The response was the yield of penicillin, as weight (the units were not given in the original text). The experimental matrix of the 22 design summarizes the experimental data and the results of each of 32 trials.
39
TABLE 3.6
EXPERIMENTAL MATRIX PENlClLLlUM CHRYSOGENUM GROWTH MEDIUM
Trial no 1 2 3 4
Factor 1 (corn liq.)
Factor 2 (lactose)
Factor 3 :precursor)
-
-
-
-
-
-
+ +
-
-
+ -
+
5
-
-
6
+
-
7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 LevelLevel +
+
+ +
+ + + +
-
-
-
+
-
-
+ +
-
-
-
+ -
-
+
-
+
+ +
+ + + +
-
-
-
+
-
-
-
+
-
+
+
-
-
+ -
+ -
+ + +
1
2% 3%
-
-
-
-
+ + + + + + + + + + + + + + + +
-
+ + + +
+ +
+
Factor 5 (glucose)
-
-
-
2% 3%
-
-
-
I
-
+ +
-
I
+ +
+ + + +
Factor 4 (sod.nit.)
1 I
0 0.05%
[
0 0.3%
I I
0
0.5%
Response 142 114 129 109 185 162 200 172 148 108 146 95 200 164 215 118 106 106 88 98 113 88 166 79 101 114 140 72 130 83 145 110
40
The effects were calculated by the standard procedure and the results are shown in the table of effects (Table 3.7). TABLE 3.7
TABLE OF EFFECTS PENlClLLlUM CHRYSOGENUMGROWTH MEDIUM
Mean
129 6
1 2 3 4 5
-17 6 06 16 1 10 20 9
12 13 14 15 23 24 25 34 35 45
-5 9 -6 1 -5 0 26 44 -1 0 30 -1 0 -10 5 22
123 124 125 134 135 145 234 235 245 345
-I 3
1234 1235 1245 1345 2345
40 26 18 42 11
12345
63
-2 9 -1 6 17 -3 2 28 -2 6 27 23 06
41
Analysis of the effects of the factors showed that two factors have no influence: Factor 2, the concentration of lactose. Factor 4, the concentration of sodium nitrate And that the effects of three factors are significant:
0
Factor I, the concentration of corn liquor Factor 3, the precursor concentration. Factor 5, the glucose concentration.
A second order interaction appears to be significant: 0
0
Interaction 35, between precursor and glucose. interaction 12345 seemed to be abnormally large. We will leave this for the time being, but come back to it later.
TABLE3.8 EXPERIMENTAL MATRIX REARRANGED PENlClLLlUM CHRYSOGENUM GROWTH MEDIUM
Trial no 1 2 5 6 17 18 21 22
3 4 7 8 19 20 23 24
9 1 1 10 12 13 15 14 16 25 27 26 28 29 31 30 32
Factor Factor 1 3 -
-
+
-
+
+ +
-
-
-
+ -
-
+ +
+ -
Results
Average -
142 129 148 114 109 108 185 200 200 162 172 164 106 88 101 106 98 114 113 166 130 88 79 83
146 95 215 118 140 72 145 110
141.25 106.50 200.00 154.00 108.75 97.50 138.50 90.00
If we look at the three factors which do influence the growth of Penicillium chrysogenum, we see that there are 32 trials, but we know that only 8 trials are required to study three factors. We can therefore group together the trials having the same levels for factors 1 , 3, and 5, regardless of the levels of 2 and 4. For example, trials 1, 3, 9, and 1 1 were carried out at the low level of factors 1, 3 and 5, so that the results of four trials should be the same, allowing for experimental error. The 32 trials are used as if four 23 designs had been performed. Table 3.9 shows the rearrangement of trials and the mean responses for each group. Thus, it appears as if a three factor design was repeated four times.
42
TABLE3.9
TABLE OF EFFECTS PENICILLIUM CHRYSOGENUM GROWTH MEDIUM
Mean
129.6i6
1
-17.6i-6 16.1 k 6 -20.9+6
3 5
-6.1 + 6 2.6*6 35 - 1 0 . 5 f 6 13 15
135
-3.2+6
The experimental domain is reduced to a cube for the three influencing factors. We can therefore introduce the mean of each response at each corner of the cube to facilitate interpretation (Figure 3.7)
138
90
/+ 108
97
GLUCOSE
200
(5) 9 . 0 (
Figure 3.7: Diagram showing the results of the trials on Penicillium chrysogenum medium.
43
A high percentage of corn liquor (factor 1) evidently reduced the yield of penicillin. At a low level of factor 1, the yield was clearly improved by the addition of precursor and the absence of glucose. The presence of glucose reduced the effectiveness of the precursor. Results:
I
Under the experimental conditions used, the best yield of penicillin is obtained
b
0
Y
with a low (2%) concentration of corn liquor
I 0 with precursor.
Q
0
without glucose, which reduces the yield and inhibits the precursor
138
108
141
200
Precursor 0%
0.05 %
Figure 3.8: Influence of precursor and glucose at a corn liquor concentration of 2%. The interaction 12345 appears to be too great; and we will look at the reason for this in the chapter on blocking (Chapter 10).
9.
COMPLETE DESIGNS WITH k FACTORS: 2k
We have seen that 22 , 23 and 25 designs can be used to study two, three or five factors. A 2k design can be used when there are more factors, with k having any desired size. The experimental matrix and the effects matrix are constructed according to the same rules as were used previously. The calculation of the k major effects and the 2k-k-1 interactions are similarly performed. There is thus no theoretical limit to the number k of
44
factors that may be studied. But in practice the number of trials needed quickly becomes very large. A total of 27 (128) trials are required to study only 7 factors. This is a considerable number, and is rarely compatible with the facilities generally available in industry or university. This brings us to a most troublesome problem. We must find a way of reducing the number of trials without reducing the number of factors studied. We will examine this problem in Chapter 6.
10. THE EFFECTS MATRIX AND MATHEMATICAL MATRIX The effects can be calculated from the experimental results using an effects matrix, which, for a 22 experiment, looks like (Table 3.10).
Trial no
Mean
Factor 1
Factor 2
1 2 3
+I
-1 +I
-1 -1
-1
+I +I
4
+I +1 +1
+1
7 Interaction
This array of numbers can be used for a calculation; it is thus a mathematical tool, a Matrix. It can be written: +I
-1
-1
+1
+I +I +I
+1 -1 -1 +1
-1 -1
+I
+1
+1
A mathematical matrix is simply a table containing elements (here they are numbers) arranged in rows and columns. When the number of rows equals the number of columns the matrix is said to be square - otherwise it is rectangular. A matrix may contain just a single row and several columns (a linear matrix) or a single column and several rows ( a column matrix, or vector matrix). We will use matrices to express experimental results. Theyi may be shown in a rather special table because its contains only one column. They response vector matrix is:
Y=
Y2 Y3 Y4
An analogous matrix can be written for the effects:
45
E=
Before we use these matrices we will examine the operations which can be performed on a single matrix, or between matrices themselves. The operations we need are transposition for a single matrix, and matrix multiplication for two or more matrices.
+1
+I
+I
+1
x t = -1
+1 -1
-1 +1
+1
+1 -1
-1
+1
-1
+l
The second operation we will need is the multiplication oftwo (or more) matrices 10.2. Matrix multiplication Any reader not familiar with matrix calculations should read Appendix 1 before continuing with this Chapter. If we multiply matrix Xt by Y we get: +I
-1 Xt Y = -1
+I
+I +1 +I -1 -1 +1
+1
-1
+1
-1
+1
+1
y1 y1
y2 y4
The first element of the matrix-product is: [+Yl
+Yz
+Y3
+Y'll
or four times the mean of the responses. Similarly the second element is: [-Y1 +Y2
-Y3
+Y,I
46
or four times El, the effect of factor 1 . The calculations for the results of the third and fourth elements of the matrix-product are similar. We can therefore write: +I -1
+1
+l
+1
+I
-1
+1
-1
-1 -1
+I
+1
~3
E,
-1
+I
y,
El,
+I
y, y 2 =4
I El
which can be condensed to X'Y = 4E or
E = -1X t Y 4
This relationship for a 22 design can be extended to all two level complete factorial designs. When n is the number of trials we have 1 n
E = -X'Y We now have, in the form of a matrix, the technique we used to calculate the effects and interaction of 2k designs. The matrix form clearly shows that the experimental responses y j have been transformed by the matrix Xt so as to be more readily interpreted. A factor increases (or reduces) the mean of responses I by a quantity equal to its effect. In the first example we examined, the yield of a chemical reaction, the four responses 6O%, 70% 80% and 90% were difficult to interpret. But when they are transformed by the matrix Xt, the effect of each factor is obtained as if it were alone. A 10°C rise in temperature increases the yield from 75% to 8O%, while a pressure rise of 0.5 bar increases the yield fiom 75% to 85%. When the responses of a 2k are examined it is impossible to distinguish the influence of each factor. But the transformation by the Xt matrix displayed the useful information in the set of responses more clearly, revealing the effect of each factor as if it were alone. As the matrix X is the mathematical translation of the location of the experimental points, it is clearly most important that these points should be optimally placed in the experimental domain. Poorly positioned experimental points obscure the information instead of highlighting it. Well positioned experimental points clarifl the information (Chapter 16). The analysis of the specific X matrices which are use in all two level factorial designs can be developed a little. These are the Hadamard matrices, and they have quite remarkable properties. Let us first calculate the reciprocal of the X matrix and then examine the product of X and its transpose, XtX. 10.3. Inverse of X
The calculation of the inverse of a matrix is complicated for the general case, requiring a computer for high order matrices. But the calculation for X matrices of factorial designs (Hadamard matrices) is greatly simplified because of the following relationship:
47
The inverse of X can be obtained by transposing X and dividing all the elements of the Xt matrix by n, the number of trials. The relationship X'Y =nE then becomes X-'Y = E
or
Y=XE This formula can be used to calculate the responses from the effects
+1
+1 +I
-1 +1 --1 -1
-1 +1 +1 +1
+1 -1
-1
+1
+1
+1 -1 -1 +1 +1 -1 +1 - 1 + I
+1 -1 -1
+I
+1
+1 +1
4 -
0
0 4
0
0
1 0 0 0
0 0 0 1 0 0 = 4 4 0 0 0 1 0
0
0
0
0 0 4
0 0 0 1
which can be condensed to XtX =41 For this design, the product of the matrix of effects by its matrix transpose is 4 times the unit matrix. The general form of the formula for all two level complete factorial designs is X'X =nI where n is the number of trials. The matrix XtX is equal to n times the matrix unit in the case of two level factorial designs. It can be demonstrated that, in this case, the precision obtained for the effects is the best than might be hoped for (see Chapter 5). Experiments in which a two level complete factorial design is used are certain to provide calculated effects with maximum precision. 10.5. Measurement units
The responses yi were measured with a unit, metre, centimetre, volt, etc., or a less usual unit, such as an index, percentage or variance. The matrix Xt does not change the unit in which yi is measured, it simply transforms the trials results into a system that is easier to interpret. As a result, the mean, the main effects and the interactions are evaluated in the unit used to measure the responses.
48
RECAPITULATION. We have used the example of bitumen emulsion stability to: Extend the concepts acquired with 22 design to a z3 design. Extend the concept of interaction. Examine the rules for calculating effects and interactions from an effects matrix. Introduce Box notation. Present the results as a table of effects. Show that both continuous variables (temperature and pressure) and discrete variables (type of bitumen) can be studied simultaneously within the same experimental design.
Using a 25 design allowed us to: Apply the principles acquired to a real case. Use the fact that some factors were without influence to construct a replicate factorial design. The mathematical matrix representation of two-level factorial designs was used to: Calculate effects, interactions and mean from the responses. Introduce transposed matrices, product matrices and vector matrices. To simplify the interpretation of results which are transformed into mean, effects and interactions. Guarantee that the effects and interactions calculated have the highest possible precision. Define the units for measuring effects and interactions.
CHAPTER 4
ESTIMATING ERROR
AND SIGNIFICANT EFFECTS
1.
INTRODUCTION
Let us now examine a problem that we touched on lightly when we discussed the bitumen emulsion stability example in Chapter 3. The problem raises two questions, which we shall attempt to answer in this chapter: When can an effect be considered significant? On what criteria can such a conclusion be based? The method generally used to answer them requires estimating the error AE in the determination of the effect E, and comparing this error with the effect itself. There are three possible situations: The effect is much larger than the error: E>> AE In this situation, there is no problem, the effect clearly has an influence.
50
The effect is smaller than the error:
E<
This situation is not so clear, the effect may have no influence or it may have a small influence. Common sense, a knowledge of the phenomenon and statistical tests are required to reach an appropriated conclusion. If the effect has no major role in the study, and if a poor decision has little or no consequence, it is not worth spending time to find out. But if there are high financial stakes or dangers linked to the decision, statistical studies should be performed and complementary tests should be considered in order to evaluate the risks. We will not describe statistical tests here, the interested reader should consult appropriate texts [12, 13, 14 and 151. But we will examine the aspects that fall within the domain of the experimenter: the definition and origin of the measurement error and how to estimate the error of an effect.
2.
DEFINITION AND CALCULATION OF ERRORS
The total error of a measure may be considered to be the sum of two errors: random error and systematic error. For example, if an investigator repeats the same measure several times under the same conditions (same method, same instrumentation, same starting materials, etc.) he will not obtain exactly the same result each time: 78.8 80.4 81.4 79.8 80.2 80.2 78.0 79.7 82.1 80.4
Although the investigator has performed ten measures, he does not record all ten of them in his report. Instead, he summarizes the information as two numbers, one reflecting the most probable true value of the measure, the other estimating the dispersion of measures around this most probable true value. The arithmetic mean is used as the best estimation of the true value, while the square root ofthe variance is generally used to estimate the dispersion.
51
2.1. Arithmetic mean If n measures are obtained, and if the individual values are indicated by yi, the mean given by the relationship:
is
For the above example, the mean of the 10 results is calculated as: 1 7= -[78.8+80.4+81.4+79.8+80.2+80.2+78+79.7-t82 1+80.4] = 80.1 10
2.2. Dispersion The dispersion of measures is rather more difficult to define. We need to find a number to estimate dispersion. But the estimate may vary depending on the assumptions made and the definitions selected. We will adopt the most widely-used definitions, variance and standard deviation. These are defined in Appendix 2 for an inftnitely large population of measurements. An estimation of standard deviation is calculated for a smaller sample (fewer than about 50 measures) as follows. First, the difference between each measure yi and the mean y is calculated, these differences are squared, the squares summed and the sum divided by the number of measures, n, minus 1. This gives an estimation of the variance. The standard deviation (denoted by the letter s) is the square root of the variance.
The divisor n-1 is used because only n-1 independent measures were used to calculate the variance, as there is a relationship between the n initial measures: the definition of the mean. We will now look at the step-by-step calculation of standard deviation, using the ten measures: 1 . Calculation of deviations and their squares
Table 4.1 shows the calculation the deviation and their squares
52
TABLE4.1 ~
Results
Mean
Deviation
Deviation Square
78.8 80.4 81.4 79.8 80.2 80.2 78.0 79.7 82.1 80.4
80.1 80.1 80.1 80.1 80.1 80.1 80.1 80.1 80.1 80.1
-1.3 0.3 1.3 -0.3 0.1 0.1 -2.1 -0.4 2.0 0.3
1.69 0.09 1.69 0.09 0.01 0.01 4.41 0.16 4.00 0.09
2. Sum of deviations squared (SDS)
The ten squared deviations are summed: SDS = [ 1.69+0.09+1.69+0.09+0.01+O.O 1 +4.4 1 +O. 16+4+0.09] = 12.24 3. Variance
The variance, 5’. is obtained by dividing the SDS by 10-1-;1 s2
12.24 =1.36 9
=--
4. Standard deviation, s, is the square root of the variance
Thus, the set of ten measures can be summarized by the meun, 80.1 and the dispersion, estimated by the standard deviation, 1.2. In this book we will use the standard deviation to estimate the dispersion of measures. But there are other methods. The uncertainty due to the dispersion of measures around the mean is called random error. In addition to this random error, there may also be variations affecting all the measures. All the results may be greater, or smaller, than one, two or several units. This error is no longer random as it affects all measures in the same way: this is a systematic error. Random error is most readily appreciated; it is also the most easily detected and estimated. Hence, the topic is well documented. Systematic error is much less evident, and its discovery often requires some detective work. The investigator must always remember that the total error is the sum of the two types of error, and that both must be kept to a minimum.
53
TOTAL ERROR = RANDOM ERROR + SYSTEMATIC ERROR Note: In this book, we will consider the random error to be equal to one standard deviation of the random dispersion around the mean.
3.
ORIGIN OF THE TOTAL ERROR
The dispersion is due to small variations in the experimental conditions, such as ambient temperature, reading errors, or small changes in electrical voltage, etc. The investigator controls certain factors very carefully when making his measurements, but all the factors influencing the results cannot be controlled. The factors whose levels are voluntarily fixed are the controlled factors, the others are uncontrolled factors. Controlled factors do not introduce experimental errors. Only variations in the uncontrolled factors give rise to the total error. Errors due to random variations in the uncontrolled factors are random eryors, The investigator may estimate them from the standard error or any other value characteristic of the dispersion. Errors due to systematic variations in uncontrolled factors are systematic errors. In general, an experimenter does not suspect the existence of this sort of uncontrolled factor and special techniques must be used to detect them. In order to analyse total experimental error in more detail, we must consider the mathematical model. A response y depends on a large number of variables, x,: y = .f(X,, x2, %...> x, ,... 1 These variables do not all have the same influence, they can be classified into the following five categories: 1, The variables studied in the experimental design to which two levels are assigned
Level +I
0
0
Level -1
0
7
0
' 8
1 2 3 4 5 6
Figure 4.1: Factor 1 studied in the experimental design.
I
54
These factors are controlled, and their levels are accurately defined. Such variables introduce no error. Their variations explain the different values of the response. For example, there are three controlled factors in a 22 design. Each of them takes level +1 or -1 (Figure 4. I), perfectly defined by the experimental matrix. If there were no other influencing factors, the response would always be the same in all the replicate trials, i.e. trials with the same levels of controlled factors. 2. Uncontrolled variables which remain invariate throughout the experiment
These variables do not change, but they do introduced a constant displacement in the measurements ofy; these give rise to systematic errors. For example, if an experiment was carried out in two separate sub-experiments, this uncontrolled variables may have different levels in each of the sub-experiments. Level a in the first sub-experiment and level b in the second (Figure 4.2).The experimenter does not know about this factor, but the responses are altered depending on the level of this factor. We will see that precautions can be taken - Blocking - against this type of uncontrolled factor.
Levela
Level b
6
6
6
6
6
Level a
Levelb
6
b
e
Figure 4.2: Uncontrolled factor which remains invariate throughout each subexperiment: level a in the first sub-experiment and level b in the second one.
3. Uncontrolled factors whose levels change in a regular fashion during the experiment.
These variables give rise to a drift in the response. The experimenter is generally ignorant of these uncontrolled factors. We will see how to combat these systematic errors later with anti-drift designs.
Level a Level b Level c Level d Level e Level f Level g Level h
0 0
. . *
* 0, 0
I
I
'
l
l
1
1 2 3 4 5 6 7 8 Figure 4.3: Uncontrolled factor whose level changes regularly.
4. Uncontrolled variables whose levels are fixed at a constant value during a trial or during several trials, but differ during the overall experiment.
They introduce systematic errors which are difficult to counteract and generally ignored by the experimenter. The best way to master these factors is to include their systematic variations in random errors by the technique of randomization.
Level a
o m .
Level b
o
Level c
o
*
o
m
/
l
/
l
l
/
I
l
1
2
3
4
5
6
7
8
Figure 4.4: Uncontrolled factor whose level is constant during one or more trials.
5. Uncontrolled variables whose levels do not change in any specific fashion either during a
trial or throughout the experiment. These are the random variables which give rise to random or experimental errors. Statistical calculations can be used for this type of error.
56
Level a Level b Level c Level d Level e Level f Level g Level h
e
1
2
3
4
5
6
7
8
Figure 4.5: Uncontrolled factor whose level changes randomly.
4.
ESTIMATING THE RANDOM ERROR OF AN EFFECT
The random error of a response has repercussions for the calculated effects and interactions. We shall see that the error of effects can be estimated in several ways, depending on what is known of the phenomenon studied andlor the time available for this estimation, There are several situations, which range from excellent to not so good (We shall begin with the best ): The experimenter knows the value of the experimental error Ay affecting the response. The experimenter does not know the value of the experimental error affecting the response, but is able to estimate it by carrying out extra experiments. The experimenter does not know the value of the experimental error of the response, and does not want to perform any extra experiments.
4.1. The investigator knows the experimental error of the response This error was established from a series of measures containing enough measures to draw the distribution curve. For sake of simplicity, we will adopt a Gaussian (or bell-shaped) distribution (Appendix 2 ) . This distribution has two characteristic parameters: the mean ~1 and the standard deviation, 0.The standard deviation can be used to define the experimental error because, odd as it may seem, the value taken as the experimental error Ay may vary, and it is left to the judgement of the investigator to establish a value appropriate to the requirements of his problem. Only the standard deviation CJ is mathematically defined, and we will refer to this value. A numerical result yi is obtained when a response is measured. This value differs from the true value, which cannot be known, by an amount which is also unknown. But we do know that the true value has certain probability of being within an interval around yi. This is the
57
cofidence interval; the smaller it is, the smaller the probability that the true value y , lies within it. The larger it is, the greater the probability that y , lies within it. Figure 4.6 shows the probability associated with the confidence interval, expressed as standard deviation, for several cases. f
f
f
1
1
I
I
Figure 4.6: Percentage associated with the confidence intervals: f
0,f
20, k 30.
y , has a 68% probability of lying within the interval yiC! 0 and a 32% probability of lying outside the interval yi f 0. y , has a 95% probability of lying within the interval yi 2 0 and a 5% probability of lying outside the interval yi f 20. y , has a 99.9% probability of being within the interval yi f 30 and a 0.1% probability of being outside the interval yi f 30.
+
+
If the investigator chooses Ay = 20, there is a 95% probability that he is correct in saying that the true value lies within the interval yi f 20 and a 5% chance that he is wrong to say so. The value that the investigator adopts for the experimental error depends on the problems, the risks and the stakes involved, knowing that there is always a certain percent chance of being wrong. It is convenient to choose plus or minus one standard deviation as experimental error, knowing that this will be right about two times out of three, and wrong once in three. Multiplying this value by a coefficient greater than one reduces the chances of being wrong. We have simply given an evaluation of the experimental error Ay of the response. But we are actually looking for the error of the effect of a factor. An effect is calculated with n responses. If the experimental error is the same for all responses, the error AE on the effect is given by the relationship (Appendix 2): A E = -AY
&
If the distribution of the error Ay is normal (Gaussian), that of AE around E is also normal. If we take the standard deviation 0 as experimental error, the error on E becomes:
58
This error depends on the number of measures, n, used to calculate E. The more measures used, the smaller the error of the effect. In the study of bitumen emulsion stability discussed in Chapter 3, the investigator used an experimental error of plus or minus two points on a response. The calculation of an effect in the complete Z3 design required eight responses. The error of the effect is thus estimated to be: 0
All the effects and interactions between -0.7 and +O 7 are very likely to have a numerical value close to zero (i.e., 0.7 and zero are statistically indistinguishable). If the effect is a little larger than 0.7, there is less chance that it is zero. We have seen how to estimate this probability with the help of statistics. The investigator must select an appropriate value for AE as a hnction of the risks and the stakes of the problem investigated. In the case of this example, the investigator chose a limit of 1 stability point, knowing that an effect of this magnitude may be negligible, while still having a little influence. The above discussion can be illustrated by a graph with the values of the effects on the abscissa and the percent probability that the effect is significant on the ordinate (Figure 4.7). This graph can be made general by measuring the effect with d& as unity, so that the effect is measured in terms of oE(standard deviation of the effect). If the effect equals 3 standard deviations, it will be without influence in 0.1% of cases and have a small influence in 99.9% of cases. If the effect equals 2 standard deviations, it will be without influence in 5% of cases and have a very small influence in 95%. Lastly, if the effect equals one standard deviation, it will be without influence in 32% of cases, and have a very very small influence in 68%.
100
90 80
70
Probability
60 50 10 30 20 10 0
0
1
2
3
J
Effects measured with standard deviation
Figure 4.7: Plot of effect measured with a standard deviation of unity against the percent probability of it being significant.
59
Clearly, it is absolutely imperative that the investigator knows the estimation of the precision of the responses measured. The most favourable case is the one we have discussed, in which the distribution of measurement errors and the corresponding standard deviation are known. Under these circumstances, it is possible to select an appropriate value for the experimental error of one, two or three standard deviations, and to evaluate when a factor is significant or not. In the second, less satisfactory situation, the investigator does not know all these elements. But we shall see that there are several ways of estimating errors, and that all is not lost even when the situation is less than ideal.
4.2. The experimental error of the response is unknown In this situation, the investigator can carry out a few supplementary trials to obtain an estimation. There are then two possibilities: repeat several measures on the same experimental point or repeat the same experimental design once more.
Several measures on the same experimental point A point in the centre of the experimental field is generally chosen, and it is assumed that the error remains the same throughout the domain. The problem is to calculate an estimate of the standard deviation with only a few experimental points. If the distribution of the whole population is Gaussian, the distribution of a small sample is a Student curve. If 4 or 5 measures are performed, an estimate s of the standard deviation can be obtained from:
In this situation the confidence interval is larger for the same risk of mistakes and the same probability of being wrong. Table 4.2 shows the multiplication coefficients employed, depending on the number of measures used, to calculate standard deviation and the desired degree of confidence. For a very large number of measures, the Student distribution is very like the Gaussian curve. The Gaussian curve is produced when N = co. For example, if the standard deviation is estimated from 5 measures, the 95% confidence interval is k 2.78s around the value measured. When the investigator knows the standard deviation CT of the population, the confidence interval for the same percent success is -t 1.96 o. The multiplication coefficient is generally called t for Student curves (Table 4.2).
60
TABLE4.2 VALUE OF t AS A FUNCTION OF THE CONFIDENCE EXPECTED AND THE NUMBER OF MEASURES PERFORMED
Number of measures used to calculate s.
, Repeat the whole experimental design
This provides a total of two responses for each experimental point, so that a mean value of the standard deviation over the whole experimental domain can be calculated. Let us use a Z 3 design camed out twice as the basis for our calculation. Table 4.3 shows the trials performed and the responses obtained. The variance of each trial is calculated first, then the mean variance of all the trials. This variance is then used to deduce the mean error of a measure. which is itself used to calculate the error of the effect. The step-by-step calculations are : TABLE4.3 CALCULATION OF THE VARIANCE OF RESPONSES IN A DUPLICATED 23 EXPERIMENTAL DESIGN
Trial no
Factor Factor 2
3
-
-
-
-
+ + -
-
+ +
-
+ + + +
--
First Second result result 60 74 49 70 52 81 46 77
56
42
84 44
18
2 2
61
1. Variance of the first and subsequent trials For the first trial, the mean value of 62 is calculated from two results, 60 and 64. The variance, s , is calculated from: : : = -[(60-62)2 s 1 2- 1
+(64-62)2]
The variance for the other trials in Table 4.3 are calculated in the same way.
Note: The difference in the meaning of the two values, s and B, is theoretical: B measures the standard deviation of a population and s is an estimate of B deduced from a random sample of that population. In practice, only s is known, and we shall refer to this standard deviation in the following pages. 2. Mean variance of a response : , is: The above calculation gives 8 variances; the mean variance s S;
1 8
96 8
= -[8+18+2+32+8+18+8+2] = - =12
The mean standard deviation for the response of a single trial is the square root of the mean variance: S , = fi=3.46 3. Variance of an effect An effect is calculated with 16 responses and the variance sg of an effect is calculated
from:
4. Standard deviation of an effect
This is the square root of the variance of an effect .sF,=
6% =0.86
62
In practice, the same experimental design is rarely repeated in order to estimate experimental error because it is much too costly in terms of time and money. If more tests can be done, it is better to study more factors. Nevertheless, the above calculation is far from useless, as it is important in the frequent situation in which there is a clearly non-influencing factor In this case, the complete 2k design run can be considered as a repeated 2k-1design. The result of a trial is unchanged if the level of the level of a non-influencing factor is higher or lower We can therefore rearrange all the trials of the 2k design in pairs according to the levels of the influencing factors This gives us two results per trial in a Zk-' design. which therefore gives two results per test We examined an example in which two factors were without influence in Chapter 2 (Penicillium chrysogenum growth medium). In this particular case, the experiments were regrouped to give 4 results per trial.
4.3. The experimental error of the response is unknown, and the experimenter does not want to perform any supplementary experiments There is a way of estimating error under these circumstances, but it must be assumed that the high order interactions are zero and their values are estimates of experimental error. Let us assume that the results o f a 24 design are those shown in Table 4 4 If the third and fourth order interactions are estimates of the effect or interaction errors, the mean of this error is obtained by calculating the variance of each interaction, followed by computing the mean variance 0: TABLE4.4 TABLE OF EFFECTS
Mean
82.30
1 2 3 4
-2.00 -10.80 30.2s -4.75
12 13 14 23 24 34
1 00 -2.50 0.30 s 7s -0.50 0.60
I23
-0.25 0 80
I24 134 234 1234
0.50 -0.30 0.10
63
1. Interaction variances:
Variance of 123 = (-0.25)2 Variance of 124 = (0.80)2 Variance of 134 = (0.50)2 Variance of 234 = (-0.30)2 Varianceof 1234= (0.10)2
= = = = =
0.0625 0.6400 0.2500 0.0900 0.0100
2. Mean variance of the effect:
o
1 [0.0625+0.64+0.25+0.09+0.0I] 5
=-
Hence, the mean standard deviation of the effect is: = 0.46
5.
PRESENTATION OF RESULTS
5. I . Numerical results We have examined the techniques for calculating the standard deviation of the effects. The results should be presented by indicating the value of the effect followed by plus or minus the standard deviation. The units for both the effect and its standard deviation should also be mentioned: 3 0.3 cm3 (effect, 0,units)
+
+
This indicates that the real value of the effect has 68% probability of being between 2.7 and 3 . 3 cm3 (provided the distribution is normal). If we wish to increase the probability of being correct, the confidence interval must be increased. Thus, if we wish to be right at least 95 times in 100, we must use two standard deviations and write: 3 rf- 0.6 cm3 (effect, k 20, units)
5.2. Illustration of results The presentation suggested by Daniel [ 161 can be used to show all the results, in addition to the table of effects presented at the end of each study.
64
All the effects and all the interactions resulting from a study are arranged in ascending order. For example, in a study on the fabrication of plastic drums (Chapter 7) the investigator arranged the 15 results in ascending order (the mean is not included here): -2.7-0.3-0.3-0.25-0.2-0.15-0.1
-0.1 -0.05-tO.l +0.2+0.5+1.9+2.5+3.2
These values are plotted on Gaussian-linear axes using graph paper which has linear graduations on the abscissa and a Gaussian scale on the ordinate. Some of the points are generally grouped around zero. If these points form a line their distribution is normal and their numerical values can be considered to be estimates of the experimental error. The other points which do not line up indicate a non-random distribution. This method of displaying the results reveals the two groups of interest: The effects and interactions having no significant influence . The influential effects and interactions .
Gaussian scale
Effects and Interactions Figure 4.8: Illustration of the result, from Daniel 1161. There are two groups of results: 1) significant effects. 2) non-significant effects which may be considered as estimations of random error.
65
RECAPITULATION Total error contains two components: random error and systematic error. Random error is relatively easy to evaluate, while systematic error is difficult to detect. But the investigator must neglect neither.
TOTAL ERROR = RANDOM ERROR + SYSTEMATIC ERROR
A response y depends on several variables or factors which can be assigned to one of 5 categories: 0
0
0
Factors controlled by the investigator during the tests. Uncontrolled factors whose levels remain fixed throughout the performance of an experimental design. Uncontrolled factors whose levels vary regularly throughout the performance of an experimental design. Uncontrolled factors whose levels remain fixed during a trial but not at the same level from one trial to another. Uncontrolled factors whose levels vary randomly during the execution of an experimental design.
The influence of a factor is determined with reference to the known experimental error. This may be evaluated by one of the following methods: With reference to the distribution of measures around a central value and an estimate of the standard deviation of the response. These are used to calculate the standard deviation of the effect. By measuring a few values to obtain an estimate of the response standard deviation, followed by calculation of the standard deviation of the effect. By using high order interactions to calculate an estimate of the effect standard deviation. The probability that the effect has an influence can be estimated by comparing the effect itself with the standard deviation of the effect. It is good practice to present results with their standard deviations, because two values cannot be considered to be the same unless both the values and the standard deviations are. For example, a result of 2 cm is not the same if the two values are 2 ?C 0.1 cm and 2 k 1 cm. The presentation proposed by Daniel gives a graphical idea of the dispersion and of dispersion normality. It is also a way of showing influencing factors.
This Page Intentionally Left Blank
CHAPTER 5
THE CONCEPT OF OPTIMAL DESIGN
1.
INTRODUCTION
It is by no means easy to choose the best strategy for performing measures. We will examine the problem using a series of weighings. Any investigator carries out an enormous number of weighings during hidher working life, but few have ever wondered if they are doing it in the best possible way The first person to study this question was Hotelling [17]. The surprising thing is that this question was not examined until relatively recently, in 1944, although people have been weighing things for centuries. Let us assume that an investigator has used an old two-pan balance to weigh two objects, A and B, having masses of ma and mb. He placed object A on one pan and read off the weight to obtain p, . As he was aware that he could not avoid an error 0,he wrote:
68
Object B was weighed in the same way to give: mb = p2
*
(J
If p, and p2 had values of 10 and 25 grams respectively, and o was 0.1 gram, then
ma=lO+O.lg and mh=25F0.1 g
Thus, the investigator made two weighings and obtained two results with an error of 5 o. This is how everyone weighs things throughout the world. But Hotelling noticed that greater precision could be obtained for the same effort if both objects were included in each weighing. He camed out a first weighing with A and B together in the same pan, and then placed A on one pan and B on the other to obtain the difference. He was then able to deduce the masses m, and mb as follows: m , + m , =P1 ma -mb =P2 Hence,
Applying the variance theorem (Appendix 2) gave:
1 1 v(ma)= - [ 0 2 + 0 2 ]= - 0 2 4 2
The error of the results of these measures is ol& and not 0,so that: ma = 10 5 0.07 g mb = 25 k 0.07 g
69
The usual weighing method could only give the same result by weighing both A and B twice, four measures rather than two. This indicates one of the reasons for including all variables, or all factors, in each trial: for the same number of trials, the precision can be improved.
2.
WEIGHING AND EXPERIMENTAL DESIGN
Let us treat the Hotelling example using the formalism introduced in Chapter 2. The factors are A and B and the responses are the weights returning the pans to equilibrium at each weighmg. We must also adopt the convention of considering the weight of the object placed in the left hand pan to be positive, and the weight of an object placed in the right hand pan to be negative (Figure 5.1).
Figure 5.1: Sign convention adopted for weighings The range of weights for object A is thus -pa to +pa, depending on whether it is on the left or right-hand pan. Using coded variables, -pa is -1 and +pa is +1, while zero is the absence of A. -
Pa
-1
0 0
+
Pa
+1
The same convention can be adopted for B, and the effect of B on the balance pan is: -
-1
ph
0
0
+
ph
+1
The two ways of weighmg, the standard method and Hotelling's method can thus be presented as experimental matrices. The calculations are performed as for an experimental design and the effects are the weights of each object.
70
2. I . Standard method First weighing: A on left-hand pan, B is not weighed The responsey, is log. Second weighing: A is replaced by B The response, y2 is 25g.
TABLE5.1
EXPERIMENTAL MATRIX
i’-tl-”i Fl Standard method of weighing two objects
25 grams
I
Effect
I
10 grams
1
25 grams
I
As there is no interaction between A and B, the experimental and effects matrices are identical. The matrix X for the standard weighing method is:
X=JLI
0
+ 01 I
It is not surprising that the effects equal the responses: the effects matrix is a unit matrix.
2.2. Hotelling method First weighing: A + B on the left-hand pan The response y , is = 3 5g. Second weighing: B on the left-hand pan, A on the right-hand pan The response y z is = 15g.
71
TABLE5.2 EXPERIMENTAL MATRIX
Hotelling method of weighing two objects
35 grams
-1
I
Effect
I
15 grams
10grams
I
25 grams
I
The effects matrix is identical to the experimental matrix:
This matrix, in which there is no zero, is characteristic of the Hotelling method, in which all objects are included in all tests. The effects are certainly the same (the same weights) as in the standard method, but the precision is greater.
2.3. Strategy for weighing four objects Different strategies can be adopted for weighing four objects a. Standard strategy
If we had not read the above section, we would weigh each of the four objects one after the other, and the experimental matrix would be: 1 0 0 0
X=
0 1 0 0
0 0 1 0 0 0 0 1
72
If the responses are 10, 2.5, 1.5 and 30 grams, measured with a precision of 0.1 gram, the measured effects are exactly as the same as the responses, with the same precision. This is not surprising as the matrix is a unit matrix. b. Intermediate strategy We can use the same approach as for weighing two objects: A + B and C + D, by performing the following weighmgs (Figure 5.2).
A
B
B
Figure 5.2: Weighing four objects using a specific strategy First weighing: A+B on the left-hand pan Responseyl = 35g. Second weighing: A on the right-hand pan, B on the left-hand pan Response yz = 15g. Third weighing: C + D on the left-hand pan. Responsey3 = 45g. Fourth weighing: C on the right-hand pan, D on the left-hand pan. Response y4 = 15g.
73
The system of equations from this is:
y , = +ma + mb y 2 = - m a + mb y 3 = + m c + md y4 = - mc + md from which can be calculated the mass of each object
1 mb = - [ Y l + Y 2 ] 2 me
= - 1[ Y ~ - Y ~ I 2
md = - 1[ Y 3 + Y 4 1 2 The theorem of variance can then be used to calculate the error of ma and the three other masses:
Thus the error for ma is o/J2
= 0.07, from
which:
1 ma = -[35-15]=10+0.07 2
1
mb
= -[35+15]
mc
= --[45-15]
md
= -[45+15]
2
g
=25*0.07 g
1 2
=15&0.07g
1 2
=30+0.07 g
74
The system ofequations can be display by using the matrix form: tl
+I
0
0
-1
tl
0
0
0
0
+1
+I
0
0
-1
tl
x=
tl
+I
0
0
-1
+I
0
0
0
0
+1
+1
0
0
-1
+1
This strategy, illustrated by the X matrix, is better than the one-object-at-a-time strategy, but is it the best? How can we obtain the smallest possible error? Could we do better than O.O7g, without increasing the number of trials or weighings? This is not possible unless we include more objects in each test. Let us try to use all the objects in each weighing (Figure 5.3)
Figure 5.3: Weighing four objects using an optimal strategy.
75
c. Optimal strategy
The following weighings are performed: First weighing: all objects on the left-hand pan. Responsey, = 80g. Second weighing: A + C on the left-hand pan, B + D on the right-hand pan. Response y2 = -3Og (weights on the left-hand pan are considered negative). Third weighing: A + B on the left-hand pan, C + D on the right-hand pan. Response y 3 = -1Og. Fourth weighing: A + D on the left-hand pan, B + C on the right-hand pan. Response y4 = Og. The resulting set of equations is:
y , = + ma + mb +m, + md y2 = + m a - mb + m, - md y , = + m a + mb - m, - md y4 = + m a - m h - m , + m d
The mass ofeach object can be calculated using the following formulae:
ma = -1[ + Y I + Y ~
+
4
1 m b = -[+YI 4
mc
1
=-[+YI
4
- ~ 2+
+
~
~ +3 ~ 4 1
~
-3 Y ~ I
-2 Y ~ - Y ~ I
1
md = -[+Y, -Y2 -Y3 + Y 4 1
4
The variance of ma is given by:
The error of ma is the square root of the variance:
76
1 2
1 2
urn, = -0 = - 0.1=0.05gram We have reduced the error by half without increasing the number of weighmgs. The precision cannot be further improved, which can be seen intuitively as all the objects were used in each test. This extremely important property is related to the X effects matrix, which in this case is:
X=
+I
+I
+I
+1
+1
-1
+I
-1
+I +I
+I
-1 -1
-1 +1
-1
This is exactly the same matrix as that for a 22 design, and therefore satisfies the relationship: X'X = nI showing that the best precision for effects is obtained and cannot be improved. We can therefore reduce the error to 0.05 g for the weighing of four objects if the precision of each weighing is 0.1 g. In general, if the error of the response is cs and n tests have been performed using X matrices such as XtX= nI, the error of the effect of a factor is OIL and it can be shown that the greatest precision is obtained in this case.
*
3.
OPTIMALITY CRITERIA
An experimental design is said to be optimal if it allows the most precise calculation of effects. It can be demonstrated that this precision is directly correlated with the X'X matrix. Optimality criteria therefore involve this matrix and, depending on the experimental conditions, one of them must be chosen by the investigator. The criteria, in order of their quality are:
3.1. Unit matrix criterion The X matrix must respect the relationship
X'X
= nI
where n is the number of experiments. X matrices which fulfil this criterion are called Hadamard matrices, after the French mathematician who studied them. This is the best criterion. It can be
77
demonstrated that the most precise possible values for the effects (or interactions) are then obtained. We have seen how this criterion was applied to the weighings problem, but it remains true when there are interactions. Unfortunately these matrices are not available for all values of n, only for n=2 n=4 n=8 n = 12 n = 4 x d (where d is a positive whole number). When the number of trials is not a multiple of 4, the optimality criterion, X'X = nI, cannot be satisfied. The problem then is to find, despite this restriction, a design that provides the best possible precision. Other criteria of optimality must be employed.
3.2. Maximum determinant criterion The value of the determinant of X'X must be as great as possible. Thus a design is required in which Det( X'X ) is as large as possible. We know that a determinant is a number and that, although it is in the form of an array, it must not be confused with a matrix. In this book, determinants are shown by Det( ). The elements of a square matrix may be considered as elements of a determinant. The X'X matrix is a square matrix; it is therefore quite possible to calculate the value of the corresponding determinant. If we use the example of the weighmg design for 4 objects, we can calculate the determinants for the three strategies. a. Standard strategy
The X matrix is 1 0 0 0
X=
0 1 0 0 0 0 1 0 0 0 0 1
The determinant of X'X is unity Det( X'X )
=
1
78
b. Intermediate strategy The X matrix is +1
+1
-1
+1
0
0
0
0
+1
+I
10
0
-1
+1
+1
-1
+I
+1
0
0
0
+1
-1
I0
0
+1
+1/
X=
0
0
The X' matrix is
x' =
0
0
0
hence:
X'X
=
+1
-1
0
0
+1
+1
0
0
+I
+I
0
0
-I
+1
0
0
0
0 0
+1 -1
+1
0
0
0
0
0
+1 - 1 +1 +1
+I
12 0 0 01 0
2
0
0
x ' x = l0 0 2 0
21
10 0 0
The value of the determinant for this matrix is: Det( X'X )
=
2'
=
16
Indicating that this strategy is already better than the standard one c. Optimal strategy All the objects are involved in each weighing, and the X matrix is + I + I +1 +1 + I -I +1 -1 X= +1 +1 -1 -1 + I -1 -1 +1
79
The X' matrix is
xt=
+1
+1
+I
+1
+I
-1
+1
-1
+1 + I
-1
-1
+1 -1
-1
+1
+1 +1
+1
+1
+1
-1
+I -1 -1
-1
hence: tl
x'x =
+1
+I
tl
-1
+I
-1 +I
tl
+1 -1
-1
-1 +1 +I + I + I -1
tl
-1
-1
+1
4 0 0 0
X'X =
0
4
0
0
0 0 4 0 0 0 0 4
The matrix X is a Hadamard matrix, giving the best precision for the effects. The corresponding determinant of X'X is: Det( X'X)
=
44
=
256
This is the highest value that the determinant can have, hence the strategy is the best.
3.3. Minimum trace criterion The trace of the matrix ( X'X) must be as small as possible. The trace of a matrix is the sum of the elements along the main diagonal. For example, the trace of the following matrix:
(X 'X) - I =
trace of ( X k - '
= tr
0.25 -0.02 -0.02 0.35 -0.03 -0.03 -0.02 0.01
-0.03
-0.02
-0.03 0.47
0.01 -0.15
-0.15
0.72
( X'X) -I = 0.25 + 0.35 + 0.47 + 0.72 = 1.79
80
This criterion can be applied to the three weighing strategies. Clearly, the same classification (for 4 objects) is obtained: standard strategy:
trace of ( X'X) -1
intermediate strategy:
trace o f ( X'X)
optimal strategy:
trace o f ( x'x)-*= I
=4
-l = 2
3.4. "The largest must be as small as possible" criterion This rather cryptic statement indicates that the value of the largest element in the main must be as small as possible. For example, a ( X'X) matrix may be the diagonal of ( X'X) following with one arrangement of experimental points:
-'=
(XtX)
0.25 -0.02
-0.03
-0.02
0.35
-0.03
0.01
-0.03
-0.03
0.47
-0.15
-0.02
0.01
-0.15
-0.02
0.72
The value of the largest element on the principal diagonal is 0.72. The investigator's problem is to choose another arrangement of experimental points to obtain a value of less than 0.72, and as small as possible. We will not continue with this problem as it requires a considerable theoretical treatment and long computer calculation (see Chapter 16). The above criteria are not mutually incompatible. For example, if a matrix satisfies the first criterion, XtX= nI, then it satisfies the other three. Two-level factorial designs all satisfy the optimality criterion, XtX= nI. We are therefore sure of always obtaining the best precision for effects when they are used.
4.
POSITIONING EXPERIMENTAL POINTS
The positioning of the experimental points in the experimental domain is a major question. There appear to be two contradictory requirements: the fewest possible tests, and the best possible precision. In other words, we must look for the best compromise, or as we saw in the weighmg experiment, the best strategy. The solution to this problem can be very complex, sometimes requiring extensive calculation, and a great deal has been published on it [ 181. A general treatment is outside the scope
81
of this book. We shall restrict ourselves to two level factorial designs, first examining a single factor, then two, and finally the general case for k factors.
4.1. Positioning experimental points for one factor If a response y depends only on a single factor x, how can the experimental points be chosen so as to give an optimal experimental design? The variable x can have any value between -1 and + l . If we carry out two trials, the experimental points A and B could be at two specific locations on the abscissa, a and f3, (Figure 5.4). The experimental domain is then reduced to the straight segment of the axis Ox and limited by the points -1 and + 1 . Experimental points A and B are between -1 and +l. Our problem is to find the values for x such that points A and B produce an optimal design, i.e., they satis@the optimality criterion.
Q b
I
-1
A
B
+1
Figure 5.4: Are the experimental points A and B optimally positioned?
Assuming that the response y is the following function of x: y
=
+ al x
The two experiments give the following system of equations: y1 = % + a l a Y2 = a0 + a1 OK
P
82
+I
a a(]
IN=I+1
PI Ia,i
The response surface is a straight line, and we need to know how can we position A and B in the experimental domain to obtain and a, with the best precision. Intuitively, we can see that the response line PQ (Figure 5.4) is best defined when the points P and Q are as far apart as possible. The optimality criterion can be used to show this:
X~X = nI The X matrix is represented by:
'
The X matrix is then +1 +1 =la
We must now find the values of matrix optimality criterion, X'X = nI:
CL
and
iil
which allow the XLXmatrix to satisfy the unit
First calculate X'X
or
and calculate nl:
To satisfy the optimality criterion we must have
83
This matrix relationship is analogous to a system of two equations with two unknowns (the corresponding elements must be equal):
a2 + p2 = 2
which has two solutions:
1
(3=-1
1
a=-1
a = +1 and
p=+1
These indicate that the points must be located at each extremity of the experimental domain. The same applies to two level factorial designs in which the response depends on several factors: the experimental points must be selected so as to be at the limit of the field for each factor. Let us now apply these concepts to a specific problem, measuring electrical resistance. This example illustrates a general principle, and it is fairly easy to adapt it to suit any situation in which the response is a first degree function of the factor studied.
5.
MEASUREMENT OF AN ELECTRICAL RESISTANCE
Example: Measuring an electrical resistance The Problem: A technician has just enough time and finances to allow him to carry out ten tests in order to measure an electrical resistance as accurately as # possible. The current varies from 10 A (level -1) to 20 A (level +I). Voltage fk is measured at each level of current Where should he place the @ experimental points to obtain the most precise measure of the resistance7
Ohm’s Law states that voltage (the response) is a linear hnction of current. Thus it is not necessary to make measurements equally spaced throughout the domain, as in Figure 5.5 to check the shape of the response. We have seen that we can obtain the best precision for resistance if the experimental points are at the limits of the field. We therefore make five measures at level +1 and five at -1, as shown in Figure 5.6.
84
10
15
20
Amps
Figure 5.5: Poor method of measuring resistance.
Volts
10
15
Figure 5.6: Good method of measuring resistance.
20
Amps
85
6.
POSITIONING EXPERIMENTAL POINTS FOR TWO FACTORS
The preceding calculation was for a single factor. When two factors are involved, the positions of the experimental points must be chosen with great care. Let us first assume that the experimental points for factor 1 are placed at the extremities of the field at -1 and +1 on the x1 axis and that those for factor 2 are also at the ends of the field on the x2 axis (Figure 5.7). This disposition immediately provides the effects ofxl and x2. It also has the advantage of basing the calculation of effects on the experimental points. However, it does not allow calculation of the interaction. We are therefore faced with problem of finding the optimal disposition of the experimental points. We can use the optimality criterion of the maximal determinant. The best precision is obtained when the determinant is equal to 44 = 256.
D
t
+I x1
Figure 5.7: Poor distribution of experimental points within the experimental field.
86
The X matrix is deduced from the effects matrix (Table 5.3): TABLE5.3
EFFECT MATRIX
Point A Point B Point C Point D
+1 +I +1 +1
-1 +I
0 0
0 0
0 0
-1 +1
0 0
hence: +1 -1
0
0
+1
0
0
+1
x =+I
0
-1
0
+1
0
+1
0
+1
+1
+1
-1
0
0
0
+I
+1
0
0
+1
+1
-1
0
Calculating X'X
x'x =
+1
+1
-1
+1
0 0
0 0
0
-1 0
x'x =
0
+
1
0 0
0 2 0 0 1
00201
0 0 0 0
Det( XLX)
= 0
+
1
0
87
The value of the determinant of X'X is zero, indicating that the best precision has not been attained. But the four points are at the extremities of the domain of each factor. What went wrong? Only x, was involved in trial A, x2 was not used. Similarly for points B, C and D. Thus placing the points at the extremities of the domain of each factor is not sufficient, all the variables must contribute, and must therefore be involved in all the trials. If we adopt a configuration in which the points A, B, C and D are at the corners of the domain, they are all involved in each trial, and we get the system shown in Figure 5.8, to which we can apply the same optimality criterion.
Figure 5.8: Good positions for experimental points in the experimental domain.
88
The X matrix is deduced from the effect matrix (Table 5.4) TABLE5.4 EFFECT MATRIX
Trial no
I
XI
x2
"1x2
Point A Point B Point C Point D
+1
-1 +1 -1
-1 -1 +I +1
+1 -1
+I +I +1
+1
-1
+1
And the mathematical matrix X is: I+l
-1
-1
+1(
/+I
+I
+I
111
Calculating X'X
x'x
=
+I +I
-1
-1
+I
+1 +1
+1
-1
-1
-1
+1 +1 +1 -1
-1 +1
+1 +1
-1
-1
+I -1
+1 +1 +1 -1
-1 +1
+1 +1
+I
I
xtx=l0
4 0 0 0 0 4 0
Det( X'X )
=
256
The determinant of X'X is 256, the largest value possible. Thus, this configuration gives the maximum precision, and the disposition of points is the best. This is the disposition used for two level factorial designs.
89
7.
POSITIONING THE EXPERIMENTAL POINTS FOR K FACTORS
If we use the same reasoning for all two-level factorial designs, it can be demonstrated that the experimental points must be at the comers of the domain and that they must all be involved in each trial. The experimental points for three factors must be at the eight corners of a cube. Those for four factors are placed at the sixteen corners of a hyper cube in a 4-dimensional space. Finally, for k factors the experimental points must lie at the Zk comers of a hyper cube in a k-dimensional space. The designs that we examined in Chapters 2 and 3 are optimal designs. Using them, the experimenter can be sure of having the best research strategy.
90
RECAPITULATION 1 We used the weighing design example to show the importance of including all factors in each trial.
2 We also discussed criteria of optimality of experimental designs. Four criteria were mentioned, and the most frequent two of them were applied to the weighing design and resistance measurement examples. A research strategy can be evaluated, and there is an optimum strategy. 3 There is a best strategy: the optimal strategy. The positioning of experimental points
within the experimental domain is of great importance to obtain the best precision on the effects and interactions. When only two trials are performed per factor, the experimental points must lie at the limits of the domain in such a fashion that all factors are involved in all trials. This is the case for the factorial designs covered in this book. The investigator who uses such designs can be sure of having the best possible strategy.
CHAPTER 6
TWO-LEVEL FRACTIONAL F A C T O R I A L D E S I G N S : 2k-p -THE ALIAS THEORY-
I.
INTRODUCTION
In this chapter our goal is to show how it is possible to reduce the number of trials run without reducing the number of factors. The trials themselves must be chosen carefilly, and the results interpreted with great care to ensure that the conclusions are valid. We shall begin by examining designs which involve half the number of trials of a complete design. We will introduce the TheoIy of Aliases and the Box notation with the help of an example. Then, we will continue to reduce the number of trials with more and more fractionated designs. The rules deduced from the first example will be extended to all fractional factorial designs. The reader should study this chapter with particular care, as fractional designs are extremely powerful tools for conducting fast, precise experiments.
92
2.
FIRST FRACTIONAL DESIGN: z3-I
2.1 Example: Bitumen emulsion stability(continuedfrom Chapter 3) Let us go back to the problem that we solved in Chapter 3 using a 23 design: the stability of a bitumen emulsion. The experimenter camed out eight trials. Could he, for example, have camed out only half that number? Let us assume that he ran only four trials, and that these were trials 5, 2, 3, and 8. The corresponding design matrix is shown in Table 6.1; in it, the responses are those given in Chapter 3, as this is the same experiment. We can apply the same rules for calculating the effects, but we shall label them differently because fewer responses are used in the calculation. We will record the effect of factor i as hi and indicate the mean of the responses as hM
TABLE6.1 EXPERIMENTAL MATRIX STABILITY OF A BITUMEN EMULSION (CONTINUED)
IT Trial no
I
Factor 1 (fatty acid)
I
Factor 2 (HCO
Factor 3 (bitumen)
+ -
+
~~
Level (-)
low conc
dilute
A
Level (+)
high conc
concentrated
B
Response
93
The effects calculated from this fractional design can be compared to those obtained with the complete design (Table 6.2). TABLE6.2
RESULTS COMPARISON STABILITY OF A BITUMEN EMULSION (CONTINUED)
Complete Design.
Fractional Design.
Mean
27.25
27.25
1 2
-1 .oo -6.00 -4.00
-0.75 -6.25 -4.25
3
Evidently, the results obtained from the fractional design and those of the complete design with eight trials are very similar. Somewhat surprisingly, we have obtained the same result with less effort. But nothing is free in this world, so what is the price we must pay for carrying out four fewer trials? In order to answer that question, we must look more closely at the values of effect 3 and interaction 12 obtained with the complete design.
Adding them together, we get:
This is equivalent to h,, hence:
Thus, h is equal to the main effect E, plus the interaction E,,. The effect E, and the interaction El, are said to be aliased, and the quantity h can be called the alias, or contrast or simply the effect. In this book, we will use the term contrast, but the reader should be aware that these other terms are often used.
,
It can be similarly shown that: h
=
E, + E,,
94
h,
A,=
=
E, El, I + E,,, +
The contrast h indicates that the main effect E, is aliased with interaction 23. Contrast h indicates that the main effect E, is aliased with interaction 13, and lastly contrast h is aliased with the mean I and the interaction 123. Thus, all the main effects are aliased with interactions. So this is the price that must be paid. The number of trials was reduced to half, but the effects calculated from them are no longer pure, but mixed, or aliased, with the interactions. We obtained very similar results with eight and four trials in the bitumen emulsion example because interaction 12 was negligible. Thus h3ZE,
,
It is therefore possible to obtain results with four, rather than eight trials, so long as we remain aware that the main effects are aliased with interactions. Hence, the result will be satisfactory only as long as the interaction is negligible compared to main effect. Clearly, the interpretation of fractional designs is more complex than that of complete designs. The following section describes the method presently used for interpretation. The method is convenient and generally reliable, but the reader should remember that it is always possible that there may be exceptions.
3.
INTERPRETATION OF FRACTIONAL DESIGNS
The interpretation of all fractional designs present similar problems. The working hypotheses generally used are the following: Third or higher order interactions are considered to be negligible A zero contrast could indicate: a. that the aliased effects are all zero. This is the most likely situation. We will frequently use this hypothesis to analyse the results of factorial designs. b. that the aliased effects and interactions cancel each other. This is unlikely, and we will not perused it in this book. The interaction of two small effects is also small The interaction of a small effect and a large effect is generally small There is a strong possibility that the interaction of two large effects is also large. We will make extensive use of this hypothesis to analyse fractional designs.
95
These hypotheses are useful when interpreting the results of a fractional design. But, while they are often right, they are sometimes wrong. They are just helpful in a first practical approach and a more carefbl examination is sometimes necessary to extract the right information from the results of the experiment. Clearly, before we can interpret the results of a fractional design we must know how the effects and interactions are aliased. We will now examine this important topic in more detail.
4.
CALCULATION OF CONTRASTS
The selection of trials no 5, 2, 3 and 8 may have seemed strange, but there was a good reason for doing so. Table 6.3 shows the effects matrix of a Z3 design in which the sequence of trials was chosen to display two 22 designs for factors 1 and 2. The z3 design can thus be divided into two half-designs. TABLE6.3 EFFECTS MATRIX STABILITY OF A BITUMEN EMULSION (CONTINUED).
+ 6
+
7 4
+ +
Initially, we shall consider only the upper half-design (trials 5 , 2, 3 and 8), the one used to study bitumen emulsion stability. In this half-design the column of factor 3 signs is the same as the column of signs for interaction 12. This equality of the signs columns can be represented using the notation of Box by writing that 3 and 12 are equal as they have the same sequence of signs. Hence: 3 = 12 Thus, in this half-design, a contrast is the sum of elfect and interaction having the same and- sips. We can therefore see that the notation of Box and the contrast value are equivalent. It is thus possible to go from one to the other. sequence of
96
We will frequently use this correspondence between the two relationships when interpreting and constructing fractional designs. We can use the Box notation to deduce how effects are aliased. We can deduce the Box notation from the relationship of the aliases, and hence construct appropriate designs to solve specific problems. 3 = 12 is equivalent to h
and h
=
,
=
E, + El,
E, + E,, is equivalent to
3 = 12
or
e h,= E,+ El,
3 = 12
The eight columns of signs in the first half-design lead to the following relationships: 1 = 23 2 = 13 3=12 I = 123
From which the contrasts are deduced: h , = El + E,, 1, =
E2
+
El,
A,= E,+ El,
a=,
I + E,,,
Thus the Box notation can be readily used to find the effects and interactions aliased in a contrast. This method will be used throughout this book. Returning once again to the bitumen emulsion stability example, we could have used the other, lower, half-design. Examination of the effects matrix (Table 6.2) reveals that the column of signs for factor 3 (- + + -) is the same as that for interaction 12 multiplied by -1. Thus, using the Box notation, we have 3 = -12 We can now calculate contrast half-design (trials 1, 6, 7, 4) to obtain:
A,
h',
from the column of signs for factor 3 in the lower
1
=
-[-Y, 4
-
We now need to know how the factor 3 effect and interaction 12 are aliased in this contrast. We can take the formula for the effect of factor 3 and interaction 12 used in the complete design:
97
Subtracting El, from E, , we get:
Thus
1,
=
E,- El,
Hence, in this half-design, a contrast is the difference between the effects and interactions having the opposite sequence of signs. Again, there is an equivalence relationship: 3 = - 12
is equivalent to h3
=
E,
-
El,
and
i3 = E, .
E,, is equivalent to 3 = - 12
or
13
3=-12
=
E,-
El,
The eight columns of signs in the lower half-design lead to the four relationships: 1=-23 2 = - 13 3 = - 12 I = - 123
Which give the following formulae for the contrasts: hi = El
-
i2 = E,h3
E,, El,
E,- El,
iM = I
-
El,,
Here, again, the Box notation is used to find the effects and interactions aliased in each contrast.
98
5.
ALGEBRA OF COLUMNS OF SIGNS
We can now manipulate the columns and signs in order to find all the columns of signs in a fractional design that are identical or opposite. We can then apply the equivalence relationship to obtain the aliased effects and interactions in a contrast. There are two rules governing the algebra of column signs. These calculations are valid for all fractional designs. Rule one:
A column of signs multiplied by a column of + signs does not change, e.g.;
+
~
+
multiply by
+
~
+ +
-
+
+ -
+
Or, using the Box notation 1.I= 1
and we also have 2.1 = 2 3.1 = 3
Rule two: A column of signs multiplied by itself gives a column of + signs, e.g.:
-
~
+
multiply by
+
-
-
f
+
Written in Box notation, this is: l.l=I or 2.2-1
3.3=1
We will use these two rules throughout this book
~
+ + +
+
99
Alias generators All the formulae for the upper half-design can be obtained from: I = 123 This is called an alias generator because its enables us to find all the equal and opposite columns in a 2” design. If the two sides of the alias generator are multiplied by 1 we obtain:
1. I = 1.123 applying rule one to the left hand side, we get: 1.I= I
applying rule two to the right hand side, we get: 1.123 = 1.23
hence: 1 = 1.23
Lastly, by again applying rule one, we get: 1 = 23
Similarly, starting from the alias generator, we get: 2 = 13 and 3 = 12 By applying the equivalence relationship we can obtain contrast: 1 = 23
gives
A1=
El
2=13
gives
A,=
E,+ E,,
3 = 12
gives
A,= E,+ El,
I = 123
gives
A,=
+
EZ,
I + E,z3
The two alias generators, I = 123 and I = - 123, show that the signs column of interaction 123 is divided into two parts the first contains all the + signs, and the second all the - signs The complete 23 design is divided into two half designs The upper one contains all the + signs of interaction 123, while the lower one contains all the - signs of this interaction The reader will now understand why trials 5, 2, 3 and 8 were chosen for the fractional design, they are the trials for which the 123 interaction signs were + The generator for the lower half-design is
100
I
1
-
123
From it, by applying rules one and two, we can obtain the three relationships: 1 = -23 2 = -13 3 = -12
From these, we deduce how the effects and interactions are aliased in the lower halfdesign. hi
=
El
h,
=
E,- E,,
h,
=
E,- El,
h,
=
I - E,,,
-
E,,
We could just as well have chosen trials 1, 6, 7, and 4 (the half-design generated by minus signs of interaction 123). We could have used the contrasts to calculate the effects. In our example we would have found the same values because the interactions were negligible. But is there any danger in aliasing? There is absolutely no danger, provided that we do not depart from the spirit of the interpretation hypothesis. If we have begun our study by carrying out half of the 23 design, but were then not sure about the results, we could always go on to do the second half to obtain all the information. All that we have done is taken a risk in order to obtain the results twice as fast. This is the best type of gamble: one which you cannot lose, and can often win. The main point is to evaluate the risk. We will see how to cope with this difficulty in the following example. We can therefore see that experimental designs are tools which are perfectly suited to the progressive acquisition of knowledge. An experimenter may be initially reluctant to employ fractional designs for fear of missing important information or results. He or she may believe that there is a risk and that it would be better to use a complete design. This makes little difference while only a few trials are involved. But when there is a large number of trial, the experimental cost becomes prohibitive and the risk of a mistake during the experiment increases considerably. It is therefore better to get into the habit of carrying out fractional designs and learning to analyse the results carefully. The following pages will show the reader that this is the safest and fastest approach.
6.
CONSTRUCTION OF FRACTIONAL DESIGNS (ONE EXTRA FACTOR)
If we consider Table 6.4 (which is the same as Table 6.3) and examine the columns for factors 1, 2 and 3 in the upper half-design, we see that these are exactly the same as the table
101
the 22 design to study factor 3 . TABLE6.4
We will use this similarity to explain how we can quickly construct fractional designs: 1 We start by selecting an experimental design and drawing up an effects matrix. This effects matrix, called the basic design, is the foundation on which we build the fractional design. Thus, for a 22 design we get:
2 We then select the highest order interaction, as this is most likely to be small. The only interaction in a Z2 design is 12, but if we were working with a z3 basic design we would choose 123 rather than 12, 13 or 23. 3 We use the signs of this interaction to study the extra factor by attributing the - sign to the low level and the + sign to the high level.
We can illustrate the construction of fractional designs by using a 24 basic design matrix (Table 6.5),in which there are six second order interactions, four third order interactions and one fourth order interaction. The complete design can be used to study four factors in columns 1,2, 3 and 4. If we want to study a fifth factor, we keep the four columns of signs for the four initial factors and select an interaction column for the fifth factor. For example, we shall use the 1234 interaction column and say that by the signs of this interaction are attributed to the level of the extra factor 5. 5 = 1234
102
TABLE6.5
z4 BASIC
DESIGN MATRIX
Trial no
I
I 2 3 4 5 6 7
+
-
-
+
+
-
+
-
+
8
+ + + + - + + - + - - +
9 10 11 12 13 14 15 16
+
-
+
+
-
t
-
+
1
2
3 -
4
12
.
-
-
-
t
. -
-
+
-
+
+ -
+ t
-
-
t
.
f
+
-
+ ~~
+
~
-
+
+
+
+
+
-
t -
t
+
-
-
-
+
-
+ -
-
-
+
+
t
-
+
+ +
+ +
-
_.
-
-
+
-
-
+
+
-
+
-
+ + - + + - + + - + + + - - +
-
+
+
t +
t
-
+
+
+ -
+
-
+
+
-
+ -
-
-
+
-
+
+ -
+
-
-
+
-
t
+
-
-
+
-
+
-
-
+
t
-
-
+ + + - +
13 14 23 24 34 123 124 134 234 1234
t
-
-
+
-
+
t
-
+
+
+
-
-
+
-
t
-
-
t
-
+-
-
-
-
-
t
+
+
+ +
+
t
t
-
-
+ t
t t
+ -
-
~
-
t
+ - + -~ - + + + -t + + t -t + +
+ + + + -
-
+ t
-
t t -
-
i t -
+
The experiment is carried out by attributing the high level to factor 5 each time there is a
+ sign in the 1234 column and the low level each time there is a - sign in this column. At the end of the study column 1234 will be used to calculate a contrast by applying the usual rules. We apply the equivalence relationship in order to find out what this contrast contains. S = 12340
h , = E,+ E,,,,
Contrast h is the sum of the effect of factor 5 and of the interaction 1234. If we again take the expression 5 = 1234, and multiply both sides by 5 to extract the alias generator, we get: 5.5 = 12345 1 = 12345
We can then multiply this generator by 1, 2, 3, 4 and 5 to obtain the contrasts h h,, h 4 a n d h,.
,, h
2,
103
We can also analyse the expression 5 = 1234 in order to try and understand what it represents in the complete 25 design. As in the z3 design we studied previously, this relationship divides the complete 2’ design into two parts. The first half-design contains the + signs of the 12345 interaction and the second half-design contains the - signs of this interaction. The complete 25 design has thus been divided into two half-designs, one defined by I = 12345 and the other by I = -12345. If we examine the effects matrix of the half-design defined by I = 12345, we can see that it contains thirty two columns of 16 signs and that these columns form identical pairs. As in the 23 example, the effects and interactions having the same signs are aliased.
TABLE6.6
32 columns = 16 columns identical 2 by 2
16 rows
12345
I
+ +
+ + I = 12345
+ +
+ +
1
16 rows
d
32 columns = 16 columns opposing 2 by 2
1=
~
-
12345
+ +
Similarly, when we examine the effects matrix for the half-design defined by 1 = -12345, we see that it too contains 32 columns of 16 signs and that these columns form opposite pairs. As for the z3 design, the effects and interactions having sequences of opposing signs are
104
aliased. The alias generator of this half-design, I relationship, the contrasts
=
-12345, gives, from the equivalence
In practice, there is no need to write out a complete design and select the + or - signs of an interaction in order to obtain the trials of the ftactional design. We use the basic design (effects matrix of a complete design) and select an interaction column in order to study the extra factor. But in order to understand the reasoning behind this, we must bear in mind the implications of this approach. This will be most useful when we come to examine designs with several extra factors.
7.
NOTATION OF FRACTIONAL DESIGNS
Fractional designs are indicated by a notation which shows how the design is subdivided. Thus, for a 25 that has been divided into two equal halves, each half is designated as !h 25, or 25. 2-', or 25-'. Each of the digits in this last expression has a meaning. The 2 indicates that the factors have two levels; the 5 indicates the number of factors studied; while the 1 indicates that there is one extra factor more than in the basic design. If six factors were studied using a 24 basic design, the fractional design would be written 26-2. This notation system for fractional designs provides at least three pieces of information the number of factor levels, the total number of factors studied and the number of extra factors. It also provides information on the number of trials. Thus, 24 = 25-1 = 26-2 = 16, which is the number of trials to be performed. We can therefore say that a 2k-P design contains 2k-p trials.
8. CONSTRUCTION OF FRACTIONAL DESIGNS (TWO EXTRA FACTORS) Imagine that we have 6 factors to study. We could use a 26 complete design, but that would require 64 trials, which is generally far to many. We could also use a 26-' design, which would only require 32 trials, but even this is too many. If we want to carry out just 16 trials, we must use a 26-2 design. The corresponding basic design is the effects matrix of a Z4 complete design (Table 6.5). We can study four factors in columns 1, 2, 3 and 4, and select an
105
interaction column for each of the extra factors by using the signs + and - for the high and low levels. We can illustrate this approach assuming that we have chosen interaction 123 for factor 5 and interaction 124 for factor 6. We would then write, using the Box notation: 5=123 and 6=124 If we multiply the first relation by 5 and the second by 6,we get two alias generators I=1235 and
I=1246
These two generators are called the independent generators because they were obtained independently. Multiplying these two generators together, we get 1.1 = 1235.1246
And this expression can be simplified using the two rules described earlier:
I = 1.1.2.2.3.4.5.6 I = 1.1.3.4.5.6 I = 3456 We thus have a new alias generator, called the dependent alias generator. Combining all these generators, we have:
I = 1235 = 1246 = 3456 There are four alias generators. This set of generators is called the Alias Generator Set (AGS). This AGS tells us the contents of the contrasts. Each column of the 26-2 fractional design allows us to calculate the contrasts containing the effects and interactions of the complete design. If we multiply each generator in the AGS by 1, we get: 1.1 = 1.1235 = 1.1246 = 1.3456
or 1 = 235 = 246 = 13456 We can now apply the equivalence relationship to obtain the contrast k
Instead of writing the effects and interactions with the letter E, we can use just the digits, so that the above relationship becomes:
h
1 + 235 +246 + 13456
106
When we multiply the AGS by 2, 3, 4, 5 and 6 we get the contrasts h 2, h 3, h 4, h and 6.
h 2 = 2 + 135+146+23456 h = 3 + 125 + 12346 + 456 h , = 4 + 12345 + 126 + 356 h5=5+123+12456+346 h , = 6 + 12356 + 124 +345
We can see that each contrast contains four terms, or 22 terms, or 2 to the power of the number of extra factors. Let us look at the shorthand designations 5 = 123 and 6 = 124
to see exactly what is behind them. Consider a complete 26 design. Here, the first generator, I = 1235, groups together all the + signs of the interaction 1235 and define a 26-' fractional design The - signs of this interaction define a second 2"' design (second generator I = 1235). The 26 design has therefore been divided into two half-designs defined by the two alias generators, I = 1235 and I = -1235). Let us focus on the I = 1235 half-design and use the second alias generator, I = 1246. The + and - signs of the 1246 interaction divide this half-design into two new parts (Table 6.7), or two quarter-designs with respect to the original complete 26 design. The first quarterdesign is defined by the two independent alias generators 1 = +I235 and I = +1246
The second quarter-design is defined by the two independent alias generators I = +1235 and 1 = -1246.
The half-design I = -1235 is divided into two quarter-designs by the + and 1246. The first of these two 26-2designs are defined by:
-
signs of
I = -1235 and I = +1246,
and the second of these two 26-2designs are defined by: I = -1235 and I = -1246
The complete 26 design contains 64 columns of 64 signs. The quarter-designs each contain 64 columns and each column contain 16 signs. These columns can be rearranged in groups offour which are either identical or opposite. In the quarter-design defined by the AGS. I = 1235=1246=3456
107
TABLE6.7
64 columns identical 4 by 4
16 rows
64 columns opposing 4 by 4
16 rows
1235
1246
I
+ +
+ +
+ +
+ + + +
+ +
+ +
-
+
+ + -
64 columns opposing 4 by 4
16 rows
-
-
-
16 rows
f
-
+ +
+
+ + + +
I = + 1246
I = + 1235 I = - 1246
I = - 1235
-
64 columns identical 4 by 4
-
I = + 1235
-
+ + -
+ + + +
I = + 1246
I = - 1235 I = - 1246
108
The columns for the average and interactions 1235, 1246 and 3456 all contain 16 plus (+) signs. We can find all the columns having the same signs by multiplication of the AGS. For
example, if we multiply the AGS by 3 we get 3 = 125 = 12346 = 456
and we can deduce that columns 3, 125, 12346 and 456 have the same sequence of signs in this quarter-design. In the quarter-design defined by the AGS I = 1235 = -1246
= -3456
The average I and 1235 interaction columns contain 16 + signs, while the 1246 and 3456 interaction columns contain 16 minus (-) signs. We can find all the columns having identical or opposite signs by multiplying the AGS of this quarter-design. For example, if we multiply the AGS by 3, we get: 3 = 125 = -12346
-456
and we can affirm that columns 3 and 125 are identical, that columns 12346 and 456 are identical, while columns 12346 and 456 have a sequence of signs opposite to those of 3 and 125 in this quarter-design. It is practically impossible to write out the 26 design and look for the fractional design trials by using the interaction signs, It is much easier to use a basic design and the alias theory (Box notation and equivalence relationship) to find the trials of a fractional design. This is why we have emphasised the alias theory.
9.
CONSTRUCTION OF FRACTIONAL DESIGNS (p EXTRA FACTORS)
The concepts we have used to study one or two extra factors can be extended to p factors. A fractional design is constructed as follows: 1. The experimenter chooses a 2" basic design by writing the effects matrix for a complete 2"
design. If the total number of factors to be studied is k, and the number of extra factor is p, we can write: n=k-p For example, nine factors (k = 9) can be studied with a 24 basic design (n are five extra factors (p = 5), we still have: 4 = 9-5
= 4),
and there
109
The experimenter selects p, generally high order, interactions. He writes that the extra factors are aliased with the selected interactions. This allows him, using Box notation, to write the independent alias generators. If we take a 26-3 fractional design, we can write that the initial three factors will be studied in columns 1, 2 and 3 and that factors 4, 5 and 6 will be studied in columns 12, 13 and 23. For these last three columns:
2.
4=12 5 = 13 6 = 23
Applying the calculation rules given earlier we can establish the following independent alias generators:
I = 124 I = 135 I = 236 It is only possible to interpret the results of a fractional design if we know how the effects and interactions are aliased in each contrast. Thus the experimenter must establish the AGS and apply the equivalence relation. The AGS is calculated from the independent generators to which are added the dependent generators. These last are obtained by multiplying together pairs or threes of independent generators. Keeping the same example, we get: 124.135 = 2345 124.236 = 1346 135.236 = 1256 124.135.236 = 456
hence the AGS: I = 124 = 135 = 236 = 2345 = 1346 = 1256 = 456
The AGS contains 2P terms. The AGS can be used to calculate all the contrasts in the fractional design. Contrast h is obtained by multiplying all the AGS terms by 1 and adding the results
,
1.1= 1.124 = 1.135 = 1.236 = 1.2345 = 1.1346
=
1,1256 = 1.456
simplifling, 1=24=35=1236=12345=346=256=1456
and applying the equivalence relationship
k,
=
1+ 24 + 35 + 256 + 346 + 1236 + 1456 + 12345
110
The other contrasts, h 2, h 3, h 4, h and h are obtained in the same way. Because of the importance of these calculations we shall now look at some practical rules for going from the AGS to the contrasts and vice versa.
10. PRACTICAL RULES
10.1 Going from AGS to contrasts: Write the AGS taking care to include the signs, e.g.: +I = -124
= +235 = +1345
Multiply all the AGS generators by 1 to obtain contrast h +1.I = -1.124
= +1.235
+1.1345
1
simpli@, +I
= -24 = +1235 = +345
and remove the equals sign 1 - 24 + 1235 + 345
This expression is equal to contrast h shortest: h
= 1 - 24
I.
We can order the terms, starting with the
+ 345 - 1235
The same can be done for h 2, h 3, h etc
10.2 Going from contrasts to the AGS
,
First write the contrast, e.g. h : h , = 1 - 35 + 234 - 1245
Keep only the right hand side terms 1 - 35 + 234 - 1245
Separate the terms with equals signs
111
1 = -35 = +234 = -1245
Multiply by 1 to obtain the AGS I = -135
=
+ 1234 = -245
11. CHOOSING THE BASIC DESIGN The most appropriate basic design for a specific experiment can be chosen by considering: The total number of factors to be studied The number of trials to be performed.
11.1. Total number of factors to be studied If the experimenter wishes to study k factors, he could use the n initial columns to study the n initial factors. He could then use the k-n interaction columns for the additional factors. This is clearly only possible if there are k-n interaction columns available. For example, if the experimenter wants to study six factors, he could choose a 23 basic design, and study the three first factors in the first three columns and the three extra factors in three of the four interaction columns. But if there are eight factors, this will be impossible because there will be no column for the eighth. In this case, he must use a 24 basic design, and then study the four first factors in the initial four columns and the four additional factors in four of the eleven available interaction columns. Table 6.8 shows the maximum number of extra factors that can be studied with two level designs. In this Table, C : is the standard notation for the number of possible combinations of k objects taken q at a time: k! c q -k q !(k-q) ! In this relationship k is the number of main factors in the basic design and q the order of interaction,
112
TABLE6.8 CONSTRUCTION OF FRACTIONAL DESIGNS
Maxi number of aliised
Maxi number of studied
factors.
factors.
1
3
4
7
11
15
26
31
57
63
120
127
2k-k-1
2k-1
11.2. Number of trials to be performed The basic design indicates the number of trials that must be performed. A 22 basic design contains four trials, a basic 23 design contains 8 trials, etc. It is thus easy to choose the basic design. But we must also take into account the degree of confounding in the design. Confounding is the number of terms within each contrast. The greater the confounding, the more words there are in each contrast, and consequently, the more difficult the interpretation. For example, an experimenter wants to study 7 factors. If he chooses a 23 basic design, he will carry out a Z7" fractional design. The contrasts will contain Z4, or 16 terms. But if he chooses a 24 basic design and cames out a 27-3 fractional design the contrasts will contain only 23, or 8 terms. The risk of ambiguity will thus be reduced by half. With a Z5 basic design, the contrasts of a 27-2 design will contain only four terms. Thus, each study contains a compromise between the number of trials and the increasing difficulty of interpretation as the number of terms in the contrasts increase.
113
RECAPITULATION This chapter introduces the alias theory which is very important for understanding and preparing fractional designs. A good interpretation of the results cannot be achieved without a complete knowledge of the alias theory. We have examined:
A 23-1fractional design: the stability of a bitumen emulsion. Four trials are sufficient to obtain the effects of the three main factors. But the interactions are unknown and aliased with the main factors. Hypotheses for interpreting fractional designs. Methods for calculating contrasts. A new algebra based on the Box notation: the algebra of columns of signs. Alias generators. Notation of fractional designs. Methods for constructing fractional designs: One extra factor. Two extra factors. p extra factors. Practical rules for moving between AGS and contrasts. How to choose the basic design, taking into account the number of trials to be performed and the number of factors to be studied.
This Page Intentionally Left Blank
CHAPTER 7
TWO-LEVEL FRACTIONAL F A C T O R I A L D E S I G N S : 2k-p -EXAMPLES-
1.
INTRODUCTION
Chapter 6 introduced fractional factorial designs, and we used a simple example to illustrate the theory of alias. This example was useful for showing how to construct fractional designs, how to calculate contrasts and how to interpret results. But we need a little more information in order to be able to use the powerful tool of fractional designs. In this chapter we will apply our knowledge to three examples: Minimizing the colour of a product using a 25-2 design. Optimizing spectrofluorimeter settings using a 274 design. Plastic drum fabrication using a 284 design. These examples show us how to use experimental design methodology, how to choose the initial design, and how to find the complementary design which resolve ambiguities between main effect and interactions.
116
2.
25-2 FRACTIONAL DESIGN
2.1. Example: Minimizing the colour of a product The problem: The product must have as little colour as possible. This colour is @ measured using a coiour index. Fabrication of the product involves & several factors. The makers believe that the factors which influence the colour of the final product are.
!! 9 i# -
0
Factor 1 reaction temperature. Factor 2: origin of the raw material (2 suppliers)
0
Factor 3: mixing rate
0
Factor 4: storage time.
0
Factor 5. type of additive.
All five factors must be studied. A 25 complete factorial design involves 32 trials. This is far too much for the budget, and could take too long time to execute. The technician in charge of this study decides to use a fiactional factorial design. He begins by running 8 trials (initial design) which may then be followed, if necessary, by eight trials to resolve any ambiguities (complementary design). The initial design is easily constructed. Two interactions of a basic design are selected and aliased with the extra factors. For example: 4 = 123 5 = 13
He could just as easily have selected: 4 = 13 and 5 = 12, or 4 = 23 and 5 = 12
For the one he has chosen, the contrasts can be calculated using alias generators. The two independent generators are: I = 1234 I = 135
Multiplying them together gives the dependent generator:
I = 1234.135
= 245
The three generators, with the mean, give the AGS:
117
I = 135 = 245 = 1234
The AGS will allow him to calculate the contrasts, each of which has four terms. The terms of the AGS are multiplied by 1, they are then added together to give the contrast h The same terms of the AGS are multiplied by 2 and added together to give the contrast h 2, etc. For h 1. I 1.135 1.245 1.1234
= 1 35 = 1245 = 234
=
hence:
h , =1+35+234+1245 and the other seven contrasts: h, h, h, h, h,,
=
h23 h,
=
=
= = = =
2 +45 +134+ 1235 3 + 1 5 +124+2345 4 + 25 + 123 + 1345 5 + 1 3 + 24+12345 12+34 +145+235 23+ 14 + 125+345 I +135+245+1234
Table 7.1 shows both the experimental matrix and the effects matrix. The columns for factors 1, 2, 3, 4 and 5 form the experimental matrix and the same columns are used to calculate the contrasts h h 2, h 3, h and h 5. The interaction 12, 23 and the mean I columns are used to calculate the contrasts h 12, h 23 and h M. The results are analysed by applying the interpretation hypothesis indicated previously: Interactions of orders above 2 are ignored. If a contrast is zero, each of its terms is assumed to be zero. All results smaller than the calculated error are considered to be zero.
,
The non-significant contrasts are: h2 = 2+45= 0 h, = 4+25- 0 h , , = 1 2 + 3 4 ~0 h , , = 2 3 + 1 4 ~0
or or or or
2=0 4=0 12 = O 23=0
and and and and
45=0 25=0 34=0 14=0
The following contrasts appear to be significant: h, h, h,
= =
=
-2.18 -3.33 -4.55
or or or
1 + 35 3 + 15 5 + 13 +24
=
= =
-2.18 -3.33 -4.55
118
Factor 1 is aliased with interaction 35. We do not know if the value of -2.18 is due to factor 1 alone, interaction 35 alone, or to a combination of the two
TABLE 7.1
EFFECTS MATRIX: INITIAL DESIGN COLOUR OF A PRODUCT
---
-
remp. R.Mat Mix.R Stor. 3 4=123 1 2
Add..
Response
5=13
27.4 31.1 26.6 32.4 31.4 16.5 27.5 15.5
Level -
low
1
ILeveI+)
I high 1
2
Contr. 26.05 -2.18
-0.55
slow
1
fast
-3.33
short
I
long
0.10
A
1
B
-4.55
I 0.63
-0.68
According to the interpretation hypothesis indicated earlier ( Chapter 6), interaction 3 5 could be zero if the contrasts h and h are zero, but this is not so. The result is thus ambiguous, and we cannot reach any conclusion. We can use the same reasoning for factor 3 and interaction 15. If we examine contrast h we see that factor 5 is aliased with two interactions. One, interaction 24, is probably zero, but the other, interaction 13 may be influent as h and h are not zero. We must always be aware of the risks we take in interpreting the results of a design. Saying that an interaction is zero implies that this interaction is very likely to be statistically equivalent to zero when compared to the standard deviation. Saying that an interaction is zero when the effects contributing to it are zero is a risky hypothesis which is often true, but can sometimes unfortunately be wrong. We must therefore not automatically use statistical tests to interpret the results. We should use them intelligently. We should above all rely on our experience and knowledge.
,
119
Provisional conclusion:
1
~
L'
Factors 2 and 4 have no influence, but effects 1, 3, and 5 seem to be I influent, and there may be interactions between them. A second series of trials is required to resolve these ambiguities This new set of experiments should allow the experimenter to calculate effects 1, 3 and 6 5 alone, I e , without the influence of interactions 13, 15, and 35 The main effects are then said to be dealiased from the second order interactions I
2.2. Techniques for dealiasing main effects from interactions The contrasts h
,
,, h
, and h are, if we neglect interactions above order 2:
h,
= 1
h,
=
1,
=
+
35
3 + 15 5 + 13
We need to calculate new contrasts, such as:
h', h'?
=
1
-
35
=
3
-
15
=
5
h,
13
~
We can obtain the main effects 1, 3, and 5 and interactions 13, 15 and 35 by a simple
,
calculation. The sum of h and
h', gives: 1= -[h, 1
+h',]
2 and their difference:
"1,-h ,' l
35=2
The sum h +
,
h',
, h'? give 3 and 15. The sum h ,+ h', and the
and the difference h -
difference h - h', give 5 and 13. We must now find an experimental design from which we can calculate the contrasts:
h', h'?
=
1 - 35
=
3
-
15
120
TABLE7.2 DESIGN
FOUR2"' FRACTIONAL DESIGNS IN A COMPLETE
I 25-2 Fractional Design.
1234
135
1
+
+ +
+ +
+ +
+ +
+
8 rows
I = + 1234 1=+135
+ + + 25-2Fractional Design.
+
-
+ +
1 = + 1234
8 rows
I = - 135
+ + -
25-2 Fractional Design.
-
-
+ +
+ + + f
1 = - 1234
8 rows
1=+135 -
-
2s-2 Fractional Design.
-
+
+ -
+ t
+ + 11-
1234
8 rows
I = - 135 -
-
-
-
+
+
121
If we apply the equivalence relationship in the opposite sense, i.e. from contrasts to the AGS, we see that these contrasts come from a design that has the alias generator I = - 135. We used one of the four 25-2 fiactional designs of the complete 25 design in our first design. This was defined by the two independent generators (Table 7.2):
I = + 1234
and
I=+135
There are three other 25-2fractional designs, and two of them contain the alias generator
I = -135.We could therefore take either one of them: I=+1234
and
I=-135
I=-1234
and
I=-135
or The experimenter has chosen the I = + 1234 and I = -135 fiactional design. These two independent generators give the dependent generator: I = 1234. -135 = -245
Which gives the AGS of the complementary 25-2 design that can be used to dealias effect 1 and interaction 35, effect 3 and interaction 13, and effect 5 and interaction 15. We can now construct the complementary design and calculate the contrasts.
2.3. Construction of the complementary design The two independent generators give: 4 = 123 and 5=-13 In the basic design from which the complementary 25-2design is constructed, the 123 interaction column is used to study factor 4. The signs of interaction 13 are changed to study factor 5. The resulting complementary design is shown in Table 7.3.
122
TABLE 7.3 EFFECTS MATRIX: COMPLEMENTARY DESIGN COLOUR OF A PRODUCT
Temp. R.Mat Mix.R
Trial no '
9 10 11
12 13 14 I5 16
1
+ + + + + + + +
3
4=123
-
-
-
-
+
-
-
+
+ +
-
-
+
-
-
-
+
+
Response
+
24.8 34.6 26.0 26.7
-
-
+
low
1
slow
short
A
Level +
high
2
fast
long
B
-05
-I 00
3 18
-1 84
-3 13
2485
Inter.
-
Level -
Contr
Inter.
+ +
+ + + +
+
-
J-JT Add..
Stor.
2
-045
-068
2.4. Contrast calculation The alias generators set is: I = 1234 = -135
= -245
We can obtain the contrasts by multiplying the AGS successively by 1, 2, 3, 4, 5, 12, 23 and I
h', h', h'3 h, h5
=
1
- 3 5 + 234- 1245
=
2
-45
=
3
= 4 =
5
+
+ h',
E
1 - 35
1
2 - 45
134-1235
j
h,
-
1 5 + 124-2345
4
h',
-
2 5 + 123-1345
-
13- 2 4 + 12345
+
3-
15
h,
z
4 - 25
h5
E
5 - 13- 24
123
12
h’l2=
+ 3 4 - 145-235
h2,= 23
+
1 4 - 125-345
h ’ ~ =I
-
135
-
iI2 5 1 2 + 34 + i23 z 1 4 + 23 +
245+ 1234-
h’M
I
The combined initial and complementary designs can now be used to remove the ambiguities. If we neglect the interactions of order higher than 2, we have from contrasts h
,
and
A,: h
, + i,I + 35 + 1
h,
=
-
-
35 =two times 1
i, =1+35-1+35=twotimes35
Substituting the numerical results of the two designs in the above formula gives: h, +
h,
-
i1 = -2.18-0.5 h‘, = -2.18+ 0.5 1 = -1.34
35 = -0.84 The same calculation can be done with the other contrasts to obtain: The five main effects dealiased from second order interactions The second order interactions, either alone, or aliased together, but dealiased from the main effects. We can then draw up the table of effects (Table 7.4) a,nd use it to interpret the results of the two experimental designs.
2 . 5 Interpretation The experimenter estimates that any effects smaller than k 1 colour index units are negligible. Using this criterion, there are only two influencing factors to be considered. Factor 1 - the reaction temperature Factor 5 - the type of additive. We can also see that there is a very strong interaction between these two factors. The high value of contrast h found in the initial design is due to interaction 15 and not to factor 3. Similarly, the high value of contrast h is due to factor I and not to interaction 35. The high value of contrast h is due to factor 5 and not to interaction 13.
,
124
TABLE7.4 TABLE OF EFFECTS COLOUR OF A PRODUCT
I + 1234 1 2 3 4 5 15 25 35 45 12 + 34 13 + 24 14 + 23
25.45 f -1.34 -0.78 -0.07 -0.87 -3.84
1
1 1 1 1 1
*
f
1
1 1
-3.26 f 0.97 k --0.84 1 0.22 -
*
1
1 1
1
0.09 k 1 -0.71 1 -0.68 f 1
*
The ambiguities are now resolved. As there are only two influencing factors we could have camed out only a 22 design of four trials. In fact, we have camed out sixteen trials during the experiment. The sixteen responses can be arranged in four groups of four trials as if we had performed the same 22 design four times. We can see that factors 2, 3 and 4 have no influence. Thus the response, the colour index, does not change whether they are at high or low levels. TABLE7.5 EXPERIMENTAL MATFUX REARRANGED COLOUR OF A PRODUCT
I I I Trial no
5 2 1
6
7 4 3 8
9 1 1 14 16 13 15 10 12
+
+
Results
Average
31.4 27.5 27.0 23.6 31.1 32.4 34.6 26.7 27.4 26.6 24.8 26.0 16.5 15.5 17.0 19.1
31.2 26.2 17.0
125
We can then reconstruct a 22 design taking into account the levels of factors 1 and 5. There are four trials (numbers 5, 7, 9 and 11) in which factors 1 and 5 are both at low level. We can group together the four corresponding responses (31.4, 27.5, 27 and 23.6) and use their mean value, 27.4. The same can be done for the other combinations of levels of factor 1 and 5. Table 7.5 shows the results ofthis rearrangement. The results can be conveniently presented as a diagram (Figure 7. I), which is easier to read and analyse than a set of numbers.
27.4 26.6
16.5 15.5 17.0
Ternperature
23.6
26.7
A.
Additif
DB
Figure 7.1: Colour of a product. Graphical display of results. Examination of this figure shows: 1 . The additive type has no influence at low temperature, the colour changes very little, from 27.4 to 26.2.
2. Additive A gives a more highly coloured product at high temperature (31.2) than at low temperature (27.4). 3. At high temperature, additive B gives a slightly coloured product, while additive A gives a highly coloured product. 4. Additive B gives a coloured product at low temperature, but it gives a colourless product at high temperature. We can therefore conclude that we can obtain a colourless product by using a combination of additive B and working at high temperature.
126
However, we must not neglect the non-influencing factors. They are often the source of most usehl savings or simplifications.
.Factor 2 Origin of the raw material: As this factor has no influence on product colour, we can select either of the two suppliers. For example, we could choose the least expensive, or the one with the best delivery schedule. We could also stimulate competition between them without endangering the quality of the final product.
.
Factor 3 . Stimng speed: As stimng speed has no influence, we can choose the lowest speed, and economise by using less energy We could also check to see if any stimng at all is required. Perhaps it could be dispensed with. It is impossible to decide immediately because it is outside the domain of the study. But we could propose a complementary study to see if stimng could be dispensed with. This would save on energy, maintenance and capital investment. Factor 4. Storage time: Storage time has no influence. We need therefore not worry about this storage time (within the limits examined in the experiments) on product colour.
Clearly, the analysis we have carried out is only relevant for product colour. The final conclusion could be very different if other responses were to be considered. Once the results are interpreted, the experimenter may publish his conclusion. They will be short and include only the essentials of the analysis. Conclusion: Only two effects influence product colour the reaction temperature and the additive These two factors interact strongly Additive A gives a F dark colour, regardless of the reaction temperature Additive B gives a a light colour if the reaction is carried out at a high temperature Thus the reaction should be performed at high temperature using additive B The stirring speed has no influence, the slowest speed should be chosen so as to save energy The study could be extended to see whether stirring can be abandoned, saving energy, reducing capital investment and running costs The origin of the raw material has no influence, hence the least expensive supplier should be chosen
127
High
17.0
26.2
0
4b
c
-0
Temperature
Low
27.4
31.2
Additif
A-
CB
Figure 7.2: Reaction should be performed at high temperature using additive B.
The next example shows how it is possible to satisfy simultaneously the constraints imposed by each of the responses studied to attain the desired objectives.
3.
27-4 FRACTIONAL DESIGNS
This example is taken from a four-laboratory co-operative study to develop an assay for the suspected carcinogen, benzo-a-pyrene. The experimental designs were directed by Total, and were published in Analusis [ 191.
3.1. Example: Settings of a spectrofluorimeter The Problem:
k
We needed to define the optimum conditions for the quantitative analysis of benzo-a-pyrene. The technique used was r spectrofluorimetry, and the tests were carried out to determine the best gsettings for analysis of low and high concentrations of benzo-a-pyrene B in mixtures. The experimenters required. - high sensitivity - high 1 ai selectivity - low background noise
128
Response selection
Figure 7.3 shows a typical fluorescence emission spectrum for benzo-a-pyrene. The peak height at 481 nm, 4 was chosen for sensitivity, the width at half-height, B, of peak A as an index of selectivity, and the parameter D as background noise.
481 nm
Figure 7.3: Definition of the responses.
Factor selection The seven factors chosen are all critical parameters in spectrofluorimetry. As can be seen in Figure 7.4, the beam of a xenon lamp passes through a monochromator. This light is diffracted by the excitation monochromator, and a specific wavelength hits the sample. The sample absorbs part of this monochromatic light and emits fluorescent light in all directions. The fluorescence emitted at 90" is passed through the emission monochromator and the emission spectrum is collected by a photomultiplier. The seven factors studied were:
.. .. .
Factor I: Excitation slit width. Factor 2: Emission slit width. Factor 3: Sample temperature. Factor 4:Scan speed. Factor 5: Recorder gain. Factor 6: Photomultiplier voltage. Factor 7: Recording pen amplitude.
129
excitation monochromator
emission monochromator
Figure 7.4: Spectrofluorimeterschematic
Experimental domain High and low levels were set for each factor, as follows: Level Factor 1: Excitation slit width (nm). Factor 2: Emission slit width (nm). Factor 3: Sample temperature (“(2). Factor 4: Scan speed (ndmin). Factor 5: Recorder gain. Factor 6: Photomultiplier voltage (V). Factor 7: Pen amplitude.
2.5 2.5 20 20 1
310 2
Level + 7.5 7.5 40
100 10 460 4
Choice of pxperimental design It was decided to use a fractional design based on a 23 basic design. The four interactions were used to study the extra factors. Thus the design was saturated. The extra factors were aliased as follows: 4 = 123 5 = 12
6 = 23 7 = 13
I30
3.2. Calculation of contrasts The contrasts were calculated from the AGS. This was established by writing the independent generators and then calculating the dependent generators of the 27" design used. From the aliases listed above, the four independent alias generators are: I
=
1234 = 125 = 236 = 137
The dependent generators were calculated by multiplying the independent generators in twos, threes, and fours. Multiplication in twos ( C i
= 6)
1234.125 1234.236 1234.137 125.236 125.137 236. 137
= = =
= = =
345 146 247 1356 2357 1267
Multiplication in threes ( C j = 4) 1234.125.236 = 2456 1234.125.137 = 1457 1 2 5 . 2 3 6 . 1 3 7 = 567 1234.236.137 = 3467
Multiplication in fours ( C i = 1) 1234.125.236.137
=
1234567
The 27 design contains 128 effects and interactions, the 27-" design allows calculation of only eight contrasts. There will thus be sixteen terms or words in each contrast, and these will
be obtained using the alias generators set we have calculated: I =1234=125=236=137= 345=146=247=1356 =2357=1267=2456=1457= 567=3467=1234567
The complete calculation of a contrast is shown here for factor 1 . Each term of the AGS is multiplied by 1 and the resulting sixteen terms are added together to give contrast h
,
h
,
=
I+ 234 + 25 + 1236 + 37 + 1345 + 46 + 1247 + 356 + 1 2 3 5 7 + 2 6 7 + 12456+457 +1567+13467+234567
If we neglect the interactions greater than second order, then: h , = 1 + 2 5 + 3 7 + 4 6 +...
13 1
And the other six contrasts h , = 2+ h 3 = 3+ h , = 4+ h , = 5+ h,=6+ h , = 7+
15 + 36 + 47 +... 17 + 26 + 45 +.,. 16 + 27 + 35 +.,. 12 + 3 4 + 6 7 + . . . 1 4 + 2 3 + 5 7 +... 13 + 24 + 56 +,.,
Once the contrasts have been calculated, we can set up the experimental design, carry out the trials and enter the results in an effects matrix. In this case, the effects matrix is identical to the experimental matrix because all the interactions were used. Table 7.6 shows the experimental design and the results obtained for each trial. TABLE 7.6
EFFECTS MATRIX: INITIAL DESIGN SPECTROFLUORIMETER
Trial no
Excit 1
-
1 2 3 4 5
-
6
+
7 8
Scan Gain 4=123 5=12 -
+
+ +
-
+ -
-
P.M. 6=23
Sensi Selec Noise tivity tivity
+ +
1.22 5.5 -1.47 0.90 9.0 -1.47 5.33 20.0 2.30 5.64 12.0 -0.69 3.89 7.5 0.69 3.88 8.0 0.40 2.82 13.0 0.26 2.33 23.0 -3.91
-
+
+ +
-
-
-
-
-
+
+
+
+ +
+
-
-
-
Level
-
2.5
2.5
20°C
20
1
310v
2
Level
+
7.5
7.5
40°C
100
10
460v
4
132
3.3. Interpretation of the initial design The results of these eight trials are interpreted by examining one response at a time Sensitivity
TABLE7.7
TABLE OF EFFECTS
SPECTROFLUORIMETER (Sensitivity)
Mean
3.25
I 2 3 4
-0.06 0.78 -0.02 -0.14 0.02 -1.43 -0.06
5
6 7
Sensitivity appears to be influenced by two factors - the emission slit width (2) and the photomultiplier voltage (6). Interaction 26 is the only one which could be influent. Let us see if the contrast containing this interaction is significant. h = 3+ 17 + 26
+ 45 +...= -0.02
This is one of the smallest contrasts. The interpretation hypothesis we have adopted assumes that in this case all the terms of the contrast are zero. Thus, interaction 26 is also zero. We conclude that there are only two factors (2 and 6 ) that influence sensitivity, and that there is no interaction between them.
133
Selectivity
TABLE7.8 TABLE OF EFFECTS SPECTROFLUORIMETER (Selectivity) Mean
12.25
1
0.75 4.75 0.63 2.63 -0.25
2 3 4 5 6 7
0.38 1.88
Three contrasts appear to influence selectivity: h *, the emission slit width (2); h , scan speed (4); and h 7, recorder pen amplitude (7).
h , = 2 + 1 5 + 3 6 + 4 7 + ...
A, = 4+ h
= 7+
16 + 27 + 35 +,.. 13 + 24 + 56 +..,
Examination of the contrasts shows that the main effect of factor 2 is aliased with interaction 47. Contrasts h and h are large. The interpretation hypothesis adopted assumes that, in this case, interaction 47 could be significant. We cannot, therefore, reach any conclusion, as the high contrast h may be due to either a large main effect, 2, or to a strong interaction, 47, or to the effect and the interaction. The same ambiguity appears with factor 4 and interaction 27 in contrast h 4; and with factor 7 and interaction 24 in contrasts h
,
Background noise
There seem to be three influencing factors: excitation slit width (l), recorder gain (5) and photomultiplier voltage (6). Factors 1, 5 and 6 are not aliased with interactions 15, 16 and 56. There is thus no apparent ambiguity for this response. The results for the three responses show that we should consider a complementary design to obtain more information on selectivity. We must establish a complementary design in order to study this response more precisely. For each of the eight new trials we will measure the three responses, sensitivity, selectivity and background noise, and make our interpretation based on all sixteen trials.
I34
TABLE 7.9 TABLE OF EFFECTS
SPECTROFLUORIMETER (Noise) Mean
-0.48
1 2 3 4 5 6 7
-0.93 -0.02 -0.15 -0.11 -0.86 -1.61
-0.18
3.4. Construction of the complementary design We defined the initial Z7-' design by the independent alias generators: I
=
+ 1234 = + 125 = + 236 = + 137
This is only one of the sixteen fractional 274 designs forming a complete 27 design. The sixteen 27-4 designs are defined by all the combinations of 8 generators:
+
1234 l=+_125 1 = k 236 I = + 137 1=
Our problem is to select one of the I 5 remaining fractional designs which will allow us to dealias the effects of factor 2 from interaction 47. We could choose a fractional design giving the contrast
h'; = 2 + 15 + 36
-
47
This contrast associated with I , gives the system: h2=2+15+36+47
(,
=
We could then calculate:
2
+ 15 + 36
-
47
135
L2 = 2 + 15 + 36 + 47 + 2 + 15 + 36 - 47 - h'; = 2 + 1 5 + 3 6 + 4 7 - 2 - 1 5 - 3 6 + 4 7
h, +
=t w o times (+ 2 + 15 + 36 )
h,
=twotimes 47
And this would give us factor 2 associated with interactions 15 and 36. But it is preferable to have factor 2 completely dealiased from interactions. We must therefore try to obtain the contrast: h2
+2-15-36-47
and hence the system: h , =+2+15+36+47 h2
=
+ 2 - 15 - 36 - 47
which gives:
h , +- h2 1 2
~
i 2
+ 2 + 15 + 36 + 47 + 2 - 15 - 36 - 47= two times + 2 = + 2 + 15 + 36 + 47 - 2 + 15 + 36 + 47=two times(+ 15+ 36 + 47) =
The 274 fractional design which allows us to calculate h'; must contain the generators:
1 = - 125 I = - 236 I = - 137 Two fractional designs have these generators: The first one is I = + 1234=- 125 = - 236 = - 137 and the second is 1 = - 1234 = - 125 - 236 = - 137
We could choose either. Let us use the design: I = + 1 2 3 4 ~ -1 2 5 = - 2 3 6 = -
137
The eleven dependent generators of the complementary design are obtained from the four independent generators using the same technique as was used in the initial design. Multiplication in twos (C:
= 6)
1234. ( - 125 ) = - 345
136
1234.(-236) =-146 1234. ( - 137 ) = - 247 - 125. ( - 236 ) = + 1356 - 125. ( - 137) =+2357 -236.(-137) =+1267
Multiplication in threes ( C i 1234. 1234. -125. 1234.
Multiplication in fours (C: 1234. (
-
= 4)
( - 1 2 5 ) . ( - 2 3 6 ) =+2456 ( - 125 ) ( - 137 ) = + 1457 ( - 236). ( - 137) = - 567 ( - 236 ) ( - 137 ) = + 3467
.
.
=
1)
.
.
125 ) ( - 236 ) ( - 137 ) = -1234567
From this we obtain the sixteen AGS terms of the complementary design:
I
1234 = - 125 = - 236 = - 1 3 7 ~ 345 = - 146 = - 247 = 1356 =2357=1267=2456=1457=-567=3467=-1234567
The contrasts, ignoring interactions greater than second order, are then obtained directly fi-om the AGS:
h', L2 h'?
=
1 -25
-
37-46
=
2 -15
-
36-47
=
3
-
1 7 - 26-45
4
-
16- 27-35
h,
h'5
5
-
12 - 3 4 - 6 7
h6 = 6
-
14 - 2 3 - 5 7
7
-
13 - 24-56
h,
=
Using both the initial and the complementary designs gives the main effects dealiased from the second order interactions: 1= -[A, 1 + i,] 2
137
1 2= - [ k 2 2
+ h;] etc.
and the associated second order interactions:
etc.
TABLE7.10 EFFECTS MATRIX: COMPLEMENTARY DESIGN SPECTROFLUORIMETER
P.M. 6= -23 7= -13
Sensi Selec Noise tivity tivity 4.14 3.18 2.82 2.74 2.44 0.98
t-
+-
+
+-
5.66
5.63
Level
-
2.5
2.5
20°C
20
1
310v
2
Level
4
7.5
7.5
40°C
100
10
460v
4
3.5 12.0 14.0 14.0 8.0 6.0 14.0 14.0
3.56 -2.41 1.39 -1.90 -0.62 -3.22 0.02
-1.61
138
The complementary design is constructed from a basic 23 design with the independent generators: 1 = f 1234 = - 125 = - 236 = - 137
These generators can then be used to establish the columns of signs for each of the factors. Multiplying + 1234 by 4 gives: 4 = 123
Thus, the column of signs drawn up for factor 4 are those for column 123 of the initial design. Multiplying - 125 by 5 gives: 5 = -12
So the column of signs attributed to factor 5 is column 12 multiplied by -1. It is thus the opposite, all the signs are changed. We can continue by multiplying - 236 by 6 and then - 137 by 7. 6=-23 7=-13
The signs of columns 6 and 7 of the original design are changed to give the columns of the complementary design. Thus, the signs ofthe columns for factors 1, 2, 3, and 4 remain unchanged, while those for factors 5, 6 and 7 are changed. Table 7.10 shows the trials performed for the complementary design together with the experimental results and the contrast values. The results of all sixteen trials can be used to calculate the main effects dealiased from second order interactions. If we now go back to the problem of selectivity encountered for factors 2, 4 and 7, we see: 2 = -1 [ h z +&I=-[4.75+3.31]=4.03 I 2 2
h2 - A 2' ] = -[4.75-3.31]=0.72 I 2 4 =-[A4 1 2
+~q]=~[2.63+1.31]=1.97
1 I 2 7 - -[h4 2 -h,]=-[2.63-1.31]=0.66 2
139
1
[
7 = - h7 2
+h'7]-:[- -
1.88+1 31]=1.59
We can calculate the effects and interactions for all the factors in this way. The results are shown in the table ofeffects (Table 7.11)
TABLE7.11
TABLE OF EFFECTS
SPECTROFLUORIMETER (Initial and Complementary Experimental Designs) ~
Sensitivity
~____
Selectivity
Noise
Mean
3.35
1I .47
-0.46
1
-0.19 0.77 0.10
0.78 4.03 0.22 1.97 0.28 0.10 1.59
-0.99 0.11 -0.03 -0.20 -0.97 -1.26 -0.38
-0.03 0.72 0.41 0.66 -0.53 0.29 0.28
-0.06 -0.13 -0.12 0 08 0.12 0.10 0.20
2
3 4 5 6 7 25 + 37 + 46 15 + 36 + 47 17 + 26 + 45 16 + 27 + 35 12 + 34 + 67 14 + 23 + 57 13 + 24 + 56
-0.03 -0.13 -1.32 0.00
0.13 0.0 1 -0 12 -0.10
0.15 -0.11 -0.06
140
3.5 Interpretation of the initial and complementary designs We use all sixteen trials for the interpretation. In this case, the precision of effects and interactions is better than that from the results of the initial design having only eight trials. Let us examine each response in turn: Sensitivity The results of the initial design are confirmed. There are two influencing factors excitation slit width (2) and photomultiplier voltage (6). There is no interaction 26 as the corresponding contrast is small. Selectivity The results in Table 7.10 show that there are three influencing factors: excitation slit width (2), scan speed (4) and pen amplitude (7). The contrasts containing interactions 47 and 27 are a little high, but the experimenters considered them to be too small to be taken into account. They therefore decided that there is no significant interaction. Background noise The results of the initial design are confirmed, there are three influencing factors and no interactions The complementary design removed the ambiguities and the results are clear enough to give the conclusion of the study. Conclusion: There is no significant interaction . The influencing factors are not the @
- same for the three responses. The results can be summarized in a
B
table that also shows the optimal setting levels.
TABLE7.12
SPECTROFLUORIMETER Factors influencing the selectivity, sensitivity and background noise in setting up a spectrofluorimeter
I
-
Sensitivity
I
Selectivity
I
Noise
I
~
Excitation ( I ) Emission (2) Temperature (3) :\
+
+ I
-
I
I
I
141
Rules for settina UD the spectrofluorimeter: 1. The experimenters were not concerned with sample temperature.
2. The spectrofluorimeter should therefore be set up as follows: 0
excitation slit width (1)
level +
0
scan speed (4)
level -
0
recorder gain (5)
level +
pen amplitude (7)
level -
3. Two factors are difficult to control: the emission slit width (2) and the photomultiplier voltage (6). It would be best to adjust them for specific assay conditions, i.e., the emission slit should be wide for high sensitivity when determining traces, and narrower for high selectivity when assaying mixtures. Photomultiplier voltage should be adjusted last so as to minimize background noise without reducing sensitivity.
These recommendations were followed and the chemists obtained extremely precise determinations (Figure 7.5)
I
L 481 nm Figure 7.5 Fluorescence spectrum of benzo-a-pyrene, sensitivity/selectivity ratio.
obtained for an optimum
142
We have used all the techniques presented so far to analyse a 274 design. We studied seven factors using just eight trials in the initial experiment. This is called a saturated design as it does not allow evaluation of interactions. This type of design is extremely usefbl in the initial phase of a study. Its particular strength is that it provides the influencing and the noninfluencing factors within the experimental domain. If there are ambiguities in interpreting the initial design, the technique of progressive knowledge acquisition can be applied by running a complementary design to obtain hrther information, and hence resolve the problem. The theory of aliases is indispensable in looking for ambiguities. This is the only way of detecting the main factors which could be aliased with non-negligible interactions.
4. STUDYING MORE THAN SEVEN FACTORS The 274 design is constructed from a basic z3 design having three columns of main factors and four interaction columns. It therefore cannot be used to study more than seven factors. For eight or more factors, we must use a basic 24 design to build the design we need. With eight factors, we use the four initial columns to study four factors and select four interactions for the other four factors. The choice of interactions requires some care. We must: a. Take into account anything we know of the phenomenon, so as to choose the interactions that are most likely to be zero or small. b. Take the precaution of aliasing on interactions of the highest possible order, as these are the one that, without any particular knowledge of the phenomenon, are most likely to be small or zero.
We would use a 284 design to study eight factors, and a 29-s design to study 9 factors. The basic 24 design can be used to study up to 15 factors (215-'* ). Once we get to 16 factors, we must use a basic Z5 design. Table 6.8 can be used to select the basic design in connection with the number of factors to be studied. This number could, in theory, be as large as wanted by the investigator. c
5.
Avoid aliasing the main factors together or with low-order interactions, especially second order interactions. This precaution is automatically taken if we are carehl to use a high resolution design.
THE CONCEPT OF RESOLUTION
5.1. Definition of resolution When we are constructing a fractional design, we must consider the matrix of effects of a basic design and select columns of interactions for studying the extra factors. We generally choose high order interactions as these are likely to be small or zero. But this is not enough. We must also be sure that the main factors are aliased with interactions having the highest
143
possible order. It is better to alias the main factors with third order interactions than with second order ones. The concept of resolution is based on this idea. A resolution 111 design is a fractional design in which the main factors are aliased with second order interactions. This is the case with a 23-Ldesign defined by 1 = 123. When the contrasts h h and h are calculated we have: h1=1+23 h2=2+13 h3=3+12
A resolution IV design is a fractional design in which the main factors are aliased with third order interactions, and never with second order interactions. For example, a 24-1 design having the alias generator I = 1234: hl=1+234 h , = 2 + 134 h = 3 + 124 h = 4 + 123
In a resolution 1V design the second order interactions are aliased between themselves or with higher order interactions: h , 2 = a 34 = 12 + 34 h 1 3 = h 24 = 13 + 24 h ,1 = h23= 14 + 23
Let us look at an example in which several extra factors are aliased with interactions. If we wanted to study eight factors in sixteen trials, we would select a basic Z4 design, so that we could write 5 = 1234 6 = 123 7 = 124 8 = 234
We can now find the resolution of this 28-4 design. The independent alias generators are: 1= 12345 = 1236 = 1247 = 2348
We can then deduce the dependent generators
I =456=357=158=3467=1468=1378=12567=23568=2678=24578=1345678 These independent and dependent generators are used to write the AGS
144
I =12345=1236=1247=2348=456=357=158=3467=1468 =1378=12567=23568=2678=24578=1345678
When the terms of the AGS are multiplied to obtain the contrasts (equivalence relationship), the three-component generators give second order interactions, for example in contrast h in this example: h , = 4 + 5 6 + ...
The resolution is 111: a main effect is aliased with a second order interaction. The resolution of a design is equal to the number of components in the shortest alias generator in the AGS. The shortest generators in the AGS of the present example have three components: 456,357and 158. Let us return to the 2x4 and use other interactions to alias the four extra factors: 5 = 234 6 = 134 7 = 123 8 = 124 The alias generators set is: 1 =2345=1346=1237=1248=1256=2578=2345=1678 =1346=3478=3567=4568=2467=1457=12345678
The shortest generator contains four components, hence the resolution of the design is IV. The main effects are aliased with third order interactions, while the second order interactions are aliased between themselves. This way of aliasing main factors 5, 6 , 7 and 8 with interactions is better than the first method because the resolution is higher. Notation: The resolution is given in roman numerals, and is shown as an index. We
would therefore write 2:i4, and call it a two to the power eight-minus-four, resolution four design.
5.2. An example of a 2;
design: Plastic drum fabrication
The problem:
An investigator desires to rigorously control the volume of the plastic drums The design volume of 2 litres was found to vary too much = around this value. Careful examination of the fabrication conditions showed that eight factors could slightly alter the capacity of the drums z A complete factorial design was not possible because it requires running 256 trials The investigator decided to carry out only sixteen
e
-
145
"*
Factors
trials, and then, depending on the results, carry out either a D complementary sixteen-trial design or just a few complementary trials. p
Factor injection speed. ...Factor 2: injection temperature. Factor injection pressure. ..Factor moulding temperature Factor feedstock input speed. ..Factor feedstock supplier. Factor mixing speed. .Factor dwell time. 1:
3: 4: 5: 6: 7: 8:
Response The objective is to obtain a volume of 2000 cm3. The response chosen is the difference between the true volume and 2000 cm3. The responses could be positive or negative. The experimenter must propose a solution with a volume never smaller than 2000 cm3 and if possible not greater than 2002 cm3. Design The initial design is thus a :2; constructed from a basic Z4 design. Factors 1, 2, 3 and 4 were studied in the first four columns of the design, and factors 5, 6, 7, and 8 were associated with high-order interactions, so as to obtain a resolution IV design as follows: 5 = 234 6 = 134 7 = 123 8 = 124
There are thus four independent alias generators: 1=2345=1346=1237=1248
The eleven dependent generators were calculated by multiplying the independent generators in pairs, threes and fours. The fifteen contrasts were calculated from the AGS: I =1237=1248=1256=1346=1358=1457=1678=2345=2368 =2467=2578=3478=3567=4568=12345678
Each contrast contains sixteen terms, but we shall write only the main effects and the second order interactions. Hence: h, h2 h,
= L + ... = 2 +... =
3
f...
146
h, h, h, h, h,
=
h,, h,, h,,
=
A,,
=
h,, h,, h,,
= =
= = =
=
= =
=
4 +.. 5 +... 6 +... 7 +... 8 +.. 12+ 37 13+ 27 14+ 28 15+ 26 16+ 25 17+ 23 18+ 24
+ 48 + 56 + 46 + 58
+ 36 + 57 + 38 +- 47 + 34 + 78
+ 45 + 68
+ 35 + 67
The sixteen trials were carried out according to the design shown in table 7.13, the responses were the differences in volume from a reference volume of 2000 cm3. The resolution IV of this design provides the main effects unaliased with second order interactions. This means that we can probably obtain a good estimate of the main effects and immediately detect the influencing factors. However the second order interactions are aliased together, so that if one of the corresponding contrasts is high, we would be unable to decide whether second order interaction was responsible. TABLE7.13 28-4 1"
DESIGN
(I = 2345 = 1346 = 1237 = 1248) PLASTIC DRUMS
Response 47 35 -0 2 -1 0 05 10 7 15 0 72 -0 5 -0 1 66 26 15 9 73 08 94
147
TABLE7.14 TABLE OF EFFECTS PLASTIC DRUMS
Mean
5.15
1 2
-0.20 -0.10 3.20 0.10 -2.70
3 4 5 6 7 8
cm3 11 I1 I, I1 ,I I,
-0.05
0.20 1.90
12 13
,I ,I
I1
-0.30
0.50 -0.25 2.50 -0.10 -0.15 -0.30
14 15 16 17 18
II
II 11 ,I
I1 I1
Three factors seem to influence the final volume (Table 7.14): injection pressure (factor 3), feedstock input speed (factor 5) and the dwell time (factor 8). VOLUME VARIATION
A
(cm3 )
c -1
0
+l
INJECTION PRESSURE
Figure 7.6: Influence of the factor 3: injection pressure.
148
VOLUME
VARlATlON
(m3)
+I
0
1
FEEDSTOCK INPUT S P E E D
Figure 7.7: Influence of the factor 5: feedstock input speed.
VOLUME
4
3
2
/
-1
0
+I
DWELL TIME
Figure 7.8: Influence of the factor 8: dwell time. We can also see that contrast h ,5 is high. We will therefore have to do another study to identify the second order interaction responsible for this high contrast value, 15, 26, 38 or 47:
I,,
=
15+ 26
+
We will cover this question in Chapter 12.
38
+
47
149
Provisional conclusion: zi
The average excess drum volume found during the trials was about
5 cm3 This average value can be reduced by lowermg the injection pressure (factor 3), increasing feedstock input speed (5) and reducing the dwell time (8) These emergency measures will reduce the excess volume, but a complementary study will be necessary to explain the high value of the contrast h ,5, and to define the set-up values to be used to come closer to the specifications.
150
RECAPITULATION The three examples in this chapter have shown: The use of alias generator sets (AGS) for calculating contrasts The method for constructing and choosing a complementary design. How to dealias the main factors from second order interactions One major point of interpretation should be emphasised: it concerns the use of factors which do not influence the response They can often provide savings (e.g., stirring speed, supplier) and provide considerable flexibility when a compromise must be found between several responses. A factor may influence one response but not another. The choice of interactions to use in studying extra factors must take account of the resolution concept to avoid, whenever possible, associating the main effects with second-order interactions. The higher the resolution, the better the choice of design.
CHAPTER 8
TYPES OF MATRICES
1.
INTRODUCTION In the preceding chapters we have introduced three types of matrices: The experimental matrix. The effects matrix. The basic design matrix for constructing fractional designs
These three types of matrices can be confused, especially in the early stages, because they are so similar, and sometimes even identical. This Chapter makes it easier to identie each of these matrices by examining the principal similarities and differences between them. It will be easier to identifjr each of these matrices if we know more about them.
2.
THE EXPERIMENTAL MATRIX The experimental matrix is a table containing all the trials to be performed. It includes:
a. all the factors to be studied. b. the number of trials to be run. c. the levels assigned to each factor in each trial
152
The standard table can be drawn up in two different, but equivalent ways. In the first, the factor levels are given in usual or normal units (bar, cm, hour, etc.). In the second, these levels are expressed as coded units, with the significance of the -1 and +1 being shown in an appendix table. The first type is simpler and should be used if the trials are to be run by anyone who is not familiar with experimental designs.
TABLE8.1
EXPERIMENTAL MATFUX STABILITY OF A BITUMEN EMULSION I
Trial no
Factor 1 (fatty acid)
Factor 2 (HCI)
Factor 3 (bitumen)
1
low conc. high conc. low conc. high conc. low conc. high conc. low conc. high conc.
diluted diluted concentrated concentrated di1uted diluted concentrated concentrated
A A A A B
2 3 4
5 6 7 8
B B B
Let us return to an example we looked at in Chapter 3, the complete z3 design adopted for the bitumen emulsion stability example. The experimental matrix shown in Table 8.1 has been prepared using the first style. We have not adopted this form of presentation in this book as it is not suitable for developing the theoretical background of experimental designs and the general calculations involved. All the experimental designs are presented using coded units. The low levels of factors are indicated by -1 and the high levels by + l . This type of table contains only numbers that are easily read and remain valid regardless of the absolute level expressed in the usual units. A complementary table containing the specific values, in normal units, of the high and low levels of each factor, must be added in order the carry out the trials. Table 8.2 shows the complete 23 design for the bitumen emulsion stability example. It is the same as Table 3.1 in Chapter 3 Tables 8.1 and 8.2 are the same experimental design presented in two different ways. These two tables show that the experimental matrix is simply a working plan. This plan is used by the experimenter to perform the programmed trials.
153
TABLE 8.2 EXPERIMENTAL MATRIX STABILITY OF A BITUMEN EMULSION
3.
Trial no
Factor 1 (fatty acid)
I 2 3 4 5
Factor 2 (HCI)
Factor 3 (bitumen)
-
-
-
+
-
-
-
+
+ +
-
-
6
+
-
-
-
7
-
8
+
+ +
+ + + +
Level (-)
low conc.
diluted
A
Level (+)
high conc.
concentrated
B
THE EFFECTS MATRIX
This matrix is used to calculate the effects. It is no longer a working plan, but a mathematical tool for interpreting all the responses measured during the experiment. The signs +1 and -1 are no longer the levels, but are real figures, and these figures are used in the calculations. It is no longer possible to replace the +1 and -1 signs by the values in normal units as these figures no longer represent the levels.
TABLE8.3 1
2
3
-1
-1 -1
-1 -1 -1 -1 +1
+I -1 +1 -1 +1 -1 +1
+I +1 -1 -1 +1
+I +I
+I
+1
154
The effects matrix for factorial designs is a Hadamard matrix. It can be constructed using the columns of the experimental matrix (second form), assuming for the calculation that the signs representing the levels are real numbers (Table 8.3), to which can be applied the signs rule to obtain the other columns of the matrix. This subterfuge allows us to quickly and easily find all the columns to be used in calculating the effects and interactions. We must add a column of + signs to obtain the X Hadamard matrix. This column of + signs is used to calculate the mean (Table 8.4).
TABLE 8.4
Hadamard matrix derived from the experimental matrix by applying the signs rule 2
3
12 13 23 123
-1 -1 +1 -1 -I +l +1 + I -1 -1 +1 -1 -1 +1 +1 +I
-1 -1 -1
+1 -1 -1
1
x
=
+1 +1 -I +I +1 -1 +I -1 -1 +1 -1 -1
-1
-1 +I +I
-1 -1
-1 +1 +1 -1 +1 +I -1 -1 +1 +1 + I
+I +I -1 +I
+I
I
+1 +1 +I +I +I +1
+1 +I
This X Hadamard matrix links the response vector-matrix Y to the effects vector-matrix E according to the matrix equation:
Y=XE The experiments give the elements of the vector matrix Y. The rules set out in Chapter 3 are used to caIculate the elements of the vector matrix E (i.e., the mean, main effects and interactions) from Y and X.It can be seen that the X matrix is really a mathematical tool.
4.
THE BASIC DESIGN MATRIX FOR CONSTRUCTING FRACTIONAL DESIGNS
This matrix is used to construct a fractional factorial experimental design. It is therefore, by its nature identical to the experimental matrix.
155
The problem is to construct this type of matrix taking into account the number of trials to be carried out and the number of factors to be studied. Lists of all the principal matrix cases have been published in tables, and the reader can refer to them. However, the experimenter must be able to readily find the matrix that he needs for a specific study. The method of finding such a matrix is closely linked to the theory of aliases, and is of considerable help to the reader in interpreting results. A fiactional factorial design is easily obtained from an effects matrix. The first step is to choose an effects matrix having a number of lines equal to the number of trials to be performed. In the case of factorial designs this number is not any number, it is equal to 2" : 2, 4, 8, 16, 32, etc. trials. We have called this matrix the basic design because the - 1 and +I of the effects matrix are considered as levels of factors and not figures. The second step is to choose as many columns from the basic design as there are factors to be studied. The experimenter thus obtains a rectangular matrix having as many lines as there are trials and as many columns as there are factors. For example, we can construct a design with eight trials to study five factors as follows:
I . We first write out a Z 3 experimental matrix ( three columns, 1, 2 and 3) for eight trials (Table 8.5). TABLE8.5 1
2
3
-1 +I -1 +I -1 +I -1 +I
-1 -1 +I +1 -1 -1 +I +I
-1 -1 -1 -1 +1 +1 +I +I
Levels are considered as figures and the signs rule is applied to calculate the interaction columns, giving an effect matrix without the mean column I (Table 8.6). This is the 23 basic design. This matrix seems to be the same as an effects matrix, except that the -1 and +I are no longer figures, but experimental levels.
I56
TABLE8.6
Z3 BASlC DESIGN I
2
3
12 13 23 123
+ + + + - + - + + + + +
+ + + - +
+ - + + + + + + +
2 We then select five columns from the seven in the 23 basic design (Table 8.7). Theoretically we could use any column for any factor. But in practice, we select the columns so as to obtain the greatest possible resolution. We also take into account the experimental constraints. For example, if we know that an interaction is important, we do not choose its column to study an extra factor. TABLE8.7 25-2 EXPERIMENTAL DESIGN 1
2
3
4=12 5=13
+ -
+ + -
-
+ +
-
+ + + +
+ + -
+
3. The numbers -1 and + 1 are considered to be the low and high factor levels. Then, in order to define the trials to be performed, we can either add an extra table (Table 8.8), or replace the + I and -1 by their values in normal units. Lastly, the trials are
numbered.
157
TABLE8.8
EXPERIMENTAL MATRLX 25-2 FRACTIONAL DESIGN
Trialn"
I
1
+ + + + + + + +
2 3 4 5
6 7 8
1
2
-
-
+
-
+
+ +
-
-
+
-
-
-
+
+ +
iResponse
The fractional experimental design matrix can be transformed into a calculating tool as explained for the effects matrix. But this time only the contrasts can be determined, and the results correctly interpreted by using the alias theory. This topic was covered in Chapters 6 and 7, and we will examine it in more detail in the following chapters.
I58
RECAPITULATION
The significance of -1 and +1 in the different types of matrices are summarized in Table 8.9. TABLE 8.9
THE DIFFERENT TYPES OF MATRICES
Type of matrix
Significance of -1 and +I
Experimental matrix of a complete design
Levels
To perform experiments
Effects matrix of a complete design
Values
To calculate main effects and interactions
Basic design
Levels
Experimental matrix of a fractional design
Levels
To perform experiments
Effects matrix of a fractional design
Values
To calculate contrast.
To construct fractional designs. To choose levels of extra factors
CHAPTER 9
TRIAL SEQUENCES: RANDOMIZATION AND ANTI-DRIFT DESIGNS
1.
INTRODUCTION
Does the order in which trials are carried out make any difference to the results? In order to answer this question we must examine several aspects of experimental design, some practical and others theoretical. The sequence of trials is selected bearing both these aspects in mind. A sequence may be determined by experimental constraints. For example, let us assume that a fragile component of some equipment must be studied at two levels. To avoid breaking this component, it would be wise to carry out the low level trials first, followed by the high level trials. Another possible constraint is that setting up a system could be long and complex, and the experimenter may not wish to repeat it several times. This occurred during an experimental design run in a glass works. The experimenters wanted to study the influence of
160
furnace temperature. But changing the temperature has a certain inertia, so that it is impossible to go from a low temperature to a high temperature and back again between each trial. They decided to cany out all the low temperature trials first, and then the high temperature trials. There is only one column of signs which allows this strategy in a factorial design. If there are two factors that are difficult to set up, the trial order must be rearranged to reduce the number of changes. The first factor can be studied on the one-change column:
+ + + + and the second factor on the two-change column
+ +
This arrangement is easily obtained with several trial orders, e.g.: Trial no
factor 1
3
-
7
~
factor 2
+ +
5 1 2 6 4 8
There are many examples in which the experimental conditions themselves determine the sequence in which the trials are performed; they may be material, temporal or other constraints. Experimental constraints are often more important than statistical constraints. Systematic errors must also be taken into account when the order of trials is being decided. We shall analyse two types of systematic errors: drift errors and block errors.
161
1.1 Drift errors
The experimenter may suspect that there is a regular change over time or space in the phenomena that he is studying. This change leads to systematic variations in the response. For example, the ageing of a catalyst which progressively becomes less active. The yield of the chemical reaction will be different if the same trial is carried out at the beginning of the series or at the end. But this type of error does not prevent us using factorial designs. On the contrary, we will see that it is possible, despite any response drift, to obtain effects and interactions whose values are correct. 1.2. Block errors
This type of error is mainly due to uncontrolled factors which remain fixed throughout a series of trials, but have different values for another series. This type of error can be seen in the fertility of two plots of land. An experimenter wanting to study the intluence of rain, fertilizer, temperature and sowing date on crop yield may be obliged to run his trials on two plots whose fertilities are not necessarily the same. We will see that, despite this handicap, it is possible to obtain correct values for the effects of the factors studied. The reader will undoubtedly be able to find other examples, such as experiments carried out at two different times (night and day, summer and winter) or at two sites (two regions or two laboratories). Lastly, the variations may be due to causes that are not regular, periodical, or organised. These variations are random variations. Random variations are not treated in the same way as systematic ones. We will examine examples of several strategies that can be adopted depending on what we know of the perturbations that may occur during the course of an experiment. But before doing so, we must be sure that altering the sequence of trials does not change the effects calculated. Let us examine the effect of factor 1 in a z3 design.
Each term in this effect is obtained by multiplying +1 or -1 in the effects matrix by the response yi. Permutating two responses, i.e., two trials, 3 and 6 for instance, does not change the sum of these products, only the order of the calculations:
As the addition is commutative, the value of E, does not change when the yis are permuted. Consequently, the effect calculation does not depend on the order in which the trials are carried out. Let us now return to the subject of the sequence in which the trials must be performed. We shall begin by learning how to deal with small uncontrollable systematic errors, then examine systematic drift errors and, lastly (in Chapter 10) block errors.
162
2.
SMALL UNCONTROLLABLE SYSTEMATIC VARIATIONS
The experimenter knows that small variations due to uncontrolled factors can slightly alter the values of the responses he wants to measure, but he does not know exactly what these variations are. These small variations will introduce errors into each measurement. If they are purely random, the trials may be carried out in any order as they will introduce no systematic error. But if they are not, they will introduce systematic errors, and then statistical tests will no longer be valid, as they assume that errors are randomly distributed. However, if we can randomize the small unknown systematic errors, then we can use statistical tests. For this we must carry out the trials in a random order. This is called randomizing the trials. The technique of randomization is very simple. The trial numbers are written on pieces of paper, mixed well, and randomly selected; the first number drawn is the trial performed first, and so on. A more sophisticated way is to use a spreadsheet program which can generate random sequences of numbers. Randomization should be used whenever there is any suspicion of an uncontrolled variation in the levels of factors which cannot be detected, measured or controlled. This is often the case in agricultural trials in which soil fertility varies from one area to another, but the factors giving rise to it cannot be controlled. These uncontrolled factors lead to both random and systematic errors (Chapter 4). Randomization of trials makes the distribution of the systematic errors random, so allowing the application of statistical tests. We should not forget the importance of these tests for determining the influence of a factor by comparing the effect to the random error. Randomization is a way of obtaining really random error. But randomization can lead to serious misinterpretation as it tends to increase the value of the random error. With randomization, the systematic errors inflate the random error. This is the opposite of a good strategy for the experimenter, who must look for the smallest possible random error in order to best evaluate the influence of the factors studied. Hence, randomization must be used carefhlly: it must allow statistical tests to be used, but must not inflate the random error too much. These two objectives are not incompatible, and we shall see that the best approach is to first control the systematic errors - whenever that is possible - and then randomize those trials which may still be randomized. The experimenter can reduced and obtain a good estimate of the experimental error by choosing the appropriate trial sequence. When trials are randomized, the order in which they are carried out does not conform to their original numbering. We must therefore distinguish between the name of the trial and the order in which it is performed. We use the convention of indicating the trial by its number and the order of trial execution by a number in brackets following it (see Tables 9.4 or 9.6).
3
SYSTEMATIC VARIATIONS: LINEAR DRIFT
Let us look at drift, which is a form of systematic variation. Drift occurs when the response of each trial increases or decreases by an increasing amount over the drift-free response. This is shown more clearly by examining the theoretical case of linear drift. The response of the first trial is y , when there is no drift. When there is drift it is yl, where y ; = y , + h and h is the incremental increase in drift.
163
The drift-fiee response for the second trial is y2, and the response with drift is y ; , where
y ; = y2 + 2h. Table 9.1 summarizes all the responses for a Z3 design. TABLE9.1
Linear Drift
Responses without drift.
Responses with drift.
y' = y + I h 1
1
y',=y + 2 h 2
y' = y + 3 h 3
3
y' = y + 4 h 4
4
y' = y + S h 5
5
y' = y + 6 h 6
6
y', = y , + 7 h y' = y + 8 h 8
8
The effects matrix of a 23 design allows us to calculate the influence of the drift on each effect and each interaction. Table 9.2 shows such a calculation. TABLE9.2
Influence of drift on the effects and interactions of a Z3 design
Trial no
I
Response
+ + + +
Ih 2h 3h 4h 5h 6h 7h 8h
-
+ + + +
Drift
4.5h
0. 5h
Ih
2h
Oh
Oh
Oh
Oh
164
We can see that the effects are modified as follows: E',
=
El
El2
=
E,
E',
=
E,
Ell2
=
El2
Ell,
=
El,
El23
=
E23
E'l2, =
I'
=
El23 I
+ + +
0.5 h
+
4.5 h
1.0 h 2.0 h
The four interactions are not biased by the drift, while the main effects are. Clearly, it would be much better if we could arrange things so that the main effects are not biased by drift. It can be seen that the order in which the + and - signs appear in the columns of the interactions cancels the influence of drift. We must therefore organise the trials so that the main effects use this special order of signs. Written in Box notation, three of the four interactions in the original design, 12, 23, and 123, are chosen to give their columns of signs to the three main effects, l', 2' and 3', of the new design. 1' = 123 2 ' = 12 3 = 23 The first column of signs, l', will contain the - and + signs of interaction 123; the second column contains those of interaction 12, while the third column contains those of interaction 23
TABLE 9.3 Order of trials to obtain drift-free main effects Old Design
Trial no
1
2
New Design 3
2'
3'
Trial no
+
+ +
-
7 6 2 3 4 1
+ +
5 8
-
+ + ~
-
+
-
-
165
This produces the new experimental matrix, in which the trials are the same as those of the original matrix, but they are in a different order (Table 9.3). According to the hypotheses we have adopted, we must use the trial sequence: 7 6 2 3 4 I 5 8. The main effects are then independent of drift, but the interactions still contain an error, as shown by the effects matrix of the new design (Table 9.4)
TABLE9.4
Influence of drift on the effects and interactions of a Z3 design 2'3'
4 .5 h
Oh
Oh
Oh
2h
0.5h
Response
-
+
-
-
+
-
6h
+
+ +
8h
Oh
lh
-
Drift
1'2'3'
The reader should remember that the results are affected by the order of the trials when there is drift; this order should therefore be chosen to provide pertinent conclusions about the main effects and interactions. We should be particularly careful about those interactions which, in this case, do not represent true interactions or an estimate of experimental error, but also include an estimate of drift. This gives us a way of detecting drift, as there are three values which are proportional to each other in the ratio 1: 2: 4. The order of trials shown above is not the only one which provides the main effects unaffected by a linear drift. A total of 144 different orders will give the same result for a 23 design. Table 9.5 shows some of these, and the complete list is given in Appendix 3 [20]. Provided that we have taken the precaution of arranging the trials according to oneof the orders given in Table 9.5, we can detect drift (by examining the interactions) while still obtaining a good estimation of the main effects.
166
TABLE9.5 Main effects free from the influence of linear drift
z3 Factorial Design 7 6 4 7 4
6 7 7 4
2 3 5 2
3
4
1
2 2 5
4
1
1 1
6
5
6
4 5 3 2 2
3
6
3
7 4 6 6 4
3 5 2 2 5 3 3 5
6 6 7 7
7
6
1
3 5 3 5 2 2 2
4 6 7 7 4 6 4
5
7
4 6 4 7 7 5
1 1
5 5 3 3 2 2 1 1 1 1
8 8 8 8 8 8 8 8 8
1
8 8 8
1 3
8
etc
4.
WHEN SHOULD TRIALS BE RANDOMIZED?
We can understand the consequences of randomization and the choice of anti-drift designs a little better with the aid of an example. We shall call this the powder mill example. Three investigators working in three separate laboratories carry out the same study. They choose three different strategies - simple randomization, a specific order to take into account linear drift and a complete set of experiments to measure effects, interactions and drift.
Example: The powder mill The problem:
*
Three investigatorswant to increase the amount of powder produced
I by their mills They have exactly the same kind of mill and all must
4
grind the same product The factors to be taken into account by the three experimenters are 0 The rotation speed (factor I) 0
The crushing pressure (factor 2) The grinder head clearance (factor 3)
They are aware that grinder wear could introduce drift The response ~
ISthe mass of powder (grams) of the correct particle size produced by
Ibi
ii
167
the mill after each trial of the same duration. But each investigator chooses a different experimental strategy Let us examine these three !f strategies. ~
7r:
4.1. First investigator's strategy The first investigator analyses his problem and decides to use a 23 design. He decides to randomize the trials in order to overcome the systematic error introduced by grinder wear. The sequence drawn is: 1 5 8 7 4 6 2 3. He carries out the trials and the results are shown in Table 9.6.
TABLE9.6 POWDERMILL First experimenter Trial no
Factor 1 (rotation)
Factor 2 (pressure)
Factor 3 (air clear.)
Response
433 337 332
+ Level - 1
40
4
0.20
Level +1
60
8
0.30
The eight responses are used to calculate the effects and interactions shown in the effects table (Table 9.7)
168
TABLE9.7 TABLE OF EFFECTS
POWDERMILL First experimenter Mean
369
1 2 3
25
9,
8 9
II
12 13 23
51 3
123
-13
22
grams
11
, ,I
,, II
The first investigator interprets his results as follows: he knows that the error of each response is k 1 gram, so the standard error of the effects and interactions is then slightly less than 0.5 gram. If he take three times the standard deviation as experimental error, then the interpretation is:
- one major effect - rotation speed (1). - two minor effects, crushing pressure (2) and grinder clearance (3). - one very large interaction (1 2). - one large interaction (23) and one smaller interaction ( 1 3).
4.2. Second investigator's strategy The second investigator also decides to use a Z3 design, but to guard against the systematic error introduced by grinder wear he uses a trial sequence that eliminates the drift error from the main effects. To take into account statistical constraints, he selects at random one of the trial orders in Appendix 3 - order number 107, which is: 6 7 3 2 4 1 5 8. The results of his experiments are shown in Table 9.8.
169
TABLE9.8
POWDERMILL Second experimenter Trial no
Factor 1 (rotation)
Factor 2 (pressure)
+
-
-
+ +
Response
+ +
+ +
-
Level - 1 Level +1
Factor 3 (air clear.)
-
-
-
+
-
-
-
-
-
-
+
+
+ +
40 60
4 8
0.20 0.30
216 338
TABLE9.9 TABLE OF EFFECTS POWDERMILL
Second experimenter Mean
371
1 2 3
48 26 -6
9,
12 13 23
-58 3 -16
,I
123
-29
8,
grams I,
,, I1
The interpretation takes drift into account. Order number 107, the anti-drift sequence (Appendix 3), shows how drift modifies interactions. We therefore know that the main effects
170
are not corrupted by drift, and that interaction 13 is not corrupted, while the other three are biased as follows:
- interaction 23 contains 0.5 x the drift error. - interaction 123 contains 1 .O x the drift error
- interaction 12 contains 2 x the drift error.
If we examine these three interactions we can see that they are almost in the ratio 1:2:4, so that it is very likely that the interactions are negligible and that they measure a systematic drift error of about 30 grams between each trial. The most important influencing factors are: - The rotation speed ( 1 ) and
- the crushing pressure (2). TABLE9.10 POWDER MILL
Third experimenter
Response 500 493 460 410 426 402 393 410 363 434 335 26 1 309 216 285 338 263
40
Level - 1 Level 0 Level +1
I
50 60
4
I
6 8
0.20
I
0.25 0.30
I
171
4.3. Third investigator's strategy The third investigator has the time and budget to study the drift by carrying out about 20 trials. Like the first two investigators, he decides to use a 23 design. He studies the driR by sandwiching between each trial of this design a trial at the centre of the experimental domain. If there were no drift, the response of this point should always be the same. If, however, there is drift, the response will vary in a regular fashion. These trials at the centre of the experimental domain are numbered with a 0 before the trial number. The investigator thus plans 9 + 8 = 17 trials. Table 9.10 shows the trials and the responses. These trial results are interpreted in two steps. The first examines the drift and the second corrects the responses and calculates the effects and interactions. Examination of Drift The drift can be followed by the changes in the central point response (trials 01, 02, 03, 04, etc.) as the experiment progresses. The curve in Figure 9.1 shows this change. The drift is marked and non-linear, being greater at the start than towards the end.
Grams 500 400
300 200
01 02 03 04 05 06 07 08 09
Trial number
Figure 9.1: Powder mill study. Ordinate: yield at each trial. Abscissa: chronological sequence of trials.
172
Correction of the responses The results of the 2j design trials are shown on the same graph, we can therefore refer their position to that of the drift curve. The point for trial number 6 (fist executed trial) is placed between trials 01 and 02. The point for trial number 7 (second executed trial) is placed between trials 02 and 03. The remaining points are arranged in the same fashion (Figure 9.1). The effects and interactions are calculated by eliminating the influence of drift &om each response. Thus, for trial number 6, the mean variation in the yield between points 01 and 02 is 500 - 460 = 40 grams
We assume that this variation is linear between the two trials, 01 and 02, so that the value on the drift curve at trial number 6 is 480 grams. The measured response of trial number 6 is 493 grams, or 13 grams greater than the value on the drift curve. If there were no drift the value of the response at the central point would have remained at 500 grams. We must therefore add these 13 grams to the 500 to obtain the drift-free response of trial number 6, 513 grams. The difference between 500 grams and the response calculated for the central point is 57 grams for trial number 7. This must be added to that of trial number 7 to obtain the corrected response. 410 + 57 = 467 grams.
This process must be repeated for all eight responses in the design. The results are shown in Table 9.1 1.
TABLE 9.11
TABLE OF EFFECTS
POWDER MILL Responses corrected from drift Trial number
Corrected responses
6
513
7
467 492
3 2 4 1 5 8
530 584 438
420 566
173
These corrected responses can be used to calculate the effects and interactions as if there were no drift, as shown in Table 9.12.
TABLE9.12 TABLE OF EFFECTS
POWDERMILL Third experimenter Mean
501
1
47
II
26 -10
II
2
3
grams
11
12 13 23
-1
I,
123
1
0
$9
1
,,
1
This clearly shows that the interactions are non-significant and that they may be considered as zero. The three factors studied all influence the results; their effects are shown in Figures 9.2, 9.3 and 9.4 GRAMS
550 500 450
-1
0
+l
ROTATION SPEED (1)
Figure 9.2: Influence of factor 1 on powder mill yield.
174
GRAMS
550
/
500 -
I
450
-
, b
Figure 9.3: Influence of factor 2 on powder mill yield.
GRAMS
A
t
550 500 -
450
I
< I
-
I
I
I 1
-1
0
+I
GRINDER AIR CLEARANCE (3)
Figure 9.4: lnfluence of factor 3 on powder mill yield.
175
The third investigator can therefore conclude: Conclusion: The three factors studied influence the powder mill yield. The most d important is rotation speed, which must be at least 60 rpm Crushing pressure must be as great as possible, 8 in this case, while the grinder clearance must be 0 20, the lowest setting. These factors do not interact There IS a large drop in yield from one trial to another due to grinder wear
Recommendation. A different type of grinder should be used to avoid loss of yield. The experimenter could propose a new study in which a higher rotation speed and greater crushing pressure are examined to see if they will increase yield
5.
8
I
RANDOMIZATION AND DRIFT The approaches used by the three investigators can be compared by regrouping effects
and interactions in Table 9.13. TABLE9.13 POWDERMILL
Comparison of the effects obtained by the three methods First Experimenter
Second Experimenter
Third Experimenter
Randomization
Special Order
Corrected
Mean
369
371
501
1 2 3
25 8 9
48
47
26 -6
-10
12 13 23
51 3 22
-58
1
3 -16
1 -1
123
-13
26
1 76
We can now examine the results of each strategy, knowing that those of the third investigator are the best. Simple randomization
The effects and interactions are incorrect. The investigator’slack of foresight could result in a catastrophe. Randomization is a useful technique that should be used as much as possible, but it provides no protection from experimental traps. The problem is not due to randomization, but to the investigator not thinking enough about the problem. Randomization is no substitute for reflection. In this example randomization obscured the systematic drift error. Specific order to obtain the main effects free of drift.
The investigator was not sure that there was drift, but he could only run eight trials. He selected the best experimental conditions, and obtained good results for the main effects, but could not come to any conclusion about the interactions. This strategy is usem when only a few trials can be run. Complete correction of all responses for drift
The effects and interactions are correct, but the price paid is a large number of trials.
I77
RECAPITULATION 1, The order in which trials are carried out is most important. 2. If only random errors are suspected, the trials can be run in any order. 3. Randomization of trials allows errors due to uncontrolled factors whose level
variations introduce systematic errors that are not controlled to be considered as random. Hence statistical tests can be used despite the presence of these systematic errors. 4. If drift is suspected, a sequence of trials must be selected that provides drift-free
main effects, but gives incorrect estimations of the interactions and average. The list of these sequences for a 23 design is given in Appendix 3. 5 . Drift can be measured by carrying out regular measurements at the central point,
using this to correct the responses and thus obtain drift-free effects. This method requires a few more trials but has the advantage of giving accurate main effects, interactions and average. 6. If systematic errors due to drift are suspected, and the investigator wishes to randomize the trials, it is always possible to choose at random one of the special orders (for 8 trials) shown in Appendix 3.
This Page Intentionally Left Blank
CHAPTER 10
TRIAL SEQUENCES BLOCKING
1.
INTRODUCTION
Chapter 9 dealt with the systematic error due to drift. The present chapter deals with the second type of systematic error, block errors. This type of error is mainly due to uncontrolled factors that remain fixed throughout a series of trials, but have different values in different series of trials. One of the best examples is the fertility of two plots of land. An experimenter wanting to study the influence of rain, fertiliser, temperature and sowing date on crop yield may be obliged to run his trials on two plots whose fertilities are not necessarily the same. We will see that, despite this handicap, it is possible to obtain correct values for the effects of the factors studied. The reader will undoubtedly know of other examples, such as experiments carried out at different times (night/day, summer/winter), at two different sites (regions or laboratories) or by two different persons.
180
2.
BLOCK VARIATIONS
The investigator could cany out his research in two or more groups of trials. Factors act similarly on the responses, but there may be a systematic difference because the trials were not run in the same batch. For example, an agronomist may be studying the yield of cereal fiom two similar plots having different fertilities. Similarly, a technician may be studying the setting up of some apparatus whose response is influenced by the difference between the morning and evening ambient temperatures. These systematic differences may be overcome by arranging the trials in groups generally called blocks. We will examine three examples of blocking - which is the art of arranging trials to eliminate the influence of a troublesome factor, such as differences in fertility, ambient temperature, etc.
3.
BLOCKING
Example: Preparation of a mixture The problem:
The investigator cannot prepare all the mixture he required to carry
p out a z3 design at one time He was obliged to prepare the mixture in
two batches, and the two were not completely identical. Let us assume
f that they were very similar, but that one was a pale red (r) and the other @
*
was a deep pink (p) Can the planned experiment be performed despite this difference?
s And if so, how should the trials be run to ensure that the effects
s calculated would be the same as if the two mixtures had been & identical7 A way must be found by which the deviations due to the different mixtures cancel each other during the calculation of the main effects and interactions. We assume that the red mixture introduces an error into the response with reference to the homogeneous mixture, and that the pink mixture introduces an error E ~ . Had a single, uniform mixture been used, the effect E, in a z3 design would be:
When the pink mixture is used for trial i, yi becomesy'i = yi+ E When the red mixture is used for trial i, y, becomes y', = yi+ E,.
~ .
The arrangement of trials which leaves E, with the value of the homogeneous mixture should incorporate E, and E~ with + signs as many times as with - signs. We could, for
181
example, use the pink mixture €or trials 1, 4, 6 and 7 and the red mixture €or trials 2, 3, 5 and 8. -
Pink mixture: trials: I , 4, 6, 7 Red mixture: trials: 2, 3, 5, 8.
We can then calculate the effect E‘, of factor 1 under these experimental conditions:
simplifylng
then
E; = E, The effect of factor 1 is thus the same as if there was a single mixture for all eight trials The effects of factors 2 and 3 can be similarly obtained. The effect of factor 2 is:
This can be simplified to:
or
The effect of factor 2 calculated with the two mixtures, red and pink, is equal to the effect that would be obtained using a single homogeneous mixture in all trials. The effect E’3 of factor 3 is as follows:
simplifying
I82
or E3 = E3
Here again the effect calculated using the red and pink mixtures is the same as that which would have been obtained with a single homogeneous mixture for all 8 trials. The choice of trials is thus quite suitable, as we have obtained the three main effects as if there had been a single mixture. These results can be explained graphically, with the experimental domain being a cube, as there are three factors (Figure 10.1). The red and pink mixtures are arranged so that there are two types of mixture on each side of the cube.
I
I
7
8
3
4
Figure 10.1: There are two red and two pink mixtures on each side of the cube
We know that the effect of a factor is half the difference between the average response at the high level of a factor and the average response at the low level of the same factor. The mean of the responses at the high level of a factor always includes two red trials and two pink trials (Figure 10.1) The mean of the low level responses of this factor also includes two red trials and two pink trials When the difference of the means is taken, the error E, appears twice with + signs and twice with - signs, and the same for E*, so that the effects are the same as if the mixture had been homogeneous We now calculate interaction 12 from the trials arranged as in Figure 10 1
I83
Once more, the error due to the two mixtures does not corrupt the value of the interaction. The arrangement of the experimental points provides the same value for the interaction as would be obtained if a homogeneous mixture have been used. The reader can check to make sure that the same holds true for the second order interactions, 13 and 23. But what about interaction 123?
So El,, is not equal to El,,. El,, measures the interaction plus half the difference between E, and E ~ What . does this half-difference mean? E, and E measure the variation in the response due to the different natures of the two mixtures. The {alf-difference between them is thus a measure of the effect of the type of mixture It is just as if we had introduced an extra factor, with the high level being the red mixture and the low level the pink one. This fourth factor was studied on interaction 123.
The experiment is said to have been performed in two blocks: First block, the red mixture, trials 2, 3, 5 and 8, i.e., using the + signs of interaction 123 Second block, the pink mixture, trials 1, 4, 6 and 7, i.e., using the - signs of interaction 123 We can use the concepts introduced during our study of fractional designs (Chapters 6 and 7) There were three initial variables, 1, 2 and 3 and a fourth was introduced, variable 4 The blocking resulted in the column of signs of interaction 123 being used to study variable 4
184
This is equivalent to studying four factors using a basic Z3 design, i.e. there is an extra factor the type of mixture. We can therefore apply the alias theory to blocking. A
E,.
/, I EFFECT OF MIXTURE
&P
II 1
I
I
c
+I Mixture
0
-1
RED
PINK
1 Figure 10.2: -[cr - e p ] measures the effect of the type of mixture. 2
Blocking can be expressed in Box notation as: 4 = 123
This introduces the alias generator I = 1234
We carried out a 24-1 design in which factor 4 is aliased with interaction 123. We therefore have: El,, = 4+123
The technique of blocking is used each time systematic variations between groups of trials are suspected. This technique allows us to calculate the effects and interactions free from
the influence of these systematic variations.
185 Conclusion: Despite the difficulty posed by using two different mixtures instead of one homogeneous one, it was possible to measure the main effects and all the interactions except one without bias. This could be done because the experimenter used the technique of blocking in choosing the trials of each block. Only one interaction is incorrect, it is aliased with the blocking factor
ti
Note: Blocking assumes that the effects are the same in the two blocks. Here, the effects of factors 1, 2 and 3 were assumed to be the same in the red and pink mixtures. Blocking cannot be used unless this assumption is verified. Blocking is easy to perform. In a design, a high order interaction is chosen as this is likely to be almost zero, and the + signs are used to form one block of trials, while the - signs are used for the other. The design is thus divided into two half-designs. 0
Blocking can be done on more than one factor. The same technique is repeated choosing several interactions. We shall examine an example of this in the next section. Blocking and randomisation are not incompatible. It is always possible to randomise the trials within each block. Blocking and drift are not incompatible. A special order can be selected within each block in order to eliminate the influence of drift on the main factors (see Appendix 3 for an example).
4.
BLOCKING ON ONE VARIABLE
Example: Penicillium chrysogenum growth medium (continued) We examined the experiments of Owen L. Davies [ 1 1 1 on the nutritional medium of Penicillium chrysogenum in Chapter 3 . One point was not clarified at that time - interaction 12345 appeared to be too great. This was because the investigator had not run all his trials at once, but performed them at two different times and in two separate sub-experiments. As he suspected systematic errors between each sub-experiment, he camed out a blocking using interaction 12345. He thus introduced an extra factor, factor 6, the sub-experiment. The design used can then be thought of as a ffactional 26-' design, with the alias generator:
186
J
=
123456
Interaction 12345 is thus aliased with factor 6, and contrast h, is measured. This is equal to: h, = 6 + 12345 If we assume that interaction 12345 is zero, then contrast h6 measures the effect of carrying out the two batches of trials at different times. The levels of several uncontrolled factors could change. It is interesting to take a closer look at the way in which these trials were run. Dr Davies chose the - signs of interaction 12345 to select the trials for the first batch; so he ran a first block of trials as follows: 1 4 6 7 10 11 13 16 18 19 21 24 25 28 30 and 31
This block is half the 2"' independent alias generators:
design, or a sixteen-trial 26-2 design (Table 10.1) with the
1 = 123456
and I=-6
The AGS is thus I
= -6 = -12345=
123456
This can be used to calculate the contrasts. For the main factors, we get: h,
=
1
-
h,
=
2
-26
h,
=
3
-
36 + 12456
-
h,
=
4
-
46+12356
-1235
h,
=
5
-
5 6 + 12346
-1234....
h,
=
6
-
I +123456
-
16 +23456
+
13456
-2345 -1345 1245
12345
+ h, + h, + h, + h, + hj _..+ h,
I
1-16
E
2-26
F
3 - 36 4 - 46
I
5 - 56
I
6-
I
I87
TABLE10.1
EFFECTS MATRIX: FIRST BATCH PENlClLLlUM CHRYSOGENUM GROWTH MEDIUM
----
Trial no
7om Iq. 1
Lactose Precurs 2
3
;od.
nit.
4
3ucose Batch 1 5 5 = -12345
Response 142 109 162 200 108 146 200 I18 106 88 113 79 101 72 83 145
1
4 6 7 10 11 13 16
18 19 21 24 25 28 30 31
Contr
-18.6
-3.6
14.2
-1.6
-24.9
-123.2
Dr Davies chose the + signs of interaction 12345 for the second batch of trials. Thus the second block contained trials: 2 3 5 8 9 12 14 15 17 20 22 23
26
27 29 32
These sixteen trials were organised in a 26-2design (Table 10 2) having the AGS: 1=6=12345=123456
188
TABLE10.2
EFFECTS MATRIX: SECOND BATCH PENlClLLlUM CHRYSOGENUM GROWTH MEDIUM
Lactose Precurs
Trial no
2
2 3 5 8 9 12 14 15 17 20 22 23 26 27 29 32
Sod. nit. Glucose 4
3
5
Batch 1
+ + + + + + + + + + + + + + + +
-
+ -
+ -
+ -
+ -
+ -
+ -
+ -
+
Contr.
-16.6
4.75
17.87
3.6
Zeesponse
i = +12345
-16.9
114 129 185 172 148 95 164 215 106 98 88 166 114 140 130 110
135.87
The contrasts of the main factors were deduced from the AGS:
al
=
1
+16
h2 = 2 + 2 6
h',
= 3
+36
h,
= 4
+46
h,
= 5
+56
h,=
6
+ I
+ +
2345+23456
+ h', z
1 + 16
1345+13456
+
h2
2 + 26
+ L3 E + h, z + hs I + h, z
3 + 36
+ 1245+ 12456 + 1235+12356 + 1234+12346 + 12345+ 123456
The results of the two designs can be interpreted in two ways
4 + 46 5 + 56
6+ I
189
1. By using the addition and subtraction formulae and neglecting high order interactions : =-17.6
16
+ h’,] = +[+18.6-165]
= -[-Al 1
2
=+1.1
-3.624.75] =+0.6
+3.62+4.75] = 4 . 2
3 =-[+A, 1 2
+h’,]=~[+14.25+17.87]=+16.1
-14.25+17.87] =+1.8
-A,
+h,]=2[+1.6+3.6]=+2.6 * 1
-24.9-16.91 = -20.9
+24.9-16.91~+4.0
-123.25+135.87] =+6.3
I = -[-h, 1 2
+h’,]
= $+123.25+135.87]=+129.6
This approach is inconvenient, but it has the advantage of showing that there may be interactions between the blocking variable (in this case number 6) and the main factors.
I90
2. By treating the 32 trials as a single unit and calculating the effects directly from the responses. This was the technique used in Chapter 3, but it has the shortcoming of
masking the second order interactions between the blocking variable and the main factors, except if we note that: 16 = 2345 26 = 1345 36 = 1245 46 = 1235 56 = 1234
The fourth order interactions (1234 and 1345) which appear to be high, are thus only second order interactions between factor 6 and factors 2 and 5 . It is certain not worth correcting the responses by +6.3 or -6.3 in order to recalculate the effects, as the appropriate choice of trials in each of the two blocks allowed the effects of the main factors to be obtained free from the effect of running the two batches of trials at different times.
5
BLOCKING ON TWO VARIABLES
5.1. Example: Yates' bean experiment The Problem: example is one of the oldest published experimental designs. It i$ - wasThis performed in 1935 by Yates, and published in 1937 [21]. He was interested in the influence of five factors on the yield of a specific species of bean. i2 9
The triat results should form the basis of advice to growers to help I them obtain the best harvest at the lowest cost We shall retain the .@ original units used by Yates for the sake of historical accuracy, but we @ will use the calculation and notation which we have adopted throughout D this book a
Factors and domain
The five factors studied were: level Factor 1 : Space between rows Factor 2: Amount of manure Factor 3 : Amount of nitrate,
-
18 inches, 0 tonslacre 0 Iblacre,
level + 24 inches I0 tonslacre 50 Iblacre
191
level Factor 4: Amount of superphosphate Factor 5: Amount ofpotash
0 lblacre, 0 Ib/acre,
level + 60 lblacre 100 Ib/acre
Response
The response was the weight of beans harvested per trial. Yates decided to use a complete factorial 25 design, but was carefil to divide the experiment into four blocks of 8 trials, each block on a separate plot. He knew that there could be differences in the fertilities of the plots, and attempted to obtain more homogeneous results by separating the trials into the 4 blocks. The underlying reasoning was that, although there are systematic differences in plot fertility, the effects of the factors studied would be the same regardless of the plot used. Blocking thus allowed him to measure the effects directly without extra calculations. It also allowed him to assess the differences in the fertility of the plots
BLOCK
BLOCK
I
II
124 135 -
124 + 135 -
BLOCK
BLOCK
IV
111
124 + 135 +
124 135 +
-
Figure 10.3: Yates' bean experiment-arrangement of the four blocks on the plot of land.
Yates chose interactions 124 and 135 for the blocking, i.e., the signs of the 32 trials of interaction 135 were divided into two blocks; the first half contained all the - signs of 135 and the second contained the + signs. Interaction 124 then divided these blocks into two. Table 10.3 shows the assignment of trials to the four blocks, labelled I, 11, 111 and IV. This introduced two extra factors:
192
- Factor 6, which measured the effect of the difference in fertility between blocks I1 and IV for one part, and blocks I and 111for the other.
- Factor 7, which measured the effect of the difference in fertility between blocks I11 and IV for one part, and blocks I and I1 for the other.
Block I Block I1 Block I11 Block IV
6 = 124
7 = 135
-
-
+ ~
+
-
+ +
Figure 10.4 shows how blocking was carried out and the arrangement of trials in each block. The trials within each block were randomised.
,
Figure 10.4: Random distribution of trials within each block.
193
TABLE10.4
EXPERIMENTAL MATRIX YATES' BEAN EXPERIMENT
Trial no
Factor 2 (manure)
Factor 3 (nitrate)
Factor 4 (s. phos.)
Factor 5 (potash)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
LevelLevel +
+ + + +
+ + + +
-
-
+
-
-
-
-t
-
-
+ + + +
+ + + +
+ + + + + +
-
-
-
-
-
-
-
+ +
-
-
-
-
+ + -
-
-
-
+
-
-
-
-
-
-
-
+ + + + + + +
+ + +
+ +
+ + + +
+ + + + +
+ + + +
0 10
0 50
0 60
0 100
+ -
18 24
-
+ + + +
-
I I
-
-
-
+ + + + +
Response
66.5 36.2 74.8 54.7 68.0 23.3 67.3 70.5 56.7 29.9 76.7 49.8 36.3 45 7 60.8 64.6 63.6 39.3 51.3 73.3 71.2 60 5 73.7 92.5 49.6 74.3 63.6 56.3 48.0 47.9 77.0 61.3
194
Experiment
Table 10.6 summarizes the experimental parameters plus the responses measured in each trial.
5.2. Interpretation of experimental results We shall carry out an overall analysis of effects and interactions of the factors studied by Yates (Table 10.5). Three factors appear to influence the yield: Factor 1 : distance between rows Factor 2: amount of manure. Factor 5 : amount of potash. The results for the main factors are shown in Figure 10.5, 10.6 and 10.7
66 64 62
60 58 56 54
52
-1
0
+I
ROW SPACING (1)
Figure 10.5: Influence of row spacing (Factor 1)
195
TABLE10.5 TABLE OF EFFECTS YATES' BEAN EXPERIMENT
Mean
58 91
1 2 3 4 5
-3 90 7 85 162 -2 75 3 80
12 13 14 15 23 24 25 34 35 45
2 52 1 66 147 4 37 2 57 -0 24 -1 98 -2 58 2 17 -0 21
123 124 125 134 135 145 234 235 245 345
0 99 -5 85 -0 76 0 45 -3 08 - 1 74 0 54 112 -0 87 -2 42
1234 1235 1245 1345 2345
-0 31 -1 01 -1 86 -3 17 1 56
12345
2 38
196
66 64
62 60 58 56 54
52 -1
0
+l
MANURE (2)
Figure 10.6: Influence of manure (Factor 2)
A
66
-
64
-
62
-
60
-
58
-
56
-
54
-
52
0
-1
POTASH (5)
Figure 10.7: Influence of potash (Factor 5)
+I
197
Factors 3 and 4 may have a small influence. It is difficult to judge from the data available. There are several high values among the interaction: 15 = 4.4 124 = -5.85 135 = -3.1 1345 = -3.2
The value of interaction 15 is not surprising as factors 1 and 5 are influencing. Factors 6 = 124 and 7 = 135 measure the effects of blocks. This explains the high values of interactions 124 and 135. The fertilities of the four plots were very different, and this difference can be shown in Figure 10.8 in which each corner of the square represents a plot and the response is the average harvest from each.
IV
111 60.1
B4 51.5
135
54.6
-4 124 I
4
Figure 10.8: Differences in the fertility of the four plots.
’
,+
II
198
TABLE 10.6 YATES' BEAN EXPERIMENT
Experimental matrix rearranged as four blocks using interactions 135 and 124
rrial no
1 26 11
20 21 14 31 8 9 18 3 28 29 6 23 16 17
10 27 4 S 30 15 24 25 2
19 12 13 22 7 32
-
Factor 3 nitrate
;actor 4 5. phos.
;actor 5 potash
Lteraction
iteraction
135
124
199
It is not easy to explain the high value of interaction 1345. It is aliased with interaction 47. This may indicate that superphosphate (factor 4) does not have the same influence on blocks I11 and IV as it does on blocks I and I1 (factor 7). But this is just an assumption which could be verified if necessary. This analysis shows that: - Three factors influence the yield: 1, 2 and 5.
- One interaction must be taken into account:
15.
- The interactions representing the effects of blocks are large
A Z3 design can be constructed ifwe consider only factors 1, 2 and 5 . As there are 32 trials, this is equivalent to having run this design four times. Table 10.9 shows this interpretation. It is thus possible to enter the average responses into the experimental domain (Figure 10.9).
TABLE10.7 YATES' BEAN EXPERIMENT
j i j i
Results for the three factors influencing yield, arranged by blocks
ITrial number
--
II
II
Factors and interactions
111 IV
-
21 29 26 18 31 23 20 28
I
5 10 15 4 17 30 27 24
--
Effect
I
I 11 111 ---
13 2 7 12 25 22 19 32
Response
+ - +
- - + + + - + + L
I
i
I
+
I
66.5 45.7 76.7 70.5 71.2 74.3 77.0 73.3
-
56.7 23.3 74.8 64 6 48.0 39.3 73.7 56.3
68.0 29.9 60.8 54 7 63.6 49 6 47.9 60.5 63.6 51.3 92.5
I ( ~ 3 . 9 1 7 . 8 1 3 . 8 1 2 . 5 1 4 . 4 1 - 1 . 9 1 - 1 . 9 1 5 8 . 9 169.4154.6160.1151.51 1
We can interpret these results as follows: When the distance between rows was great and no fertilizer is added, the yield was poor (34 Ib/plot). Almost the same improvement was obtained by reducing the distance between rows (57 Ib/plot), adding manure (60 Ib/plot), or adding potash (55 Ib/plot).
200
0
The yield could be increased still hrther by:
- adding manure and planting the rows 18 inches apart. In this case potassium makes no improvement and may even be deleterious. - adding manure and potash and keeping the row spacing at 24 inches. In this case,
a single fertilizer is insufficient: both are needed.
71
66
600) 55
Potash
I
,*-~
70
60
0
,-'
{18)-+41 ~
1
Row Spacing
Figure 10.9: Experimental results entered in the experimental domain to show that the effects are not additive.
Yates could therefore make the following recommendations. Conclusion: Bean yield can be improved by 0
either reducing the distance between rows and using a single fertilizer, manure.
f
il
20 1
k3 B
0
or spacing the rows 24 inches apart and using two fertilizers, manure and potash.
Z
4
The yields from these two growing methods are equivalent. The farmer must thus choose the most convenient andlor economical one.
6. BLOCKING OF A COMPLETE DESIGN One of the inconveniences of complete factorial designs is that all the trials must be run before the results can be interpreted. However, the investigator can sometimes use an elegant solution to satisfl his curiosity. For this, the complete design is divided into several fractional design and blocking is used. For example, if we wish to run a 25 design, we can divide it into four fractional 25-2designs. This organisation has two advantages: We can calculate the contrasts at the end of the first 25-2design to obtain an indication of the influence of the main factors. The second 25-2design is then selected to dealias the interactions and main effects that contain ambiguities. Running the trials in this way also follows our original concept of the progressive acquisition of knowledge. The results of the first design may make it possible to avoid running the initially planned 2’ design. Even if it becomes necessary to run the thirty-two trials, we will be able to more closely monitor the experiment and obtain partial results before completing the design.
202
RECAPITULATION 1. The technique of blocking is analogous to that used for fractional designs. High
order interactions are aliased with the blocking variable. The sign of this interaction divides the design into two half-designs: one with the - interaction signs and the other with + interaction signs. Alias theory applies to blocking. 2. The Yates bean yield experiment shows that we can use the technique of blocking on two variables. This technique introduces two extra factors into the original study. In addition to five original factors, Yates studied the difference in the fertilities of four plots of land. The advantage of blocking is that the thirty two responses are used to calculate the effects of five factors without the differences in ground fertility corrupting the results. 3. Blocking can be performed on any variable. As the alias theory developed for
fractional designs can be applied to blocking, it is always possible to know how the effects and interactions are aliased with the blocking variables. 4.
The Yates' bean yield experiment also shows that the presence of noninfluencing factors allows all the responses to be used to reconstruct duplicate or quadruple designs. This possibility is important because it shows that we should never initially plan to duplicate a design. It is better to study extra factors.
5 . This chapter has shown how important it is to select the order in which trials are run. This must be carefully defined before beginning the experiment. If there is a risk that the responses may be affected by overall variations, the investigator must consider blocking. A specific order should be chosen if drift is suspected. We can even consider carrying out measurements at the central point of the experimental domain (or at another point if the central point cannot be used). Lastly, the trials should be randomised to transform small systematic errors into random errors. The three techniques of blocking, anti-drift and randomisation are partly compatible; the investigator should use them because they help him overcome systematic errors and obtain the lowest possible random error.
6 . We saw in Chapter 4 that high order interactions could sometimes be considered as measurements of experimental error. The present chapter on the order of trials shows the influence of systematic errors on the results of the main effects and interactions. As a result, there is some risk in determining the experimental error by examining interactions. This method is usehl, but it must be applied with great care.
CHAPTER 11
MATHEMATICAL MODELLING OF
FACTORIAL 2k DESIGNS
1.
INTRODUCTION
The designs that we have discussed in the preceding chapters and all the concepts that we have developed so far are based on a mathematical model. Although we have implicitly made use of this model, we have not yet explored its implications. Now it is time to lift the veil and examine the mathematical model underlying all that we have discussed so far. We will begin this chapter by outlining the mathematical model on which the simplest design is based, one-factor designs: 2l. This will allow us to identify the basic hypotheses and show how the average and effect of the studied factor are taken into account by the mathematical model. We will then examine the model for two- and three-factor designs, 22 and 23, and see how interactions are expressed by the mathematical model. Lastly, we will extend the model to k factors. The mathematical model for a factorial design allows us to widen the field of application. We will therefore examine several applications of the model, each of which can give rise to important developments for all investigators. We will use several examples to clearly demonstrate the range of applications of the mathematical model. These complement and
204
extend the preceding chapters. It provides a deeper interpretation, more fruitful predictions and safer decisions on the directions of future research. The experimental results are better presented and the available information is fully extracted (wherever possible).
2.
MATHEMATICAL MODELLING OF FACTORIAL DESIGNS
Let us assume that we need to study only one factor in order to solve a problem. We set the experimental points at levels -1 and +1, and we measure the corresponding responses: y, and y2 (Figure 11.1). But what happens between -1 and + I when the factor studied covers the whole variation interval? We have assumed that the change in the response is linear, Le., we have adopted a mathematical model with the form: y = a. + a,r
where
.y .
is the response and x is the level of the factor studied (it varies continuously from -1 to +1 and can thus have all the values between these two limits). and al are coefficients
Y Y2
'Q
YO Yl
B -1
0
X
+I
Figure 11.1: Factor x varies continuously between -1 and +1. The response y varies linearly with x.
205
and a, are readily seen. Let us apply the mathematical model to The significance of experimental point A (x = -1) we get:
Y1 = a0 - a, If we do the same for experimental point B (x = +1), we get: Y2 =
+
a,
8This gives us a system of two equations with two unknowns,
and a,.
Y , = a0 - a, Y2 = a0 + a1
Solving this system for
and a,,
1 ao = - [ + Y l + Y z ] 2
al
=
-1[ - Y ~ + Y Z ] 2
We can see that:
..
.
is equal to yo, the average of the two responses y1 and y2 a, is the slope of the line PQ (Figure 11.1). But this latter relationship also defines the effect of the factor. Coefficient a, is thus the effect of the factor studied.
We can write the mathematical model in a new form: y=I+Ex
where I is the mean of the responses and E is the effect of the factor studied. If two factors are studied, we assume that each of them acts linearly and additively on the response. If the two variables x1 and x2 are continuous, the mathematical model takes the following form when there is no interaction:
y
=
a. + al xl + 3 x2
and when there is interaction. the mathematical model becomes: y
=
ao+ a, x1 + 3 x 2 + a I 2 x 1x2
where: .y is the response. x, is the variation in the level of factor 1 between -1 and +1
.
206
..
x2 is the variation in the level of factor 2 between -1 and + I a,,,, al and aI2are coefficients.
What is the significance of these coefficients? When the experimenter carries out trial number 1 of a 22 design he sets x1 at - I , x2 at -I and measures the response y and finds y,. If this value is inserted in the mathematical model for a 22 design we get:
The experimenter then runs trials 2, 3 and 4. He now has a system of four equations with four unknowns:
- a0
Y4
From which we get
+
al
+
a2 a12 +
1
a0 = -[+h +4’2 +Y3 4
+Y4
1
These relationships show that:
.a,,al isis the mean the effect offactor 1 . is the effect of factor 2 .a12is the interaction between 1 and 2 The mathematical model for a 22 design can thus be written as Y
or using the Box notation
~
I + El
XI
+ E2 ~2 + El2 XI ~2
207
I' I + 1 x,
t
2x2 + 12x1x2
It is thus particularly easy to produce a mathematical model from the results of a factorial 22 design, because the model coefficients are given directly by using the effects and interaction. This modelling can apply to all Zk designs. The model for these designs includes:
.. k
a constant term, aa, which is the mean I of all the responses, coefficients for the factors, these coefficients are the main effects of the corresponding factors,
.C: coefficients for the interactions of order q, these coefficients are the values of the interactions calculated from the effects matrix. The response for each trial may be written with reference to the 2k mathematical model. For a 23 design we have: .Y
I + El
~
XI + E2x2 + E,
x3
+
E 1 2 X 1 x2
+
E I ~Xi~+ IE23X2 X? + E123x1 x 2 X 7
The mathematical model for 2k designs thus opens the way to a number of interesting applications, and we will examine examples of these. The model should be used whenever the investigator wishes to interpret experimental results with more than a simple table of effects. We will use four examples to examine these applications:
0
Paste hardening (new example), Yield of a chemical reaction (see Chapter 2), Sugar production (new example), Yates' bean experiment (see Chapter 6).
We will use these examples to:
0
0
0
0
explain the formation of the effects matrix, evaluate the responses throughout the experimental domain without running new trials. test the validity of the linear model and, if necessary, to develop a more suitable model. select a working program for orienting the ongoing study (steepest ascent) towards the desired solution. select complementary trials for a fractional design and dealiasing certain interactions. compare factorial designs and analysis of variance. introduce residual analysis. find the error distribution over the response surface.
208
3.
FORMATION OF THE EFFECTS MATRIX
As the experimental points are at the limits of the domain, the factor levels have only the values -1 and + I . The experimental matrix indicates the levels assigned to each factor for each trial. The effects matrix is obtained from the design matrix by applying the signs rule. The model clearly shows this approach. Let us use a 22 design as an example, and write the system of equations as a matrix: Y=XE
developing this condensed form, we get: I El E2
El2
where xi. is the level of xi for the jth trial. For example, x23 is the level of factor 2 in trial number 1. We can readily find the value of this level by examining the experimental matrix. In this case, x23is equal to + I . We have seen that, for factorial designs, the X-' matrix, the inverse of X,is simply the X' transpose of X divided by the number of trials, n. We can write:
E = -1 X ' Y n
This relationship can also be developed: I El
1 -
E2 El2
1
1
1
Yl
1
xll
x12
x14
Y2
n
x21
x22
x23
x24
Y3
x11x21
x12x22
x13x23
x14x24
Y4
The mean I is obtained by multiplying the first line of the X' matrix by the Y matrix, or (as n
= 4):
I
1
=
~ [ + . Y+I ~ +2 ~ 3 + ~ 4 ]
The effect El of factor 1 is obtained by multiplying the second line of the X' matrix by the vector matrix Y:
209
When the are replaced by their numerical values, i.e., by the levels of factor 1 in the experimental design, we have XI,
=-1
XI2
=+I
XI?=
-1
XI3 =
+1
hence:
The effect E2 of factor 2 is obtained in the same way, by multiplying the third line of X‘ by Y and replacing xij by their numerical values: E2 = -[-4’l 1 4
-1’2 f Y 3 +y4I
Lastly, El? is obtained by multiplying the fourth line of X’ by the elements of Y and giving the xi, their numerical values:
One of the advantages of using the matrix form is that it combines all the relationships in a single formula. We can also see that the product of the levels of the two factors, 1 and 2, are shown in the X‘ matrix, and that this product is taken into account in calculating the interactions.The levels x, and x2 take the values -1 and +1, and the product xIx2 is either -1 or +1, the interaction signs. This satisfies the signs rule that we have used to establish the interaction signs.
4.
EVALUATION OF RESPONSES THROUGHOUT THE EXPERIMENTAL DOMAIN
The experimental points were placed at the limits of the domain, but the mathematical model may be used to calculate y for any value of x. If we assume that the model is valid when the factors vary fi-om -1 to +1, we can calculate responses for all points within the domain. All the responses make up the response surface, and it is possible to plot isoresponse curves on the experimental domain. The following example of paste hardening shows how the mathematical model can be applied.
210
4.1.
Example: study of paste hardening
The problem: A paste must be usable for at least one hour after the tube is opened. Certain tubes are known to contain paste that hardens in 30 minutes, i.e , much too fast. The person in charge of this study must : provide instructions such that clients can be sure of having at least 60 minutes to work the paste before it becomes too hard
#
8 g
Factors The experimenter assumes that the following factors are among those that influence the usable time: Factor 1: ambient temperature. Factor 2: water content. Factor 3 : mixing time during manufacture Factor 4: time in storage.
-.
.
TABLE11.1
EXPERIMENTAL MATRIX STUDY OF PASTE HARDENING
Trial no
remperature Water cont. Mixing time 2 3 1 -
~
-
-
+
-
+ + + +
-
+ -
+ (-1
Level (+)
Response
-
+
Level
Storage 4= 123
33 68
15°C
0 1 Yo
short
6 months
25°C
05%
long
a week
21 1
Response
The response is the time (in minutes) required for the paste to reach a defined consistency. The error of the response is +2 minutes. The experimenter decides to use a 24-' design with I = 1234 as alias generator. The experimental conditions and the responses are given in the design matrix (Table 1 1.1).
4.2.
Interpretation
The calculated effects are shown in the table of effects, and the experimenter estimates that these results are sufficient, so that no extra trials are required. Only two factors are influencing, water content (2) and storage time (4), and there is a strong interaction between them. Factor 1 has only a slight influence, and can be neglected. As the error of the response is k 2 minutes, the error of the effects is k 2& = -f 0.7 minutes. TABLE11.2
TABLE OF EFFECTS STUDY OF PASTE HARDENING
Mean
60.4
k 0.7 minutes
1 2 3 4
-1.4 -9.9
0.4 14.4
i k k k
12 + 34 13 + 24 14 + 23
-0.6 5.1 -0.4
i 0.7 minutes k 0.7minutes -f 0.7 minutes
0.7 minutes 0.7 minutes 0.7 minutes 0.7 minutes
The mathematical model is easily written, but the experimenter can choose one of several solutions, depending on the factors selected and the way in which the results are rounded off Without rounding off and keeping all the factors, 1, 2 and 4 plus interaction 24, the model is: y = 60.4 - 1.4 XI 9.9 ~2 -
+ 14.4 ~4 + 5.1 XZ ~4
Rounding off the numbers to the nearest half-minute gives: J'=
6 0 . 5 - 1 . 5 ~ l- 1 0 ~ , + 1 4 . 5 ~ ~ + 5 ~ ~ ~ ~
Eliminating factor 1 which has a slight influence gives:
2 12
J=
60.5 - l o x * + 1 4 . 5 ~ , + 5 ~ , ~ ,
We will use this model to produce a graphical representation of the results. As there are now only two factors, the experimental domain becomes a square. The isoresponse curves for y can be projected onto this square to give the isoresponses. The response values vary from 3 1 to 80 minutes, and we can draw isoresponse curves for 35, 40, 45 min., etc. From this, we can simply read off the setting time for the paste for a given water content and storage time (Figure 11.2).
STORAGE TIME (MONTH)
0.1
0.3
0.5
WATER CONTENT
Figure 11.2: Changes in paste setting time as a function of water content (2) and storage time (4).
The isoresponse curves are segments of a hyperbola when there is an interaction and straight lines when there is no interaction. Conclusion:
I-
The experimenter can use the results diagram to inform the Management of the criteria to be used for policy selection
.
d
If they wish to impose no constraint on fabrication (increased costs) by allowing the water content to rise to 0 5%, the tubes should carry a use-by date. tubes that have been in stock for over a month and a half will have working times of less than 1 hour
.
If production constraints can be imposed to keep the water content below 0 3%, the storage time can be extended to 3 months
213
.
The acceptable storage time will be 4.5 months at a water content of 0.2%, and 6 months for a 0.1% water content. Given this technical knowledge, the final decision will be based on
4
9 commercial and economic criteria, according to the industrial strategy
Y
d
& of the Company
5.
i~i
j,
TEST OF THE MODEL ADOPTED
Let us stay with the example of paste hardening and use it to illustrate this application of the mathematical model of factorial designs. The experimenter had run eight trials and is sure that the response surface lies close to the experimental values, i.e. close to the points at the extremes of the domain. He assumes that responses corresponding to all the other points within the domain could be calculated using a mathematical model in which the factor effects are linear and additive. The validity of this hypothesis, and the model derived from it, must be checked. He does this by running experimental points within the domain and comparing the measured response to the response calculated from the model. The validity of the model can be conveniently checked by running trials at the centre of the experimental domain. In the paste hardening example, the experimenter runs a trial with the temperature set at 20°C, water content at 0.3% and uses a paste that has been stored for 3 months. He obtains a hardening time of 62 min. If we compare this value to that given by the model - 60.5 minutes - we see that there is good agreement given the error of the response. The model is therefore valid. If the experimenter had found a value very different from the model mean, i.e differing by several standard errors, he would have to:
. either question the measurements at the centre of the domain. This is why he generally out not just one, but two or three measurements at this point. . carries or question the model adopted. In this case he must consider another more complicated model, e.g., a model in which factors are of second degree or a more complex mathematical fimction. We will not go into this extremely important topic here, but this is the logical extension of factorial designs in the progressive acquisition of knowledge.
6.
SELECTION OF A RESEARCH DIRECTION
We can illustrate this application by going back to the first example we studied, the yield of a chemical reaction (Chapter 2). The yield of this reaction depends on two factors,
214
temperature and pressure. The limits of the experimental domain, the design matrix and the trial results are shown in Table 1 1 . 3 .
Trial no
Temperature
Pressure
1
-1
-1
2 3 4
+1 -1
-1 +1
60 % 70 Yo 80 %
+I
+I
90 Yo
Level (+)
80°C
2 bar
Interpretation of the results produces the following table of effects:
TABLE11.4 TABLE OF EFFECTS THE YIELD OF A CHEMICAL REACTION
Mean
75%
1
5%
2
1 0%
12
0%
We can use these results to set up a mathematical model and draw the isoresponse curves in the experimental domain.
215
6.1. Mathematical model Using coded variables, the temperature xl varies from -I to + I , as does the pressure x2. The yield y is then given by:
v=
75+5Xl+IOX2
It is sometimes convenient to use normal values of temperature and pressure (degrees Celsius and bar) for the interpretation. The relationships between coded variables and normal variables are.
0 = 0,)+ (Step,) x1 P = Po + (Step,) x.2 where
- 0 is the temperature in "C it varies from 60°C to 80°C .6, is the mean temperature in "C at the mid point,
0,
-
=
70°C
xI is the temperature measured as a coded variable: it varies from - 1 to + I
Step, is the step selected for converting normal units to coded units; the temperature step is 10°C
.p and po are the pressures measured in bar. p varies from 1 to 2 bar and po is
.
x2 is the pressure measured in coded units; x2 varies from -1 to + 1
Step, is the pressure step, 0.5 bar in this example Substituting numerical values in the above equations gives:
8 =70+ loxl p = 1.5 + 0.5 x.2 or 1 -7 10
XI = -8
1 5 bar.
216
Introducing these values into the model equation gives a relationship in which the yield p is a hnction of temperature in "C and pressure p in bar: 1 2
p = 1 0 + -0 +20 p
This relationship is valid in the experimental domain studied
6.2. Isoresponse curves Once we have checked the validity of the model by comparing measured and calculated values at the centre of the experimental domain, we can draw the isoresponse curves. If we now want to know how to alter the temperature and/or pressure to approach a 100% yield, we must leave the experimental domain. We are thus obliged to develop hypotheses which must be later verified. Let us assume that the model remains valid outside the experimental domain. We have the greatest chance of finding a 100% isoresponse in a direction perpendicular to the isoresponse lines within the experimental domain (Figure 11.3). We can even draw 95% and 100% isoresponse curves and calculate the pairs of temperature/pressure points which could give a 100% yield. There is an infinite number of solutions. The experimenter must choose those which are compatible with the constraints of his installation or products, and which permit him to attain his aim at the lowest cost. Let us assume, for example, that the temperature cannot be increased, but that the installation can withstand a pressure of 2.75 bar. It is then easy to calculate possible experimental points: for 0 = 80°C 1 100 = 10 + - 80 + 20p 2 p = 2.5 bar for 0
=
70°C 1 100 = 10 + - 70 + 20p
2
p = 2.75 bar The experimenter can thus identifl the points giving him the best chance of success. He must, however. check the hypotheses adopted by running trials.
217
2 bar
PRESSURE
1 bar
80°C
60°C TEMPERATURE Figure 11.3: lsoyield curves for the chemical reaction example
6.3. Steepest ascent vector The reader will have noticed that, in the preceding example, yields increased in a direction perpendicular to the isoresponse lines. This point is important, and this direction is given by a vector, the steepest ascent (V). The projections of V along the axes 0 x1 and 0 x2 are v , and v2. The general equation for isoresponse curves is: (setting the value ofy constant). or
This is the general equation for a straight line and the coefficients of x1 and x2 are the direction cosines of the straight line which is at right angle to the isoresponse curve. E, and E2 are therefore, allowing for a proportionality coefficient K, equal to the direction cosines of the steepest ascent vector. This is only true when the unit of measure is the same on the two axes. This shows the importance of coded variables. We have:
218
+I
v , =I
Figure 11.4: Construction of the steepest ascent vector. For the chemical reaction yield example, we have:
orwith
K
=0.2
We will now see how the steepest ascent vector is constructed (Figure 1 1 4). The relationships giving the values of the components of V are only valid for coded variables, the V vector must therefore be constructed retaining these units on the factor axes. In the present case, axis x1 is one coded temperature unit, so v1 =. 1 and two coded pressure units on axis x2, or v2 = 2. The steepest ascent vector V is thus readily constructed. It is useful to return to the original variables when presenting results or calculating new experimental points. But with the original variables we must always remember that the units on the temperature and pressure axes are no longer the same. The coefficients of variables 8 and p in the isoresponse line equation are no longer the direction cosines of the steepest ascent vector. This equation therefore cannot be used to establish the direction of V. The coded variable units must be retained to obtain the direction cosines which are then transformed to the original units. We therefore multiply the values of projections v I and v2 by the step of the corresponding coded variable. Thus: v1 or
= K
El (stepH)
219
$1,
= K
5 x 10 = 50 K
in degrees Celsius
and v2
= K
E2 (step,,)
or v2
= K
10 x 0.5
=5 K
inbar
But as the choice of K is arbitrary, with K = 0.2 we have: I),
=
0.2 x 50= 10°C
vz
=
0.2 x 5
=
1 bar
This indicates that the temperature 6 must be increased by 10°C and the pressure p by 1 bar simultaneously to move along the steepest ascent (Figure 1 1.4). This relationship for two factors can be extended to any number of factors. For three factors, the isoresponse surfaces are planes, and the steepest ascent vector is perpendicular to these planes. This vector indicates the direction in which the responses increase (or decrease) most rapidly. For four factors, the isoresponse surfaces are hyperplanes in a four-dimensional space, the normal to this hyperplane is defined as above and the steepest ascent vector is defined by four components. The isoresponse hyperplanes for k factors have a steepest ascent vector with k components. These components may be expressed in coded variables or in normal variables. When coded variables are used, the components of the steepest ascent vector V are, within an arbitrary constant, the effects of the corresponding factors:
L’k = K
Ek
When normal variables are used, the components of the steepest ascent vector V are, within a arbitrary constant, the effects of the factors multiplied by the step chosen for each of them when coding the variables v1
=K
E, (step,) (step2) (stepl)
v2 = K E2 vi = K El
vk
=K
Ek (stepk)
It is thus possible to calculate the direction in which there is the greatest chance of improving the results, for any number of influencing factors.
220
7.
CHOICE OF COMPLEMENTARY TRIALS
There are generally ambiguities that remain to be resolved when a fractional design is interpreted. This can be done either by running a complementary design, or simply by running a few extra trials. The choice of these extra trials and the way in which the calculations are done depends on alias theory and on the mathematical model for the factorial designs. The following chapter (Chapter 12) is devoted to this question.
8.
ANALYSIS OF VARIANCE AND FACTORIAL DESIGNS
In this section we will analyse a single set of data by two methods: factorial design and analysis of variance. This will allow us to compare the results and recognize analogies between these two techniques for interpreting data. Let us first assume that the experimenter has only a single response per experimental point. In this way we will develop a formula illustrating the similarity between the two methods. We will then assume that the experimenter has two responses per experimental point, These results will be used to calculate a value for the error of the effects. A comparison of the results will show that the two methods are identical. A new example, sugar production, will be used to examine this problem.
Example: Sugar production The problem: The weights of sugar obtained using two treatments, treatment A and treatment B, are measured while varying the temperature between 5°C and 15°C. The investigator wants to know the conditions providing the greatest yield of sugar.
8.1. Analysis of the problem by factorial design (one response per trial) Factor 1 is the type of treatment and factor 2 is the temperature. A 22 design is run. The experimental data and the results are shown in Table 1 1 . 5 .
22 1
TABLE 11.5
Trial no
Treatment
Temp.
Interaction
1
2
12
I
+ +
+ + + +
-19.5
142
I
.-
-
2
+
-
3 4
-
+
f
Level (-)
B
5°C
Level (+)
A
15°C
Effect
-8.5
28
Response
~
+
-
125 198 142
Factor 1 has a negative effect, i.e., treatment B gives a greater yield than treatment A (Figure 11 5 ) SUGAR
(Grams)
150.5 142
131.5
-1
0
+1 -
B)
i
a \I
TREATMENT
Figure 11.5: Effect of treatment type on sugar production.
222
Factor 2 has a positive effect, i.e., increasing the temperature favours sugar production (Figure 1 1.6). SUGAR
(Grams)
170
/
142
114
+’
-1
TEMPERATURE
15 o c
5oc
Figure 11.6: Effect of temperature on sugar production. And there is a very strong interaction between factors 1 and 2 -19 5 grams (Figure 1 1 7)
15 O
198
142
C
lb
103
0
b
125
Treatment
Ba
,A
Figure 11.7: Influence of treatment and temperature on sugar production.
223
Conclusion: iii 3
The investigator chooses the conditions giving the best yield treatment B and a temperature of 15°C These settings take advantage of the strong interaction between the two factors
8.2 Analysis of the problem by analysis of variance (one response per trial) The same trials were run, treatments A and B were both run at 5°C and 15°C. The responses, given in Table 1 1.6, show: - The means for each treatment are: 150.5 and 133.5. - The means at each temperature are: 170 and 114. - The mean of all results. or the overall mean is: 142.
TABLE11.6 ANALYSIS OF VARIANCE SUGAR PRODUCTION
Non-duplicated design
B I
TEMPERATURE
A
Mean
I
15°C
198
142
170
5°C
103
125
114
where
.c y :
is the sum of the squares ofthe responses
Cf
= (198)’+(142)* +(103)2+(125)2 = 85 602
227
2. Hence the mean variance of the effects is: 1
2
2
1 8
CTE = - O y , = - 35
n
3 . and the standard deviation of the effects is:
The calculated effects and interaction are therefore significant because they are several times the standard deviation, For example, for temperature we have:
TABLE11.8 EXPERIMENTAL DESIGN SUGAR PRODUCTION
Duplicated design
Trial no
4
First result
Second result
108 120 194 I44
98 130 202 140
Fi Deviation
198 142
Variance
8.4 Analysis of the problem by analysis of variance (two responses per trial) We can carry out an analysis of variance on the eight responses by calculating the sums of squares from the table of results (Table 11.9) and applying the analysis of variance formula. The analysis of variance formula contains an extra term to allow for the duplicated trials.
CYt =
Y,
Y
228
TABLE11.9
ANALYSIS OF VARIANCE SUGAR PRODUCTION
Duplicated design
TREATMENT
15°C
TEMPERATURE
202
140
150.5
133.5
5°C
Mean
142
Numerical calculations give the following values:
cyz
= 171344
c”J
-161312
c(+yl - y ) 2 =578 z ( j 7 0- y )
2
= 6272 2
x(j-jo -JT +y,) = 3042 c(j7-,y , ) 2 = (198-194)2 +(198-202)2 +(142-144)2 +(142-140)2
(103-98)2 +(103-108)2 +(125-120)2 +(125-130)2
We thus have all the elements required to construct the analysis of variance table (Table 11.10).
229
TABLE11.10 ANALYSIS OF VARIANCE TABLE SUGAR PRODUCTION
Duplicated design Variance due to mean
Sum of Squares
treatment temperature interaction dispersion
161312 578 6272 3042 140
Total
171344
Mean Squares
Degrees of Freedom
Mean Squares computed with the effects
161312 578 6272 3042
8 x (1 42)2 8 x (8.5)2 8 x (28)2 8 x (19.5)2
35
I
I
8
The significance of a mean square is generally evaluated as the ratio of the mean square itself to that of the response dispersion. If the ratio is large, then the mean square is significant. For an experimental design, this involves comparing the mean variance of the effects (obtained by duplicating the trials) to the square of the effects themselves, the ratio:
E?
__ 0;
In order to understand the similarity between analysis of variance and experimental designs, let us look the effect E, of temperature. The mean square, derived from the temperature variations, is 6272, or eight times the square of the temperature effect E,. 6272 = 8x(28) 2 = 8 x E i = nEi
The dispersion of responses gives a mean square of 35 which is, in reality, the mean variance
0:
of the responses. As there are eight responses for calculating an effect, the
variance of the effects 0; is given by:
230
The ratio of the mean squares is again the comparison of the square of an effect to the square of the error of the effect:
If the errors have a normal distribution, this ratio follows Fisher's law, and there are tables showing the probability that the effect is significant. If the value of F from the table is high, the effect is significant, otherwise it is not. We can also take the square root of this ratio and use Student's t test to estimate the probability that an effect is significant. The effect is compared directly with its standard error. Applying this to the temperature, we have:
3= 1 3 . 3 (JE
This value is the same as the one we calculated by the experimental design method
9.
INTRODUCTION TO RESIDUAL ANALYSIS
The mathematical model for experimental designs focuses on synthesising trial results. The information in the results has been transformed into a mathematical formula. Even if the investigator decides that the model is valid, all the information in the trial results may not be entirely expressed by the model. Any information that remains to be extracted from the experimental data is contained in the residuals. But what are residuals? A residual ri is the difference between the measured value of a trial y , and the value calculated from the model y,.
We can see how residuals are used by examining the Yates' bean growing experiment. We will see that the choice of mathematical model is not without consequence, and that the conclusions can depend on this choice. But the choice is not automatic and it is the investigator who must decide. He must take great care to choose the model that best reflects the phenomenon studied. We will examine two models (1 and 2). The residuals for these two models are shown in Table 1 1.11.
23 1
TABLE11.11 YATES'
BEANGROWING EXPERIMENT
Calculation of residuals for model 1 and 2 Model
1
Model
2
Trial no
neasured value
Calculated Value
Residue
Calculated Value
Residue
I 2 3 4 5
66 5 36 2 74 8 54 7 68 0 23 3 67 3 70 5 56 7 29 9 76 7 49 8 36 3 45 7 60 8 64 6 63 6 39 3 51 3 73 3 71 2 60 5 73 7 92 5 49 6 74 3 63 6 56 3 48 0 47 9 77 0 61 3
55.62 39.08 71.32 54.78 55.62 39.08 7 1.32 54.78 55.63 39.08 71.32 54.78 55.62 39.08 71.32 54.78 54.48 55.42 70.18 71.12 54.48 55.42 70.18 71.12 54.48 55.42 70.18 71.12 54.48 55.52 70.18 71.12
10.88 -2.88 3 -48 -0.08 12.38 -15.78 -4.02 15.72 1.08 -9.18 5.38 -4.98 -19.32 6.62 -10.52 9.82 9.12 -16.12 -18.88 2.18 16.72
66.07 3 1.63 67.07 56.03 56.87 34.83 63.87 65.23 51.37 40.33 81.77 47.33 48.17 49.53 72.57 50.53 55.73 51.17 62.73 81.57 64.93
5.08
47.97
3.52 21.38 -4.88 18.88 -6.58 -14.82 -6.48 -7.52 6.82 -9.82
65.93 72.37 47.03 65.87 71.43 66.87 50.23 56.67 80.63 63.67
0 43 4 57 7 73 -1 33 11 13 -11 53 3 43 5 27 5 33 -10 43 -5 07 2 47 -11 87 -3 83 - 1 1 77 14 07 7 87 - 1 1 87 -11 43 -8 27 6 27 12 53 7 77 20 13 2 57 8 43 -7 83 -10 57 -2 23 -8 77 -3 63 -2 37
6
7 8 9 10
11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
232
Model 1 We have seen that only three factors and one interaction have major effects
. .
row spacing ( I ) .manure (2) .potash (5) interaction (15)
-3.9 7.8 3.8 4.4
The overall mean is 58.9, and we can write the equation for model 1 as follows: y
=
58.9 - 3.9 x1 + 7.8 ~2 + 3.8 xg + 4.4 XI xg
We can use this model to calculate all the responses for the thirty two trials and obtain the residuals (Table 11.1 l), i.e., the differences between the yields actually measured and those calculated from the model. If this difference is positive, the yield was better than that predicted by the model. The significance of these differences can be appreciated by entering them in the area of the corresponding trial (Figure 11.8) We can now see that almost all the positive differences lie around a continuous band of ground. So there are two parts in this plot of land: a high fertility area and a low fertility area.
Figure 11.8: Analysis of residuals reveals a high fertility zone when model 1 is used.
233
So, studying the residuals has revealed hidden information that was not apparent from analysing the effects and interactions. Model 2
But Model 1 does not take into account the effects of blocking, and we can discover whether there are high fertility zones by studying the residuals obtained with a second model that allows for variations in fertility between blocks. We must add the terms for the interactions 124, 135 and 2345, which are linked to differences in fertility between blocks. Model 2 can be written:
y
= 58.9 -
3.9 XI+ 7.8 ~2 + 3.8 ~5 + 4.4 XI ~5
-
5.8 ~ 1 . ~x42 - 3 . 1 XI~3
xg
+ 1.5 12 ~ 3 . x5 ~ 4
The residuals are calculated in the same way as before and entered on the ground plan in the plots for each trial (Figure 1 1.9). Block I1 does not seem to uniform and there is a band of high fertility running through blocks I1 and 111. This more detailed analysis reveals a high fertility area. If the investigator wishes, he could carry out further research to explain this phenomenon, which may be due to a band of different soil or an unsuspected layer of subsoil water.
Figure 11.9: Analysis of residuals reveals a stripe of high fertility when model 2 is used.
234
10. ERROR DISTRIBUTION In a 22 design, the response surface is defined by the following relationship:
YPaf)+a,xl+a2~2+a12XlX2 The coefficients 3 in this relationship are equal to the effects and interactions. They are thus known with a certain imprecision. These errors of the model coefficients influence the calculated response y. The response is therefore affected by an imprecision which we can evaluate. We assume that the levels xiare accurately determined and introduce no error. The surface is defined with a margin of error that must be determined and which, as we will see, is not the same throughout the experimental domain. Let us apply the variance theorem to the response surface:
We can now calculate each variance of the right hand side, remembering that the error of the measured response o,,is assumed to be the same throughout the experimental domain. In order to calculate the variance of coefficient a,as a hnction of the error of each response, we must write the relationship defining
a.
= - 1[ + ~ 1
n
+ ~ +2 ~ +3 ~ 4 1
and apply the variance theorem
If we assume that the error is the same for all the measured responses, we can simplifL this relationship as follows:
or by changing the notation
23 5
1 . V ( ao) = - 0; n
We can calculate the other variances in the same way, remembering that the xi are constants which introduce no error. V ( a , x , ) = x,2 v ( a l ) = x: 1
2
v ( a 2 x 2 ) = x;v(a2) = x22 -0 1 2
n y
Entering these values in the variance equation gives the response variance calculated from the mathematical model: v(y) = -1O y 2 n 1 n
v(y) = -0;
+ X I2 -1O y 2
+ x22 -0; 1 + x12 x22 ,Gr 1 2
n [I+ x:
n
+ x; + x:;.]
and if we make f 2 ( x ) = [l+x;+x;+x:x;]
we obtain
The function f2(x) varies with the values of xi, hence the error o f the response calculated from the model also varies and depends on the coordinates of x, and x2. The levels of these factors vary from -1 to + l ; the hnction f 2(x) thus varies from 1 to 4.it is minimal when x, and x2 are zero, i.e. at the centre of the domain. Table 1 1.12 shows the values o f f (xi and the position of the points where this hnction is constant within the experimental domain are shown in Figure 1 1.10. We can see that the responses calculated from the model become less and less precise with their distance from the centre of the domain.
236
TABLE11.12
f
2(x>
I
12
1.5
2
3
4
f
(4
1
1 09
1.22
1.41
1.73
2
In the paste-hardening example the standard deviation of the responses was estimated as & 2 minutes. Eight trials were run. If we apply the above formula, we get:
1
v(y)= -D; n
[I+ x;
4
+ x4‘ + x:xqz]
=8 f2
(x)
1 2 f 2 (x)
-
The standard deviation of the responses calculated from the mathematical model,oyc, is the square root of this variance: 1 0% --- ff(ix ) We only need to divide all the values of f(x) by in the model for the paste hardening example.
fi to know the confidence we can have
Figure 1 1.10 shows the standard deviation curves cYcof the responses calculated for the paste hardening example. These deviation curves can be superimposed on the isoresponse curve network. f(x) = 2.0
- T
x2
0
f(x)= 1.41
b
’
* ’
f(x) = 1.22
\
Xl
/
a
Figure 11.10: The precision of calculated responses is not the same throughout the experimental domain. It is best at the central point. The further from this point, the less accurate the model.
237
RECAPITULATION 1. The mathematical model associated with factorial designs is a first degree polynomial
for each of the factors taken independently. 0
0
0
The effect of a factor is assumed to add algebraically to the effects of other factors. The mathematical model for factorial designs is established with coded variables. The polynomial coefficients are thus simply the mean, main effects and interactions. The model is valid for continuously varying variables. The yield of a chemical reaction example has provided us with formulae which can be used to go fi-om values in coded units to values in the more usual physical units, and vice versa. The sign rule is used to establish the + and - sign columns for the interactions derives from the mathematical model for factorial designs.
2 . The paste hardening example showed that, despite the fact that the trials were run at the extremities of the domain, they provide predictions for the whole of the experimental domain.
The validity of the model must be checked. Supplementary trials should be run, if possible, at the centre of the experimental domain. The calculated and measured responses at this point are then compared. We have also used this example to show how the mathematical model can be used to draw isoresponse curves within the experimental domain, or even outside it. 3 . The main effects of factors are the direction cosines of the steepest ascent vector
(coded values). 4. Isoresponse curves and the steepest ascent vector can be used to predict the regions
in which there is the greatest likelihood of success. Confirmatory experiments are required to verify the assumptions made. 5 . The sugar production example revealed the analogies between analysis of variance
and factorial designs: the same mathematical model, same results, and the same way of determining significant effects. These similarities are summarized in the formula: YtY
=
n E'E
The main inconvenience of analysis of variance is that this method uses squares: the signs are lost and comparisons made difficult. Factorial designs give the effects themselves, together with their signs, making interpretation much easier
23 8
6. Residuals analysis is used to extract information still contained in the responses after model-fitting. Interpretation is often delicate and relies on the good sense and
intelligence of the investigator. 7.
The mathematical model does not have the same precision throughout the experimental domain. It is most accurate at the central point.
CHAPTER 12
CHOOSING COMPLEMENTARY TRIALS
1.
INTRODUCTION
In the preceding chapters we have seen that a problem is first studied by testing several factors using a fractional experimental design. But this approach has the disadvantage of providing contrasts in which the effects are aliased. This results in ambigpities which must be resolved by carrying out more trials. The setting up of the spectrofluorimeter example showed how a complementary design was constructed to dealias the contrasts giving difficulty. It is sometimes possible to run just a few extra trials rather than a complete complementary design. This chapter shows how to make such a choice. Any doubts remaining after the initial fractional design can be eliminated by running one, two, three or four extra trials.
240
2.
A SINGLE EXTRA TRIAL
Example: Clouding of a solution The problem: ti -r;
The experimenter wishes to know what is causing a slight cloudiness in a solution containing several components. The factors chosen for study were:
.Factor 1: +.
3 9
F
. .
#
temperature.
Factor 2:
product A.
Factor 3:
product B.
Factor 4:
stirring speed.
The response is an index of opacity which accurately reflects the way in which the cloudy appearance of this solution varies with the factors studied.
The experimenter has carried out a fractional Z4-' design using I generator. The contrasts are therefore: 1 + 234 2 + 134 3 +124 4 +123 I , , = 12 + 34 h,,=13+24 3Llq = 14+ 23 h = I +1234
h, = h,= h,= h4=
=
I
1234 as alias
24 1
TABLE12.1 EXPElUMENTAL MATRIX CLOUDING OF A SOLUTION
Response
17.0
The contrasts are calculated from the experimental results and entered in the table of effects (Table 12.2)
TABLE12.2 TABLE OF EFFECTS CLOUDING OF A SOLUTION
Mean
11.01
1 2
3
3.86
4
12 + 34 13 + 24 14 + 23
1.96 0.34 0.34
1
242
There are two influencing factors.
.
Temperature : factor 1 .Product B : factor3
We can see that the sum of interactions 12 + 34 is large. Which is larger, 12 or 34? To find out, we must dealias 12 from 34. The design chosen (I=1234) gives the contrast: h,,
= h,, =
12 + 34
We must find a trial which gives
h',,
=h'34
= 12-34
This trial is in the complementary design I = -1234 (equivalence relationship), and was therefore not run during the first set of trials. Table 12.3 shows the eight possible trials.
TABLE12.3 EXPERIMENTAL MATRIX CLOUDING OF A SOLUTION
Trial no
remperature 1
l 2
Product A
9 10
I1 12 13
Product B 3
Stirring Spd 4 = - 123
-
+
-
-
-
-
-
+
+ + + +
14 15
16
-
+ + -
Level (-)
15°C
0 Yo
0 Yo
I00 rpm
Level (+)
30°C
I Yo
0.5 Yo
300 rpm
r Respo
If we assume that only influencing factors are 1 and 3 and the influencing interactions are 12 and 34, the mathematical model of the responses from this design is given by the formula:
243
yi
= 1+1+3f(12-34)
The difference 12 - 34 can be determined using any of the trials in Table 12.3, e.g., the easiest one to run under the particular experimental conditions, Let us assume that the experimenter has chosen trial number 10. The response y,, is obtained when factor 1 is at the high level (+) and factor 3 at the low level (-), The mathematical model gives the value of this response as:
ylo = l + l - 3 - ( 1 2 - 3 4 ) The numerical values of 1, 1 and 3 were calculated from the initial design and the value ofy,, was determined by the extra trial.
I = 11.01 1 = 4.34 3 = 3.86 ylo = 13.03 We get, 13.03 = 11.01 + 4.34 - 3.86 - (12 - 34)
or 12 - 34 = -1.54
The first set of eight trials gave: 12 + 34 = 1.96
and hence the system: 12 + 34 = +1.96 12 - 34 = -1.54
adding this two equations 12 = 0.21
and subtracting them: 34 = 1.75
Interactions 12 and 34 were dealiased and we can conclude that interaction 34 makes the greatest contribution to 12+34.
244
The extra trial was not run at the same time as the eight trials of the initial design. It could thus produce a distortion due to the fact that the non-controlled factors were set at different levels. This then results in a change in the mean of the mathematical model of factorial designs. When we set up the system of equations to calculate 12 and 34, we assumed that the mean I was the same as that of the initial trials. This may be a bit risky. If we want to avoid any risk, we need an additional equation to measure the mean I' of the mathematical model of the extra trial. Hence we must run two extra trials rather than one.
3.
TWO EXTRA TRIALS
Example: Clouding of a solution ( block effect) The experimenter suspects a shift in the mean. He has available the eight trials of the initial design, to which he adds the results of new two extra trials. These two trials were chosen from the eight trials of the complementary design I = -1234. The choice is very wide as 12 - 34 can be calculated from all the trials. We will use as an example the high level of factor 1 and the low level of factor 3 . This means we must run trials 10 and 12 (Table 12.3). The results of these two extra trials are: y,, y,,
=
13.03
=
9.73
These results are all shown in Table 12.4 as two blocks, one with the eight initial trials having a mean I, and the other with the two extra trials having a mean 1'. The second block may my considered as a Z4" design containing two trials and having the alias generator set:
From which we can calculate the value ofthe contrast h I 2
iI2 = 12- 34 + 2 - 123 - 134 + 4 - 23 +14 But we know that: 2=123=134=4=23=14
Thus:
h',2 = 12-
34
0
245
TABLE12.4
EXPERIMENTAL MATRIX CLOUDING OF A SOLUTION
Initial design + complementory trials
Trial no Temperature
Product A
Product B
Stirring Spd
1
2
3
4 = 123
1
-
-
-
-
2 3 4 5 6 7 8
+
-
-
-
-
-
+
+ +
+ + + +
+
5.2 9.5 1.1 12.8 11.8 17.0 8.6 22. I
Product A
Product B
Stirring Spd
Response
2
3
4 = -123
-
-
-
+
-
+
+
+ +
-
-
+
-
-
-
+
Trial no Temperature 1
10 12
+ +
Response
-
+ + -
13.03
We obtain a system without the means. In fact, they appear the same number of times with + signs and - signs in the expressions giving h,, (first block) and h,, (second block): 1 ~ 1 2= 12 + 34 =
1
-[ +YI 8
- Y2 - Y3
f
Y4 + Y s - Y6 - Y7 + Y s
Substituting the experimental data:
h,,
I 8
= 12+34 = -[+5.2-9.5-1.1+12.8+11.8-17-8.6+22.1]
]
246
15.7 h,, = 12+ 34 = __ = 1.96 8
1 -3.3 i, =, 12-34=-[-13.03+9.73]=-==1 2 2
65
hence the system: 12 + 34 = 1.96 12-34:-1.65
giving: 12 = 0.15 34 = 1.80
Again, interaction 34 clearly makes the greatest contribution to the sum 12+34. The small difference between the values of 12 and 34 obtained with a single extra trial and two extra trials indicates a shift in the mean of the two blocks from 1 1.01 to 1 1.38. The experimenter can thus come to the following conclusions: Conclusion: Product A has no influence on cloudiness , and can thus be used between 0% and 1% Clouding always occurs when the temperature is set at the high level. The low temperature (15°C) must therefore be used. At low temperature, there is always cloudmess when product B is present. Cloudiness can be avoided by I
-
3
1. Using product A 2 Working at low temperature 15°C
J
b -
1 3$
4. THREE EXTRA TRIALS Let us now leave the solution clouding example and return to the fabrication of plastic drums discussed in Chapter 7.
247
The problem was: &
The plastic drums must have a volume of two litres We need to find the fabrication condittons which provide a volume of at least two Iltres, 4, but not more than 2 002 litres At least two lttres to give clients full value, and not more than 2 002 lltres to avoid being too generous
A 284 design was run and there were four significant contrasts
h, = 3+127+146+158+245+268+478+567 h , =5+126+138+147+234+278+367+468 h8 =8+124+135+167+236+257+347+456
h,, = 15+26+38+47+.. We can see that the main effects are aliased with third order interactions. The interpretation hypotheses we have adopted allow us to assume that these interactions are negligible. There are thus only three significant effects, and their values can be calculated from the initial design (Chapter 7, section 5.2):
h, = 3 = +3.2 c m 3
hi = 5 = - 2 . 7 c m 3 h, = 8 = + 1 . 9 c m 3 h 1 5= 1S+26+38+47 = + 2.5 c m’
his.
We do not know what interaction is responsible for the large value of contrast We must therefore run extra trials in order to calculate each of the four interactions making up contrast This requires three extra trials. These three extra trials and the contrast A,,, obtained from the initial design, lead to a system of four equations and four unknowns. The signs of the iiiteractions are selected so that the system is a Hadamard matrix: -15-26+38+47 +I5 -26 - 38 + 47 -15+26-38+47 +15+26+38+47
Many of the 240 trials that were not run satisfy these conditions. We can write only those interactions having the assigned levels and deduce the signs of the main factors from them.
248
1 2 3 4 5 6 7 8
signs to deduce
15
26
-
-
C
-
+
+ +
38
+
-
47
+ +
- + + +
We can choose at random, or we could impose extra constraints. For example, we could choose the levels of influencing factors so that the new trials can be included in the calculation of the effects of these factors. To do so, factors 3, 5 and 8 should have the signs of a Hadamard matrix, while 3 and 8 should correspond to the signs of interaction 38. 1 2 3 4 5 6 7 8
+
+ -
-
-
+
+
-
-
+ +
The signs for columns 1, 2,4, 6 and 7 are deduced from those of columns 3, 5, 8, 15, 26, 38 and 47, again with a certain freedom, producing the following table:
TABLE12.5
ITriinOI
1 ;I ;;1 - + + + + -
+ +
-
Response
8
- + - + - - + + + + - + - +
3+8 471
We can now write the three responses y,,, y,, and y,, and the contrast mathematical model:
7.6
4.4
A,, using the
Y , =~1 - 3 + 5 - 8 - 15 - 26 + 38 + 47
y,, = I + 3 - 5 - 8 + 15 - 26 - 38 + 47 J J ,= ~
115 =
I- 3-5
+ 8 - 15 + 26- 38 + 47 + 15 + 26 + 38 + 47
These four relationships form a system of equations. The unknowns are the second order interactions. The known elements are the mean I and effects 3, 5 and 8, that were calculated from the results of the initial design. The value of contrast A,, is also obtained from the initial design.
249
h, h3 hj h, hi5
I = 3 = = 5 = = 8 = = 15 + 26 + 38 + 47 = =
=
+S.ISC~~ +3.2cm3 -2.7cm' +1.9cm3 +2.5 cm3
with these values the system of equations becomes:
~ 1 = 9
5 . 1 5 - 3 . 2 - 2 . 7 - 1.9- 15-26+38+47 5.15 + 3.2 + 2.7 - 1.9 + 15 -26 - 38 + 47 5 . 1 5 - 3 . 2 + 2 . 7 + 1.9- 15+26-38+47
h,,
2.50
~ 1 = 7
yi,
=
=
or Y , =~-2.65 yi8 =
- 15 - 26
+ 38 + 47
+9.15 + 15 - 26- 38+47
yI9 = +6.55 - 15+26- 3 8+47 2.5
=
+ 15 + 26-
38 +47
Replacing the responses by their numerical values: -15 -26 + 38 + 47 = +2.65 + 0.20 = +2.85 +15-26-38+47=-9.15 + 7 . 6 0 = -1.55 -15 +26 - 38 + 47 = -6.55 + 4.40 = -2.15 +15+26+38+47= = +2.50
Lastly.,resolving the system: 15 = +0.06 26 = -0.23 38 = +2.26 47 = +0.4 1
Thus, interaction 38 is significant. But we have assumed that the mean of the second block was the same as that of the first block. If we wish to take into account any shift in this mean we must run one more trial and carry out the calculations on twenty trials, sixteen initial trials and four extra ones.
250
5.
FOUR EXTRA TRIALS
In addition to trials 17, 18 and 19, we shall run trial 20, whose mathematical representation contains the expression: +15 +26 + 38 + 47
The four extra trials are shown in the following table (Table 12.6).
TABLE12.6 Trial no
1
17 18
-
19 20
+ +
2
3
4
5
6
8
15
26
38
47
- - + + + - + - - + - + - + + + - + - + + + + + + + + + + +
+ - + + - + + + - -
+ +
7
Response
+
10.2
We can write the four responsesyl, ,y,,, y,9 and y20using the mathematical model:
y,, = 1'-3+5-8-15-26+38+47
y,8 = I ' +3-5-8+15-26-38+47 y,!, = 1 ' - 3 - 5 + 8 - 1 5 + 2 6 - 3 8 + 4 7 y20 = 1' + 3 + 5 + 8 + 1 5 + 2 6 + 3 8 + 4 7 This system can be resolved by assuming that the effects and interactions are the same as those given by the initial design: A3
=
3
=
hj h, h,,
=
5
=
= =
8 = 15 + 26 + 38 + 47 =
+3.2cm3 -2.7 cm3 +19cm3 +2.5 cm3
Entering these values in the system ofequations, we get: yI7 = 1 ' - 3 2 - 2 7 - 1 9 - 1 5 - 2 6 + 3 8 + 4 7 .yI8 = 1 ' + 3 2 + 2 7 - 1 9 + 1 5 - 2 6 - 3 8 + 4 7 y,, = 1'-32+27+19-15+26-38+47 4'20 = 1 ' + 3 2 - 2 7 + 1 9 + 1 5 + 2 6 + 3 8 + 4 7
25 1 The fourth equation allows calculation of I' 10.2 = 1' + 4.9
I'
=
5.3
Substituting this value in the four equations gives a system analogous to that of the previous section, and from which the values of each interaction can be deduced. -15 -26 + 38 + 47 = + 2.50 + 0.20 = +15 -26 - 38 + 47 = - 9.30 + 7.60 = -15 +26 - 38 + 47 = - 6.70 + 4.40 = +I5 +26 + 38 -t 47 = - 7.70 + 10.20 =
+2.70 -1.70
-2.30 +2.50
or 15 = -0.10 26 = -0.20 38 = +2.30 47 = +0.30 Again, the result is similar, with interaction 38 being significant Comments
The results of four extra trials may be added to the sixteen results of the first set of trials to calculate the effects of 3, 5 and 8. The contrasts then contain twenty terms and there is, theoretically slightly better precision. Take, for example, contrast h, : 1 ---[-y, - 20
-)'2
- ) ' 3 -)'4
+)'5 + y 6 +?'7 +)'8 -)'9 -1110 -yll -)'I2
+)113 +yl4 +yl5 + y 1 6 -)'17
'y18
->'I9 +).20]
If the standard deviation of the response y is o whatever the value of the response, then the standard deviation of the contrasts is of&, where n is the number of responses. When the four extra responses are included in the calculation of contrasts, the standard deviation changes from o ~ f to i o1J20. As there are only 3 influencing factors we can:
reconstruct the experimental design as a duplicate 2, design as there are sixteen trials. We can thus calculate the effects and interactions, and adopt a model. present the results on a cube (experimental domain) with the response surface of interest y = 0.
252
5.1 Reconstruction of the experimental design All the trials with the same levels for factors 3, 5 and 8 (Table 12.7) are grouped together, and the effects and interactions calculated from the sixteen initial trials. The extra trials are shown in brackets, and not used to calculate the effects and interactions.
TABLE12.7
EXPERIMENTAL MATRIX REARRANGED PLASTIC DRUMS Reconstruction of the effects matrix from the results of the initial 2u design
Trial no
2 7 3 6
10 15 12 13 9 16
(17)
(19)
(20)
Effects
5.15 3.2 -2.7
1.9 -0.3 2.5
0.5 -0.2
The model is thus readily obtained by rounding off and neglecting small interactions: y
5.2 +3.2 x3 - 2.7 x5 + 1.9 x8 + 2.5 x3 XS
5.2 Presentation of results The values of the responses at each corner are calculated from the model, and the isoresponse surface drawn: The zero isoresponse surface, which is the limit that must not be crossed. The +1 cm3 isoresponse surface.
253
The best settings that can be suggested are chosen as follows: Changes in influencing factors have little effect on changes in drum volume. Changes in influencing factors around their settings that avoid crossing the zero isoresponse surface. Examination of Figure 12.1 shows that these conditions are satisfied when factor 5 is high and factor 8 low (segment AB in figure 12.1). Factor 3 can thus vary without causing excessively large changes in volume. An injection pressure ( 3 ) set to the middle of the selected segment AF3 should be suitable: point R in Figure 12.1. We can see that, for this point, changes in factors 5 or 8 never produce volumes less than 2000 cm3. However, factor 3 must vary within strictly defined limits to avoid producing drums that do not meet the specifications.
4.1
15.5
(3.05)
(15.45)
Dwell
Time
(8) 6.7
5.3
Y
Flow Rate
(5.65)
Injection Pressure (3) Figure 12.1: Isoresponse surfaces were drawn using the mathematical model. The calculated point R is only a first approximation which must be refined by supplementary trials. The setting point has been determined by the mathematical model giving the drum volume under specific conditions. But we must be careful, because if the mean experimental responses from the trials are recorded at the comers of the cube representing the experimental
254
domain (shown in parentheses in Figure 12 I), then the zero response area is slightly different from that obtained &om the mathematical model. The recommended point R could be too close to the real zero surface. A complementary study near point R is required to accurately define the best settings. We can thus extend to the first provisional conclusion fiom this study of plastic drum fabrication given in Chapter 7 as follows: Conclusionfrom the study of plastic drum fabrication: The volume of the plastic drums is generally too great, due to three
. Injection pressure (3)
. . .
I
g
Feedstock flow rate (5) Dwell time (8) There is a strong interaction between injection pressure (3) and dwell time (8)
c
The settings must be:
. . .
Feedstock flow rate (5) at high level Dwell time (8) at low level. Injection pressure, selected between levels 0 and +I in the domain defined for this factor.
These settings ensure that no drum has a volume less than 2000 om3, and should produce no drums larger than 2002 cm3 The sensitivity of volume to changes in the three influencing factors around the set point R should be studied to optimize fabrication conditions
8
255
RECAPITULATION 1. It is not always necessary to run a complementary design when there are ambiguities in interpreting the results of a fractional design. It is sometimes possible to carry out just a few extra trials. 2. Block effects are better taken into account by running two or four extra trials and using a Hadamard matrix than by running one or three trials. 3 . The extra trials for dealiasing doubtful interactions are selected on the basis of the mathematical model for the factorial design. This can resolve several problems. But it
is nevertheless wise not do this blindly. It is absolutely necessary to go back to the experimental results for the best interpretation and check them with confirmatory trials.
This Page Intentionally Left Blank
CHAPTER 13
BEYOND INFLUENCING FACTORS
1.
INTRODUCTION
All the examples we have described in the preceding chapters were used to detect influencing factors, but we have seen that a complementary study is often usefbl (e.g., in the study of plastic drum volume). This complementary study can be used to obtain three types of information: to identi@ the domain of interest. to find an optimum. to find the minimum response sensitivity to external factors
1.1. Identifying the domain of interest The choice of experimental domain is important. The desired solution may well be outside the domain selected for the initial design. This is useful for selecting influencing factors and producing an approximate model of the phenomenon. It is possible that the initial trials may not meet the experimenter's requirements. In this case, the initial results can be used by the
258
experimenter to define a new domain in which there is a good chance of finding the solution to the problem. This new part of the experimental domain that best fulfils the requirements of the experimenter is called the domain of interest.
1.2. Looking for an optimum Factorial designs are not suitable for optimization studies, because they use only two levels for each factor, so that the model contains no second degree terms. We have only covered the search for influencing factors in this book. This is thus only the initial stage in Experimentology. Once the influencing factors have been identified, it is often necessary to run an optimization design. These designs are no more complicated that those we have already studied, but the calculations are much longer and require a microcomputer. We will not study these designs, but simply indicate that they are readily obtained by adding extra experimental points to the original fractional design. All the results obtained in the first experimental phase are reused for optimization. This satisfies one of our original objectives: the progressive acquisition of knowledge by running the fewest possible trials. We will examine how best to verify that the mathematical model of the factorial designs is valid or invalid for previsions within the experimental domain.
1.3. Finding the minimum response sensitivity to external factors Application to Quality An important application of experimental design is in Quality improvement. It can be used for both product design and for manufacturing process development. The Japanese expert, Taguchi, has been responsible for applying factorial designs to the concept of quality in industrial development. In order to persue this and obtain clear results of all the tools of Experimentology must be used: influencing factors, optimisation and modelling, etc. However, it is possible to understand the fimdamental concepts of this method using a simple example. Even a search for influencing factors can resolve many of the problems of Quality. Modelling and optimization provide yet more power and efficacy.
2.
IDENTIFYING THE DOMAIN OF INTEREST
2.1. Example: Two-layer photolithography There are several critical steps in the production of the microchips bearing the thousands of transistors required for integrated circuits. High performance two-layer photolithography is one of the key steps in the fabrication. Two-layer photolithography involves several operations which must be closely controlled to obtain the submicroscopic sculpture of microchips.
259
The following example is quoted with permission from a study carried out by RTCCompelec (France). Some of the data have been changed for the sake of confidentiality. The problem: *
Photolithography is used to prepare two types of microchip: lift off and etching. Many steps are involved in this process, but just two of them can be used to understand the problem. Deposition of two layers of photosensitive resin (resin A and resin 6) onto a semiconductor substrate.
B
Photographic engraving of a groove in the resins
k
This groove must have a specific profile for each of the two applications. the lift off profile and the etching profile. This study was carried out to define the operating conditions providing these profiles.
I ki
il
The problem was defined at a meeting. The responses were defined first. The list of factors which it was considered may influence the responses defined above was then established. ARer considerable discussion, the levels of each factor were defined, and thus the experimental domain. Lastly, the experimental design was selected. Responses The responses chosen were L2 and L3 as indicated on Figure 13.1. The experimenter is looking for two different applications, the kjit ofland the etching application. L2
7 -
Resin A Resin B
LIFT OFF
L3
L2
I
s
.--. L3
Figure 13.1: Lift off and etching profiles
i
Resin A nB
ETCHING
260
The two applications, lift off and etching, have different profiles (Figure 13.1), so that the dimensions of L2 and L3 are not the same. They are indicated in the following table:
TABLE 13.1 PHOTOLITHOGRAPHY
Selection of Factors A total of eleven factors were initiallv identified, but after examination, seven were selected for hrther study.
-.
Factor 1: thickness of resin B. Factor 2: curing temperature. Factor 3: plasma 1 time. Factor 4: W dose (millijoules) Factor 5 : development method. Factor 6: development time. Factor 7: plasma 2 time.
...
Definition of the domain It is important to define the domain of each factor, i.e., the low and high levels. The combined levels define an experimental domain in a seven-dimension space. Level Factor 1: thickness of resin B Factor 2: curing temperature ("C) Factor 3 : plasma 1 time (min.) Factor 4: UV dose (millijoules) Factor 5 : development method Factor 6: development time (min) Factor 7: plasma 2 time (min)
Level +
thin 150 0.5
thick 200 3
500
1000
dip 1 1
stir 4 4
Choice of initial design As so often happens, the experimenter was not certain that the defined experimental domain contained the solution to the problem. He preferred to begin with an exploratory
26 1
design. He selected a Z74 design with only eight trials. He feared that certain factors that had been abandoned would have very slight influences, but as he did not wish to monitor them later, he decided to consider them as background noise and include them in experimental error. The trials were randomized. The planned trials were run according to the design shown in Table 13.2. Unfortunately, some responses were not measurable, so that three trials were unusable. Nevertheless, measurements could be made with the thin resin layers. The experimental domain chosen was not appropriate for these resins, but was suitable for the thick resin layers. TABLE13.2
DESIGN NO1 PHOTOLITHOGRAPHY
--
Temp.
Metd.
Time
Plm 2
2
5=12
6=23
7=13
Level-
thin
150°C 30sec
500
dip
1 min
1 min
Level +
thick
200°C
1000
stir
4 min
4 min
3 min
It was decided to choose different experimental domains for each of the two resins. One study was run on thin resins and one on thick resin layers, so that only six factors - 2, 3, 4, 5, 6 and 7 - were studied. It was decided that: The limits of factor 4 (UV dose) were changed only for the thin resin layers. 0
A completely different domain was defined for the thick resin layers. As the experimental domains for the thick and thin resin layers do not overlap, two separate designs must be used, one for each thickness.
We will confine our attention to the study of the thin resin layer. A design was run with the new domain (slightly smaller for factor 4, 350-750 millijoules rather than 500-1000). To
262
avoid any confusion in the overall analysis of results, the numbering of the factors was not changed and the trials were numbered subsequently. (Table 13.3) The new design is a 26-3 as there only six factors to be studied. We know that there are 23 terms in the AGS, and the columns of the design are aliased as in the initial 274 design: 4 = 123 5=12 6 = 23 7 = 13
We may therefore be tempted to write the independent generators as: 1= 1234 = 125 = 236 = 137
but we must take into account the fact that the first column (factor I, the resin factor) no longer represents a factor and thus we must remove it from the alias generator. We have: I
=
123.23 = 4.6
Column 1 now represents interaction 46. If we replace 1 with 46 in the alias generators we have: I = 46234 = 4625 = 236 = 4637
SimplifLing and ordering, 1=236=2456=236=3467
Two independent generators are equal, hence the four remaining independent alias generators are: 1=236=2456=3467
Multiplying them by twos and by threes gives the dependent generators, and we can write the AGS. 1=236=2456=3467=345=247=2357=567
From which the contrasts are: h, = 2 + 3 6 + 4 7 + ... h, = 3 + 26 + 45 + .. _ h, = 4 + 2 7 + 3 5 + ... h, = 5 + 34 + 67 +... h, = 6 + 23 + 57 f ...
263
h,= 7 f 24 + 56 i- ... h, = & = 46 + 25 + 37 +
The eight trials were run as indicated in design number 2 (Table 13.3). The results and the effects calculations of the two responses L2 and L3 are shown in the same table. This time all the responses can be measured, the domain is better defined but we must still be sure that the values required for L3 and L3 lie within the domain examined.
TABLE13.3
DESIGN NO2 PHOTOLITHOGRAPHY Plm 3
-22
45
150°C 30 sec
350
dip
1 min
1 rnin
Level + 200°C 3 rnin
750
stir
4 rnin
4 rnin
Level-
L,
3.75
11.25
5.5
0.75
17
14.25
2.5
65
L,
0.5
8
9.25
-2.5
13.25
33.5
-0.75
46.75
2.2. Examination of the results for response L2 The influencing factors are:
.. .
Plasma 1 (3). The development time (6). Plasma 2 (7).
264
0 0 0
Factor 4 may have a slight influence. Curing temperature (2) has no influence. The development method (5) has no influence.
Examhation of the contrasts shows that there is no significant interaction between the factors. If we adopt this hypothesis we can use the following model (neglecting factor 4): L 2 = 6 5 + l l x 3 + 17x6+14x7 where L2 is measured in hundredths of microns. This relationship can be represented by drawing the isoresponse surfaces in the experimental domain (Figure 13.2).
85
107
Plasma 2 (7)
23
45 *
c
Time (6)
Plasma 1 (3) Figure 13.2: lsoresponse curves calculated for L2 values of 0.30, 0.50 and 0.70 micron. We can see that the planned dimensions for L2 (0.3 p) is located in one comer of the domain.
265
2.3. Examination of the results for response L3 0 The influencing factors are the same as for L2 with the UV dose (4) having a larger, non-negligible influence. The curing temperature (2) and the development method (5) are both without influence. Examination of the contrasts shows that there is no ambiguity and that there are no interactions.
We will therefore adopt the following model: L3 = 4 7 + 8x3+ 9x4 + 13 x6 + 35 x7
It is difficult to plot the isoresponse surface within the experimental domain because we cannot draw them in a four-dimensional space. However, we can get a geometric representation of L3 by setting one of the factors at a given value. For example, we can compare L2 and L3 by setting x4 to 0. This is a convenient value, but another could be chosen. The model is then written: L3 = 47 + 8 x3 + 13 x6 + 35 x7
And this allows us to draw Figure 13.3
a7
103
61
Plasma 2
(7)
@ @ -9
33 ,
7
7 ,/Development Time
Figure 13.3. Isoresponse curves calculated for L3 values of 0.1 and 0.3 micron (with the UV dose set at level zero).
266
The interesting values of L3 (from 0.1 to 0.3 p) lie in the region of the domain where we found the appropriate dimensions for L2. The problem may thus be resolved We must find the region where L3 varies from 0.1 to 0.3 p and L2 remains below 0.3 p. The isoresponse surfaces indicate that this region exists, and is probably slightly outside the domain studied. We can look for this region by using mathematical models. But as we approach the domain of interest we must not neglect the influence of factor 4 on L2. The models are therefore: L2 = 65 + 11 x3 + 5 x4 + 17 x6 + 14 x7 L3 = 47 + 8 x3+ 9 x4 + 13 X6 + 35 x7
t,~~T": 1 1 -1
-2
x3
x4 x6
0
+I
Plasma 1 (sec) 750 UVdose
Development time (min) Plasma 2 (min)
x7
0
'
4
Figure 13.4: Definition of theoretical study domain.
This theoretical search will allow us to define a new domain in which there is a good chance of finding the required solution. We can illustrate the approach by setting x3 and x4 and studying the isoresponse curves in the plane x6 x7. The variable x3 is set at 30 seconds (level 1 ) and variable 4 at 150 (level-2) (Figure 13.4) L 2 = 4 4 + 17x6+ 14x7
L 3 = 2 l +13x6+35x7 Naturally, this must be confirmed experimentally. These extrapolations are only guidelines. We can draw the isoresponse curve for L2 and L3 in this domain using the above formulae (Figure 1 3.5).
267 4'
Plasma 1
2'30"
1'
1'
2'30'
Development time Figure 13.5: Region in which the experimental condition providing the lift-off (L) and etching (E) profiles will probably be found. Examination of figure 13.5 shows that the operating conditions providing the required lift off and etching profiles can be defined. The following table (Table 13.4) indicates one possible solution.
TABLE13.4 PHOTOLITHOGRAPHY
Plasma 1
w.
Development time Plasma 2
Lift off
Etching
30 sec 150 15 sec 3 min 40"
30 sec 150 78 sec 2 min 30"
But it would be unwise to consider these values as certain. We have employed several assumptions for calculating them and they may be questioned. They simply permit us to roughly define a region in which the solution to a problem may be found. It is now time to confirm that we can really obtain the lift off and etching profiles by running an extra set of trials. The new experimental domain will be reduced to four factors, as the curing temperature (2) and the development method (5) are without influence. The new domain for the remaining factors may be: Level + Level Factor 3: plasma 1 (sec) Factor 4: UV dose Factor 6 : development time (sec) Factor 7: plasma 2 (sec)
10 100 10
120
70 500 130 240
268
As the domain has been clearly defined and probably contains the required solutions, the experimenter can plan two or four measures at the central point to check the validity and quality of the model. We will not show the final results of the study. We assume that the experiments were carried out as indicated in the partial conclusions set out below.
Partial conclusion: ~
7
The thin and thick resin layers must be studied separately. For thin resin layers, there are two non-influencing factors, curing temperature and development method. We will choose:
. The lowest curing temperature (energy saving) . Dipping development method (simplest operation)
8
The trials run allowed definition of a small domain probably containing the solutions required to produce the lift off and etching profiles. The operating conditions given by the calculations must be confirmed by an experiment studying four factors: plasma 1, plasma 2, UV dose and development time. A 24 design will be run with four central points to check the quality and validity of the model.
We will now examine a case in which the methods that we have studied (factorial designs) are not powerfbl enough to provide the desired solution. This is finding an optimum.
3.
FINDING AN OPTIMUM
3.1. Example: Cutting oil stability This unpublished study was carried out in the Total laboratories. It shows how, despite their power, factorial designs are sometimes not suficient for solving a problem, and that studying the domain of interest sometimes requires even more complex methodological tools than those we have discussed so far.
The problem: Cutting oil is used to facilitate metal machining: it lubricates the machined metal and cools the cutting tool. A cutting oil is a milky f looking emulsion of water and oil. The emulsion must remain stable in 9 the machine shop, and this is ensured with a chemical additive. The investigator must determine the quantity of additive necessary to keep
1
269 '"
the emulsion stable under normal working conditions. He a stability of at least 100.
IS
looking for
*
Factors studied The investigator selected two factors:
.-
Factor 1: temperature. Factor 2: additive concentration
Response The response is the index of cutting oil stability, measured with a precision of 2 . The higher the index, the greater the stability.
Domain Low temperature level: 5°C. High temperature level: 45°C. Low additive concentration: 0.4% High additive concentration: 0.8%
Design The investigator decided to use a 22 design, but he adds two points at the centre of the experimental domain to check the validity of the model.
TABLE13.5 EXPERIMENTAL MATRIX CUTTING OIL STABILITY
Trial no Temperature
Add. conc.
I
2
1
-
-
2
+
-
3 4
-
+
+ +
5
0 0
0 0
6
Response
100
270
3.2. Interpretation The first four trials were used to calculate the effects of temperature and additive concentration. their interaction and the mean. The error on the effects was: +2
-=
A
k 1 stability point
TABLE13.6.
TABLE OF EFFECTS CUTTING OIL STABILITY
Mean
point
107.5
*
2
-3 12.5 k
point point
12
-4 k
point
1
It is wise to determine whether the calculated mean can be considered equal to the measured mean (central point) before beginning to build a model. The average of the two trials at the centre was 99 and the standard deviation was
With a 95% probability of this being true (i.e., k two standard deviations) we have: model mean: measured mean:
107.5 5 2 99 k 2.8
These two means are clearly different (Figure 13.6), making it impossible to use the factorial design model. How can we explore the domain of interest in more detail? We do not have enough information to answer this question, so we need more experimental points. But: Where should these new experimental points be placed? How can we define an optimal design? How do we do the calculations? How can we use and present the results?
271
99
107.5
Figure 13.6: The two means are clearly different.
All these questions require detailed answers, but they exceed the original objective of this book, which is to find influencing factors. Let us use this example to begin our study of modelling and optimization. These two problems fall within the range of questions treated by Experimentology. We can use matrix and statistical techniques that are as powerful as those for factorial designs to solve the problems. These techniques and their application will be the subject of a new book.
Provisional conclusion: The results show that the objective of a 100 point stability can be achieved For example, at the high additive concentration (0 8%) level the stability is over 100 at both the temperatures tested But this result is incomplete because the additive concentration is too high - it is not optimized. The economic aspect most be reconciled with demands of quality, I e , the lowest additive concentration which ensures a stability of 100 points between 5°C and 45°C The difference between the calculated and measured means makes it impossible to use the mathematical model associated with factorial designs to plot the isoresponse curves The investigator cannot make any recommendations with the information available Complementary experiments are necessary to establish a second
t degree model and plot isoresponse curves for the phenomenon
4.
FINDING A STABLE RESPONSE
4.1. Example: thickness of epitaxial deposits Taguchi [22] proposed an interesting approach for finding a robust solution to a problem. When factor levels are chosen so that the response of interest is minimally influenced
272
by factor variations, the response is said to be robust. The fundamental concept of robustness can be illustrated by a quality study carried out at AT&T in the USA, as reported by Kackar and Shoemaker [ 2 3 ] . The equipment used for preparing epitaxial deposits was set up to give thicknesses of 1415 microns. The epitaxial deposits are formed on wafers in a heated chamber. A total of 14 wafers are arranged on a support, the susceptor, that can be rotated or oscillated (Figure 13.7).
Wafer
Figure 13.7: 14 wafers placed on the susceptor.
The problem:
The deposits are not uniform, some are thinner than 14 microns, while others are thicker than 15 microns. Although the mean thickness of 14.5 microns is satisfactory, the number of rejects is high, resulting in unacceptable costs. The objective is to find new settings for the installation that give the smallest possible dispersion around the B nominal value of 14.5 microns.
8
Production set-up values The pre-study production set-up parameter values were:
..
Arsenic flux Depositing temperature
5 7%
1215°C
273
.. .
Susceptor motion Deposition time Hydrochloric acid temp. Injection nozzle position Hydrochloric acid flux
oscillation short 12OOOC 4 12%
The investigators decided to use two types of wafer in the study: type 66864 and 678D4. An eighth factor was therefore added to the seven listed above, the wafer code. They also decided to stay fairly close to the normal conditions by setting the study domain around the production set-up values. Factors The factors are the set-up parameters plus the wafer code. The domain is defined by the following table: Level +
Level Factor 1: arsenic flux Factor 2: deposition temp. Factor 3: wafer code Factor 4: susceptor movement Factor 5 : deposition time Factor 6: HCl temp. Factor 7: injection nozzle position Factor 8: HCI flux
55%
1210°C 66864 rotation long 1 180°C 2 10%
59%
1220°C 678D4 oscillation short 1215°C 6 14%
Responses The susceptor carried 14 wafers in each trial, and the thickness dispersion was measured at 5 points on each wafer.
Figure 13.8: Arrangement of points for measuring wafer thickness.
274
There are thus seventy measurements of thickness per trial. The responses chosen were the mean thickness and the dispersion of thickness. mean thickness If e, is the measured thickness and E the mean of the 70 measurements in each trial, then 1 O' e=-Eei 70 1
-
TABLE13.7 EXPERIMENTAL DESIGN
(1=2345=1346=1237=1248)
EPITAXIAL DEPOSIT
-
Trial no
1
2
3
4
5=234 5=134 7=123 P=124
1 2 3 4 5 6 7 8 9 10 11 12 13
14 15 16
Thick.
log s*
+ + + + + + + + + + + + + +
14.821 14.888 14.037 13.880 14.165 13.860 14.757 14.921 13.972 14.032 14.843 14.415 14.878 14.932 13.907 13.914
-0.4425 -1.1989 -1.4307 -0.6505 - 1.4230 -0.4969 -0.3267 -0.6270 -0.3467 -0.8563 -0.4369 -0.3131 -0.6154 -0.2292 -0.1190 -0.8625
+ +
-
55
1210 668 Rot
long
1180
2
10
Level +
59
1220 678 Osc
short
1215
6
14
Thickn -003 -005 0 0 3 -002
-041
003
007
-004
Level
I
log s2
-0005
0052
0061
0 176
-0 124 -0035
-0282
1439
-0 05 -0648
275 thickness dispersion The dispersion was defined as the log of the variance:
[
log s2 = log -X(ei-e) ;9 :7
'1
Experimental design A resolution IV fractional factorial design :2;
was selected.
4.2. interpretation Table 13.7 shows the trials run and the responses obtained. The results were interpreted by first examining the variance of the set-up factors. This approach is emphasised by Taguchi, who used it routinely to improve the quality of products or processes. The initial objective was to select the levels of factors which give the smallest possible variations in the response of interest. Once these levels have been determined for the factors influencing the variance, the nominal deposit thickness is then adjusted using the factors that influence thickness but not the variation in thickness dispersion. This provides a set-up with the correct thickness and minimal dispersion. Under these conditions the number of rejects is very small, and may even be zero if the dispersion is sufficiently small.
Thickness dispersion The factois influencing the thickness variance are shown in Table 13.8
TABLE13.8 TABLE OF EFFECTS
EPITAXIAL DEPOSIT Thickness variance Mean
-0.6484
1 2 3
-0.005
4 5
6 7 8
0.052 0.061 0176 -0.124 -0.035
-0.282 -0.050
276
The three influencing factors are, in order of importance:
..Factor 7: injector nozzle position Factor 4: susceptor rotation
.
Factor 5: deposition time.
Figu :s 13.9, 13.10 and 13.11 show that the thickness variance will be redu :d if
.
Factor 7 is set at the high level, position 6, Factor 4 is set at the low level (continuous rotation) Factor 5 remains unchanged (short deposition time).
Log s2
- 0.366 - 0.648
- 0.930 -1
(.;i /,
0
+I
i(6') ,
INJECTION NOZZLE (7)
Figure 13.9: Influence of injection nozzle position (Factor 7) on thickness variance.
277
Log
s2
t
-1 0 +I Rotation Oscillation SUSCEPTOR MOVEMENT (4)
Figure 13.10: Influence of susceptor rotation (Factor 4) on thickness variance.
s2
t
-1
Long
0
+I Short
DEPOSITION TIME (5)
Figure 13.1 1: Influence of deposition time (Factor 5) on thickness variance.
278
All that remains is to interpret the results of the experimental design for the thickness itself
Thickness Table 13.9 summarizes the effects and interactions of the factors studied
TABLE13.9 TABLE OF EFFECTS
EPITAXIAL DEPOSIT Thickness Mean
14.39 micron
1 2 3 4 5 6 7 8
-0.03 micron -0.05 micron 0.03 micron -0.02 micron -0.41 micron 0.03 micron 0.07 micron -0.04 micron
12 13 14 15 16 17 18
-0.01 micron
0.02 0.00 -0.01 0.02 0.01 -0.03
micron micron micron micron micron micron
Only one factor is influent:
.
Factor 5 : the deposition time.
There is no apparent interaction. Figure 13.12 shows the influence of deposition time on the mean thickness of the epitaxial deposit. The exact deposition time providing a thickness of 14.5 microns is readily calculated if the values of the short and long levels are known in minutes and seconds. These values are not available for obvious industrial reasons. We can however do the calculation using coded values. The mathematical model is: y
=
14.39 - 0.41 x
279
14.80
14.39 14.50
13.98
--\ -0.27 Long
Short
DEPOSITION TIME ( 5 )
Figure 13.12: Influence of deposition time (Factor 5) on the mean thickness of the epitaxial deposit.
setting y = 14.5, we get: 14.5 = 14.39 - 0 . 4 1 ~ hence: X=
-14.5+14.39 0.4 1
-
0.11 - -o,268 0.41
This value can be used to calculate the optimal deposition time in coded variable (Figure 13.12) and given as a recommendation. The information provided by the experimental design allows: Reduction of the dispersion of deposit thickness by changing the injection nozzle position and using continuous rotation instead of oscillating. The nominal thickness of 14.5 microns can be obtained by changing the deposition time. This slightly increases the thickness variance, but fortunately factor 5 is not the most important for dispersion.
280
Before giving the results and making recommendations for setting up the industrial production, the interpretation must be verified. A series of trials was therefore run with the new settings - the confirmatory trials. The standard deviation was found to be 0.24 micron, the thickness was 14.5 microns and there were almost no rejects. Conclusion:
i The dispersion of epitaxial deposit thickness can be reduced by adjusting the set-up factors as shown in the following table.
6
I
Factor Arsenic flux Deposition temperature.
1
Original setting 5 7%
I
New setting 5 7%
1215°C
1215°C
Susceptor movement
oscillation
rotation
Deposition time HCI temperature
1200°C
variable 1200°C
Injection nozzle HCl flux
12%
12%
I
The deposition time will be set to obtain the required mean thickness of 14.5 microns. The recommended settings guarantee a standard deviation of k 0.25 microns around this thickness.
This example illustrates an important concept emphasized by Taguchi: there is one setting, among all the possibilities, that minimizes the variance in the target response. In order to ensure the Quality of a product or process the stable or robust settings must be found. But Taguchi went further. Not only did he study the factors influencing fabrication, he also studied the factors that could influence the life of the product after it had left the factory. He examined all the conditions, from product design, to manufacture and subsequent client use. This approach to research and development, coupled with cost control, illustrates a particularly interesting application of experimental design to quality improvement.
28 1
RECAPITULATION 1 . The photolithography example has emphasized one of the key points in experimental
design: the search for the domain of interest. A combination of a progressive approach and detailed analysis at each stage will invariably lead to a solution whenever such a solution exists. Only experimental design can provide a rapid, reliable solution to a problem involving several factors. We must therefore again emphasize that non-influencing factors are not necessarily of no importance. No change in the response can produce savings (see also the examples on the colour of a product, bean-growing experiment and epitaxial deposits) and facilitate the production of optimal settings. 2. The cutting oil example shows that we must be extremely carefil before adopting a
model, even for a very simple case. It is vital to carry out trials at the centre of the domain to test the model. Central point trials must be run as soon as the investigator believes he is within the domain of interest and wishes to begin model-building. They are immediately usehl for estimating the standard deviation, and will remain usehl for finding a second degree model, if required. Model-building and optimization are almost always the experimenter's goal. Identifylng influencing factors and using the first degree model are often only the initial phase of the study. 3. An important application of experimental design is the use of the variance as the
response. It is possible to identifl factors that minimize the dispersion of responses so as to make them less sensitive to external factors. This is an effective way of improving the quality of a product or process, Applying this technique right fiom the design of a product, i.e. during the R&D phase, is the surest method of ensuring quality, reduced product control costs and the widest market for the product.
This Page Intentionally Left Blank
CHAPTER 14
PRACTICAL METHOD
OF CALCULATION USING A QUALITY EXAMPLE
1.
INTRODUCTION
It is much easier to interpret experimental designs and allied methods if the calculations, outlines and isoresponse curves are prepared quickly and accurately. This can only be done with a microcomputer. But this does not imply that it always requires expensive, dedicated software. All the examples in this book were prepared using a simple spreadsheet, Lotus 123. In general, the calculations involved in searching for influencing factors and evaluating variance are simple. However, those for optimization designs and identifirlng the best experimental points when preparing special optimal designs (mixture designs or designs with constraints) are not. Specific softwares are then necessary. These dedicated programs (see Nachtsheim [24] vary in complexity and ease of use, and require careful selection. The experimenter must choose carefully at all phases: design selection, aliase selection, factor
284
selection, mathematical model, residuals calculation, domain selection, etc. The software should help the experimenter make decisions and not make them for him! It is a good idea for the experimenterbeginning to use factorial designs to do the detailed calculations himself, so that he can better understand the si@cance of each result. The quality of his interpretation and conclusions depend on this. This is why we will now go step by step through an example. The presentation has been made more accessible by separating the calculations from the descriptive section. The first part of this chapter describes the problem to be studied, and references are given for each detailed calculation. The second part of the chapter covers the details of each operation; these can be reproduced by anyone with a copy of the Lotus spreadsheet or an equivalent. In this way the reader can follow the reasoning and check the calculations.
2.
A QUALITY IMPROVEMENT EXAMPLE
Example: Study of truck suspension springs This example is taken from Pignatiello and Ramberg [25].It was carried out by the firm of Eaton Yale to improve the manufacture of truck suspension springs. It is the type of design that provides quality improvement, as defined by Taguchi, by reducing response variance and then adjusting the response to the required value.
The problem Truck springs are made up of leaves having a precisely defined curvature. The curvature must be exactly eight inches, with a very small variation around this value. The leaves undergo several treatments during manufacture, including: 0
Heating to high temperature in a furnace.
0
Immersion in an annealing oil bath.
Bending in a special forming machine. The study was carried out to determine the factors influencing the curvature and the dispersion of the curvature during the three phases described above. This information was used to advise on how: 0 0
The mean curvature could be kept at 8 inches, The dispersion of curvature around 8 inches could be as small as possible.
It is important to clearly define the factors to be studied, the experimental domain and the responses, before beginning any experiments.
285
Factors The fabrication engineers believed there to be four influencing factors that may show important interactions. This was kept in mind when choosing the experimental design and the aliases. They also believed that a fifth factor (the temperature of the annealing bath) could be influencing,,but its control during fabrication would require extra equipment. They therefore preferred to consider it as a non-controlled factor contributing to the experimental error; this factor could also be said to increase the background noise. Nevertheless, steps were taken to give it two levels during the study (low and high), but these levels were defined approximately because temperature was not accurately regulated.
.
Factor 1 : furnace temperature. .Factor 2: heating time. Factor 3 : transfer time between leaving the furnace and placing in the bending machine. Factor 4: bending time. Factor 5: annealing bath temperature.
.. .beInteractions neglected.
12, 13 and 23: The experimenters believed that they could not
The level of factor 5 is difficult to keep constant because there is no regulation of the bath temperature. This factor could be considered as background noise, and the experimenters preferred ta treat it as part of the experimental error. This factor is not, therefore studied in the initial interpretation (step l), and the six responses of each set of trials will be treated as equivalent.
Domain The experimental domain for the five factors studied is shown below (Table 14.1).
TABLE14.1
TRUCK SUSPENSION SPRINGS
Factor 1 (OF) Factor 2 (sec) Factor 3 (sec) Factor 4 (sec) Factor 5 (“F)
1840 25 12 2 130-150
1880 23 10 3 150-170
286
Experimental design The engineers had defined four factors and wished to carry out only eight trials. They therefore chose a 241 design. Factor 4 is aliased with interaction 123. The reader can see that this is a resolution IV design. For each trial, three experiments are run at the low level of factor 5, and three at the high level. This provides an indication of the background noise introduced by this factor for each of the eight trials. The engineers want to know the values of interactions 12, 13 and 23, in addition to those of the four factors. The three remaining columns of the 2&' design are therefore assigned to these three interactions.
Responses The responses must be chosen so as to reflect both the value of the curvature and the dispersion ofthe curvature around the mean value. The engineers selected one response for the curvature and three responses for the dispersion: Curvature The mean curvature is selected as the sole response for curvature Dispersion of curvature The three responses are: 1 . The variance of the curvature for each set of trials. This variance will be indicated by s", with a subscript indicating the set of trials.
2. A variance fhction, Z, defined by z=10 logs2 3. The signallnoise hnction proposed by Taguchi, Z'
Z' = 10 log1 Y 2 s
Experiments Table 14.2 shows the experimental design and the six responses obtained per trial The results are interpreted in two steps: Step 1: There are four main factors, factor 5 was set at two levels but is not taken into account. It is treated as an uncontrolled factor. The six responses from each trial are therefore equivalent and are analysed together.
287
Step 2: As the influence of factor 5 could not be ignored, the experimenters use the trial results to construct a design including all five factors. As factor 5 is a controlled factor, its effect on the responses is determined.
TABLE 14.2 EXPERIMENTAL MATRIX
TRUCKSUSPENSION SPRINGS Responses ~~
~~
~
5-
6 7
7.78 8.15 7.50 7.59 7.94 7.69 7.56 7.56
+ +
3.
7.78 8.18 7.56 7.56 8.00
8.09 7.62 7.81
~
5+ 7.81 7.88 7.50 7.75 7.88 8.06 7.44 7.69
7.50 7.88 7.50 7.63 7.32 7.56 7.18 7.81
7.25 7.88 7.56 7.75 7.44 7.69 7.18 7.50
7.12 7.44 7.50 7.56 7.44 7.62 7.25 7.59
INTERPRETATION, STEP 1
3.1. Calculation of responses 3.1.1.
Mean curvature
For trial number 1, the six values are used to calculate the mean 7,
yl = -[1 6
7.78+7.78+7.81+7.50+7.25+7.12]
The same calculation is performed for the seven remaining sets of trials (see calculations, screen 14.1, p. 310).
288
3.1.2.
Dispersion of curvature
(see calculations, screen 14.2, p. 3 11)
1. Variance
For trial number 1, the variance 2 is given by the formula: (7.78-7.54) 2 +(7.78-7.54)2 +(7.81-7.54)2 +(7.50-7.54)2 +(7.25-7.54)2 +(7.12-7.54)2]
6-1 S:
= 0.0900
TABLE 14.3
EFFECT MATRIX TRUCK SUSPENSION SPRINGS
+ +
+ +
Level - 1840
25
12
2
Level+ 1880
23
10
3
289
2. Eunction Z
(see calculations, screen 14.3, p. 3 12)
For trial number 1, the Z hnction is Z, : Z, = 10 logsf = 10 log 0.09
Z,= -10.45 3. Function Z'
(see calculations, screen 14.3, p. 312)
For trial number I, the Z' hnction is Z;:
z;=
-2
10 log% = 10 log s1
~
(7.54)2 0.09
Z;= 28.00 These four responses (7,9, Z and Z') calculated from the raw experimental results can be used to calculatethe effects of each factor. The effects matrix is shown in Table 14.3, which also contains the effects and interactions. (see calculations, screens 14.4 - 14.8, p. 313, 314, 315) We can now analyses these results knowing that the influence of factor 5 remains to be examined. We will take this factor into account in step 2 of the interpretation.
3.2. Analysis of results (interpretation, step 1) 3.2.1. Mean curvature The mean standard deviation of one trial is 0.2147 (screen 14.2, p. 3 11). Each effect is calculated from the 48 experimental results, giving a standard deviation of 0.2147 OE =--
J48
0.2147 6.928
- -= 0.031
The results can be summarized in a table of effects, Table 14.4. (see calculations, screen 14.5, p. 314)
290
TABLE14.4 TABLE OF EFFECTS
TRUCKSUSPENSION SPRINGS
1
First interpretation
Mean 1
2
3 4 12 + 34 13 + 24 14 + 23
7.64 f 0.03 0.11 -0.09 -0.01 0.05
f f L f
0.03 0.03 0.03 0.03
-0.01
& 0.03
-0.01 -0.02
f 0.03 5 0.03
It appears that = Factor 1 (furnace temperature) is influent.
-
Factor 2 (heating time) is influent.
= Factor 4 (bending time) has a small influence.
Factor 3 (transfer time) and all the interactionshave no influence.
3.2.2.
Dispersion of curvature
The results can be summarized in a table of effects, Table 14.5 (see calculations, screens 14.6, 14.7 and 14.8, p. 314, 315)
29 1
TABLE14.5 TABLE OF EFFECTS TRUCKSUSPENSION SPRINGS Dispersion of spring curvature First Interpretation Effect Mean
variance
Z
Z'
0.0460
-16.02
33.67
1 2 3 4
-0.0088 -0.0300 0.0037 -0,001 1
0.29 -4.73 2.27 -1.41
-0.16 4.63 -2.28 1.47
12 + 34
0.0054 -0.0057 0.0079
1.14 -1.73 2.57
-1.15 1.72 -2.59
13+24 14 + 23
Factor 2 (heating time) had a great influence on the variance of curvature. The fbnctions Z and Z' confirm the influence of factor 2, and suggest that factor 3 and interaction 23 could be influent. 3.2.3.
Effect of factor 5 on curvature
Factor 5, which has been considered as background noise until now, remains to be examined. As this factor was studied at two levels we can calculate its effect: the average of the low level is 7.76, and the high level average is 7.50 (see calculations, screen 14.9, p. 316). 1 E5= -[7.50-7.76]= - 0.13 2
This factor has the greatest influence on spring leaf curvature! The experimenters consider it to be unreasonable to leave it unregulated during fabrication. But, before deciding to make the investment required to control it, they check that the objective could be attained. They therefore carry out a further analysis of the results.
292
Curvature 7.76
\
7.63
7.50
.
-1
+I
140°F
160°F
-0.13
TEMPERATURE (5)
Figure 14.1: Influence of annealing bath temperature (Factor 5) on curvature
4.
WHAT IS A GOOD RESPONSE FOR DISPERSION?
We will begin by studying the dispersion of curvature in order to identi@ the settings of factors that minimize it. The reader may well ask why three responses were used to define this one property. One would have been enough, but it would have to accurately define the dispersion. Unfortunately, this ideal response does not exist. In this situation we generally try to substitute quantity for quality. But as we will see, it is a vain hope, and each of the responses has a weakness. The three responses, variance, logarithm of variance and signal-tonoise ratio will be examined individually.
4.1. Variance This is, a priori, a good response as it measures the dispersion of a set of measures around the mean. But variance must always be positive, like all algebraic squares. This property may not be respected when the mathematical model of factorial design is used. There is a risk of having a negative variance. To avoid this problem, statisticians use the logarithm of variance.
293
4.2. Logarithm of variance The logarithmic function, log x, has a great advantage. It can be positive or negative, but it always gives a positive x (Figure 14.2). Using it, therefore avoids any problems of impossible variance. But, the problem with the log hnction is that it distorts the original information. It emphasises small differences in the variance when the variance is very small. As a result, the effects depend more on the difference between small variances than on the variances themselves.
Figure 14.2: Plot of the logarithmic function.
4.3. Comparison of variance and logarithm of variance Let us assume that we have to interpret the results of a 22 experimental design. We study the dispersion and have two responses, the mean variance and the log of the mean variance. The mean variance of the high level of factor 1 (indicated as s z ) is 0.30 and the mean variance
of the low level is 0.10 (indicated as s!). We can calculate the effects of the factor with the two responses .? and 10 log .?: Variance 9
EsZ
1
= -[0.30-0.10]=
2
4.10
294
z = 10 log s2-
1 2
E, =-[101og0.30-10log0.10]
E,
1
= --[-5.23-(-lo)]
2
1 = - 4.77 = 2.36 2
The mean variance for the high level of factor 2 is 0.01, and the mean variance at low level is 0.001. We can also calculate the effects of factor 2 with the two responses, variance and log of the variance. Variance 9 1 Es2 = --[0.01-0.001] = +0.0045 2
z = 10 log .9 E,
1
= 2[10
log 0.01-10 log 0.0011
1 E L = -[-20-(-30)] 2
=5
Figure 14.3 compares the two methods, showing the effects of factors on dispersion. With variance .-?,factor 1 has the greatest influence, while with 10 log ,>?- factor 2 has the greatest influence. Thus, interpretation is not easy in this case.
10 Log s2
S2
E1=0 1000 E2=0.0045
I -1
b
+I
E2= 5 E l = 2 36
-1
+I
Figure 14.3: The response selected (9or log s2) may influence the evaluation of effects.
295
4.4. The signal-to-noise ratio The signal-to-noise ratio, Z', is no better. We can write:
z' = 10 log-Y2 = 10 logy2 -10
logs 2
s2
As J 2 varies little, log p2 can be considered to be constant. We can calculate the effects E zr of a factor with the function Z': EZ,=,((1010gJ2 1 -1010gs:)-(1010gY2
E,,=
L
L
1
[(-10 logs: )
-lOlogs?)] J
-
(-10 logs-
or simply
Thus the function Z' has the same advantages and disadvantages as Z , except that the signs are inverted, providing a further risk of error in interpretation. In our study of spring leaves, we will use 2 and log 2.Let us now continue with the second stage of interpretation.
5.
INTERPRETATION, STEP 2
5.1. Calculation of reponses The results are completely reanalysed as if there are five factors and sixteen trials. A 25-1 design is constructed by adding the two levels of factor 5. The preceding 24-1design is used, first with the low level of factor 5 (trials 1-8 in Table 14.6), and then a second time using the high level of factor 5 (trials 9-16 of Table 14.6) This new design may be considered as a 24+1-1 design, which is more simply written as Z5-'.The alias generator is I = 1234, which can be used to calculate the contrasts. The four responses ( y , 9,Z and Z') are calculated: screen 14.9
296
p.316, screen 14.10 p. 317 and screen 14.11 p. 318. The results of these calculations are shown in Table 14.6.
TABLE 14.6
EXPERIMENTAL MATRIX TRUCK SUSPENSION SPRINGS
Responses
7.78 8.15 7.50 7.59 7.94 7.69 7.56 7.56 7.50 7.88 7.50 7.63 7.32 7.56 7.18 7.81
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
-
7.78 8.18 7.56 7.56 8.00 8.09 7.62 7.81 7.25 7.88 7.56 7.75 7.44 7.69 7.18 7.50
-
7.81 7.88 7.50 7.75 7.88 8.06 7.44 7.69 7.12 7.44 7.50 7.56 7.44 7.62 7.25 7.59
--
L 7.79 8.07 7.52 7.63 7.94 7.95 7.54 7.69 7.29 7.73 7.52 7.65 7.40 7.62 7.20 7.63
3 273 12 104 36 496 84 156 373 645 12 92 48 42 16 254
Z
Z'
-35.22 -15.64 -29.21 -19.81 -24.44 -13.04 -20.76 -18.06 -14.28 -11.90 -29.21 -20.34 -23.19 -23.73 -27.87 -15.94
53.06 33.77 46.73 37.43 42.43 31.04 38.30 35.77 31.54 29.67 46.73 38.01 40.57 41.37 45.02 33.60
-
The reader will note that controlling factor 5 reduces the standard deviation of one trial from 0.21 to 0.13. In general the more factors that are controlled, the smaller the experimental error (see calculations screen 14.11 p. 3 18).
5.2. Analysis of results (second step of interpretation) The experimental matrix can be used to construct the effects matrix, whose results for variance of curvature are shown in Table 14.7 and for curvature in Table 14.8. 5.2.1.
Dispersion of curvature
The results are shown in Table 14.7 (see calculations, screen 14.17 p. 324, screen 14.18 p. 325, screen 14.19 p. 325)
297
TABLE14.7
TABLE OF EFFECTS TRUCKSUSPENSION SPRINGS Dispersion of curvature Second Interpretation Effect
variance
Z
z'
0.0165
-2 1.40
39.07
1 + 234 2 + 134 3 + 124 4 + 123 5 + 12345
0.0092 -0.0074 -0.0023 0.0014 0.0020
4.10 -1.23 0.54 0.47 0.60
-3.90 1.13 -0.55 -0.41 -0.75
12 + 34 13 + 24 14 + 23 15 + 2345 25 + 1345 35 + 1245 45 + 1235
-0.0032 0.0003 0.0060 -0.00 19 -0.0017 -0.0071 0.0040
0.00 -0.92 1.45 -1.28 -1.30 -2.41 0.28
-0.01 0.91 -1.48 1.33 1.39 2.38 -0.26
125 + 345 135 + 245 235 + 145
0.0038 -0.00 18 0.0076
2.36 0.94 1.85
-2.37 -0.91 -1.88
Mean
These data show that the influent factors are different, depending on w..:ther is used for interpretation. This is not surprising.
z = 1Olon s2The influent factors and interactions are: .Factor 1 Interaction 35 Interaction 125 Interaction 235
...
4.10 2.41 2.36 1.85
2 or log 2
298
Variance .2 Factor I is again influent, while factor 2 appears to have a negative influence. Interactions 35 and 235 are influent, while interaction 125 has little influence. The high value of 125 given by Z is due to a large difference between two small values. The influent factors and interactions are thus: .Factor 1 Factor 2
.
+0.009 -0.007
.Interaction 35 Interaction 23 5
+0.007
-0.007
The influence of factor 1 on curvature variance is shown in Figure 14.4; the variance is smaller if the low level of factor 1 is selected. The influence of factor 2 is shown in Figure 14.5; here, in contrast, the high level should be chosen
S2
0.0257
0.0165
0.0073
/
i
+ 0.0092
+ 1840 F
1880 F
FACTOR (I)
Figure 14.4: Influence of furnace temperature (Factor 1) on the variance of curvature.
299
S2
0.0239
0.0165
- 0.0074 0.0091
+
-1
25 sec
23 sec
FACTOR (2)
Figure 14.5: Influence of heating time (Factor 2) on the variance of curvature.
3 00
5.2.2
Curvature
The results are shown in Table 14.8 (see calculations screen 14.16 p. 323)
TABLE14.8 TABLE OF EFFECTS TRUCKSUSPENSION SPRINGS Second interpretation Mean
7.63 k 0.02
1 + 234 2 + 134 3 + 124 4 + 123 5 + 12345
0.11 -0.09 -0.01 0.05 -0.13
f 0.02 f 0.02
12 + 34 13 + 24 14 + 23 15 + 2345 25 + 1345 35 + 1245 45 + 1235
-0.01 -0.01 -0.02 0.04 0.08 -0.03 0.01
f 0.02
125 + 345 135 + 245 235 + 145
*
0.02
k 0.02 f 0.02
*
0.02
f 0.02 f 0.02 f 0.02
k 0.02 f 0.02
*
-0.005 0.02 0.02 k 0.02 -0.02 k 0.02
It is hardly surprising that the effects are the same as those previously (first interpretation) calculated for curvature: they are derived from the same data and treated in the same way. The inclusion of factor 5 allows calculation of sixteen contrasts instead of eight and reveals new interactions. There are three influencing factors: 1, 2 and 5 , and one interaction that cannot be ignored, interaction 25.
30 I
Curvature
7.74
i
7.63
+ 0.11
7.52
-1 1840 F
+I 1880 F
FACTOR (1) Figure 14.6: Influence of furnace temperature (Factor 1) on spring curvature.
Curvature
7.72 7.63 7.54
-1
\
-0.09
+I
23 s
25 s
TIME ( 2 )
Figure 14.7: Influence of heating time (Factor 2) on spring curvature.
3 02
Curvature 7.76
7.63
1
-0.13 I
7.50
-1
+I
140°F
160°F
TEMPERATURE (5) Figure 14.8: Influence of annealing bath temperature (Factor 5) on spring curvature.
6.
OPTIMIZATION
As the average curvature is low, 7.63 inches, the levels of factors that increase it as much as possible should be selected. But we must also keep the variance of curvature to a minimum. Table 14.9 shows the elements of the discussion that we will use to choose the factor levels, listing the significant factors and interactions for the corresponding responses.
TABLE14.9
TRUCK SUSPENSIONSPRINGS Factors and interactions significantly influencing the responses
Factorsor interactions
1
2
Variance of curvature
+3
0
Curvature
.:. .:.
3
4
5
23
25
0:.
.:. .:.
35 0:.
.:.
235
*:.
303
The high level of factor 4 may be chosen to increase curvature. It is more difficult to choose the levels of the four other factors as the interactions must be taken into account. The easiest way to resolve this problem is to write mathematical models for the two responses. Curvature = 7.63 + 0.1 1 x1 - 0.09 x2 + 0.05 x4 - 0.13 xs + 0.08 x2 x5 Variance = 0.0165 + 0.009 x, - 0.007 x2 + 0.006 x2 x3 - 0.007 x3xs + 0.007 x2 x3xS The two responses, curvature and variance of curvature, are calculated (see calculations screen 14.20 p. 326, screen 14.21 p. 328, screen 14.22 p. 329, screen 14.23 p. 330) for all the possible combinations of the four factors 1, 2, 3 and 5 (Table 14.10). Factor 4 is held at the high level.
TABLE14.10 TRUCK SUSPENSION SPRINGS Calculation of curvature and dispersion of curvature for all combinations of x1 x2 x3 and xs ~~
Case no 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
-1 +1 -1 +I -1 +I -1 +1 -1 +1 -1 +I -1 +1 -1 +I
-1 -1 +1 +1 -1 -1 +1 +1 -1 -1 +1 +1 -1 -1 +1
-1 -1 -1 -1 +1 +1 +1 +1 -1 -1 -1 -1 +1 +1 +1 +1
-1 -1 -1 -1 -1 -1 -1 -1 +1 +1 +1 +1 +1 +1 +1 +1
+I ---
Dispersion of curvature
Curvature
65 245 -55 125 225 405 65 245 345 525 -5 5 125 -55 125 65 245
7.87 8.09 7.53 7.75 7.87 8.09 7.53 7.75 7.45 7.67 7.43 7.65 7.45 7.67 7.43 7.65
The results in the table show that an eight-inch curvature is obtained in two cases:
3 04
1. Case 2: I + 2- 3- 4+ 5- produced a curvature of 8.09 inches with a variance
of 0.0245 2. Case 6: I + 2- 3+ 4+ 5- gives the same curvature (8.09 inches) but the variance of curvature is greater, at 0.0405. It would thus be best to select the low level of factor 3 to minimize the variance of curvature. This selection has no influence on the curvature itself, as curvature is independent of xj. Examination of Table 14.6 shows that Case 2 is the same as Trial number 2. The experimental values confirm the values calculated from the model. The study could therefore be stopped at this point and these setting used. But perhaps we could try to reduce the variance of curvature still further. For this purpose, we adopt a special representation of a four dimensional space. The levels of factors 1 and 2 are represented as in a 22 design (Figure 14.9). We then plot the isoresponse curves for curvature for each level of the pair of factors 1 and 2 in the plane of factors 3 and 5.
160°F
160°F
(5)
(5)
140°F
140°F
160°F
U / \
12"
(3)
10"
12l
(3)
10"
(2)
(l)
(5) 7.8
140°F
12"
140°F (3)
lo"
Figure 14.9: Isoresponse curves of curvature variance.
305
This diagram shows that a curvature of 8 inches can only be obtained in the region of comer where factor 1 = +1 and factor 2 = -1. If we assume that the smallest possible value of factor 5 is -1, we can calculate the location of points where curvature is 8 inches within the space of factors 1 and 2. For this we write that curvature is eight inches. 8 = 7.63 + 0.11xl - 0.09 x2 + 0.05 ~4 - 0.13 ~5 + 0.08 ~2 ~5
The trajectory where the curvature is 8 inches when x5 = -1 and x4 = +I is given by: 8 = 7.63 + 0.1 1 x1 - (0.09 + 0.08) ~2 + 0.05 + 0.13
or 0.17 x2 = 0.1 1 x1 - 0.19 or
11x1 17
x2 =---
19 17
this relationship is represented by a straight line (Figure 14.10). It shows all the possible settings of factors 1 and 2 to obtain a curvature of eight inches. The final choices of setting will depend on the variance, which must be as small as possible. We have:
V
0.0165 + 0.009 x1 - 0.007 x2 + 0.006 x1 x3 - 0.007 ~3 x5 + 0.007 ~2 ~3
~5
Ifwe apply the preceding hypothesis x3 = -1, x5 = -1 and x2 = 11x1/17-19/17, we get: V = 0.0162 + 0.0051 x1
This relationship can be used to place a scale on the straight line found previously. The different values of the variance of curvature are shown on Figure 14.10
306
Factor 2
A;;l*;
+I
Factor 1
v = 0.0213
-1
v= 0.0171
Figure 14.10: Optimizing the variance of curvature. The smallest variance is given when xz is at its low level and x1 equals 0.18. If we remain within the domain studied, the settings providing a curvature of eight inches with the smallest possible variance are (in coded units): x, = x2= x3= x4= x5=
0.18 -1 -1
+1 -1
In real-world units these are: .Factor I Factor 2 Factor 3 Factor 4 Factor 5
.. .
1863.6 deg F 25 seconds I2 seconds 3 seconds I30 - 150 deg F
If the manufacturing constraints allow us to move outside the experimental domain, the investigators should be able to reduce the variance of curvature still further while maintaining an average curvature of 8 inches. They could: Reduce the furnace temperature (factor 1). Increase the heating time (factor 2). More closely regulate and reduce the annealing bath temperature (factor 5).
307
Whatever the results of the theoretical study, trials must be carried out to continn that the predictions are accurate and that the objective could be attained. Conclusion
a
rli iii II
The dispersion of the curvature of truck suspension spring leaves can be reduced by setting: 0
The furnace temperature at 1863 deg F.
0
The furnace heating time to 25 sec.
0
fi 0
I! fr
41 B
The transfer time between furnace and bending machine to 12 sec. The bending time to 3 sec.
The annealing bath temperature cannot be considered to be background noise as it strongly influences the degree of curvature. If it is not regulated all the other efforts are worthless Therefore the required investment must be considered. Thus, the annealing bath temperature should be set at the lowest temperature used in this study, 130 deg F. Confirmatory trials should be run to verify the predicted settings
308
RECAPITULATION This study of the curvature of truck spring leaves has highlighted several points. 1 . The difficulty and the importance of interpretation. The experimenter must make the results speak, which requires considerable calculation to extract their useful information content. Interpretation is never automatic; it requires intelligence and imagination. The experimenter must select the most appropriate hypothesis in order to present clear, useful conclusions. 2. The use of variance as a response. This technique is often used to improve quality.
The first step is to reduce dispersion by choosing the appropriate levels that influence it; the other factors are then adjusted to reach the target value. 3. The selection of the right response is often tricky, and requires a great deal of care.
The variance example clearly illustrates this. 4. The setting of a factor influencing the response studied can reduce experimental error.
The more factors that are controlled, the more the experimental error is reduced. The next part of the chapter covers the details of calculation. It shows how to obtain the results used in the first part of the chapter. We recommend that this second section should be accompanied by the calculations run on a microcomputer.
CHAPTER 14 (CONTINUED)
DETAILED CALCULATIONS FOR
THE TRUCK SUSPENSION SPRINGS
EXAMPLE
The calculations are readily performed using a spreadsheet. This example was prepared with Lotus 123 running on a PC compatible microcomputer. The reader can transpose the calculations to hidher particular softwarehardware system.
1.
CALCULATION FOR THE FIRST INTERPRETATION
a) Calculating the mean curvature for each trial Open the first worksheet : sheet 1, and enter the 48 trial results (screen 14.1). The mean is calculated by placing the instruction :
@AVG(B9..G9)
310
in cell 19. The other means are calculated by copying this instruction to cells I10 to 116.
SCREEN 14.1 Calculation of mean curvature for each trial
Trial no Y1
Y2
Y3
Y4
Y5
Y6
7.78 8.15 7.50 7.59 7.94 7.69 7.56 7.56
7.78 8.18 7.56 7.56 8.00 8.09 7.62 7.81
7.81 7.88 7.50 7.75 7.88 8.06 7.44 7.69
7.50 7.88 7.50 7.63 7.32 7.56 7.18 7.81
7.25 7.88 7.56 7.75 7.44 7.69 7.18 7.50
7.12 7.44 7.50 7.56 7.44 7.62 7.25 7.59
Mean
8
1 2 3 4 5 6 7 8
7.540C 7.9017 7.520C 7.640C 7.670C 7.785C 7.3715 7.660C
18 19 2 0"
b) Calculating the curvature variance There are several ways of doing this calculation. This is one way. The data on screen 14.1 are used to construct the deviation squared table ( y , -Fil2. The term in cell B25 in screen 14.2 is obtained from the instruction : (B9 - $19)*(B9 - $19)
or
(B9 - $19)*2
and copying it to cells B25 to G32. The variance, .$, of a trial is obtained by adding the squares of the differences and dividing the sum by 6 - 1. For trial number 1, the instruction : @SUM(B25..G25)/5
is placed in cell 125, and copied to cells I26 to I32 (screen 14.2)
311
SCREEN 14.2 Calculation of curvature variance, SZ A
VARIANCE CALCULATION
Variance
0.0576 0.0617 0.0004 0.0025 0.0729 0.0090 0.0355 0.0100
0.0576 0.0775 0.0016 0.0064 0.1089 0.0930 0.0617 0.0225
0.0729 0.0005 0.0004 0.0121 0.0441 0.0756 0.0047 0.0009
0.0841 0.0005 0.0016 0.0121 0.0529 0.0090 0.0367 0.0256
0.0016 0.0005 0.0004 0.0001 0.1225 0.0506 0.0367 0.0225
mean
0.1764 0.2131 0.0004 0.0064 0.0529 0.0272 0.0148 0.0049
0.09004 0.07073 0.0009E 0.00792 0.09084 0.05291 0.03801 0.01728
variance
0.046088
standard deviation
0.214681
These calculations can be used to obtain the mean variance of all the trials : the variances in column I are added together and divided by 8 using the instruction @SUM(I25..I3 2)/8
placed in cell 135. The square root of the mean variance gives the standard deviation of an individual response. It is obtained by placing : @SQRT(I35) in cell I37 (screen 14.2)
c) Calculating the Z function The variance in cell I25 is used to calculate the Z hnction for trial number 1 by placing the instruction : lO*@LOG(125)
312
in cell L25 (screen 14.3). The instruction is copied to obtain the eight Z hnctions (L25 to L32).
SCREEN 14.3 Calculation of the responses s2, Z and Z'
z
Variance 0.09004 0.07073 0.00096 0.00792 0.09084 0.05291 0.03801 0.01728
-10.4556 -11.5035 -30.1772 -21.0127 -10.4172 -12.7646 -14.2002 -17.6242
Z' 28.003 29.458 47.701 38.674 28.113 30.590 31.551 35.309
d) Calculating the 2' function For trial number 1, js, (cell 19) and sf (cell 125) are used to calculate Z' with the instruction : 1 O*@LOG(I9*19/125)
in cell 025. The other values for Z are obtained by copying the formula to cells 0 2 6 to 0 3 2 (screen 14.3).
e) Calculating the effects A new worksheet is opened (sheet 2) and the matrix of effects is entered (cells B9 to 116, screen 14.4). The responses from worksheet 1 are copied to cells f9 .. M16, alongside the effects matrix (screen 14.4).
313
SCREEN 14.4 Effects matrix of the z4-' design plus the four calculated responses
EFFECT MATRIX 4 =
1
2
3
-1 1 -1 1 -1 1 -1 1
-1 -1 1 1 -1 -1 1 1
-1 -1 -1 -1
1 1 1 1
12
13
23
123
1
1 -1 1 -1 -1 1 -1 1
1 1 -1
-1 1 1 -1 1 -1 -1 1
-1 -1 1 1
-1 -1 1
-1 -1 -1 1 1
I CURV. 1 1 1 1 1
1 1
1
1.54 7.90 7.52 1.64 7.67 7.79 7.37 7.66
82
0.090 0.071 0.001 0.008 0.091 0.053 0.038 0.017
z
Z'
-10.5 -11.5 -30.2 -21.0 -10.4 -12.8 -14.2 -17.6
28.00 29.45 47.70 38.67 28.11 30.58 31.55 35.30
The data are then used to calculate the effects of each factor as follows. Using the mean curvature as an example, the instruction : +$J9*B9
in cell B26 gives the product of the mean curvature of trial 1 multiplied by - 1. This instruction is copied to the cell range B26 _ .I33 to give all the products. The columns are then added and divided by eight to give the effect (screen 14.5). The instruction in cell B35 :
gives the effect of factor 1. The effects of the other factors are obtained by copying this instruction to cells B35 to I35 (screen 14.5).
314
SCREEN 14.5 Calculation of the effects of factors on the curvature
+
B26 :
$J9*B9 CURVATLTRE 2
12
13
-7.54 -7.90 7.52 7.64 -7.67 -7.79 I.37 7.66
4=123
23
7.54 7.54 1.54 -7.90 -7.90 7.90 -7.52 7.52 -7.52 7.64 -7.64 -1.64 -7.67 -1.67 1.61 7.19 -1.79 -1.19 -1.31 -7.37 1.31 1.66 1.66 1.66 ____--__----------_ - - - - - - - -. .- - - - - - - - _ - _ 0.1106 -0.0881 -0.0143 -0.0085 -0.0098 -0.0177 -7.54 7.90 -7.52 1.64 -7.67 1.79 -7.31 7.66
I
3
-7.54 -7.50 -7.52 -1.64 7.67 7.79 1.31 7.66
1
-7.54 7.90 7.52 -1.64 7.67 -7.79. -7.31 7.66
-- - - _ _ _
I.54 7.90 7.52 I.64 7. 67 1.19 1.37 1.66
--
0.0518
7.63601
35
The effects of factors for the three other responses are obtained in the same way using an analogous series of instructions. These instructions are indicated in the top left-hand cell of each screen.
SCREEN 14.6 Calculation of effects of factors on curvature variance B44 :
i$K9*B9
VARIANCE OF THE CURVATURE
44 45 46
7
1 2 3 4 5 6
I 8
-0.0900 0.0707 -0.0010 0.0079 -0.0908 0.0529 -0.0380 0.0173
-0.0900 -0.0707 0.0010 0.0079 -0.0908 -0.0529 0.0380 0.0173
-0.0900 -0.0707 -0.0010 -0.0079 0.0908 0.0529 0.0380 0.0173
0.0900 -0.0707 -0.0010 0.0079 0.0908 -0.0529 -0.0380 0.0173
0.0500 -0.0707 0.0010 -0.0079 -0.0908 0.0529 -0.0380 0.0173
0.0900 0.0707 -0.0010 -0.0079 -0.0908 -0.0529 0.0380 0.0173
-0.0900 0.0707 0.0010 -0.0079 0.0908 -0.0529 -0.0380 0.0173
0.0900 0.0707 0.0010 0.0079 0.0908 0.0529 0.0380 0.0173
315
SCREEN 14.7 Calculation of the effects of factors on function Z
B62 :
+
$L9*B9
EQNCTION Z
1 2 3 4 5 6 8
10.46 -11.50 30.18 -21.01 10.42 -12.76 14.20 -17.62
10.46 10.46 11.50 11.50 -30.18 30.18 -21.01 21.01 10.42 -10.42 1 2 . 7 6 -12.76 -14.20 -14.20 -17.62 -17.62 _ _ _ _ _ _ - _ _ - _ - - _ -_----0 . 2 9 3 -4.734 2.268
-10.46 11.50 30.18 -21.01 -10.42 12.76 14.20 -17.62 -------
1.142
-10.46 11.50 -30.18 21.01 10.42 -12.76 14.20 -17.62
--_____
-1.736
23
4=123
I
-10.46 -11.50 30.18 21.01 10.42 12.76 -14.20 -17.62 __----2.573
10.46 -11.50 -30.18 21.01 -10.42 12.76 14.20 -17.62 -1.411
-10.4E -11.5C -30.18 -21.01 -10.42 -12.7E -14.2C -17.62 -___---16.019
4=123
I
-------
SCREEN 14.8 Calculation of the effects of factors on function Z'
B80 :
+
$M9*B9
F"CT1ON
Z'
23 1 2 3 4 5 6 7 8
-28.00 29.46 -47.70 38.67 -28.11 30.59
-31.55 35.31
______ -0.167
-28.00 -29.46 47.70 38.67 -28.11 -30.59 31.55 35.31
-28.00 -29.46 -47.70 -38.67 28.11 30.59 31.55
35.31
------ -------
4.634
-2.284
28.00 -29.46 -47.70 38.67 28.11 -30.59 -31.55 35.31
28.00 -29.46 47.70 -38.67 -28.11 30.59 -31.55 35.31
28.00 29.46 -47.70 -38.67 -28.11 -30.59 31.55 35.31
-28.00 29.46 47.70 -38.67 28.11 -30.59 -31.55 35.31
28.00 29.46 47.70 38.67 28.11 30.59 31.55 35.31 - - - - - - _ ------- - - - - - - _ - - _ _ _ _ _ ---_-_-1.150 1.726 -2.595 1.470 33.675
316
f) Calculating the mean curvature at the low level of factor 5 The data are contained in columns B, C and D of worksheet 1 (screen 14.1). They are used to calculate the mean curvature by placing the instruction : @AVG(B9..D16)
in cell El8 (screen 14.9)
g) Calculating the mean curvature at the high level of factor 5 Columns F, G and H are treated in the same way using the instruction @AVG(F9..H16)
in cell I18 (screen 14.9) SCREEN 14.9 Calculation of mean curvatures at levels 5+ and 5-
CURVATDRES AT
LEVELS 5- AND 5+ 5+
1 2
3 4 5 6
7 8
2.
1.78 8.15 1.50 1.59 7.94 7.69 7.56 7.56
7.78 5.18 1.56 1.56 8.00 8.09 7.62 7.81
1.81 7.88 1.50 7.15 7.88 8.06 7.44 1.69
1.19 8.01 1.52 7.63 7.94 1.94 7.54 7.68
1.50 7.88 1.50 7.63 7.32 7.56 7.18 7.81
1.25 1.88 1.56 7.15 7.44 7.69 7.18 7.50
1.12 7.44 1.50 7.56 1.44 1.62 1.25 7.59
1.29 7.73 1.52 7.64 1.40 7.62 7.20 7.63
Mean
1.54 7.90 1.52 7.64 1.67 1.78 7.37
7.66
CALCULATION FOR THE SECOND INTERPRETATION
Two new worksheets are opened; worksheet 3 is used to calculate the elaborated responses from the raw responses, while worksheet 4 is used to calculate the effects. The calculations themselves are analogous to the ones camed out for the Z4-'design.
317
a) Calculating the mean curvature for each trial Each trial now contains only three results. The data in worksheet 1 are copied to worksheet 3. The 48 results on 16 lines occupy the range B9..D24 (screen 14.10). The mean curvature for trial number 1 is calculated with the instruction : @AVG(B9..D9) in cell F9, and this instruction is copied to cells F10..F24.
SCREEN 14.10 Calculation of mean curvatures for the trials in the
$'-'design
Trial
no
Y1
Y2
Y3
Mean
1 2
7.78 8.15 7.50 7.59 7.94 7.69 7.56 7.56 7.50 7.88 7.50 1.63 7.32 7.56 7.18 7.81
7.78 8.18 7.56 7.56 8.00 8.09 7.62 7.81 7.25 7.88 7.56 7.75 7.44 7.69 7.18 7.50
7.81 7.88 7.50 7.75 7.88 8.06 7.44 7.69 7.12 7.44 7.50 7.56 7.44 7.62 7.25 7.59
7.79 8.07 1.52 7.63 7.94 7.94 7.54 7.68 1.29 1.73 7.52 7.64 7.40 7.62 7.20 7.63
3
4 5 6 7 8 9 10 11 12 13 14 15 16
b) Calculating s2,2 and 2' The squares of the differences are calculated in cells H9..J24 using the instruction (screen 14.11) :
copied to all the cells of this range (screen 14.11). The curvature variance of trial number 1 is obtained by placing the following instruction in cell L9 : @SUM(H9..J9)/2 This instruction is copied to cells L9 to L24 (screen 14.1I).
318
SCREEN 14.11 Calculation of variances
square deviation 1 2 3 4 5 6 7 8 9 10 11
12 13 14
15 16
7.78 8.15 7.50 7.59 7.94 7.69 7.56 7.56 7.50 7.88 7.50 7.63 7.32 7.56 7.18 7.81
7.78 8.18 7.56 7.56 8.00 8.09 7.62 7.81 7.25 7.88 7.56 7.75 7.44 7.69 7.18 7.5
7.81 7.88 7.50 7.75 7.88 8.06 7.44 7.69 7.12 7.44 7.5 7.56 7.44 7.62 7.25 7.59
7.79 8.07 7.52 7.63 7.94 7.94 7.54 7.68 7.29 7.73 7.52 7.64 7.40 7.62 7.20 7.63
0.0001 0.0064 0.0004 0.0019 0.0000 0.0659 0.0004 0.0160 0.0441 0.0215 0.0004 0.0003 0.0064 0.0040 0.0005 0.0312
0.0001 0.0121 0.0016 0.0054 0.0036 0.0205 0.0064 0.0152 0.0016 0.0215 0.0016 0.0107 0.0016 0.0044 0.0005 0.0178
0.0004 0.0361 0.0004 0.0136 0.0036 0.0128 0.0100 0.0000 0.0289 0.0860 0.0004 0.0075 0.0016 0.0000 0.0022 0.0019
Variance 0.00030 0.02730 0.00120 0.01043 0.00360 0.04963 0.00840 0.01563 0.03730 0.06453 0.00120 0.00923 0.00480 0.00423 0.00163 0.02543
Mean variance
0.01655
Standard deviation
0.12866
The mean variance is obtained by placing : @SUM(L9..L24)/ 16
in cell L26, and the standard deviation ofeach response using the instruction BSQRT(L26)
in cell L28 (screen 14.1 1) The hnctions Z and Z' are calculated from the variance using the instructions (screen 14.12) : 10*@LOG(L9) in column P to obtain Z
3 19
I O*@LOG(F9*F9/L9) in column R to obtain Z'.
The expression 10 log y 2 can also be calculated to check that it varies little (column N in screen 14.12) SCREEN 14.12 Calculation of 9,Z and Z'
0.00030 0.02730 0.00120 0.01043 0.00360 0.04963 0.00840 0.01563 0.03730 0.06453 0.00120 0.00923 0.00480 0.00423 0.00163 0.02543
17.83075 18.13747 17.52436 17.65428 17.99641 18.00370 17.54743 17.71476 11.25455 17.76733 17.52436 17.66944 17.38463 17.64290 17.15067 17.65428
-35.22879 -15.63837 -29.2 0819 -19.81577 -24.43697 -13.04227 -20.75721 -18.05948 -14.28291 -11.90216 -29.20819 -20.34641 -23.18759 -23.73318 -27.86925 -15.94597
53.05954 33.77584 46.73254 37.4700: 42.43335 31.04597 38.30463 35.77425 31.53746 29.66945 46.73254 38.01586 40.51222 41.37607 45.01992 33.60025
320
c) Preparing the effects matrix The effects matrix of the 25-1 design is entered into worksheet 4, cells B9..Q24. The signs ofthe interaction columns are calculated according to the signs rule (screen 14.13).
SCREEN 14.13 Effects matrix of the 2'-' design
no 1 3 5
7
9 10 11 12 13 14 15 16
1
2
3
-1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1
-1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1
-1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1
123
5
12
13
-1 -1 1 1 1 -1 -1 -1 1 -1 -1 -1 1 -1 -1 -1 -1 -1 1 -1 -1 1 1 1 1 1 -1 1 1 1 -1 1 -1 1 1 1
-1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1
1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1
14
15 25
1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1
1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1
1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1
35
45 125 135 235
1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1
1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1
-1 1 1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 1
-1 1 -1 1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1
-1 -1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1
I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32 1
The experimental results are copied to R9..T24 fiom worksheet 3, and the means of the trials entered in column U (screen 14.14).
SCREEN 14.14 Calculation of mean curvatures
Mean
8.00 8.09 7.62 7.81 7.25 7.88
7.88 8.06 7.44 7.69 7.12 7.44
7.75 7.44
7.56 7.44
7.94 7.94 1.54 7.68 7.29 7.13 7.52 1.64 7.40
3 22
The variance is shown in column AA, Z in column AC and Z in column AE (screen 14.15).
SCREEN 14.15 Calculation of variance, Z and Z'
deviation square 0.0001 0.0064 0.0004 0.0019 0.0000 0.0658 0.0004 0.0160 0.0441 0.0215 0.0004 0.0003 0.0064 0.0040 0.0005 0.0312
0.0001 0.0121 0.0016 0.0054 0.0036 0.0205 0.0064 0.0152 0.0016 0.0215 0.0016 0.0107 0.0016 0.0044 0.0005 0.0178
0.0004 0.0361 0.0004 0.0136 0.0036 0.0128 0.0100 0.0000 0.0289 0.0860 0.0004 0.0075 0.0016 0.0000 0.0022 0.0019
Variance 0.00030 0.02730 0.00120 0.01043 0.00360 0.04963 0.00840 0.01563 0.03730 0.06453 0.00120 0.00923 0.00480 0.00423 0.00163 0.02543
man variance
0.016554
atandard deviation
0.128663
Z -35.2 -15.6 -29.2 -19.8 -24.4 -13.0 -20.8 -18.1 -14.3 -11.9 -29.2 -20.3 -23.2 -23.7 -27.9 -15.9
53.1 33.8 46.1 37.5 42.4 31.0 38.3 35.8 31.5 29.7 46.1 38.0 40.6 41.4 45.0 33.6
323
d) Calculating the effect of factors on curvature The effects are calculated by multiplying the effects matrix by the mean curvatures using the instruction: +$U9*B9
copied to the range B32..Q47, then adding the columns and dividing this sum by 16 using the instruction (cell B49): @SUM(B32..B47)/16
which is copied to cells B49 - 4 4 9 (screen 14.16).
SCREEN 14.16 Calculation of the effects of factors on curvature
1
2
1-1.79-1.19 2 8.01 -8.07 3 -1.52 1.52 4 1.63 1.63 5 -7.94 -1.94 6 1.95 -1.95 1 -1.54 1.54 8 1.69 1.69 9 -1.29-7.29 10 1.13-7.73 11-1.52 7.52 12 1.65 7.65 13-1.40-1.40 14 1.62 -7.62 15-7.20 7.20 16 1.63 1.63
3
123
5
12
13
14
15
25
35
45
125 135 235
-1.19-1.79-1.19 1.19 1.79 7.19 1.79 1.19 1.19 1.19-1.79-1.19-1.79 1.1 -8.07 8.01 -8.01-8.07-8.07 8.01-8.07 8.07 8.07 -8.07 8.07 8.07 -8.07 8.0 -1.52 1.52 -7.52-7.52 1.52-7.52 7.52-1.52 1.52-7.52 7.52-7.52 1.52 1.5 -1.63-1.63-7.63 1.63-1.63-1.63-7.63-1.63 1.63 1.63-7.63 1.63 7.63 1.6 1.94 7.94 -1.94 1.94 -1.94 -1.94 1.94 1.94 -1.94 -1.94 -1.94 1.94 1.94 7.9 1.95 -1.95 -1.95 -1.95 1.95 -1.95 -7.95 1.95 -1.95 1.95 1.95 -1.95 7.95 7 . 9 1.54-7.54 -1.54-1.54-7.54 7.54 7.54 -7.54-7.54 7.54 1.54 7.54 -1.54 7 . 5 7.69 1.69-7.69 1.69 1.69 7.69-7.69-1.69-1.69-1.69-1.69 -1.69-7.69 1.6 -1.29-7.29 7.29 1.29 1.29 7.29-1.29-1.29-1.29-7.29 1.29 1.29 1.29 1.2 1.13 1.1 -7.73 1.13 1.13-1.13-1.13 1.13 1.73-1.13-7.13 1.13-7.13-1.73 -1.52 7.52 1.52-1.52 1.52-7.52-1.52 1.52-1.52 1.52-1.52 7 . 5 2 -1.52 1.5 7.65 1.65-7.65-7.65 1.65-1.65-7.65 1.6 -1.65-1.65 7.65 1.65-1.65-7.65 7.40 1.40 1.40-7.40-7.40 7.4 1.40 7.40 1.40 1.40-1.40-7.40-1.40-1.40 7.6 7.62-7.62 7.62-7.62 1.62-1.62 7.62-1.62 1.62-1.62-7.62 1.62-7.62 1.20-7.20 1.20-1.20-1.20 1.20-7.20 1.20 1.20-7.20-1.20-7.20 1.20 1.2 1.63 1.63 1.63 7 . 6 3 1.63 1.63 1.63 1.63 1.63 1.63 7.63 1.63 1.63 7.6 - _ _ _ - _ - _ _ _ _______ _ _ - - _ - _ -- _ _ _ __- _ _ _---- ---- ---- ---- - - _ _ ---- ---- ----
3 24
e) Calculating the effect of factors on curvature variance The effects of factors on curvature variance are calculated in the same way as the effect of factors on curvature itself The effects matrix is multiplied by the variance using the instruction (in cell B55): +$AA9*B9 copied to all cells in the range B55 ...Q70. The elements of each column are then added and the sum is divided by 16 (instruction in cell B72): @SUM(BSS...B70)/16
SCREEN 14.17 Calculation of the effects on curvature variance
1-0.0003 -0.0003 -0.0003 -0.0003 2 0.0273 -0.0273 -0.0273 0.0273 3-0.0012 0.0012 -0.0012 0.0012 4 0.0104 0.0104 -0.0104 -0.0104 5-0.0036-0.0036 0.0036 0.0036 6 0.0496 -0.0496 0.0496 -0.0496 1-0.0084 0.0084 0.0084 -0.0084 8 0.0156 0.0156 0.0156 0.0156 9-0.0373 -0.0313 -0.0373 -0.0313 10 0.0645 -0.0645 -0.0645 0.0645 11-0.0012 0.0012 -0.0012 0.0012 12 0.0092 0.0092 -0.0092 -0.0092 13-0.0048 -0.0048 0,0048 0.0048 14 0.0042 -0.0042 0.0042 -0.0042 15-0.0016 0.0016 0.0016 -0.0016 16 0.0254 0.0251 0.0254 0.0254 ..--. . . . .
~----
-0.0003 0.0009 0.0003 -0.0213-0.0273-0.0273
0,0003
0.0003
0,0003
0.0003
0.0003-0.0003
-0.0003 -0.0003 0,0003
0.0213-0.0213 0.0273 0.0273 -0.0273 0,0273 0.0273 -0,0273 0.0273 -0.0012-0.0012 0.0012 -0.0012 0.0012 -0.0012 0.0012 -0.0012 0,0012 -0.0012 0,0012 0.0012 -0.0101 0.0104 -0.0104 -0.0104 -0.0104 -0.0104 0.0104 0.0101 -0,0104 0.0104 0,0104 0.0104 -0.0036 0.0036-0.0036 -0.0036 0.0036 0.0036-0.0036-0.0036-0.0036 0.0036 0.0096 0.0036 -0.0496-0.0196 0.0496 -0.0496-0.0496 0.0496-0.0096 0.0496 0.0996 -0.0496 0.0496 0.0496 -0.0084 -0.0084-0.0081 0.0084 0.0084 -0.0084 -0.0084 0.0084 0.0081 0.0084 -0.0084 0.0084 -0.0156 0.0156 0.0156 0.0156-0.0156-0.0156-0.0156-0.0156-0.0156 -0.0156-0.0156 0.0156 0.0313 0.0373 0.0373 0.0373~0.0313-0.0373-0.0373-0.0373 0.0373 0.0313 0.0373 0.0173 0.0645-0.0645-0.0645 0,0645 0,0645 -0,0645-0.0645 0,0645-0.0645 -0,0645 0.0645 0.0645 0.0012-0.0012 0.0012 -0.0012-0.0012 0.0012-0.0012 0.0012-0.0012 0.0012 -0,0012 0.0012 0.0092 0.0092-0.0092 -0,0092 0.0092 0.0092-0.0092 -0,0092 0.0092 -0,0092-0.0092 0,0092 0.0048 0.0048-0.0018 -0.0048-0.0048 -0.0048 0.0018 0.0018 0.0048 -0,0048 -0.0048 0.0048 0.0042-0.0012 0.0042 -0.0012 0.0012-0.0042 0.0042 -0.0042-0.0042 0.0042 -0.0012 0.0042 0.0016-0.0016-0.0016 0.0016-0.0016 0.0016 0.0016 -0.0016-0.0016 -0.0016 0.0016 0.0016 0.0251 0 . 0 2 5 1 0.0254 0.0254 0.0254 0.0254 0.0254 0.0251 0.0254 0.0254 0.0254 0 . 0 2 5 4
. . . . .
..._. ..~. _... . . . .
. . . .
. . . .
. . . .
. . ~ . . . . ~~ ~ ...~ . .~~.~
0.00925 -0.0074 -0.0024 0.00140 0.00199-0.00320.00031 0.00601 -0.0019 0.00176-0.00710.00403 0.00386 -0.0018 0.00766 0.0165
The effects of factors on the fbnctions Z and Z are calculated in the same way, using the instructions: +$AC9*B9 and +$AE9*B9
325
SCREEN 14.18 Calculation of the effects of factors on the function Z
1 1
35.23 -15.64 29.21 -19.82 5 24.44 6 -13.04 7 20.76 8 -18.06 9 14.28 1 0 -11.90 11 2 9 . 2 1 1 2 -20.35 13 2 3 . 1 9 1 4 -23.73 1 5 21.87 1 6 -15.95
2 3 4
2
3
123
5
12
13
35.23 35.23 35.23 3 5 . 2 3 - 3 5 . 2 3 - 3 5 . 2 3 15.64 15.64 -15.64 15.64 15.64 15.64 -29.21 29.21 -29.21 29.21 29.21 -29.21 -19.82 19.82 19.82 19.82 -19.82 19.82 24.44 -24.44 -24.44 2 4 . 4 4 -24.44 24.44 1 3 . 0 4 -13.04 1 3 . 0 4 1 3 . 0 4 13.04 - 1 3 . 0 4 -20.76-20.16 20.16 20.76 20.16 20.16 -18.06 -18.06 -18.06 18.06 -18.06-18.06 14.28 14.28 14.28 -14.28 -14.28 -14.28 11.90 11.90-11.90-11.90 11.90 11.90 -29.21 2 9 . 2 1 - 2 9 . 2 1 -29.21 29.21 -29.21 -20.35 20.35 2 0 . 3 5 - 2 0 . 3 5 - 2 0 . 3 5 20.35 23.19-23.19-23.19-23.19-23.19 23.19 23.73 -23.73 23.73 -23.13 23.13 -23.13 - 2 7 . 8 1 -27.87 27.87 - 2 1 . 8 7 27.87 27.87 -15.95 -15.95 -15.95 -15.95 -15.95 -15.95
14
0 . 4 6 8 0 0 . 6 0 6 9 0.0034 - 0 . 9 2 2
25
35
45
125 135 235
-35.23 -35.23 -35.23 -15.64 15.64 -15.64 29.21 -29.21 29.21 19.82 19.82 19.82 24.44 -24.44 -24.44 1 3 . 0 4 13.04 - 1 3 . 0 4 -20.76 -20.76 20.76 -18.06 18.06 18.06 -14.28 14.28 14.28 -11.90-11.90 11.90 29.21 29.21-29.21 20.35 -20.35-20.35 23.19 23.19 23.19 23.13 -23.13 23.13 -27.81 27.87 -27.87 -15.95 -15.95 -15.95
-35.23-35.23 35.23 35.23 35.23 -35.23 15.64 -15.64 -15.64 15.64 -15.64 -29.21 29.21 -29.21 29.21 -29.21 -29.21 -19.82 -19.82 19.82-19.82-19.82 -19.82 24.44 24.44 2 4 . 4 4 - 2 4 . 4 4 - 2 4 . 4 4 -24.44 1 3 . 0 4 -13.04 - 1 3 . 0 4 13.04 -13.04 -13.04 20.16-20.16-20.76-20.76 20.16-20.76 18.06 18.06 18.06 18.06 18.06 -18.06 14.28 14.28 - 1 4 . 2 8 - 1 4 . 2 8 - 1 4 . 2 8 -14.28 11.90-11.90 11.90 11.90-11.90-11.90 29.21 -29.21 29.21-29.21 29.21 -29.21 20.35 20.35-20.35 20.35 2 0 . 3 5 - 2 0 . 3 5 -23.19-23.19-23.19 23.19 23.19-23.19 -23.73 23.13 2 3 . 7 3 - 2 3 . 1 3 23.73 -23.7 -27.87 21.87 2 7 . 8 1 2 1 . 8 7 - 2 7 . 8 7 -27.8 -15.95 -15.95-15.95-15.95-15.95 -15.9
1.4559 -1.218 -1.298
-2.412 0.2804 2 . 3 6 5 3 0 . 9 3 9 1 1 . 8 5 3 5
- _ _ _--- - - - - - - _ _ _ _ _ _ _ - ---__ - - - ____ _---
4.1059 -1.2350.5314
15
--__ ____
-15.64
_ _ _ -----
-_______
----
---
-21.41
SCREEN 14.19 Calculation of the effects of factors on the function Z'
1
2
3
123
5
12
13
14
15
25
35
45
125 135 235
1 -53.06 - 5 3 . 0 6 - 5 3 . 0 6 - 5 3 . 0 6 - 5 3 . 0 6 53.06 53.06 53.06 53.06 5 3 . 0 6 2 33.78 - 3 3 . 7 8 - 3 3 . 7 8 33.78-33.78 -33.78 -33.78 33.78-33.18 33.78 3 -46.73 46.73-46.73 46.13-46.73-46.73 46.73-46.73 46.73-46.73 4
5 6 7
8 9 10
11 12 13
14 15 16
37.47 3 7 . 4 1 - 3 1 . 4 7 - 3 7 . 4 7 - 3 7 . 4 1 37.47 -42.43-42.43 42.43 42.43-42.43 42.43 31.05-31.05 31.05-31.05-31.05-31.05 -38.30 38.30 38.30-38.30-38.30-38.30 35.77 35.77 35.71 35.77-35.77 35.77 -31.54 -31.54-31.54-31.54 31.54 3 1 . 5 4 29.67 -29.67-29.67 29.67 29.67-29.67 -46.73 46.73-46.73 46.73 46.73-46.13 38.02 3 8 . 0 2 - 3 8 . 0 2 - 3 8 . 0 2 38.02 38.02 -40.57 -40.57 40.57 40.57 40.57 40.51 41.38 -41.38 41.38-41.38 41.38-41.38 -45.02 45.02 45.02-45.02 45.02-45.02 33.60 33.60 33.60 33.60 33.60 33.60
- - _. - -- - -
----
-3.9791.1362
-0.554-0.408-0.751
-37.47 -31.11-37.47 -42.43-42.43 42.43 31.05-31.05-31.05 -38.30 38.30 38.30 35.77 3 5 . 7 7 - 3 5 . 7 1 31.54 31.54-31.54 -29.67 29.67 29.67 46.73 -46.73-46.73 -38.02 -38.02 38.02 -40.57 -40.57-40.57 41.38 - 4 1 . 3 8 41.38 -45.02 45.02-45.02 33.60 33.60 33.60
53.06 5 3 . 0 6 - 5 3 . 0 6 - 5 3 . 0 6 - 5 3 . 0 6 33.78-33.78 33.78 3 3 . 7 8 - 3 3 . 1 8 46.13-46.73 46.73-46.73 46.73 -31.47 31.41 37.47-37.47 37.47 37.47 42.43-42.43-42.43-42.43 42.43 42.43 31.05-31.05 31.05 31.05-31.05 31.05 -38.30-38.30 38.30 38.30 38.30-38.30 -35.77-35.77-35.77-35.71 -35.71-35.17 - 3 1 . 5 4 -31.54 - 3 1 . 5 4 31.54 31.54 31.54 -29.67 -29.67 2 9 . 6 7 - 2 9 . 6 7 -29.67 29.67 46.73-46.73 46.73-46.73 46.13-46.13 38.02-38.02-38.02 38.02-38.02-38.02 -40.57 40.57 40.57 40.51 -40.51-40.57 -41.38 41.38-41.38-41.38 41.38-41.38 45.02 45.02-45.02-45.02-45.02 45.02 33.60 33.60 3 3 . 6 0 33.60 33.60 33.60
53.0 33.7 46.1 37.4 42.4 31.0 38.3 35.7 31.5 29.6 46.7 38.0 40.5 41.3 45.0 33.6
- - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -0.012 0.9123 -1.4171.32891.3904
2.3808-0.263-2.372
-0,916-1.881
39.0
326
3. a)
CALCULATION FOR OPTIMIZATION Experimental design for curvature and curvature variance
Open worksheet 5 and enter the values shown in column B,C, D and E of screen 14.20. This table contains the levels of factor 1, 2, 3 and 5 in the range B9..E24, arranged in a Z4 design. Two columns are reserved for the results of the calculations of curvature (column G9..G24) and curvature variance (column F9..F24)
SCREEN 14.20 The calculated curvature and variance of curvature
-1
1
1
-1
variance
curvature
0
0
The instruction : +K53 is entered in cell F9 and copied to the rest of the cells F9..F24. The worksheet will initially show zeros in this column.
327
The instruction : +K77 is entered in cell G9 and copied to cells G9..G24. Again the column will initially contain zeros.
b) Effects matrix The values for curvature and the curvature variance are obtained from the effects matrix as follows (screen 14.21) : a. +1 is entered in cells B3 1..B46. b. The range B9..E24 is copied to C3 I..F46. c. The interaction products are calculated in cells G3 1..J46 by placing :
.
Instruction D3 1*E3 1 in cell G3 1 and copying it to G3 1..G46. Instruction D3 1*F3 1 in cell H3 1 and copying it to H3 1. .H46.
. .
Instruction E3 1*F3 1 in cell I3 1, copying it to I3 1..I46 Instruction D3 1*E3 1*F3 1 in cell J3 1, copying it to J3 1 . . J46.
328
SCREEN 14.21 Effects matrix
11 %42 12
1 1
13
1
14
1
-1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1
16
1
1
1
4 5
1 1 1
6
1
I
1 1 1
3
8
1
I
I
5
23
25
35
235
1 1 1 1 -1 -1 -1 -1 -1
-1 -1 1 1
-1
1 -1 -1 1 1
-1 -1 1 1
-1 -1 -1 -1
1
1
-1 -1 1 1 -1 -1 1 1 -1 .1 1 1 -1 -1
-1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1
-1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1
1 1 -1 -1 -1 -1 1 1 1 1
-1
1 1 -1 -1 1 1 -1 -1 --1 -- 1 1 1 -1 -1
1
1
1
1
1
-1 -1 -1
1
c) Modelling the variance of curvature
The curvature variance is then calculated (screen 14.22). The values for the coefficients of the model are entered in cells B5 1 ..J51and all the coefficients are multiplied by 10,000. Cell B5 1 Cell C5 1 Cell D5 1 Cell E5 1 Cell F5 1 Cell G5 1 Cell H5 1 Cell I5 1 Cell J51
+I65 +90 -70 +O +O +60
+O -70 +70
329
The instruction +B3 1*B$51
is entered in cell B53 and copied to cells B53.. 568. The instruction @SUM(B53.. J53)
is then entered in cell K53 and copied to K53.. K68.
SCREEN 14.22 The calculated variance of curvature
DISPERSION
90
90
165
10 11 12 13 14 15 16
d)
165 165 165 165 165 165 165 165
90 -90 90
-90 90 -90 90 -90 90 -90 90
70 -70
70
-70 -70 70 70 -70 -70
70 70 -70 -70
0
0
60
0
0
-60
0 0 0
0 0
-60 60
0
0 0
0
0 0 0
0
0 0 0
0 0 0 0 0 0
60 60 60 -60 -60 -60 -60
0
-70
-70
0 0
70
-70
70
-70
70 70 -70
-70 -70 -70 -70
0 0
0 0
-70
60
Modelling the curvature
The curvature is calculated in the same way. The values of the coefficients of the model are entered in cells B75 - J75. The value in cell B75 (7.68) is obtained by adding the influence of factor 4 (0.05) to the mean (7.63).
330
Cell C75 Cell D75 Cell E75 Cell F75 Cell G75 Cell H75 Cell 175 Cell 575
+o. 1 1 -0.09 +O -0.13
+O +0.08 +O +O
The instruction +B3 1 *B$75 is entered in cell B77, and copied to cells B77..J92. The instruction @SUM(B77..J77)
is then placed in cell K77 and copied to cells K77..K92. SCREEN 14.23 The calculated curvature
I
CURVATURE 7.68
0.11
-0.09
0
-0.13
0
0.08
0
0
7.68 7.68 7.68 7.68 7.68 7.68 7.68 7.68 7.68 7.68 7.68 7.68 7.68 7.68 7.68 7.68
-0.11 0.11 -0.11 0.11 -0.11 0.11 -0.11 0.11 -0.11 0.11 -0.11 0.11 -0.11 0.11 -0.11 0.11
0.09 0.09 -0.09 -0.09 0.09 0.09 -0.09 -0.09 0.09
0 0
0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.13 -0.13 -0.13 -0.13 -0.13 -0.13 -0.13
0
0 0
0 0
0 0 0 0
0.08 0.08 -0.08 -0.08 0.08 0.08
0
0 0
0
-0.08
0
0
0 0 0 0 0
-0.08 -0.08 -0.08 0.08 0.08 -0.08 -0.08 0.08 0.08
0 0 0 0 0
0 0 0 0 0
0
0
0
0
0
0
0
0
curvat. 1 2 3 4
5 6 7 8
9 10 11 12 13 14 15 16
0
-0.09
0 0 0 0 0 0 0 0 0
0.09
0
0.09
0
-0.09
0 0
0.09 -0.09
-0.09
0
-0.13
0 0 0
-0.13
0
0 0 0
0 0
65 245 -55 125 225 405 65 245 345 525 -55 125 -55 125 65 245
33 1
The values for curvature variance and the curvature itself thus calculated are automatically copied to cells F9..F24and G9..G24 on screen 14.20, as shown in screen 14.24.
SCREEN 14.24 The calculated curvature and variance of curvature
Case
X1
x2
x3
x5 variance
curvature
332
RECAPITULATION 1. This calculation exercise has shown that a spreadsheet is adequate for fairly simple calculations (effects, variance or standard deviation). In general, the experimenter needs no more than this tool for his early designs. But he may eventually find it limiting. 2. This limit is reached quite soon, as it requires a lot of calculation time to test all the potential hypotheses. Graphic outputs, which are a great help, are very slow to prepare. This is why, while it is quite possible to do the calculation with a basic spreadsheet, specialized software can greatly reduce the time required for interpretation. But such software must help the experimenter to check all the hypotheses, results and conclusions. Unfortunately, this ideal program is not yet available. 3. Interpretation requires many hours of work. Therefore, anything that speeds things up is welcome. The perfect program should be transparent enough to give the experimenter the maximum freedom to use all his creativity, fast enough to prepare the calculations and graphics, and powerful enough to handle complex cases.
CHAPTER 15
EXPERIMENTAL DESIGNS AND COMPUTER SIMULATIONS
1.
INTRODUCTION
In the preceding chapters we have seen how experimental designs are used in the area of their original application: experimental science. But we will shortly see that they are equally suitable for use in computer simulation of phenomena or processes. This type of application is beginning to spread, heralding a new and promising area of application for Experimentology. The techniques can be used to organise computer manipulations so as to reduce the number of runs and provide simplified mathematical models that are much more easily used than the original simulation software. This simplified model is only valid within a limited domain, defined by the minimal and maximal values of each factor. A computer simulation is a complex program based on general scientific laws. It provides output values as a hnction of the input data. In this respect a computer simulation run is completely analogous to an experiment. The specialist who runs computer-based mathematical
334
simulations may be compared to the experimenter carrying out an experiment. As the two activities are so analogous, we can set up the following parallels (Figure 15.1):
0 0
Input variables correspond to factors. Output values correspond to the responses. A computer simulation run corresponds to an experiment. The computer calculation software is comparable to the experimental phenomena. The calculation domain within which the model is valid is analogous to the experimental domain within which the experimental design model is used. The scientific laws used to establish the mathematical simulation correspond to the natural laws governing the experimental process.
-+
+
Factors -+
-+
+ + Input -+
n Natural laws.
Definition of experimental domain.
+
R~~~~~~~~
Scientific laws software.
4
Definition of the calculation domain.
-+
-+
+ -+
-+ used in simulation
+
+ -+
output
Simplified mathematical modelling valid in experimental domain.
Simplified mathematical modelling valid in calculation domain.
Figure 15.1: Analogy between experimental designs and computer simulations. These analogies imply that the methods of experimental design are readily applied to computer-based mathematical simulations. However, there are several peculiarities in the use of experimental design methodology for mathematical simulations. These are:
0
The responses contain no random error. A computer calculation always gives the same output for the same input data. There is no response drift. There is no block effects.
The immediate consequence is that the order in which the run are performed is no longer a problem. Blocking is not required, neither are anti-drift designs, and the computer runs need not be randomised. We can thus program the whole set of runs for a design right at the start of a study. In most cases, the standard order is adopted and the computer is set up so that all the calculation are performed sequentially. The calculating power of the computer also allows calculation of the mean, effects and interactions.
335 As there is no random error analogous to experimental error it is more difficult to differentiate between the significant and non-significant effects. Discarding weak coefficients leads to differences between the initial simulation model and the simplified model. The user must judge, on the basis of his experience, whether these differences are acceptable in the context of his application. Let us now examine the application of experimental designs to three examples of simulation:
Propane remover optimizing. This study was carried out by the refining and distribution division of Total. Optimization of a hydroelastic motor suspension system. This study was camed out by the engineers at Paulstra. Optimization of industrial gas production. This was run by the explorationproduction division of Total [26].
2.
EXAMPLE 1, PROPANE REMOVER OPTIMIZING
The major component of a propane remover is a distillation tower that separates propane from butane in oil refineries. Figure 15.2 shows the main components of a propane remover. The distillation tower is fed by a propane-butane mixture. This mixture can be introduced either at plate 19 or at plate 21. The separated propane is taken off at the top of the tower while the butane is taken off at the bottom. The propane contains only traces of butane, and these are removed in an air condenser. The traces of propane in the butane taken off at the base of the tower are removed by redistillation. 2.1. The problem: The study was designed to identify the propane remover operating conditions providing the greatest possible energy economy. The engineer responsible for the study used a powerful simulation program. This software allows many of the parameters to be varied and a wide range of thermal yields to be accurately calculated. A single study could take several months if all the available options were used. But as this study was specific to a particular refinery and several parameters were fixed, the study domain may be reduced to three main factors and a single response. The engineer decided to use the methods of experimental design to organise the simulation runs.
336
C Tern
Figure 15.2: Propane remover.
Response The only element of the propane remover whose energy consumption can vary significantly is the condenser. The selected response was therefore the condenser energy consumption, which should be minimised. Factor The three factors selected are: m
Factor 1: condenser pressure. Factor 2: charge input temperature. Factor 3: the plate number at which the charge is introduced.
Domain definition
Level Factor 1 : Factor 2: Factor 3:
12.1 bar 70°C 19
Level
+
16.1 bar 80°C 21
337
The two accessible plates are chosen for charge input. It is not possible to introduce the charge at any other level with this particular installation. Thus, the choice is between plate 19 and plate 2 1. Selection of experimental design
As there are two levels for each factor a standard 23 design was selected.
2.2. Simulation The experimenter ran eight simulation runs in the standard order of the trials in a 23 design (Table 15.1) and the results are entered in the response column. TABLE 15.1 EXPERIMENTAL MATRIX PROPANE REMOVER OPTIMIZING
Run
Pressure
XI0
1
Temperature Plate numbei 2 3
Response
1 2 3 4 5
25.2 38.1 24.4
6 ‘7 8
I Level(-) I
12.1 bar
1
7OoC
I
19
I
2.3. Interpretation The results are interpreted in exactly the same way as for laboratory trials. The effects and interactions are calculated by the usual methods. The calculations for this simulation example were programmed directly into the computer using the responses in Table 15.1. The effects and interactions are shown in Table 15.2.
338
TABLE15.2 TABLE OF EFFECTS PROPANE REMOVER OPTIMIZING
Mean
34.23
1 2 3
8.33 -0.40 -2.98
12
0.00
13 23
-1.87
123
0.00
0.00
Thus, two factors are influent, condenser pressure (factor 1) and plate number (factor 3), with a small interaction between them. The results are illustrated in Figure 15.3
21
Plate number
19 12.1 bar
16.1 bar
Condenser pressure
Figure 15.3: Influence of condenser pressure (1) and plate number (3) on condenser energy consumption.
339
The energy consumption will be minimal if the charge is introduced at plate 21 and condenser pressure is set at 12.1 bar. 2.4 Conclusion:
The main conclusions of the interpretation are: Low condenser pressure favours energy saving. 0
The charge should be input at plate 21.
The charge temperature does not influence energy consumption. The recommendations are thus:
3.
0
Charge temperature:
70-80 "C.
0
Charge input:
Plate 21.
0
Condenser pressure: 12.1 bar.
EXAMPLE 2: OPTIMIZATION OF A HYDROELASTIC MOTOR SUSPENSION
Automobile motors are not mounted directly on the car chassis. Instead, they are mounted via a suspension system known as a hydroelmtic motor suspension. The trade name of the Paulstra system is Strafluid. This system replaces the old "silent block" system. It minimizes vibrations between the motor and the bodywork, and it is clearly of vital importance for the life of the vehicle and the comfort of the driver. The motor vibrations are filtered out by the Strafluid so that they are not transmitted to the chassis or occupants of the car. Every time a new car model is produced, this essential component must be designed, calculated and optimised for the new vehicle and its occupants. The quality of hydroelastic motor suspensions depends largely on the rubber used and its shape. Experienced engineers can design the shape of the component most likely to be suitable for a new motor and new bodywork. But the main characteristics of this preliminary approximation must be checked. Although this can be done using real prototype units, a computer simulation is cheaper and faster than component fabrication. Paulstra have therefore developed a finite elements simulation program that calculates most of the characteristics of these units. The unit features likely to be best adapted to all the requirements can then be selected. Once this unit has been defined, a prototype component must be made to ensure that it satisfies all the requirements of the car maker and will perform satisfactorily throughout the life of the vehicle. This initial prototype unit generally satisfies the majority of the requirements, but there may be one or two specifications that are not complied with. The unit must therefore be
340
slightly modified to optimize performance. Here again, the computer avoids random searching, the small alterations are evaluated by the simulation software. We will examine this phase in the development of a Strafluid which the designers optimized using experimental design methodology. The example thus begins at the stage when the Paulstra engineers have prepared the first prototype designed on the basis of their experience and calculations. Laboratory tests were carried out and the prototype appeared satisfactory on all counts, except for endurance. It correctly filtered out the whole range of undesired vibrations and it was sufficiently robust. But its working life was limited to less than 400,000 cycles in stress tests, while the required working life was over 1,500,000 cycles. The engineers knew that this poor performance was due to the poor distribution of forces within the component. They also knew that this was due to the shape of the outer surface of the rubber component. Proper stress distribution depends on a small change in the shape of the component. In this way they try to retain the original properties, but with a better energy distribution throughout the new unit. The shape of the original prototype was divided into five profiles (Figure 15.4), which we will call the existing profiles. The engineers produced slightly different geometries for each of these profiles, and these will be called the proposed profiles. There are thus five existing profiles and five corresponding proposed profiles. Profile 4
n
Figure 15.4: The five profiles of the initial prototype.
3.1 The problem:
8 4 k! 8 @
3g
The new prototype may be built using a combination of existing profiles and proposed profiles. There are 32 possible combinations of the two types of profiles. The problem is to choose the best $$ combination, which produces a component with a working life of over 1,500,000 cycles. All the combinations can be calculated, but as the $8 < .<
8
341
computer analysis is time consuming and expensive, the engineers preferred to use experimental design methodology to limit the computer simulations to eight runs, thus saving 24 computer simulations. The interpretation of the results from these eight runs should show the influence of each profile on the number of cycles that the component will withstand before destruction. They thus hope to be able to directly obtain the most durable component.
Level +
Profile 2
Profile 4
n
Profile 5
Figure 15.5: Strafluid study domain.
Level -
342 Factors to be studied and experimental domain
The factors are the profiles originally defined. They may have two levels: the levels designated + are the existing profiles, while the levels designated - are the proposed profiles. The study domain is shown in Figure 15.5. The experimental domain is unusual in that the variables, the profiles, are neither continuous nor measurable quantities. A component is defined by a set of five profiles, each of which is selected from the existing or proposed profiles. It is assumed that the position of a particular profile cannot be changed and that each component comprises one profile 1, one profile 2, one profile 3, one profile 4 and one profile 5 .
Response Several characteristic values for the component can be calculated from the finite elements simulation program for each set of profiles. One of these values may be selected as the response if it corresponds to the number of cycles that can be run. The engineers have shown that this property is related to the density of deformation in the rupture zone, Although the definition of this criterion is complex it is readily calculated by the simulation program. The smaller the response calculated by the simulation software, the better. The objective of the study is thus to choose those levels of the factors that minimize this deformation density.
Calculation design
The engineers studied five factors and carried out only eight computer runs. As each factor had only two levels, a 25-2 fractional factorial design was adopted (Table 15.3). Factor 5 was aliased with interaction 12 and factor 4 with interaction 123. The alias generator set was therefore: I = 1234 = 125 = 345
343
TABLE15.3
EXPERIMENTAL MATRIX
STRAFLUIDOPTIMIZATION Run no
Profile 1
Profile 2
Profile 3
Profile 4
Profile 5
1
2
3
4 = 123
5=12
Level (-)
Proposed profiles
Level (+)
Existing profiles
3.2. Calculations The calculations were made on a Sun 4 workstation using software developed in house by Paulstra. It uses the method of finite elements which allows evaluation of the principal elastic characteristics of the main rubber component. We will only use the value of the chosen response, the deformation energy in the rupture zone. Calculation number 1 was run on a component having the following five profiles: Profile Profile Profile Profile Profile
1-
2345+
The shape of this component is shown in Figure 15.6. The results of the computer calculation give the deformation energy in the rupture zone, 1534 in this case. This value is found in the response column of the effects matrix (Table 15.4).
3 44
TABLE15.4 EFFECT MATRIX
STRAFLUIDOPTIMIZATION Profile 1 Profile 2 Profile 3 Profile 4 Profile 5
Run no
I
1
2
3
4=123
1 2 3 4
+ + + + + +
-
-
-
-
-
+ +
5
6
+ -
+ -
+
+ + -
-
1897
204
134
13
23
+ +
+
+
-
-
-
+
-
-
+ +
+
+ +
-
-101
-
Effects
5=12
150
-
-
-
-
+
-
-
+
13
8
3
2188
-
2210 2060
The computer program ran the eight calculations sequentially for each set of profiles defined in the design and the calculated values of the response are shown in Table 15.4. Profile 4-
n
Profile 1-
Profile 3-
Figure 15.6: The combinations of profiles used for run number 1.
345
3.3. Interpretation The results shown in Table 15.4 can be interpreted by calculating the effect of each factor on the response selected. This calculation is done automatically by the computer, according to the principles of experimental design theory. The results are shown in Table 15.5. Examination of this table shows: 1. Factor 5 has very little influence on the deformation energy. We may therefore adopt
either the existing or the proposed profile without causing any great change in the response. In the absence of any other information, the figures available indicate that the proposed profile is slightly better. 2. The interactions are small enough for them to be neglected without any problem 3. Only the first four factors are influent. This indicates that profiles 1, 2, 3 and 4 must be carefhlly chosen so as to reduce the response as much as possible. Figure 15.7 will help make this selection.
TABLE15.5 TABLE OF EFFECTS STRAFLUIDOPTIMIZATION Mean
1897
1+25 2 + 15 3 + 45 4 + 35 5 + 12 + 34
204 134 150 -101 13
13 + 24 23 + 14
8 3
The deformation energy can be made as small as possible by choosing the set of profiles that gives the smallest value. As the effects are assumed to be additive, the - levels of factors 1, 2, 3 and 5 are selected, plus level + for factor 4. Thus the optimized component comprises the proposed profiles for 1, 2, 3 and 5 and the existing profile for 4. This component is shown in Figure 15.8. The reader will see that this component was not included in the 25-2 design. The interpretation of the experimental design allowed immediate identification of the shape that is probably the best from among the 24 possible sets of profiles. A computer analysis was run on the component having the five selected profiles. The results confirmed that this set of profiles was excellent. The deformation energy in the rupture zone was found to be very low; it was
346
2101 2031 1a97
1897 1763
1693 -1
+l
-1
PROFILE 1
2047
1998 1897 1796
1897 1747
-1
+1
PROFILE 2
+l
PROFILE 3
l’i
-1
+1
PROFILE 4
Figure 15.7: Influence of the profiles on the deformation energy in the rupture zone. much lower than for all the previously calculated sets of profiles. It is thus very likely that the hture prototype made on this basis will survive a great number of cycles without damage. All that is required is to make one and confirm this. This component was made and underwent all the quality checks required to show that it conformed to the specifications set out by the car maker. In particular its endurance turned out to be a remarkable 4,000,000 cycles, and it remained in perfect condition. 3.4. Conclusion:
The shape of the new Strafluid motor suspension should have the configuration given by the set of five profiles shown in Figure 15.8: Proposed profiles for 1, 2, 3 and 5. Existing profiles for 4.
347
Figure 15.8: Shape of the new Strafluid motor suspension: Proposed profiles for 1, 2, 3 and 5. Existing profile for 4.
The new Strafluid so defined satisfies all the car maker's specifications.
4.
EXAMPLE 3: NATURAL GAS PLANT OPTIMIZATION
This third example describes the identification of the best operating conditions for a plant producing two products from natural gas: sale gas (SG) and liquefied gas (LG). The two products must meet quality specifications that are obtained by optimal adjustment of three manufacturing parameters. The properties of the SG and LG are estimated using a mathematical model based on fluid thermodynamic properties. The simulation calculations are performed with a computer program. The normal approach, i.e., not using experimental designs, is a random search for a functional point that meets the required specifications. In this type of case, it is not unusual to cany out thirty or so runs before obtaining a satisfactory result. We will show how, using experimental design methodology, it is possible to not only reduce the number of trials, but also to identi@ all the possible setting and choose from them the most appropriate.
4.1 The gas production system and the problem to be solved The natural gas arriving at the plant is a mixture of hydrocarbons which cannot be sold without processing. It also contains pollutants such as COz and H2S. It pressure, temperature, composition and flow rate may all vary. The plant function is to generate end products of constant quality that conform to the required specifications despite all these variations. We will now look briefly at what happens to the gas in the plant. The main features are outlined in Figure 15.9.
348
%p, -
Factor 2:
E PRODUCTION GAS
a
Factor 1: W,
> 7 SALE GAS
Ca
3-,
-
I Factor3: OTB
LIQUEFIED GAS
Figure 15.9 : General outline of gas production plant. a) The natural gas is cooled to about 10°C in a heat exchanger E and then sent to a unit, separates the liquid and gas phases.
U1, that
b) The pressure of the gas phase is lowered in a turbo-expander, TE. The drop in pressure is accompanied by a drop in temperature that causes the heaviest constituents to liquefy. c) The liquid phase from the turbo-expander passes to unit U 2 where the uncondensed gases are separated off. The liquids flow out of the base of the tower and the gas from the top. d) The gas phase from the top of U2 is the SG. This is warmed in the heat exchanger E and compressed in compressor C1, which is also coupled to the turbo-expander. An auxiliary compressor, Ca, can be used if the outlet pressure at C1 is too low. e) The liquids from U1 and U 2 pass through a stabilisation tower, U3, to extract the stabilised liquefied gas (LG). Any remaining SG is recovered at the top of U3 and fed back into the system. A mathematical simulation is used to calculate the properties of the SG and LG as a function of the incoming natural gas and the processing steps. The computer analyses are long and expensive. It is these calculations that we will try to reduce using experimental design methodology.
349
4.2. Choice of responses The responses chosen by the engineer are the values for which specifications are provided. Each of the two products, SG and LG, from the plant must conform to the following rigid specifications :
Sale gas (SG) There are two specifications : the Wobbe index and the gas pressure P a) The Wobbe index This is a major commercial specification. At a given Wobbe index the flame always has the same appearance and the same consumer properties. The domestic user is considered to have the same product if the Wobbe index is between 48.2 and 5 1. 48.2 < Wobbe Index < 5 1 b) The gas pressure PI The gas pressure PI at the outlet of compressor C1 must be below 30 bar absolute in order to be compatible with compressor Ca. The delivery pressure is adjusted by compressor Ca to the pressure required by the distributor. The pressure P I must be less than 30 bar absolute. P I <30bar
Liquefied gas (LG) Three specifications must be met : a) C2/C3 ratio The liquefied gas must have a low ethane content (designated by C2). This property is checked by measuring the C2/C3 ratio, i.e., the ethane/propane ratio. The ratio must be : C2/C3 < 0.03% by weight b) C02/C2 ratio The LG must also contain very little carbon dioxide. The CO, content is estimated with respect the: ethane, which is itself at a very low concentration in LG. The ratio must be : C02/C2 < 0.1 % by weight c) The true vapour pressure (TVP)
350
The true vapour pressure of the LG must be below 18 bar absolute
;
TVP < 18 bar absolute Choice of factors The engineer in charge of the study can vary three factors to mod@ the properties of the two end products to the required specifications :
..Factor .Factor Factor
1 - The power of the auxiliary compressor : Wca, 2 - The turbo-expander gas inlet temperature : 8 TE. 3 - The temperature at the bottom of the stabilisation tower (unit U,) : 8 TB.
Calculation domain The limits by which the factors can vary are defined as follows 1
- For economic reasons, the power of the auxiliary compressor Ca must be between
4
h4W and 10 W .The lower the power the better the solution.
2 - The turbo-expander gas inlet temperature should be between -10°C and 0°C . 3 - The temperature at the bottom of tower U3 should probably be adjusted to between 60°C and 80°C. It is always possible to increase the temperature up to 100°C if necessary.
These six limits define the calculation domain (Table 15.6).
TABLE15.6 TABLE OF EFFECTS NATURAL GAS PLANT OPTIMIZATION Factors : Variation limits Level +
Level -
I
Power of the auxiliary compressor (Factor 1) Turbo-expander gas inlet temperature (Factor 2) Temperature at the bottom of stabilisation tower (Factor 3)
I
I 60°C
o~~ 80°C
I
351
7
8
5
Factor 3
3 1
/* 2 Factor 1
Figure 15.10 : Selected calculation domain. The calculation points are the corners of the cube.
4.3. Choice of calculation design
An approach analogous to that of experimental design is adopted, i.e., the computer runs are illustrated by the eight points that are at the corners of the calculation domain (Figure 15.10). The calculation matrix is shown in Table 15.7. TABLE15.7 EXPERIMENTAL MATFLIX
NATURAL GAS PLANT OPTIMIZATION Run no
Compressor Power 1
Level (-)
4Mw
Level (+)
10 MW
Turbo-expander temperature 2
Stabilisation tower temperature 3
10°C
60°C
0°C
80°C
-
352
4.4. Calculations Each computer run calculates the responses for each trial. There are five responses in this example : Wobbe index Pressure PI of the gas leaving compressor C COzlC2 ratio TVP C2lC3 ratio The simulation software gives the following results for run number 1 (Wca TE = - 1ooc,e
=
4 MW, 0
= 60°C) :
Wobbe index 49.56 PI 47.34 bar CO2/C2 0.244 % wlw TVP 26.45 bar C2IC3 0.791 Yo WIW The other runs gave results for each specific response; these results are shown in Table 15.8.
TABLE15.8 EXPERIMENTAL MATRIX
NATURAL GAS PLANT OPTIMIZATION Results of eight computer simulation runs Gas pressure
C02/C2
TVP
index 49.56 49.84 48.97 49.20 49.55 49.84 49.20 49.42
47.34 42.74 26.46 22.29 47.34 42.74 26.36 22.74
0.2440 0.2260 0.0046 0.0012 0.2260 0.2140 0.1090 0.0006
26.45 23.46 10.06 8.14 18.33 15.87 6.34 5.02
0.699 0.538 0.351 0.775 0.686 0.160 0.063
353
4.5 Interpretation The choice of eight simulations allows interpretation of the results according to experimental design theory. Let us now examine each of the five responses in turn.
The Wobbe index The computer simulations provided eight values for the Wobbe index. These eight values can be used to calculate the effects of each of three factors and their interactions (Table 15.9).
TABLE15.9
TABLE OF EFFECTS NATURAL GAS PLANT OPTIMIZATION Wobbe index Mean
49.45
1 2 3
-0.25 0.06
12 13 23
-0.02 0.00 0.06
123
0.00
0.13
(factor 2) is -0.25. The Thus the effect of Wca (factor 1) is 0.13 and the effect of 8 effect of 0 TB (factor 3) and the interactions between factors are very small and can be neglected. Clearly, the Wobbe index depends on only two factors, auxiliary compressor power (1) and the turbo-expander gas inlet temperature (2). There is no interaction. The simplified model is easily constructed as it uses the effects previously calculated : Wobbe index = 49.5 + 0.13x, - 0 . 2 5 ~ ~ where :
. xI .
is the auxiliary compressor power in coded units
x1 is the turbo-expander gas inlet temperature in coded units
354
The requirement is 48.2< Wobbe index < 5 1
We can learn a great deal by drawing the lines representing these limits and finding the domain within which the specifications are satisfied (Figure 15.11). It can be seen that the specifications for the Wobbe index are fblfilled throughout the selected calculation domain. If the Wobbe index were the only constraint, we could : choose any temperature at the base of tower U3 between 60°C and 80°C as this factor has no influence. choose the turbo-expander gas inlet temperature from a wide range of values. choose the lowest auxiliary compressor power, or even consider removing it But unfortunately there are other specifications to be met and we will see that they are more demanding than the Wobbe index.
Figure 15.11 : Specification compliance: Wobbe index.
355
The compressor C1 outlet gas pressure PI
Effects and interactions of factors on the outlet gas pressure P1 are shown in Table 15.10.
TABLE15.10
TABLE OF EFFECTS NATURAL GAS PLANT OPTIMIZATION Outlet gas pressure PI
1 2 3
-2.12 -10.29 0.04
12 13 23
0.18 0.07 0.04
123
0.07
Analysis of the results shows that the pressure P I of the gas leaving compressor C, depends on the same factors as the Wobbe index. Simplified modelling is performed in the same way as for the Wobbe index. We have : gas pressure P1 = 34.75 - 2.10 x, - 10.3 xz This pressure must be below 30 bar, the line separating the zones of compliance and noncomplianct: with the specifications is given by : 30=34.75-2.10~,- 1 0 . 3 ~ ~
Figure 15.12 shows the results obtained. This time the constrain is greater as part of the experimental domain (or calculation domain) is forbidden. We can still freely choose the temperature at the base of tower U3, but not that of the turbo-expander input. The low temperature zone is forbidden.
356
t
x2
10°C I
Figure 15.12. Specification compliance: compressor C1 outlet gas pressure PI. Gas carbon dioxide I C2 hydrocarbon ratio
Effects and interactions of factors on the CO,/C2 ratio are shown in Table 15.1 1
TABLE15.11 TABLE OF EFFECTS
NATURAL GAS PLANT OPTIMIZATION CO,/C2 ratio
Mean
0.13
1 2 3
-0.02 -0.10 0.01
12 13 23
-0.01 -0.01 0.02
123
-0.01
357
Analysis of the results shows that this ratio depends on three factors. It will thus be necessary to draw the isoresponse curves in a three dimensional space. As the C02/C2 ratio
0.109 1
0.006
0.226
Factor 3 I /
0.0012
Factor 2
0.244 0.226 Factor 1 Figure 15.13 : Specification compliance: C02/C2 ratio.
must be below 0.01, the corresponding surface is plotted. This separates the zone where this specification is hlfilled from that where it is not (Figure 15.13). If the values found for the C0$2 ratio are entered at the corners of the cube, we can see that the surface CO$C2 = 0.10 separates the calculation results into two parts according to whether or not they meet the specifications.
True Vapour Pressure
The effects and interactions of factors for True Vapour Pressure are shown in Table 15.12. All three factors influence TVP. There is also an interaction between the two temperatures, and this makes the model a little more complicated : TVP = 14.2 - 1.1 X, - 6.8 x2 - 2.8 x3 + 1.1 ~2 ~3 where :
..
x, and x2 are the same as for the previous factors xj is the temperature at the bottom of tower U3.
358
TABLE15.12 TABLE OF EFFECTS NATURAL GAS PLANT OPTIMIZATION
TVP
pir-
14.21
-1.09 -6.82 -2.82
1
2 3
As the TVP must be below 18 bar, the separation between the allowable and forbidden zones will be indicated by an 18 bar isoresponse surface defined by the equation : 1 8 ~ 1 4 . 2 -1 . 1 ~ , - 6 . 8 ~ 2 - 2 . 8 ~ : , + 1 . 1 ~ 2 ~ ~ This is shown in Figure 15.14. The T W < 18 bar constraint divides the studied domain into two zones. Fortunately, the allowable zone is situated in the region where the three previous specifications are complied with. There is thus no incompatibility between the four specifications.
18.33
Factor 3
26.45 23.46 Factor 1
Figure 15.14 : Specification compliance : TVP.
359
C2K3 ratio The effects and interactions of factors on the C2K 3 ratio are shown in Table 1 5 . 1 3 .
TABLE15.13 TABLE OF EFFECTS
NATURAL GAS PLANT OPTIMIZATION C2/C3 ratio
2 3
-0.06 -0.23 -0.09
12 13 23
-0.01 0.01 -0.08
123
0.01
1
Here again the three factors are influent and there is interaction between the two temperatures. The C2lC3 ratio should be below 0.03 %w/w. The isoresponse curve for C2lC3 = 0.03% can be drawn using the model based on the experimental design : y = 0.03 =0.51 - 0.06 XI - 0.23 x2 - 0.09 x3 - 0.08 x2 ~3 This surface does not divide the experimental domain, but that does not prevent us from plotting it if we also consider the volume surrounding the initially selected domain. Figure 15.15 shows the position of this surface, The reader will see that the permitted zone (C2lC3 < 0.03%) is located outside the originally selected domain. Again, fortunately, the permitted domain lies within the zone where the four previous specifications are met. It will thus be possible to find suitable setting points. They will be slightly outside the experimental domain initially chosen by the operator. But this is not important, as this domain was chosen as that having the best chance of containing the solution, and there is nothing to prevent us going outside it. There are two options for defining the region where all the specifications will be met. The operator may either define a new study domain and repeat the calculations analogous to
3 60
the above, or he can extrapolate outside the domain of validity defined by the previous empirical model. The first solution is clearly the better and safer in terms of reliability. The new study domain selected by analysing the first set of results could be that shown in Table 15.14.
TABLE15.14
NATURAL GAS PLANT OPTIMIZATION Possible new limits for variations of factors
I
Power of the auxiliary compressor (Factor 1 )
t
I
I
Turbo-expander gas inlet temperature (Factor 2) Temperature at the bottom of stabilisation tower (Factor 3)
-,ooc 80°C
I
I
I
0°C
100°C
As four calculations have already been run, just four additional calculations are needed to obtain new empirical models and redefine the zones meeting the specifications. The calculations and interpretation will be similar to the ones just completed, and there is little new insight to be gained by going through them again. The second solution is more risky, but it has the advantage of requiring fewer calculations. It consists of assuming that the empirical models found during the initial analysis remain valid outside the studied domain and that they are still sufficiently accurate in a region close to the studied domain. The precision and risk evaluation adopted depends on the specific case. The good sense and experience of the operator play an important part in the final decision. As the hypotheses adopted are a little risky, at least one confirmatory calculation should be run to ensure that the solution adopted really solves the problem. We will adopt this strategy. The choice between the two approaches must be dictated by the difficult balance between the risk taken and the cost. It all depends on the factors at stake, and no single rule is appropriate.
361
Factor 3
Factor 1
Figure 15.15. Specification conformity: C2K3 ratio
4.6. Optimization The five specifications must all be met simultaneously. It is thus convenient to superimpose the five allowed domains and identify the zone in which the five specifications are all met. In this case, we can maie sections in the plane x3 = constant., i.e., at constant temperature 8 TB. As the desired turbo-expander inlet temperature and the auxiliary compressor pressure must both be as low as possible, examination of Figures 15.1 1, 15.12, 15.13, 15.14 and 15.15 show that there can be, for example two sections for xg (0 TB) at 1 (SOOC), 1.5 (SSOC) and 2 (90°C). These sections are shown in Figure 15.16. The investigator may thus make an informed choice of the gas plant operating conditions that he considers most economical. For example, he may select : Auxiliary compressor power TE eTB
7Mw 0°C 90°C
This set of plant operating setting is indicated by point MI in figure 15.16. But the figure also shows that a multitude of other operating conditions may also be selected without exceeding the specifications provided we remain within the permitted zone.
3 62
co21c2
-
C2IC3
Gas Pressure
I
= 80°C
co21c2
Gas Pressure
@
= 85°C
T0
CO2lC2
Gas Pressure
0
= 90°C
TB
Figure 15.16 : Permitted and forbidden zones as a function of temperature 8 TB at the base of tower Us
363
The reader will also see that this graphic solution makes the operation of the plant easier to understand and remember. It is an intelligent display of the raw data produced by the computer simulation. If the physical or economic constraints change from the present ones, they can be taken into account by referring to the results already obtained. For example, if the power of the auxiliary compressor must be reduced, the lowest power possible that meets the present specifications is easily calculated. If the specifications are changed, the demarcation lines between the accepted and forbidden zones will be established without new computer simulations. 4.7. Conclusion: The five specifications may be respected simultaneously by setting up the following gas plant operating conditions :
0
Auxiliary compressor power
7 MW
Turbo-expander gas inlet temperature eTE
0°C
Bottom temperature of tower U, 8TB
90°C
If other operating conditions are selected, the mathematical modelling can be used to estimate the Wobbe index, the outlet gas pressure P, , the C02/C2 ratio, the TVP and the C2/C3 ratio.
364
RECAPITULATION The experimental design methodology used in the laboratory may be adopted to organise computer calculations. The three examples we have examined have shown how they can : 1. Reduce the number of computer runs and give an overall view of the phenomena within the domain studied.
2. Determine all the possible solutions rather than finding one at random. 3. Provide simple mathematical models that can be run on a microcomputer.
Experimental designs are applicable to computer calculations, and they are just as p o w e h l in this area as in their normal applications in the experimental laboratory. The examples examined here have only made use of simple first degree mathematical models. It is certainly possible, provided some appropriate supplementary computer runs are performed, to obtain second degree or more complex mathematical models. But these will still be simpler and less complex than the original simulation software. We hope that these examples will be enough to stimulate engineers and technicians to explore the possibilities offered by Experimentology for mathematical simulations.
CHAPTER 16
PRACTICAL EXPERIMENTAL DESIGNS
1.
INTRODUCTION
Iri all our preceding discussions we have assumed that the factors were fixed at exactly -1, 0, or + I . When this is the case, the designs are optimal, and the calculations are easy. But in the real world, these levels may have other values. This can happen because of experimental constraints or errors in execution. For example, the experimenter setting up an experiment plans an optimal factorial design based on a Hadamard matrix. But when the experiments are run, the level of one of the variables is set at 0.8 instead of +1 for one trial. The problem then is to find a way of using the results obtained in this trial in the final calculation. Two solutions are possible: 1 . The trial can be re-run using the original +I level, so changing nothing from the original experimental design.
2. The result can be used from the trial which is not so well located in the experimental domain. But if we do this, we must find a method of handling the data to obtain the right effects and interactions.
366
Let us now examine this question and develop a general method for resolving it. The solution is contained in matrix calculus. Experimenters who wish to master practical experimental designs should take the time to learn how to manipulate this type of calculus. The long and rather arduous calculations are now much easier when done on a microcomputer. We have kept this explanation of the theory of "poorly-located'' point designs as clear as possible by using a 22 factorial design.
2. CALCULATION OF EFFECTS AND INTERACTIONS WHEN AN EXPERIMENTAL POINT IS MISPLACED Let us use the example of a catalysed chemical reaction that we covered in Chapter 2 . We shall make one important change - the experimenter has poorly set up one of the levels. As a result, one of the experimental points is shifted, so that it no longer lies at the corresponding corner of the experimental domain.
Example: The yield of a catalysed chemicai reaction (ONE POINT DISPLACED) The problem: An experimenter must improve the yield of a chemical reaction. He has placed a catalyst in the reaction mixture, and decides to determine how the two factors, temperature and pressure, that he has selected influence the yield of the reaction. He wishes to optimize operating conditions for maximum yield. But the level of the temperature was poorly controlled during the fourth trial. It remained constant, but at 78°C instead of 80°C. The pressure remained set at 2 bar throughout the fourth experiment.
eYi t
Choice of factors The two factors selected were:
.
Factor 1: .Factor 2:
reaction bath temperature pressure.
Experimental domain The installation allowed the reaction temperature to be varied from 60°C to 8O"C, and the pressure between 1 and 2 bar. The experimental domain is thus defined as: 60" C
80"C
A Ilbar
{ l bar
60" C
12b.r
80" C
bbar
367
The position of one point no longer coincided with the high level of factor 1 (temperature). We shall use primed letters to distinguish it from the points at the corners of the domain. 60" C 80" C 60" C 78" C A' B' C' D' I bar 1 bar 2 bar 2 bar
{
1
{
{
The ordinate of point D' is at 2 bar, i.e., it corresponds to the +1 pressure level. The abscissa of this point is a temperature of 78"C, which must be transformed to a coded variable: 8 = 8,
+ (Step) X,
78=70+10x,
x, =-
78-70 lo
8 lo
= - = 0.8
There is now no problem in placing the four experimental points in the experimental domain (Figure 16.1). The values of levels x, and x2, for the four trials are shown in Table 16.1. The only difference between it and Table 2.2 (Chapter 2 ) is in trial number 4. The value of xi is 0.8 instead of 1, and the values of the response is 93.5% and not 95%. It is quite normal that the response of trial number 4 is different as the experimental conditions are not the same. Pressure
1 2bar
+I
1 bar
-1
D'
D
-1
+I
60 "C
80 "C
temperature
Figure 16.1: Experimental domain and positions of experimental points. A, B, C and D are the corners of the experimental domain. Only D' is displaced with respect to D.
368
TABLE16.1 EXPERIMENTAL MATRIX THE YIELD OF A CHEMICAL REACTION
One point misplaced Pressure 70% 4
-1
80%
M.8
93.5%
Level (-) Level (f)
The effects and interactions are calculated with these four trials, rather than repeating trial number 4 with the correct levels of temperature and pressure (+1 and +l). The calculation are done using the matrix notation. The values for the effects and interactions that will be calculated must be the same as those we obtained previously. We know these values, as they were established in trials in which the experimental points fell exactly at the corners ofthe experimental domain (Chapter 2): 76.25
10
E=
-
6.25
El
11.25
El2
1.25
The effects are calculated using the general formula of the mathematical model for factorial designs. y
=
a. + al x1 + a2 x2 + a12x1 x2
where.
..
sy is the response. x1 and x2 are the coordinates ofthe experimental points in coded variables. ao, al, a2 and a,2 are the unknowns to be calculated.
369
Applying this model to point A (xl point is:
= -1
Y1 = a0 - a1 - a2
and x2 = -1) the value y1 ofthe response at this
+a12
The model can be applied to points B and C in the same way to obtain the values of responses y, and ,y3. Y I = % - a , -%+a12 y2 = + al - a2 - a 12 Y3 = a0 - a1 + a, - a12
For point D' (xl
= +0.8
and x2 = +1) the value of the response y4 is:
y4 =
+ 0.8 a, +
+ 0.8 a,,
This gives us a set of four equations with four unknowns, which we can write keeping the same order for a0 , a, , a, and aI2
y 1= + % al y2= + a , + a, y 3= + a , al+ y 4 = + a , + 0.8al +
a,+ a,-
aI2 aI2
%-
a12
a,
+ O.8aI2
This system can be written with three matrices: yl
+1 -1
-1
+1
+1 +1 -1 -1 y2 +1 -1 +1 -1 y3 +1 +0.8 +1 +0.8 y4
or in condensed form:
The column matrix Y is known, It includes the four responses measured during the experiments: 60
Y=
70 80 93.5
The X matrix is also know as it is established from placing the experimental points A', B', C' and D together with the mathematical model used for factorial designs.
3 70
Matrix A is called the coefficients matrix, because it can be used to obtain the coefficients of the mathematical model for factorial designs. It could also be called the effects matrix, because, as we have seen, the coefficients of the mathematical model are the effects and interactions. Thus the unknown is the column matrix A.
A=
I
a2
A can be calculated by inverting the X matrix and multiplying by Y:
A=X-'Y
This is readily done using a spreadsheet or other software to produce the inverted X matrix: +0.25 +0.25 M.2222 +0.2777 "-1 A
-
-0.25
+0.25 -0.2777
-0.25
-0.25
+0.2777
M.2222 +0.2777
+0.25 -0.25
-0.2777
+0.2777
and then X-'Y a0 A = a1 a2 a12
4 . 2 5 M.25 +0.2222 +0.2777 -0.25
d . 2 5 -0.2777 +0.2777 -0.25 -0.25 +0.2222 M.2777 +0.25 4 . 2 5 -0.2777 t0.2777
finally A 76.25
a()
-
A= a2 a12
6.25
11.25 1.25
by identification: an = 76.25
al
=
6.25
% = 11.25 a12= 1.25
60 70 80 93.5
371
Although one point does not lie at the comer of the experimental domain, we have the same values for the effects and interactions as were obtained previously. This is rather a remarkable result, and it shows that it is not necessary to place all the experimental points at the comers of the experimental domain to develop an experimental design and calculate the effects. But the reader should be aware that the calculation must be done by the matrix method. This is not difficult, but it does require the appropriate software, e.g., spreadsheets such as Lotus or Excel. Matrix A, containing the coefficients %, a,, % and a12 of the mathematical model of factorial designs, is thus calculated. The reader should remember that the coefficients have the following significance: al is El, the effect of factor 1
.% is E,, the effect of factor
2
al:, is El,, the interaction between factors 1 and 2 .
.
has been considered previously to be either the mean of all responses, or the value of the response at the centre of the experimental domain. We can no longer use both these definitions as one is now false. Let us remove this ambiguity.
If we go back to the general formula for the mathematical model of factorial designs we get : Y = a0 + a1 x1
+
x2 + a12 x1 x2
If we set x1 = x2 = 0, this defines the point at the centre of the experimental domain. Coefficient is the value of the response at this point. When the experimental points lie at the comers of the experimental domain, the mean of the responses is equal to the value of the response at the centre of the experimental domain. But when one of the experimental points does not lie at the comer of the experimental domain the mean of responses is no longer equal to the response at the centre of the domain. Therefore: The coefficient a, is the value of the response at the centre of the experimental domain
.
The results obtained with one point offset lead to the same values for the effects and interactions as when the experimental points are located at the comers of the experimental domain. The mathematical modelling is thus just as easy and allows the isoresponse curves to be drawn within the experimental domain (Figure 16.2).
3 72
Pressure
D' 2bar
D
+I
C' 80 -,
,75 '
70
I
B' 1 bar
-1
-1
60
+1
Temperature
80 OC
O C
Figure 16.2. Iso-yield curves within the experimental domain
3.
CALCULA 'ION OF EFFECTS AND INTERACTIONS WHEN ALL THE EXPE UMENTAL POINTS ARE MISPLACED
We will see that if the level of a factor is not set at +1 or -1, it is still possible to use the results to calculate the effects and their interactions. But what happens when all the experimental points are displaced from the corners of the experimental domain? Let us stay with the same example and examine this situation. We shall assume that the chemical reaction was checked in another laboratory, but the experimenter had so many problems that the experimental design was very different from the factorial design. In this case we must clearly differentiate between the corners of the experimental domain and the positions of the experimental points. Now they are all different. Points A, B, C and D, which define the experimental domain are located at its comers:
C
A 1 bar
6ooC 2 bar
1:s
373
and the points A', B', C' and D' indicate the positions of the experimental points: 78" C 1.lbar
Pressure
2bar
I 80°C
65°C '
D' i1.9 bar
11.95 bar
4
+I
I -+-.-
D
c' I
' D'
C'
iA' B'
1 bar
-1
B ___)
-1
+I
60 OC
80 OC
Temperature
Figure 16.3. Experimental domain and distribution of experimental points. The experimental points A', B', C', and D' are displaced from the corners of the experimental domain (A, B, C, D)
Figure 16.3 and Table 16.2 show the difference between the experimental domain and the experimental points: a) The experimental domain is given in real units (temperature in "C and pressure in bar) in the lower part of the table. b) The coordinates for the experimental points are given in coded units in the upper part. They were calculated using the following formulae derived fiom the general formula:
0 = 8, + (Step 8 ) x1 P = Po + (Step P) x2
3 74
Trial number I Temperature Pressure
60=70+ lox, 1.25= 1.5+0.5x2
thus x, = - I .Y2 =
-0.5
Trial number 2 Temperature Pressure
78=70+lOxl 1 1 = 1 5+0.5x2
thus XI =
+o 8
x2 = -0.8
Trial number 3 Temperature Pressure
65=7O+lOx, 1 95=I.S+OSx2
thus x,
= -0.5
Y,
= +0.9
Trial number 4 Temperature Pressure
80 = 70 +10 x, I 9 = 1.5+o.5x2
thus -Y 1 = + I y2 = +O
Trial no 1 2 3 4
8
Temperature
Pressure
-1
-0 5 -0 8
+o 8 -0 5
+I
+o 9
+o 8
65 00% 71 45% 82 69% 92 50%
375
We will use the matrix method to calculate the effects and interactions. Starting with the mathematical model for factorial designs, we write that each experimental point satisfies this relationship. This gives us a set of four equations with four unknowns.
y 1 =65.00=+a0y , = 71.45 =+a,+ 0.8 y , = 82.69 =+a, - 0.5 y4 = 92.50 = + a o +
0.5 + 0.50 aI2 0.8 - 0.64 a,2 al + 0.9 - 0.45 aI2 a, + 0.8 3 + 0.80 aI2 a, a,
-
-
This set of equations is then transformed to a matrix form -0.5 +0.5 +1 -1 65.00 71.45 - +1 +0.8 -0.8 -0.64 +1 -0.5 4 . 9 -0.45 82.69 92.50
+I
4 . 8 +0.8
+I
a0 a1 a2 a12
~0.1846
x-I =
M.2738 -0.2303 +0.3112 -0.3159 -0.3391 +0.4665 +0.1886 M.4003 -0.3557 -0.4863 +0.4417
-0.3547
And X-'Y calculated a0 A = a1 a2 a12
+0.2872 +0.2820 +0.2461 +0.1846 65.00 -0.3547 +0.2738 -0.2303 +0.3112 71.45 -0.3159 -0.3391 4 . 4 6 6 5 +0.1886 82.69 4 . 4 0 0 3 -0.3557
a. AZa' = a2 a12
-0.4863
76.25 6.25 11.25 1.25
hence by identification: a. al
= 76.25 = 6.25
% = 11.25 aI2= 1.25
+0.4417 92.50
3 76
Here again, we have obtained the effects and interactions despite the fact that all four experimental points were not situated at the comers of the experimental domain. The values we have calculated can be used to define the mathematical model giving the yield as a fimction of the reaction temperature and pressure (in coded units):
y
= 76.25
+ 6.25 x, + 11.25x2 + 1.25 x, x2
This model is used to calculate the yields throughout the experimental domain, and draw the iso-yield curves within the domain (Figure 16.4).
Pressure
4
2 bar
+I
1 bar
-1
r
-1
60 "C
+I
Temperature
80 "C
Figure 16.4. Iso-yield curves within the experimental domain
It appears as if an experimenter can place the experimental points where he wishes. In particular, he can take into account any constraints that may apply without abandoning the method of experimental design. We have seen an example with two factors, but it is possible to extend the process to all experimental designs, regardless of the number of factors. This result is obtained because experimental errors were not taken into account. With experimental errors experimental points must be carehlly located. So why is it important to try and place the experimental points at the corners of the experimental domain? The answer is that this particular distribution provides the coefficients of the mathematical model with the greatest accuracy. The precision drops once the points are moved away from the comers, and in extreme situations this can even ruin the whole experiment.
3 77
4.
ERROR TRANSMISSION
Let us examine the relationship linking the responses yi to the coefficients ai within the mathematical model Y=XA
In this relationship, we assume that the levels of factors are perfectly defined and introduce no error. The X matrix is thus known with inftnite precision. We also assume that all errors are due solely to the responses y+ i.e., experimental errors. These errors will influence the coefficients ai which are to be calculated. Statisticians have shown that the variance of Y. V(Y), is linked to the variance of A, V(A) by the relationship: V(Y) = X V(A) X'
where V(A)=
We are interested in the variance of A as a fbnction of the variance of Y. We can obtain this by multiplying the two sides of this equation by the same matrix X' X' V(Y)
=
X'X V(A) X'
and by the same matrix X: X' V(Y) X = X' X V(A) X'X
Even if the X matrix is not square, the X' X matrix is. It can therefore be inverted in most cases ( it is assumed that Det X' X f 0 ) and written: V(A)
=
(Xt X)-' X'V(Y) X (X' X)-'
We can simplifi this rather complex formula by assuming that the experimental error of the response y is the same throughout the experimental domain. In this V(Y) is simplified to: V(Y)
=
o2I (n,n)
V(A)
=
(X' X)-' X'02 I (n,n) X (X' X)-'
V(A)
= o2 (X'
hence or
X)-' X' X (X' X)-'
378
and as
(x'x)-'
x' x
V(A)
o2 (X'X)-'
=
I
then =
This relationship is most important as it shows how the experimental error effecting the response is transmitted to the coefficients of the mathematical model. In the particular case of factorial designs it also shows how it is transmitted to the effects and interactions. Two terms must be taken into account, the error of the response, o,which is not surprising, and the X matrix, i.e., the distribution of the experimental points within the experimental domain. This second result is much more unexpected and deserves some examination and analysis as it is familiar to few experimenters. We will use three examples of experimental point distribution that we have encountered previously in the catalysed chemical reaction study. We will then calculate how the same error of y is transmitted to the effects. For this we will calculate the (X' X)-' matrix and identie the diagonal elements with those of the V(A) matrix. Case 1
Let us begin with the experimental points located at the corners of the experimental domain. This is a classical factorial design. We can now calculate the (X' X)-' matrix from the X matrix: -1
+1
+1
+1 -1
+1
-1
+1
-1 -1
+1
+1 +1
+1
+1 -1
X=
The lines and columns of the X matrix are then exchanged to give the transposed form:
-1
+1 +1 + I +1 -1 + I
-1
-1
+1
+1
+1 -1
-1
+1
+1
x =
We must calculate the product of the transposed matrix X' and the X matrix itself
3 79
+1
+1 + I
-1
+1 -1 -1 + I -1 -1
x ' x = -1
+I
-1 +1 +1 -1 +1 -1 +1 +I +1 +1 +1 -1
+1 +1 +1
+I
+1 -1 -1
+1
Matrix multiplication gives: 1+4
x'x=
0
0
0 +
0
0
0
0
4
0 + 4 0
1nve:rting xtx we get:
I +0.25 0
0
0
0 0
+0.25
o
Applying the formula in which we have written
02
O
I
0 +0.25/
instead of& to be more precise
V ( A ) = o;(X'X)-' And developing it: 0
0
+0.25
0
0
0
+0.25
0
0
0
+0.25
+0.25
0
0
0 0
We get the variance of the coefficients by identifying the corresponding diagonal elements in the two matrices: V( a,,)
= 0.25
V( a,)
=
o',
0.25 o:,
3 80
V( aJ
= 0.25 a;
V( a12)= 0.25 o: The square roots of these expressions give the standard deviations of each coefficient as a hnction of the response errors: a (aJ
= 0.50 oy
a (a,) = 0.50 ay
a (a.J
= 0.50 a,,
o (al2)= 0.50 o, The reader will recognize the well-known formulae that are used for factorial designs. The error of the coefficients is equal to that of the responses divided by the square root of the number of trials. The standard deviation should be indicated after each calculated result, especially after the numerical coefficients of the mathematical model for factorial designs. For example, if the error is 1 yield point for each determination (ak = k I%), the coefficients will be calculated with a standard deviation of 0.5 yield point. T s value is shown in brackets below each coefficient, as follows:
*
y
= 76.5 +
(0.5)
6.25 x1 + (0.5)
11.25 x2 + (0.5)
1.25 x, x2 (0.5)
Case 2
This example is set out in section 16.2 of the present Chapter. In it, three of the experimental points lie at the corners of the experimental domain, while the fourth is displaced. The X matrix is:
x=
+1 +1 +I +1
-1
-1
+1
+1
-1
-1
-1
+1 -1
+0.8
+1 +0.8
The transposed matrix, obtained by exchanging rows and columns is
38 1
-1
+I +1 + I +1 -1 +0.8 -1 +l +I
+I
-1
+1
x t = -1
-1
+0.8
We then calculate the product of these matrices:
x'x=
+1 +1 +1 +1 +1 -1 +0.8
-1 -1
-1
+1 -1
+1 +1 -1
+I
-1
-1
+1
+1
+1
-1
-1
+1 -1 +1 +0.8
+0.8
+I
-1 +1 +0.8
or:
x'x=
+4
-0.2
-0.2 0 -0.2
+3.64 -0.2 -0.2 +4
-0.2
-0.36
+3.64
-0.2
0
-0.36
-0.2
Inverting xtx we get:
(x' x)-' =
0.2515 0.0154 0.0015 0.0154 0.0154 0.2793 0.0154 0.0293 0.0015 0.0154
0.2515
0.0154
0.0154 0.0293
0.0154
0.2793
Using the formula V(A)
= 0;
(X'X)-'
and showing only the diagonal elements, as we are only interested in the variances (or standard deviations).
2
=ov
t0.25
.
'
+0.28
.
.
+0.25
.
. +0.28
3 82
This gives us the variances of the effects and interactions by identifying the corresponding diagonal elements of the two matrices: V( ao) = 0.25
02,
V( a,)
=
0.28
V( a.J
=
0.25 o:,
V( a12)= 0.28 o; The square roots of these expressions give the standard deviations of each effect as a fbnction of the response errors: o (a,)
= 0.50 o,
o (al) = 0.53 o, o (%) = 0.50 0,
o (al2) = 0.53 o~, There is clearly a slight drop in the precision of the coefficients. The formulae generally used to calculate the standard deviations for factorial designs are no longer valid. The results are presented with the standard deviations below the numerical coefficient of the mathematical model for factorial designs. Keeping the k I yield point error for each response, we have: y = 76.5+ 6.25x1+ (0.50) (0.53)
11.25x2+
(0.50)
1 . 2 5 ~ ~ ~ ~
(0.53)
Case 3 This time we will use the distribution of experimental points shown in section 16.3, in which all four points are displaced fiom the comers of the experimental domain. The X matrix is:
X=
+1
-1
-0.5
+0.50
+1
+0.8
-0.8
-0.64
+1 +1
-0.5
+0.9
-0.45
+1
+0.8
+0.80
383
Transposed, we have +1
+1
+1
+1
-1
+0.8 -0.8
-0.5
+1
+0.9 -0.45
4.8 +0.8
x t = -0.5
+0.5 -0.64
and multiplying X by X' we get: +I +0.8 -0.8
+1 -0.5 +0.9
+1 +1
+1 +I
+0.8
+1 -0.5
+0.50 -0.64 +0.9 -0.45
+0.5 -0.64
-0.45
+0.8
+1
+0.8 +0.80
+1
-1
x ' x = -0.5
-1 -0.5 +0.8 -0.8 +1
or +4.0
Xt
x=
+0.30
+0.40
+0.210 4.013 +0.497
+0.3
+2.89
4.21
+0.4 -0.21
+0.21
+2.34
+0.013 4 . 4 9 7 +1.502
and inverting Xi X:
(xt x)-l =
0.2566 -0.0238 -0.0367 0.3505 -0.0295 -0.0238
-0.0235
-0.0295
0.4680
-0.1494
0,0100
-0.1494
0.7183
V(A)
o', (X'X)-'
-0.0367 -0.0235
0.0100
Using the formula =
and showing only the diagonal elements, as we are only interested in the variances (or standard deviations).
2 = 'SY
-+0.26 . . +0.35
.
. +0.47 . . +0.72
3 84
This gives us the variances of the coefficients by identifjmg the corresponding diagonal elements of the two matrices: V( ao) = 0.26 ot V( a,)
= 0.35 0:
V( aJ
= 0.47
05
V( a12)= 0.72 o$ We obtain the standard deviations of the effects as a fhction of the errors, ov , of the response by taking the square roots of these expressions. S I
(a,)
(3
(al) = 0.59 cry
=
0.50 oy
o (9) = 0.68 oy
cr (al2) = 0.84 or The error is even greater than in the previous example. Keeping the for each response, the mathematical model is written:
y = 76.5+ 6.25x,+ (0.50)
(0.59)
11.25x2+ (0.68)
* 1 yield point error
1 . 2 5 ~ ~ ~ ~ (0.84)
Case 4
In this last example the distribution of the experimental points is rather bad, or more accurately, disastrous (Figure 16.5).
385
C 2bar
+I
*
C'
A' I bar
-I
A/
B
I
Figure 16.5. Distribution of experimental points in case number 4.
The X matrix is: +1
-1
-0.9
+0.90
+1
+1 -0.5 +1
+0.7
+0.70
+1 -1
+I
+1
+1
+1
-0.5
+1
-0.9
+0.7
+0.4
+1
+0.9
M.7 M.2 +1
x = +1 +1
+0.4 +0.20
+1
+1
Transposed, we have
xt =
The transposed matrix Xi is then multiplied by X to give:
+4.00 +0.50 +0.40 +2.80 +0.50
x t x = +0.40
+3.25
+2.80 +0.70
+2.80 +2.46 +0.60
+2.80 +0.70
+0.60 +2.34
386
And inverting X' X:
(xt x)-'=
1.6755
-0.63 11
0.9481
-0.6311
16.1422
18.4296
0.9481
18.4296
21.5061 -1.1358
-2.0594
0.6518
-1.1358
-2.0592 0.6518 2.9876
Using the general formula, and showing only the diagonal elements.
This gives us the variances of the effects and interactions from the corresponding diagonal elements of the two matrices:
V( a") = 1.67 0; V( a,) = 16.14 0 ; V( a,J
= 21.50 0:
V( a,2) = 2.99
02
We obtain the standard deviations of the effects as a function of the error of the response by taking the square roots of these expressions. (r
(ao) = 1.29o,,
o (a,) = 4.02 0 (r
(%) = 4.63 oY
0
(al2) = 1.73 oY
The error is even greater than in the previous example
387
y
=
76.5 + 6.25 x, + (1.29) (4.02)
11.25 x2 + (4.63)
1.25 x, x2 (1.73)
The errors of the coefficients are so great that the mathematical model is worthless, it cannot be used for prediction, and no conclusion can be drawn from it. The poor quality of these experiments is not due to the precision of the measurements, which is always k one yield point, but to the poor distribution of the experimental points within the experimental domain.
Pressure
Pressure
+I
+I
-1
-1
perature CASE 1
Pressure
CASE 2
Pressure
+I
+I
-1
-1
CASE 3
CASE 4
Figure 16.6. The distribution of the experimental points in the four cases studied.
These examples show that the precision of the mathematical model coefficients depends on the location of the experimental points. Even if the response is measured with great
388
precision, a set of trials can be ruined by poorly located experimental points. Figure 16.6 shows the distribution of the points in the four examples, and Table 16.3 shows the different precisions of the effects and interactions for the same experimental response precision. The same experimental precision can give either good or bad estimations of the effects depending on the location of the experimental points.
TABLE16.3 Influence of experimental point location on the precision with which the effects are estimated
5.
EXPERIMENTAL QUALITY
There are clearly good and bad locations of experimental points within the experimental domain. How can the quality of an experiment be expressed? If we consider the four points A', B', C' and D we see that the experiment is good when they form a square and poor when they form a straight line. Experimental quality is thus linked to the area defined by the polygon A' B' C' D' (area used). The greater this area, the better the experimental quality. Conversely, the smaller the area, the poorer the experiment. To remain consistent, we must compare the area enclosed by the polygon A' B' C' D' (area used) with that obtained if the points had been distributed according to an optimal design, i.e., the useable area. This reference area is calculated from the maximal and minimal coordinates of points A', B', C' and D (Figure 16.7).
389
Pressure I
D'
I
+I
-1
-1
+I
Temperature
Figure 16.7 Comparison of the area used and the useable area. For example, we can calculate the quality of the four cases: Case 1:
quality = 1
Case 2 :
quality = 0.9
Case 3 :
quality = 0.748
Case 4:
quality = 0.296
The experimental quality varies from 1 (optimal) with points at the corners of the square defined by the coded variables, to zero, when the points form a straight line. The concept of usable area and area used may be extended to designs with more than two factors. We then speak of usable volume or hypervolume; and volume or hypervolume used. Experimental quality is always defined by the ratio of the volume used to the usable volume.
This Page Intentionally Left Blank
CHAPTER 17
OVERVIEW AND SUGGESTIONS
1.
INTRODUCTION
The research approach and the organization of experimental trials that we have covered in this book provide flexibility, precision, security, and a saving in time and money in all experimental fields. This last chapter integrates the topics we have covered in the preceding pages and examines the things that Experimentology can and cannot do. We will use this opportunity to examine some practical points and show how the approach can be expanded beyond the stage of looking for influencing factors. The approach to successful experimental studies is summarized in the flowchart in Figure 17.1. This outline shows the three main segments of Experimentology :
0
Selection of the best experimental strategy. Interpretation of results. Gradual acquisition of knowledge.
3 92
2.
SELECTION OF THE BEST EXPERIMENTAL STRATEGY
This selection must be made, before doing a single experiment, in response to a series of questions. These questions should accurately and completely define the problem.
2.1.
Defming the problem
This phase should never be neglected. Any extra time spent on this phase, including consultation of appropriate specialists, is a good investment. All information on the specific subject should be collected, analysed and critically evaluated. The problem is often changed after detailed analysis of the initial information. The most important thing is to begin the study under the best possible conditions by integrating all available knowledge. This also helps speed up the research, quickly providing correct, precise conclusion. If this phase is neglected, there is increased risk of unpleasant surprises or even major setbacks. Defining the problem can be divided into four steps. The experimenter may decide to undertake this definition either alone, or in consultation with others. The second, group approach is generally more fruitfil than one person working alone. More information is brought to bear on the problem, and each person has a slightly different opinion. The four steps are : Description of the phenomenon. Definition of responses. Selection of tactors to study. Defining the limits of the experimental domain
Description of the phenomenon This description covers the whole process, attempting to be as precise and as complete as possible. If the group approach is used, the input of the participants and their questions are all considered. This step should gather together as much information as possible, both qualitative and quantitative. The total information should provide a clear, precise, complete picture of the phenomenon to be studied.
Definition of responses The selection of the response(s) is of prime importance. The whole analysis and all the conclusions depend on it. A response which is inappropriate makes all the experiments useless and leads to irrelevant conclusions. The response selected should provide an unambiguous answer to the question asked. This choice is not quite as simple as it may appear, and is oRen the main difficulty facing the experimenter. This definition should never be rushed; time spent on it is a good investment. Sometimes it is even worthwhile running preliminary experiments to define the best response.
3 93
CHOICE OF THE BEST EXPERIMENTAL STRATEGY
Define the problem Factors Domain Responses
+
+
Preliminary questions Are there any experimental constraints? Is there any risk of systematic errors? Should the trials be randomized? Is there an estimation of experimental error? Should all the factors be studied immediately? Should measurements at the central point be planned? Choice of the design number of factors? Interactions different from zero? Complete or fractional designs? Order of trials? Measurement at the central point?
GRADUAL ACQUISITION OF KNOWLEDGE
Complementary designs Complementary trials New factors New domain New mathematical model Modelling Optimizing
-1 RUN THE EXPERIMENTS
L INTERPRETATION OF RESULTS
t
Critical examination of the results Are there any suspect values? Were the factor levels set at the right values? Calculation of effects 0 Check for ambiguities Domain of interest Comparison between calculated and measured means Is the first degree model valid?
A
t
t
t
Follow up Should the experimental domain be changed? Is a complementary design (or trial) required? Should another mathematical model be considered? Three NOS Stop the experiment One YES Continue the experiment
Figure 17.1 : How to conduct a study with the experimental design methodology.
3 94
Selection of factors
All the factors that may influence the phenomenon under investigation should be listed. No factor should be omitted; we have seen that the number of factors introduces no experimental constraint and that fractional designs can be used to reduce the number of trials run. The bad habits learned from the standard one-factor-at-a-time approach must be overcome, and any factor that may influence the response should be included. All restrictive hypotheses should be avoided. The six-M mnemonic can be used to be sure of forgetting nothing: Methods, Machines, Manpower, Milieu, Materials and Measurements Methods includes all the factors associated with the method that could influence the selected response. Machines covers the factors associated with the instruments or apparatus used that could influence the response. The factors associated with the other M headings should be identified in a similar fashion. Some investigators include a seventh M: Money Once all the factors have been listed, the most relevant ones are selected Limits of the experimental domain
The level of each factor should also be carehlly selected. The domain should not be so small as to prevent the conclusions having some general relevance, neither should it be so large as to risk making the mathematical model of the factorial design inappropriate. The best compromise is often difficult to find for the first design if there is not enough background information on the problem.
2.2.
Preliminary questions
Several questions should be asked before beginning the experiments. They should focus on the selection of the initial experimental design and any complementary designs or trials. The following questions are all worth asking: Are there any experimental constraints?
If something is difficult to regulate, for example, if the system is slow to get to the right temperature, then the trials should be planned in sequences. One series of trials run at the low level should be followed by a series of trials at the high level. A single factor can be studies in this way. Thus, in a 23 design, column 3 (the one change column, Chapter 9, page 160) should be chosen for this particular factor when the trials are run in the normal sequence. There is always a column in Hadamard matrices, but only one, which contains a series of minus signs followed by a series of plus signs (or the reverse). The position of this column depends on the order in which the trials were selected.
395
Is there any risk of systematic errors?
We have seen that systematic errors (Chapters 4, 9 and 10) have major repercussions on the evaluation of effects and interactions. It is therefore important to guard against these errors and to minimise their consequences by selecting the best sequence of trials. If the systematic error might modifi the mean of the results, blocking should be used. This is the case, for example, in a design run at two separate times, or for a design run at two different locations. Blocking allows the effects to be obtained as if the means at two periods or locations werethe same. If there is a risk of drift, then: 0
0
either a trial sequence should be chosen so that the main effects are not affected by bias due to this systematic error (Appendix 3) or the drift should be checked by measurements at a central point and appropriate correction of the responses. This second solution requires a greater number of experiments (Chapter 9 sectin 4.3). Should the trials be randomized?
Randomisation is used to obtain a random distribution of small, unknown systematic errors, thus allowing treatment of the experimental error as if it had only one component: random error. This process is therefore usehl for applying statistical tests, but has the drawback of increasing the random error. The best approach is to take into account all the constraints and make the randomization in three steps: Step one The experimental constraints are first taken into account. For example, the one-change column is chosen for a factor. There are two sequences of signs, the high and low levels. It is still possible to randomize within these sequences. Any other arrangement that simplies the manipulations is clearly possible and may be considered by the investigator before starting the experiments. Step two The systematic errors are examined and the trials are divided into blocks, or a special trial order is adopted (Appendix 3) to overcome known or suspected drift. Step threg Randomization: 0 0
The trials within each block could be randomized. One of the orders giving the main effects not affected by drift error could be chosen at random (Appendix 3)
396
If there are no experimental constraints and no reason (drift) to chose a particular trial sequence or blocking, then the trials may be randomized. Is there an estimation of experimental error?
It is important to know the experimental error in order to decide if an effect is significant or not. The experimenter should always have an estimate of this error based on a large number of measurements. Unfortunately, this ideal requirement is seldom met, and it is often necessary to obtain an estimate of this error based on a few measurements or on an hypothesis (Chapter 4).
Comparing effects and error is not always as simple as many manuals imply, and the experimenter should never forget the many hypotheses upon which the validity of this comparison is based. This can be quite difficult because many of them are not clearly stated. In fact, the hypothesis used to calculate effects should be added to those used to calculate experimental error, such as, Normal distribution of errors. There are tests to see ifthe distribution is normal Lack of systematic errors. This is rarely checked. The same error throughout the domain (homoscedasticity of the experimental error), is rarely checked. Independence of variables. Occasionally two apparently independent variables may have a hidden relationship. This hidden relationship may be introduced by a property of the population studied. 0
Calculation of the experimental error is generally based on parametric statistical methods, but this is not necessarily the best approach when there are few data. The factorial design mathematical model is first degree. If the response does not follow such a mathematical law the effects are no longer correct.
0
The factorial design mathematical model is additive. If this is not verified, the effects and interactions are false.
This list is incomplete, but it shows the weakness of the reasoning that can occur because these hypotheses are not all respected in most studies. The experimenter should therefore examine carefully the validity of his error calculations and avoid automatic methods. It should be understood that obtaining useful information from the smallest number of trials requires simplifying hypotheses. These are useful or even indispensable, but there is nevertheless a great risk of a mistake and experimenter must be very cautious. If the experimenter has no estimation of the error, but is able to repeat a measurement several times. the best solution is to run the trials at the central point or at any other specific
3 97
points if the central point is inaccessible. It is important to over-emphasise the usefulness of these measurements.
Should all the factors be studied immediately? In some instances there may be many factors and the time and budget available may be very restricted. Under these circumstances the factors likely to have the greatest influence are selected, and the other factors are temporarily assigned fixed levels throughout the experiments. For example, if ten factors have been identified, but only eight trials can be run, then a 27-4 design should be run to study seven factors while assigning a constant level to the remaining three factors throughout the experiment. There is thus always the possibility to reintroduce them later in the study without wasting the initial eight trials. They can be given levels different from those at which they were fixed in the initial eight trials, and a design similar to the first repeated. This is the reverse of fractionation: we start with a 2k design and add to it I, 2, 3 ...p supplementary factors to finally obtain a Zk+p design. The fixed factors that were not studied, but available, are called prefixed factors.
Should measurements at the central point be planned? The central point of the experimental domain is very special. We have seen in Chapter 1 1 that this is where the calculated responses are obtained with the greatest precision if the mathematical model is valid. This is also the point where the mean of all the experimental responses is equal to the model mean if, once again, the first degree hypothesis is valid. If we make a measurement at the central point, we can compare the true mean to that calculated from the model. In this way we can determine whether the model associated with the theory of factorial designs is valid. If we make several measurements at the central point, not only we can make this comparison, we can also obtain an estimate of the experimental error. It is therefore a very good idea to run several experiments at the central point as early as possible. This is a wise precaution which checks the hypothesis or orientates the remaining trials in the right direction, generally towards a second degree mathematical model.
2.3. Choice of design The selection of a design is based on the points discussed above:
0
0
The number of factors. The interactions that may be significantly different from zero. Selection of a complete or fractional design, depending on the experimental conditions, or on the number of trials. The order of trials must be defined. A central point is often included, with or without repeats.
398
This design may be divided into two sub-designs (blocking, fractional design) so as not to use up the whole budget (time and/or money) on a single avenue of research. This approach abides by the principle of the progressive acquisition of knowledge. If a factor is not immediately studied, but may be included later, it should be kept at a fixed level (prefixed factor), in preparation for any subsequent trials. AII that needs to be done then, as we have seen, is to assign it another level and run the new trials to obtain a design having one or more extra factors. There can be as many prefixed factors as required. The setting up of the initial design will be made easier by the presentation we have used for all the examples in this book. All the data, factors, domain and responses are included. This presentation makes it easy to monitor the experiment and analyse the results.
3.
RUNNING THE EXPERIMENT
This phase is not included in Experimentology, but we should not forget that it is the most important part of the study. Thus it is quite normal for experimenters to take particular care over this phase. The quality of the whole study and of the resulting recommendations depend on it. Experimental designs cannot compensate for poor experiment execution, quite the contrary. The more powerful the tool, the greater the risk when using it.
4.
INTERPRETATION OF RESULTS
Before performing the calculations, the experimenter will examine the correctness of the experimental conditions and the quality of the results of each trial. Analysis of results can be divided into two phases: critical examination of the results obtained, and follow-up.
4.1.
Critical examination of the results a. Are the results coherent? Are there any suspect values?
A poor result will influence all the effects, and particularly the interactions. It is therefore prudent to check the quality of the results before beginning the calculation and interpretation steps. A suspect result may be due to an error in transcription, setting up, defective instrumentation, incorrectly set level, etc.
b. Were the factor levels set at the right values? This question has been covered in Chapter 16, and we have seen that it complicates the calculations. It sometimes happens that the high and low levels for a particular factor are correctly set in some trials but not in others. For example, the high level of factor 3 was set at +1.2 in trials 2 and 5. These trials are not wasted, but the usual calculations cannot always be used and matrix calculus is required.
3 99
When the critical examination of the experimental results is complete, and everything has been checked, we can begin to calculate the effects, interactions and the experimental error. c. Calculation of effects, checking for ambiguities. This is a multi-step analysis. The experimenter should never forget that fractional factorial designs give contrasts, and not effects, although the term effects rather than contrasts is generally used. The preliminary calculations and examination of aliases should prevent any hasty conclusions. Analysis is a long, critical phase during which the results are examined in depth to answer such questions as:
0
0
What are the influencing factors? What are the s i m c a n t interactions? Are there any ambiguities in the calculated contrasts? Does the linear mathematical model of the design explain the experimental results? Does comparison of the calculated and measured means at the central point validate the first degree model? Are there any interactions that are too large to be compatible with the mathematical model? Was it reasonable to assume that the effects are additive? Does the solution to the problem lie within the experimental domain? The steepest ascent method (Chapter 11) can be used to indicate the direction of hrther studies. Would a residuals analysis extract any remaining information?
These questions, in addition to the calculations, graphs, interpretation of results and comparisons, all form part of a critical, in depth examination of the whole experiment. The phenomenon studied may have given up most of its secrets, perhaps even enough to allow the study to be stopped after the initial design. But supplementary trials are generally needed to provide a better answer to the original question. These can follow several possible, sometimes independent, avenues of research. d. Domain of interest The experimental domain could be changed if the first results do not contain the solution of the problem. The initial results generally provide an indication of where this solution could lie. The steepest ascent vector and the mathematical model are used to redefine the region of interest. e. Complementary designs This type of design can be considered when the main effects are aliased with interactions that should not be neglected. A few extra trials are often enough.
400
f. Complementary trials A couple of well chosen trials can remove ambiguities between interactions (Chapter 12) or between main factors and interactions. g. Is the first degree model valid?
The mathematical model of factorial designs is no longer valid if the interactions are too great or the measured and calculated means differ too much. A good example of this is the cutting oil example in Chapter 13. A more complete model must then be used for interpretation. The choice of experimental points leading to an optimal design and the calculation of the model coefficients becomes much more complicated, and they are beyond the scope of this introductory book. Fortunately, all the results obtained with factorial designs can be reused. All that is required are a few weil chosen extra experimental point in order to produce a second degree model. Here again, the principle of progressive acquisition of data is complied with.
Follow up
4.2.
The experimenter may ask the following questions, depending on the conclusion of the first design: Should the experimental domain be changed? Is a complementary design required? Are complementary trials needed? Should another mathematical model be considered? Stopping the experiments
Depending on the answer to the above questions, the experimenter may: 1. Stop the experiments, as he knows enough to make recommendations to solve the original problem. 2. Require extra information, in which case he will follow the principle of progressive knowledge acquisition.
The flow chart in Table 17.1 indicates that: Three NOS are required in order to stop
A single YES requires a hrther loop, and of course this may be repeated several times.
40 1
5.
GRADUAL ACQUISITION OF KNOWLEDGE
If the experimenter wants extra information, he should define the tools and the methods to be used. The strategy is selected in the light of the initial results and the questions they raised. The choices are: Complementary design. This may be used to dealias any ambiguous main effects and interactions.
0
Complementary trials. These are used to dealias second order interactions or main factors and interactions.
0
New factors. lf the experimenter has taken the precaution of including prefixed factors, these can be included in a new design by giving them a new level. The prefixed factors then become active. 0
New domain. A new study domain may be chosen for several reasons, such as the initial domain was too large or too small, or it did not include the desired results. Mathematical modelling or steepest ascent vectors may be used to find the new domain. 0
Optimization. The study domain may be suitable, but the model may not be appropriate for explaining the experimental results. In this case the initial experiments should be supplemented with extra trials. The new experimental points should be carefully located within the experimental domain. Let us conclude this short discussion of the progressive acquisition of knowledge with a general point. The great flexibility of designs based on Hadamard matrices means that the experimenter can: 0
0
constantly add results by new trials. Complementory designs or trials enrich the information provided by the initial design; add new factors, provided that provision has been made for this by defining prefixed factors.
The greatest inconvenience of designs based on Hadamard matrices in that all the trials of the factorial design must be run in order to do the calculations and interpret the results. The above discussion indicates that, in most cases, the initial design should include the smallest possible number of trials, four or eight. This quickly provides information on which the remainder of the study can be based. Depending on the objective, additional trials can be run. They may be complementary trials or extra factors (2k* design). The principle of progressive acquisition of knowledge is derived from this discussion and fits perfectly with the first two parts of Experimentology, the selection of the best experimental strategy and interpretation of results.
402
6.
WHAT EXPERIMENTOLOGY WILL NOT DO
Experimentology is a methodological approach for successfully running a particular experiment by optimising the facilities (time and money) available to the researcher. But this method cannot do the experimenter’s work for him, and most certainly cannot replace his experience, intelligence and intuition. He must be responsible for:
Asking the right question Experimentology cannot overcome a poorly designed problem Imagination and creativity Experimentology is only a technique; it can help interpret the results, but the experimenter must formulate the hypotheses and draw his conclusions. selecting factors The experimenter is responsible for any wrong conclusions that may arise if he omits an important factor from his study. Define the experimental domain This is one of the greatest difficulties. If the domain is too small the difference between the responses could be of the same order as the random error. If it is too large, the linear mathematical model of factorial designs may not be valid. Correctly running the experiments Poorly conceived experiments, uncontrolled factors, imprecise measurements, errors of notation, errors of transcription, etc. cannot be overcome by experimental design. On the contrary, experimental design demands great care in running the experiments in order to be of maximum use. Choose the responses This is also vital for the quality of the interpretation. If the responses selected cannot be used to calculate the appropriate effects, or if the results are inaccurate, the whole experiment is wasted. Clearly, then,
Experimentology is a useful, powerful tool, but the experimenter is entirely responsible for correctly running the right experiments and drawing the pertinent conclusions.
APPENDIX 1
MATRICES AND MATRIX CALCULATIONS
1. INTRODUCTION The simplest and most practical way to approach matrices it to think of them as tables or arrays. We shall define these tables and list the rules for their use in calculations. But the mathematical details will not be shown, or all the possible interpretations and applications of these arrays. The interested reader should consult the appropriate textbooks listed in the bibliography.[27,28]
2. 2.1.
DEFINITIONS
General
a. Matrix
A matrix is a table or array whose elements are arranged in rows and columns, e.g.:
404
The elements are a, b, c, d, e and f
..The The first row is second row is
. .
The first column is The second column is The third column is
a, b, c. d, e, f. a, d. b, e. c, f
The array is written between brackets or vertical lines or
For simplicity vertical lines are used in this book
b. Matrix order An order (m, n) matrix is a table of m rows and n columns. The above matrix is a 2 by 3 matrix, which is written (2, 3 ) . The number of rows is given first, followed by a comma and the number of columns, and both figures are enclosed in parentheses.
c. Square matrix
This is a table with the same number of rows and columns (n, n).
d. Column matrix and Column vector This is a matrix having only a single column (n, 1). It can also be called a column vector.
e. Row matrix This is a one-row matrix (1, n). It can also be called a row vector f. Labelling elements
We will adopt the following system: the row and column of the element "a" are indicated as subscripts, the row followed by the column: aij for the element of the ith row and jth column.
405
The first matrix would then be written:
where: a=all b = alz c = aI3 d=%l e=a22 f=
a23
g. Homologous elements
These are elements that occupy the same position in two matrices of the same order. They are also called corresponding elements.
h. Nature of elements The elements of a matrix may be very different, including real numbers, complex numbers, polynomials, finctions, operators, matrices, etc. Most of our applications will contain real numbers.
2.2.
Definitions for square matrices
a. Main diagonal
The main diagonal of a square matrix is formed by the elements aii. These are shown in bold type below:
The main diagonal can also be called the principal or leading diagonal
b. Symmetric matrix
This matrix is symmetrical around the main diagonal: the aij elements are equal to the aji elements, a , . = a,. U
J1
0.25 -0.02
-0.03
-0.02
0.35
-0.03
-0.03
-0.03
-0.02
0.47
0.01
-0.15
-0.02 0.01
-0.15
0.72
c. Antisymmetric matrix The aij elements are equal to the aji elements with the sign changed: a,. = - a , . U
1'
e.g.: 0.00 -0.02
0.02
-0.03
0.00 -0.03
0.03
0.03
0.02
-0.01
-0.02
0.01
0.00 -0.15 0.15
0.00
d. Determinant of a matrix
The table of elements of a square matrix may be considered to represent a determinant. The elements of a matrix are said to be those of a determinant. This interpretation of matrices is useful but confusing, as the two mathematical entities, matrix and determinant, are different in nature. The fact that they are written similarly and have some similar rules for calculation can also be misleading. The reader should take care not to c o d h e them. The X matrix is a table, or an array of elements:
The determitant, Det X, for this matrix is a number and is written between double vertical lines: Det X=ll:
1/=6-1=5
The difference between a matrix and a determinant is thus simple. A matrix is a table which is used to perform calculations. A determinant is a number that is calculated using mathematical rules.
407
e. Trace of a square matrix
The trace of a square matrix is the sum of the main diagonal elements e.g. : 0.25 -0.02 -0.02 -0.03
-0.02
-0.03
-0.02
0.35 -0.03
0.01
0.47
-0.15 0.72
-0.03 0.01
-0.15
Trace = 0.25 + 0.35 + 0.47 + 0.72 = 1.79 f. Diagonal matrix
The diagonal matrix is a matrix in which all the elements are zero except those of the main diagonal. e.g. 0.25 0
0
0.35
0
0
0
0
0
0
0.47
0
0
0
0
0.72
3. MATRIX OPERATIONS Two types of operation can be performed on matrices, operations on the table itself, and operations between tables.
3.1 Operation on array a. Transposition
The transpose matrix A' of A is obtained by exchanging the rows and columns. The first row o f A becomes the first column of A', the second row o f A becomes the second column of A', etc.
408
all A' = a12
a21 a22
a13
a23
Matrix A is a ( 2 , 3) matrix . The transpose mathx A' is a (3, 2) matrix .
3.2 Operations between arrays a. Equality
Two matrices of the same order are said to be equal if the homologous elements are equal. The two matrices M I and M2
are equal if a11 a12
= b,, = b12 a13 = b13 a21 = b21 a22 = b22
?23 =
b23
Matrices of different orders cannot be equated b. Addition Only matrices of the same order can be added together. This produces a matrix of the same order. The elements of the matrix-sum are the sums of the homologous elements of the two matrices. Two matrices M, and M2 give the sum matrix M,:
M,+M2
=
M3
409
Using numbers:
lo 41+/0 ol=lo 41 5
3
6
0
2
1
1 1 5 1
-2
Matrix addition is associative and commutative. Subtraction is done in the same way.
Using numbers:
c. Null or zero matrix
A null matrix is one whose elements are all zero. When added to another matrix, it gives the same matrix.
d. Scalar multiplication of a matrix All the elements of the matrix are multiplied by the scalar:
Using numbers: 4 4xlO
1 =/
2
1 6 8
4/
410
e. Multiplying one matrix by another
Multiplication of matrix B by matrix A gives matrix C:
A.B
=
C
Multiplication is only possible if A has the same number of columns as B has rows. For example, if matrix A is (m, p) and B is (p, n) then C will be (m, n). Matrices are multiplied by multiplying rows by columns, so that the element cs of the product matrix is equal to:
* b2J
c,] =a,, blJ + a
+...+ aip b,
where = a l l b l l f a 1 2 b21 c12 = a l l bl, +a12 b22
'11
+
a13 b31
+
a13 b32
'21
=%I
b l l 'a22
b21
+
%3 b31
c22
=a21
bl2
b22
+
az3
+
a22
b32
Using numbers:
'I=
5 2 3 0 xi2 0 2 1 3
10 9 sx2+2xo Sr1+2x2 = 6 3 3 x 2 4 ~ 0 3xl+Ox2 2 7 Ix2+3xO lx1+3x2
Multiplication is associative and distributive, but not commutative, i.e., the product A B is generally not equal to product BA.
AB zBA
41 1
f. Unit or identity matrix
This is a square matrix in which all the elements are zero except those of the main diagonal, which are all 1, e.g.: 1 0 0
I=O
1 0
0 0 1
Multiplying a matrix A by an identity matrix gives matrix A, provided that the multiplication is possible (rule of rows by columns). A.I=A
3 2 1 1 0 0
3 2 1
1 3 2 0 1 0 = 1 3 2 2 1 3 2 1 3 0 0 1
g. Matrix inversion Only square matrices may be inverted. Matrix B is the inverse of matrix A if their product is a unit matrix A.B=I Matrix B is written A-'. Matrix A-' exists only if the determinant of A is not zero. In this case matrix A is said to be non-singular, or to be of fill rank. Matrix A is singular or degenerate, and has no inverse if the determinant of A is zero.
3.3.
Calculation of an inverse matrix The elements of the inverse matrix of A are calculated in several steps: 1. The determinant, Det A, of matrix A is first calculated. 2. Matrix A is transposed, A' 3. The minor of each element in A' is obtained by removing the line and the column of this element. The minor is a determinant whose value must be calculated. 4. The minor is given a sign depending on the position of the element. If the element is in the ith line and jth column, the sign is (-l)i+J.The minor affected by the preceding sign is called the cofactor.
412
5 . Each element in A' is replaced by its cofactor. The matrix obtained is the cofactor
matrix Ar 6. The cofactor matrix is divided by the determinant of A (step 1). This is written:
Clearly, this is a long, tedious process, and has for a long time limited the use of matrices in industrial applications. Once again, microcomputers have changed all this, and there are several programs available. Example
Let us calculate the inverse of the matrix A
1. Calculation of the determinant.
DetA = a l l
a,,, - a12
%I
2. Transposing A
3 , Obtaining the minors
The minor of a, I is obtained by removing the first line and the first column. This gives US a determinant having a single element, %2. Thus the minor of The minor of The minor of And the minor of
all aI2 a,,l
a,,,
is is is is
a,,2 %I
a12 a1 1
413
4. Calculating the cofactors.
The signs are attribute first: Coj'actor ofa, I
= (-I)]+]
a22
-
+a22
Cojhctor o f a l z = (-112+1 a21= -a,?, Cofactor o f % , = ( - 1 ) ~ + 2a12= -al2 Cofhctor ofa2,
= ( - I ) * + ~a l l =
+a,
5 . Cofactor matrix The elements of the A'' matrix are the cofactors that have been calculated
r A
a22 -812 = l - a 2 1 811)
6. Matrix inversion
The elements of A'' are divided by Det A
1
4.
a22
- a12
I a11 a22 - a12 a21
a l l a22 - a12 a21
MATRIX ALGEBRA
The definitions and operations presented above can be used to construct a matrix algebra. We will give the main rules, omitting conjugated and associated matrices.
Product transposition
(x,x,)'= x;x:
414
( X i x, X J
=
x: x: x;
Product inversion
(x1x2)-' = x;'x;' (x,x, X J i
=
x;' x; x;'
Transposing an inverse matrix
Transposing a transpose matrix t (xi)
=x
Inverting an inverse matrix
5.
SPECIAL MATRICES
So far we have defined null, unit, symmetrical, asymmetrical, and diagonal matrices. The following are also of interest.
Involutive matrix This matrix is equal to its inverse:
x x-1 x
Example:
x=i1
0
0 -1
415
Orthogonal matrix
The transpose form of X is equal to its inverse,
x' = x-' Example:
Hadamard matrix
This is a square matrix whose elements are +1 or -1, so that: X'X=nl they also satisfy the relationships:
Examples:
H2
+1 +I H4 = +1 +1
=I1 ' 1 1 -1
+1
+I
+1
-1
+1
-1
+1
-1
-1
-1
-1
+1
There are H, matrices for all values of n multiplied by 4: 8, 12, 16, etc. Or €H,,, I,, H,, etc.
416
These matrices have columns (or rows) that are orthogonal between each other. Thus if we give the columns and elements the following significance:
.
a matrix column is considered to be a matrix vector. the elements of the column are considered to be vector components.
Then, these vectors are orthogonal. Example: Column 1 of H, is vector C, with components + I , +I, +1 and + I Column 2 of H4 is vector C, with components +1, -1, +1 and - 1 . The scalar product of C, and C, is:
c, c, = 1.1 + 1.(-
1) + 1.1 +I.(- 1) = 1-1+1-1+1 = 0
showing that C, and C, are orthogonal vectors. The same result is given by the products of c, c3, c, c,, c, c3, c, c4 and c3 c,. Hadamard matrices are used in experimental designs. There are many other fascinating matrices that include complex elements, but we have limited ourselves to those that may be used in experimental designs.
APPENDIX 2
STATISTICS USEFUL IN EXPERIMENTAL DESIGNS
1.
NORMAL DISTRIBUTION The continuous random variable x is distributed normally with a mean X and a standard
deviation o if the density of probability f ( x ) is given by:
This distribution depends on only two terms: the mean X. and the standard deviation o
..
418
The curve representing the function is shown below (Figure A2.1). This Gaussian or bellshaped curve is characteristic of a normal distribution of the random variable x. As the curve is
Figure A2.1: The Gaussian curve. continuous, the mean X and the standard deviation CT are defined for an infinite number o f x values. As it is impossible to make an infinite number of measurements, we distinguish between the population and a sample.
Population This includes all possible values of x. Theoretically a population has an infinite number of individuals, but in practice this is a great number of individuals.
Sample This comprises a few individuals (several values of x) drawn randomly from the population. The larger the sample, the more representative of the population will be. The problem for the experimenter is to estimate the mean and standard deviation of the population from a sample. Statistics theory shows that:
..The sample mean is the best estimate of the population mean. The best estimate of the population standard deviation is given, from the sample, by:
419
where: of individuals in the sample, and ..xi-isXtheis number the deviation of measurement from the sample mean X. N
x1
The denominator N-1 is important when N is small, as is the case in many experiments.
Variance The variance is the square of the standard deviation u for a population; it can therefore be written 02,The variance of a sample is the square of the estimation s of the standard deviation The standard deviation is more important to the experimenter than the variance because the standard deviation is measured with the same units as the value itself But the standard deviation is calculated from the variance, according to the variance theorem.
2.
VARIANCE THEOREM
The random variables xl, x2....xn and the constants ao, at ....a, are assumed to satisfy the relationship:
+ al x, +
y=
xz + ...+a,,xn
If we also assume that the random variables xl, xz ..,xnare independent, the variance of y , Vb), is (Variance Theorem): V b ) = V(
+ V(a, XI) + V(% ~ 2 +) .. , + V(% xn)
or ~ ( y=)O+a: v ( x l ) + a:
v(x,)+... + a:
~(x,)
where:
.V( .V(
V( x,) is the variance of xI x2) is the variance of x2 x,) is the variance of x,
The above formula can be used to deduce the standard deviation in the two following cases: One random variable
The fkctiony is simply: y=ax Applying the variance theorem:
420
v(y)= a2 v ( x ) This can also be written:
=a
CT.~
( T ~
The error of the variable x is the error of y multiplied by the coefficient a.
Error of the mean If xis the mean of n values of xi 1 x=-[x, n
+...+X,]
+x2
The variance theorem allows us to write: 1 v, = -[v(
XI)
n2
+ v ( x 2 )+ ... + v(Xn)]
and if all the variances of xi are equal: 2 V(XJ = v ( x 2 ) = ... = v ( x , ) = 0,
V,
= -[1no:]
n2
2 6F
2 Ox
=-
n
The error of the mean is the error of one measurement divided by the square root of n. This is the formula that we use most frequently. It allows us to calculate the error of an effect (or interaction) from the known error of the responses.
APPENDIX 3
ORDER OF TRIALS THAT LEAVES THE EFFECTS OF THE MAIN FACTORS UNINFLUENCED BY LINEAR DRIFT: A P P L I C A T I O N T O A 23 D E S I G N
In Chapter 9 we indicated that the effects of the main factors could be determined even when there was a linear drift in the response. This drift could be due to ageing, equipment wear, tubing clogging, etc. The drift can be overcome by organizing the trials in a specific sequence. The sequences for 23 designs are given in this Appendix. We have adopted the following conventions : 1. Trials are numbered by the standard numbering, as indicated below
TABLEA3.1 Standard numbering of trials
Trial
Factor
Factor
no
1
2
-
-
+ -
+
+
+
-
-
+ -
+
+ +
71 Factor
422
2. The order of trials is indicated by a set of 8 numbers, e.g.,
1 4 6 7 8 5 3 2
indicating that trial number 1 is run first, followed by trial 4, etc. 3 . The sets of 8 trials are also numbered from 1 to 144 to identify them. 4. If the increment in drift is h, the effect or interaction may have a systematic error of -2 h -1h -0.5h 0 0.5h I h 2h. The increment h is omitted from the Table A3.2 "Anti drift designs".
5. The influence of drift is shown for the three main factors and the four interactions. It can be seen that there are always three interactions corrupted by drift.
423
Sequence
Order of trials in
number
each
Influence of drift on Main effects
sequence 1 2 3 4 5 6 7 8 9 10 11
12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27
' 1
28
1
29 30
1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2
4 4 4 4 6 6 6 6 7 7 7 7 8 8 8 8 8 8 3 3 3 3 5 5 5 5 7 7 7 7
6 7 8 8 4 7 8 8 4 6 8 8 4 4 6 6 7 7 5 7 7 8 3 7 7 8 3 3 5 5
7 6 5 5 7 4 3 3 6 4 2 2 5 5 3 3 2 2 8 6 6 5 8 4 4 3 6 6 4 4
8 8 6 7 8 8 4 7 8 8 4 6 6 7 4 7 4 6 7 5 8 7 7 3 8 7 5 8 3 8
5 5 7 6 3 3 7 4 2 2 6 4 3 2 5 2 5 3 6 8 5 6 4 8 3 4 4 1 6 1
3 2 3 2 5 2 5 2 5 3 5 3 7 6 7 4 6 4 4 4 1 1 6 6 1 1 8 5 8 3
2 3 2 3 2 5 2 5 3 5 3 5 2 3 2 5 3 5 1 1 4 4 1 1 6 6 1 4 1 6
Interactions
1
2
3
12
13
23
123
0
0
0
-1
-0.5
0
0
0
-1
0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-2 -2 -0.5 0 -0.5 0 -0.5 0 -0.5 0 -2 -2 -1 0
2 2 1 1 2 2 1 1 2 2 1
0 0
0 0
2 2 1 0.5 0.5
0 0 0 0 0 0
0 0 0 0 0 0
0 0 2 2 1 0
-0.5 0 -1 -1 -2 -2 0 -0.5 0 -0.5 -1 0 -2 -2 0 -1 0.5 0.5 0 0 1 2 2 1
0 -0.5 0 -0.5 0 -0.5 0 -0.5
0 0 0 0
0 0 0 0 0 0 0 0 0
-1 0 1
1 0 2 2
-1 -1 -2 -2 0 -1 0 -1 -2 -2 0
0 -0.5 -0.5 0 0 -0.5 -0.5 0 -1 0 -1
1 0.5 0.5 0.5 0.5 0.5 0.5 -2 -1 -1 -2 -2 -1 -1 -2 -0.5 -0.5 -0.5 -0.5
424
Sequence
Order of trials in
number
each
Influence of drif? on
-
sequence 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4
7 7 8 8 8 8 2 2 2 2 5 5 5 5 6 6 6 6 6 6 8 8 8 8 1 1 1 1 5 5
8 8 3 5 7 7 5 6 6 8 2 6 6 8 2 2 5 5 8 8 2 5 6 6 5 5 6 7 1 1
1 1 5 3 1 1 8 7 7 5 8 4 4 2 7 7 4 4 1 1 5 2 1 1 8 8 7 6 8 8
3 5 7 7 3 5 6 5 8 6 6 2 8 6 5 8 2 8 2 5 6 6 2 5 6 7 5 5 6 7
6 4 1 1 5 3 7 8 5 7 4 8 2 4 4 1 7 1 7 4 1 1 5 2 7 6 8 8 3 2
5 3 6 4 6 4 4 4 1 1 7 7 1 1 8 5 8 2 5 2 7 4 7 4 3 2 3 2 7 6
4 6 4 6 4 6 1 1 4 4 1 1 7 7 1 4 1 7 4 7 4 7 4 7 2 3 2 3 2 3
Main effects
Interactions
1
2
3
12
13
23
123
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0.5 0 0.5 0 1 2 2 1 0.5 0.5 0 0 2 2 1 0 1 0 0.5 0 0.5 0 -2 -2
0 1 0 0.5 0 0.5 0 0 -0.5 -0.5 0 0 -0.5 -0.5 0 -1 0 -1 -2 -2 -1 -1 -2 -2 0 0.5 0 0.5 0 1
-2 -2
-0.5 -0.5
-1
-2
-1 -2 -2 0.5 0.5 0 0 1 2 2 1 1 0 2 2 0
-2
-1 -1 -2 -2
I 0 0.5 0 0.5 0.5 0 0.5 0
1 0
-1 -1 -2 -1 -1 -2 -2 -1 -1 -2 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -2 -2 -1 -1 1
1 2 2 0.5 0.5
425
Sequence
Order of trials in
number
each
Influence of drift on Main effects
sequence 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90
4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
5 5 5 5 6 6 6 6 7 7 7 7 2 2 2 2 3 3 3 3 4 4 4 4 4 4 8 8 8 8
6 6 7 7 1 5 5 7 1 5 5 6 3 4 4 8 2 4 4 8 2 2 3 3 8 8 2 3 4 4
3 3 2 2 7 3 3 1 6 2 2 1 8 7 7 3 8 6 6 2 7 7 6 6 1 1 3 2 1 1
1 7 1 6 5 1 7 5 5 1 6 5 4 3 8 4 4 2 8 4 3 8 2 8 2 3 4 4 2 3
8 2 8 3 3 7 1 3 2 6 1 2 7 8 3 7 6 8 2 6 6 1 7 1 7 6 1 1 3 2
7 1 6 1 8 8 2 2 8 8 3 3 6 6 1 1 7 7 1 1 8 3 8 2 3 2 7 6 7 6
2 8 3 8 2 2 8 8 3 3 8 8 1 1 6 6 1 1 7 7 1 6 1 7 6 7 6 7 6 7
1
2
3
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0
0
0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
I
I
Interactions 12
13
23
123
-1
0 1 2 2 0 0 0.5 0.5 1 2 2 1
2 2 0 1 1 2 2 1 0 0 0.5 0.5 0.5 0.5 0 0 1 2 2 1 1 0 2 2 0
0.5
0 -1
0 -0.5 -0.5 0 0 -0.5 -0.5 0 0 0
0 -0.5 -0.5 0 0 -0.5 -0.5 0 -1 0
-1 -2 -2 -1 -1 -2 -2
1
2 2 1
0.5 0.5 0 0 2 2 1 0 1 0
1
0.50 0 0.50 0
0 0.5 0 0.5
0.5 0.5 0.5 2 1 1 2 2 1 1 2 -2 -1 -1 -2 -2 -1 -1 -2
-0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -2
-2 -1 -1
426
Influence of drift on
Sequence
Order of trials in
number
each
1
Main effects
I
sequence
I
1
2
3
I
0
0
0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0
0 0
0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0 0
0
0
0 0
0 0
91 92 93 94 95 6 97 98 99 100 101
102 103 104 105 106 107 I08 109 110 111 112 113 114 115 116 117 118 119 120
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7
1 1 1 1 3 3 3 3 3 3 4 4 4 4 7 7 7 7 1 1 1 1 2 2 2 2 2 2 4 4
3 3 4 7 1 1 4 4 7 7 1 3 3 7 1 3 3 4 2 2 4 6 1 1 4 4 6 6 1 2
8 8 7 4 8 8 5 5 2 2 7 5 5 1 4 2 2 1 8 8 6 4 8 8 5 5 3 3 6 5
4 7 3 3 4 7 1 7 1 4 3 1 7 3 3 1 4 3 4 6 2 2 4 6 1 6 1 4 2 1
7 4 8 8 5 2 8 2 8 5 5 7 1 5 2 4 1 2 6 4 8 8 5 3 8 3 8 5 5 6
5 2 5 2 7 4 7 1 4 1 8 8 2 2 8 8 5 5 5 3 5 3 6 4 6 1 4 1 8 8
2 5 2 5 2 5 2 8 5 8 2 2 8 8 5 5 8 8 3 5 3 5 3 5 3 8 5 8 3 3
~0
0 0 0 0 0
Interactions 12
13
23
123
0 05
-2 -2 -1 -1 -2 -2 -1 0 -1 0 -0 5 -05 0
05
1 1 2 2 05 05 05 05 05 05 2 1 1 2 2 1 1 2 1
0
05 0 1
0 1 2 2 0 0 05 05 1
2 2 1
0 05 0 05 0 1 0
0
-05 -05 0 0 05 0 05 0 1 0
0 05 0 1 0 2 2 0 1
1 2 2 1 0 0 05 05 -2 -2 -1 -1 -2 -2 -1
1
2 2
2
0
0
2
1
0
0 0
0 0
1
-05 -05
2
0 -1
1 2 2 05 05 05 05 05 05 2 1
427
TABLE A 3.2 "ANTI-DRIFT DESIGNS" SET OF TRIALS ELIMINATING DRIFT FROM MAIN EFFECTS
Sequence
Order of trials in
number
each
Influence of drift on Main effects
sequence 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144
7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8
4 4 6 6 6 6 1 1 1 1 1 1 2 2 2 2 3 3 3 3 5 5 5 5
2 6 1 2 2 4 2 2 3 3 5 5 1 1 3 5 1 1 2 5 1 1 2 3
5 1 4 3 3 1 7 7 6 6 4 4 7 7 5 3 6 6 5 2 4 4 3 2
6 2 2 1 4 2 3 5 2 5 2 3 3 5 1 1 2 5 1 1 2 3 1 1
1 5 3 4 1 3 6 4 7 4 7 6 5 3 7 7 5 2 6 6 3 2 4 4
3 3 8 8 5 5 5 3 5 2 3 2 6 4 6 4 7 4 7 4 7 6 7 6
8 8 5 5 8 8 4 6 4 7 6 7 4 6 4 6 4 7 4 7 6 7 6 7
1
2
3
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0
I I
Interactions 12
13
23
123
0.5 0.5 1 2 2 1 0 -1 0 -1 -2 -2 0 -0.5 0 -0.5 0 -0.5 0 -0.5 -2 -2 -1 -1
2 1 0 0 0.5 0.5 -1 0 -2 -2 0 -1 -0.5 0 -0.5 0 -2 -2 -1 -1 0 -0.5 0 -0.5
0 0 -0.5 -0.5 0 0 -2 -2 -1 0 -1 0 -2 -2 -1 -1 -0.5 0 -0.5 0 -0.5 0 -0.5
1 2 2 1 1 2 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -1 -1 -2 -2 -1 -1 -2 -2 -1 -1 -2 -2
0
428
These sequences or sets of trials may be used in several cases To overcome drift 0
To detect drift If there is no other cause of error, the interactions are like numbers (absolute values). 0 1 2 4
0
To not estimate experimental error from interactions if drift is suspected
To construct a 24-' design One of the interactions that is not corrupted by drift may be selected to alias the fourth main factor. Interaction 123 must not be automatically chosen. For blocking and eliminating drift from a 24 design. Both could be achieved at the same time for the main factors
. .
for difference in the means of two series of eight trials (blocking) for a linear drift affecting the sixteen trials in the 24 design.
This is done using two mirror series of eight trials, e.g. and
6 4 7 1 3 5 2 8 8 2 5 3 1 7 4 6
The following example shows this technique (Table A3.3)
429
TABLEA 3.3 "Experimental design allowing blocking and elimination of drift on the main factors." 'Trial no
Factor
Factor
3
4
6
-
4 7 1 3
-
5
2 8
+ + + + -
8
+
2
-
5
1
+ +
7
-
4 6
-
3
+
Blocking on 134 -
-
+ + + + + + + +
This Page Intentionally Left Blank
43 1
BIBLIOGRAPHY
ANDERSON Virgil L. and McLEAN Robert A. "Design of Experiments: a Realistic Approach" Statistics: Textbooks and Monographs Marcel Dekker. Vol nos. New-York ANDERSON Virgil L. and McLEAN Robert A. "Applied Factorial and Fractional Designs" Statistics: Textbooks and Monographs Marcel Dekker. Vol n"55. New-York BELZ Maurice H. "Statistical Methods in the Process Industries" John Wiley and Sons, New-York. 1973. BENOIST Daniel. "Notions sur les Plans d'Experiences" Societe des editions Technip . Paris. 1974 BLAQUIERE Austin. "Calcul matriciel" Tome 1 et 2 . Hachette , Paris. 1960. BOX Ge0rge.E.P and HUNTER William G . "The Experimental Study of Physical Mechanisms" Technometrics.February 1965. vol. 7, no 1.
BOX George.E.P., DRAPER N.R. "Evolutionary Operation. A Statistical Method for Process Improvement" John Wiley and Sons, New-York. 1969.
BOX Ge0rge.E.P ,HUNTER William G. , HUNTER J.Stuart "Statistics for Experimenters" John Wiley and Sons, New-York. 1971. BOX George E.P. "Studies in quality improvement: Signal to noise ratios, performance criterion and statistical analysis". Part I. Report n"l1 Centre for quality and productivity improvement University of Wisconsin-Madison. March 1986.
432
10
BOX George E.P. and RAMIREZ J. "Studies in quality improvement" Part 1I.Report n"12. Centre for quality and productivity improvement University of Wisconsin-Madison. March 1986.
11
BOX George E.P., DRAPER Norman R. "Empirical Model-Building and Response Surface" John Wiley and Sons, New-York. 1987.
12
BOX George E.P. "Signal-to-noise Ratios, Performance Criteria, and Transformations" Technomefrics.February 1988. Vol. 30,no 1.
13
CARLSON Rolf "Design and Optimisation in organic synthesis" Elsevier. Amsterdam. 1992. p.536.
14
CASTERA A., LACOSTE F. and LESPAGNE. "Optimisation du dosage du fer par atomisation electrochimique dans les corps gras raffines. Utilisation de la theorie des plans &experiences" Analusis, 1992. V01.20,p. 19-28.
15
CAULCUTT R. and BODDY R. "Statistics for analytical chemists" Ed,: Chapman and Hall. London. 1983
16
CHAPOUnLE "Planificationet analyse des experiences" Masson. Paris. 1973.
17
COCHRAN W.G.. COX G . , "Experimental design" John Wiley and Sons. 1966.
18
DANIEL Cuthbert and WOOD Fred S. "Fitting Equations to Data" John Wiley and Sons. 1971.
19
DANIEL Cuthbert "Applications of Statistics to Industrial experimentation" John Wiley and Sons. 1976.
20
DAVIES Owen L. "The Design and Analysis of Industrial Experiments" Oliver and Boyd. 1971
433
21
DEMING Stanley N. and MORGAN Stephen L. "Experimentaldesign: a chemometric approach" Elsevier. 1987.
22
DEMING Stanley N. "Quality by Design" Part 2. Chemtech, January 198. p. 52-58.
23
DEMING Stanley N. "Quality by Design" Part 4. Chemtech, August 1989. p. 504-51 1.
24
DEMING Stanley N. "Quality by Design" Part 5. Chemtech, August 1990. p. 504-5 1 1.
25
DEMTNG Stanley N., PALASOTA John A and PALASOTA Josephine M '*Experimentaldesign in chemometrics" J. Clhemometrics.5, 181-192. 1991.
26
DONEV A.N. and ATKINSON A. C. "An Adjustment Algorithm for the Construction of Exact D-Optimal Experimental Design" Technometrics. November 1988. Vol. 30, no 4.
27
DRAPER Norman R., GUTWAN Irwin. "AnIndex of Rotatability" Technometrics. February 1988. Vol. 30, no 1
28
DRAPER Norman R. and SANDERS Elisabeth R. "Designs for Minimum Bias Estimation" Technometric, august 1988. Vol. 30, no 3 .
29
DRAPER Norman R. and JOHN J. A. "Responses Surfaces Designs for Quantitative and Qualitative Variables" Technometric, November 1988. Vol. 30,no 4.
30
DUGUE D. et GIRAULT M. "Plans d'Experience" Dunod. Paris. 1969.
31
50 - FEDOROV V. V. "Theory of Optimal Experiments" Academic Press. New-York. 1972
434
32
FEINBERG M. et WIRTH P. "Aperp des methodes d'optimisation en chimie analytique " Analusis, Vol 12, n"10. 1984.
33
FEINBERG Max. "Organiser rationnellement ies essais... une necessite pour optimiser la preparation d'echantillons " Analusis, 1992, V01.20, nO1,p. M23-M25.
34
FISHER Ronald A. "On the mathematical foundation of theoretical statistics" Phil. Trans. Roy.Soc. 222A: 322. 1922.
35
FISHER Ronald A. "Statistical Methods for Research Workers" Oliver and Boyd. 1925.
36
FISHER Ronald A. "The design of Experiments" Oliver and Boyd. 1935.
37
GOWY Jacques. "La Methode des Plans d'Experiences" Dunod. Pans. 1988.
38
GOUPY Jacques. "Erreur de derive et choix de I'ordre des essais d'un plan d'experiences factoriel" Revue de statistique appliquee, Vol. 37( 1),5-22. 1989.
39
GOUPY Jacques. "Etude comparative de divers plans d'experiences" Revue de statistique appliquee, Vol. 38(4),5-44. 1990.
40
GOUPY J., SJ0GREN C. E and SEIP K. L. "Determination de l'azote organique dans les produits petroliers lourds par chimiluminescence 'I Analusis, 1990, Vo1.18, p. 524-530.
41
GOUPY Jacques, MURRAY Matthew. "Reduire les coiits de la simulation informatique grPce aux plans d'experiences: un exemple en calcul de procede" Revue IFP 37(1),5-22. 1991.
42
GOUPY Jacques. "Plans d'experiences" Techniques de l'ingenieur. Traite Analyse Chimique et Caracterisation, P230,p. 1-20, 1992
435 43
GRAYBILL Franklin A. "Anintroduction to linear statistical models" Vol.nol..Mc Graw-Hilll. 1961.
44
HAALAND Perry D. "ExperimentalDesign in Biotechnology" Statistics: Textbooks and Monographs Marcel Dekker Vol nO105. New-York
45
HADJacques. "Resolution d'une question relative aux determinants Bulletin des Sciences Mathematiques. Part 1, 17, 1893, p. 240-246
46
HAHN Gerald J. and MORGAN Carolyn B. "Design Experiments with your Computer" Chemtech, November 1988, p. 664-669
47
HALD A. "Statistical Theory with Engineering Applications" John Wiley and Sons. New-York. 1952
48
HIMMELBLAU David M. "Process Analysis by Statistical Methods" John Wiley and Sons. New-York. 1970.
49
HOTELLING Harold. "Relations between two sets of variates" Biometrika. 28, 221-277. 1936.
50
HOTELLING Harold "Some problems in weighing and other experimentaltechniques" Am. Math. Stat, 15, 297-306-4.
51
JXINTER Stuart J. "Statistical Design Applied to Product Design" Journal of Quality Technology,Vol. 17, n04. 1985.
52
JENKINS Mary W., MOCELLA Michael T., ALLEN Kenneth D. and SAWM Herbert H. "The Modelling of Plasma Etching Process Using Response Surface Methodology." Solid State Technology. April 1986. Vo1.56, nos, p. 175-182.
53
JOHN J.A.and QUENOUILLE. "Experiments:Design and Analysis " Charles Griffin and Company Limited. London. 1977.
43 6
54
JOLY A.M., LAOUT J.C.and PSYCHOYOS N. "Analyse des donnees et plans &experiences: Application a la formulation des peinture anti corrosion I' L 'actualite chimique. juin-$let 1988. p. 20 1-207.
55
KACKAR Raghu N. and SHOEMAKER Anne C. "Robust design: a cost effective method for improving manufacturing process" A.T.h T. TechnicalJournal.Vo1. 65, no 2, p.39-50. 1986.
56
KEmTHoRNE Oscar. "Design and Analysis of Experiments" Robert E. Krieger Huntington, New-York. 1975.
57
KIEFER J. "Optimum Experimental Designs" Journal of the Royal Statistical Socieq. Ser. B, Vol. 21.1959
58
LOGOTHETIS N. "Off-line Quality Control with initial Exploration of Data" GEC Journal of Research, vol5, no 1 . 1987.
59
MEAD Roger "The Design of Experiments" Statistical Principles for Practical application. Cambridge University Press. Cambridge. 1988. p 497
60
MILLER J.C. and MILLER J.N. "Statistics for analytical chemistry" Ellis Horwood limited. Chischester. 1988
61
MITCHELL Toby J. "An algorithm for Construction of D-Optimal Experimental Designs" Technometrics, vol. 16, n02. 1974.
62
MONTGOMERY Douglas C. "Design and Analysis of Experiments" John Wiley and Sons, New-York. 1984.
63
MORGAN Stanley N.,BURTON Kenneth W. and CHURCH Paul A. "Practical Exploratory Experimental Designs" Elsevier. Chemometrics and Intelligent Laboratory Systems Vol. 5. 1989. p.283-302. MOZZO G. "Plan quadratique Gigogne." Revue de statistique appliquee,Vol. 38(3),23-34. 1990.
64
437 65
MYERS Raymond H., KHURI Andr, I. and CARTER Walter H. "Response Surface Methodology: 1966-1988." Technometrics, May 1989. Vol. 31, no 2.
66
NACHTSHEIM Christopher J. "Tools for Computer-aided Design of Experiments" Journal of Quality Technology. Vol.l9,n03.July 1987.
67
NORME AFNOR X 06-080 de novembre 1989. "Application de la statistique. Plan #Experiences. Vocabulaire et indications generales"
68
PHADKE M. S., KACKAR R.N.,SPEENEY D.V. and GRIECO M.J. "Off-line quality control in integrated circuit fabrication using experimental design" The Bell System TechnicalJournul. Vol62, n"5. May-June 1983. p. 1273-1309.
69
PIGNATELLO Joseph J. and RAMBERG John S. "Discussion on Kackar's paper" Journal of Quality Technology. Vo1.17,no4.October 1985. p. 198-209
70
PILLET Maurice. "Introduction aux plans d'experiences par la methode Taguchi" Les Editions d'organisution. Paris. 1992.
71
PLACKETT R. L. and BURMAN J.P. "The design of optimum multifactorial experiments". Biometrika, n033, 1946
72
RAGHAVARAO Damaraju Tonstructions and Combinatorial Problem in Design of Experiments" Dover Publications Inc. New-York. 1988.
73
RAO Radhakrishna C. "Linear Statistical Inference and its Application" John Wiley,New-York. 1965.
74
RATZLAFF K.L. "Introduction to computer assisted experimentation" John Wiley, New-York.. 1987.
75
SADO Gilles et GOUPY Jacques L. "La methodologie des plans d'experiences appliquee a l'optimisation du reglage d'un spectrofluorimetre" Analusis, 1986, Vo1.14, n"8, p. 389-400
76
SADO Gilles et SADO Marie-Christine. "Les plans d'experiences. De l'experimentation a l'assurance qualit? *or technique, Paris. 1991.
43 8
77
SCHEFFE H. "General Theory of Evaluation of several sets of Constants and several Sources of Variability" Chem.Eng. Prog., 50 no 4, p.200-205.1954
78
SCHEFFE Henri. "The Analysis of variance" John Wiley,New-York. 1959
79
SCHIMMERLING P. "Comment accroitre la duree de vie des soufflets de transmission avec les plans d'experiences numeriques et la methodologie Taguchi". 1989.
80
SEARLE Shay1e.R. "Linear Model" John Wiley, New-York. 1971
81
SEARLE Shay1e.R. "Matrix Algebra useful for Statistics" John Wiley, New-York. 1982.
82
SILVEY Samuel David "Optimal Design" Chapman and Hall, London. 1980
83
SNEDECOR G. and COCHRAN W. "Statistical Method" Iowa State University Press,Ames,Iowa. 1980.
84
STUDENT. "The probable error of a mean". Biometrika. 6 . 1-25. 1908.
85
"Statistique appliquee a I'exploitation des mesures" Tome 1 et 2, directed by C.E.A. (France). Masson .Paris. 1978.
86
TAGUCHI Genichi and WU Yuin "Introduction to off-line quality control I' Central Japan Quality Control Association 4- 10-27 Meieki Nakamura-ku Nagaya, Japan. February 1985
87
TAGUCHI Genichi. "How Japan defines Quality " Design BNews/7-8-85/99
439
88
TAGUCHI Genichi. "Quality Engineering in Japan" %zrlI.JapanSoc.of PrecEng., Vo1.19, n"4. 1985.
89
VPJDA S. "Patterns and Configurations in Finite Spaces" Griffin's Statistical monographs and Courses ,n022 Charles Griffin and Company Limited. London. 1967.
90
V.4JDA S. "The Mathematics of Experimental Design" Griffin's Statistical monographs and Courses n"23 Charles Griffin and Company Limited. London. 1967
91
WGIER Michel G. "Methodes d'assurance qualite-fiabilite et d'experimentation" Editions Maloine. Paris. 1981,
92
VIGER Michel G. "Pratique des plans d'experiences. Methodologie TAGUCHI" Les Editions d'organisation. Paris. 1988.
93
WILLIAMS E.J. "Regression Analysis" John Wiley and Sons, New-York. 1967
94
YATES F. "The Design and Analysis of Factorial Experiments" Bulletin 35, Imperial Bureau of Soil Science, Harpenden,Herts,England.Hafher(Macmillan). 1937.
95
YOUDEN W.J. "Statisticals Methods for Chemists" John Wiley and Sons, New-York. 195 1
440
AUTHOR INDEX
GOUPY Jacques. "La Methode des Plans d'Experiences" DUNOD. Paris. 1988. 303 p. FISHER Ronald A. I'Statistical Methods for Research Workers" Oliver and Boyd. 1925. 362 p. FISHER Ronald A. "The design of Experiments" Oliver and Boyd. 1935.248 p. PLACKETT R. L. and BURMAN J.P. "The design of optimum multifactorial experiments" Biometrika, no 33. 1946. p. 305-325. HOTELLING Harold. "Relations between two sets of variates" Biometrika. no 28. 1936. p.221-277. YOUDEN W.J. "Statisticals Methods for Chemists" John Wiley and Sons, New-York. 195 1. 126 p SCHEFFE Henri. "General Theory of Evaluation of several sets of Constants and several Sources of Variability" Chem.Eng. Prog., 50 no 4. 1954. p.200-205. BOX Ge0rge.E.P ,HUNTER William G. , HUNTER J.Stuart "Statistics for Experimenters" John Wiley and Sons. 1971. 653 p. BOX George.E.P., DRAPER N.R. "Evolutionaty Operation. A Statistical Method Improvement" John Wiley and Sons, New-York. 1969. 237 p.
for
Process 7
[lo] TAGUCHI Genichi and WU Yuin.
"Introduction to off-line quality control "Central Japan Quality Control Association 4-1 0-27 Meieki Nakamura-ku Nagaya, Japan. February 1985.
7
44 1
[I 11 DAWES Owen L. "The Design and Analysis of Industrial Experiments" Oliver and Boyd. 1971. 637 p. [12] CAULCUTT R. and BODDY R. "statistics for analytical chemists" Ed.:Chapman and Hall. London. 1983
38, 184
50
[ 131 MILLER J.C. and MILLER J.N.
"Statistics for analytical chemistry" lllis HORWOOD limited. Chischester. 1988. 202 p.
so
[14] SNEDECOR G. and COCHRAN W. "Statistical Method" Iowa State University Press,Ames,Iowa. 1980
50
[IS] IWLD A. "Statistical Theory with Engineering Applications" John Wiley and Sons, New-York. 1952.
so
[ 161 DANIEL Cuthbert.
"Applications of Statistics to Industrial experimentation" John Wiley and Sons. 1976. 294 p.
63
[ 171 HOTELLING Harold . "Some problems in weighing and other experimental techniques" Am. Math. Stat, 15,p. 297-306.
67
[ 181 FEDOROV V. V. "Theory of Optimal Experiments" Academic Press. New-York. 1972
80
[ 191 SAD0 Gilles et GOUPY Jacques L.
"La methodologie des plans d'experiences appliquee a l'optimisation du reglage d'un spectrofluorimetre" Analusis, Vo1.14, nos. 1986. p. 389-400. [20] W U P Y Jacques. "Erreur de derive et choix de l'ordre des essais d'un plan d'experiences factoriel" Revue de statistique appliquee,vol. 37( 1),5-22. 1989. [21] E'ATESF "The Design and Analysis of Factorial Experiments" Bulletin 35, Imperial Bureau of Soil Science, Harpenden,Herts,England.Hafher(Macmillan). 1 937
127 165
190
442
[22] TAGUCHI Genichi. "Quality Engineering in Japan" Bull.Japan Soc.of Prec.Eng., Vol. 19, n"4. 1985
27 1
[23] KACKAR Raghu N. and SHOEMAKER Anne C. "Robust design: a cost effective method for improving manufacturing process." A.T.& T. Technical Journal.Vo1. 65,n" 2, p.39 - SO. 1986.
272
[24] NACHTSHEIM Christopher J. "Tools for Computer-aided Design of Experiments" Journal of Quality Technology. Vol. 1931'3. July 1987
283
[25] PIGNATLELLO Joseph J. and RAMBERG John S. "Discussion on Kackar's paper" Journal of Quality Technology. Vol. 17,n04,p.198-209. October 1985.
284
[26] GOUPY Jacques, MURRAY Matthew. "Reduire les coiits de la simulation informatique gr5ce aux plans &experiences:un exemple en calcul de procede" Revue IFP 37(1),5-22. 1991.
335
[27] BLAQUIERE Austin "Calcul matriciel" Tome 1: 218 p., Tome 2: 240 p Hachette. Paris. 1960.
403
[28] SEARLE Shayle R. "Matrix Algebra Usehl for Statistics" John Wiley and Sons. 1982.438 p.
403
443
EXAMPLE INDEX 1. The yield of a chemical reaction Complete two-level factorial design 22 Effect definition Experimental matrix Computing of the effects Reduced centred variables Coded variables Presentation of the results Selection of a research direction Modelling Isoresponse curve Steepest ascent 2. The yield of a catalysed chemical reaction Complete two-level factorial design 22 Interaction definition Computing of the interactions Sign rule Effect computing matrix One point displaced Four points misplaced Coded variables Modelling Error transmission Experimental quality 3. Bitumen emulsion stability
Complete two-level factorial design 23 Continuous and discrete variables. Table of effects Box notation Reconstructing two design from a 23 design Construction of complete factorial design Labelling of trials in a complete factorial design Fractional design Z3-' Alias Interpretation of fractional designs Calculation of contrasts Algebra of columns of signs Alias generator
p. 10
Chapter 2 Chapter 2 Chapter 2 Chapter 2 Chapter 2 Chapter 2 Chapter 2 Chapter 11 Chapter 11 Chapter 11 Chapter 11
p. 10 p. 11, 13 14, 21 p. 12 p. 15 p. 16 p. 17 p. 19 p. 213 p. 215 p. 216 p. 217
p. 21
Chapter 2 Chapter 2 Chapter 2 Chapter 2 Chapter 2 Chapter 16 Chapter 16 Chapter 16 Chapter 16 Chapter 16 Chapter 16
p. 21 p. 21 p. 24 p. 25 p. 26 p. 366 p. 372 p. 373 p. 376 p. 377 p. 388
p. 29 Chapter 3 Chapter 3 Chapter 3 Chapter 3 Chapter 3 Chapter 3 Chapter 3 Chapter 6 Chapter 6 Chapter 6 Chapter 6 Chapter 6 Chapter 6
p. 29 p. 31 p. 32 p. 34 p. 35 p. 37 p. 38 p. 92 p. 93 p. 94 p. 95 p. 98 p. 99
444
Construction of fractional designs Notation of fractional designs 4. Penicillium chrysogenum growth medium
Complete two-level factorial design 2’ Reconstructing four 23 designs from a 2’design Blocking: 25 design in two blocks 5. Weighing designs. (two objects) Optimal design Experimental design and standard method 6. Weighing designs. (four objects)
Research strategy Optimality criteria Positioning experimental points 7. Measurement of an electrical resistance
Positioning experimental points for one factor 8. Minimizing the colour of a product
Fractional design 25-2 Techniques for desaliasing Complementary design 9. Settings of a spectrofluorimeter
Fractional design 274 Complementary design Interpreting with several responses 10. Plastic drums fabrication
Fractional design 284 Concept of resolution Three extra trials Four extra trials Reconstructing two Z3 designs from a 24design 1I . The powder mill
Complete two-level factorial design 23
Chapter 6 Chapter 6
p. 100 p. 104
p. 38 Chapter 3 Chapter 3 Chapter 10
P 38 p. 41 p. 185
p. 69
Chapter 5 Chapter 5
p. 69 p. 70
p. 71
Chapter 5 Chapter 5 Chapter 5
p. 71 p. 76 p. 80
p. 83
Chapter 5
p. 83
p. 116
Chapter 7 Chapter 7 Chapter 7
p. 116 p. 119 p. 121
p. 127
Chapter 7 Chapter 7 Chapter 7
p. 127 p. 134 p. 140
p. 144
Chapter 7 Chapter 7 Chapter 12 Chapter 12 Chapter 12
p. 144 p. 144 p. 246 p. 250 p. 252
p. 166
Chapter 9
p. 166
44s Drift Anti-drift designs Randomization Responses corrected from driR Randomization and drift 12. Preparation of a mixture
Complete two-level factorial design 2' Blocking Fractional design 2"' 13. Yates' bean experiment
Blocking: one 25 design in four blocks Complete two-level factorial design 25 Constructing four 23 designs from a Z5 design: Residual analysis 14. Study of paste hardening
Fractional design 24-' Mathematical modelling of factorial designs Isoresponse curve Test of the model 15. Sugar production
Complete two-level factorial design 22 Analysis of variance Non-duplicated design Duplicated design Error distribution 16. Clouding of a solution
Fractional design 24-' One extra trial Two extra trials 17. Photolithography
Fractional design 214 Domain of interest Interpreting with several responses 18. Cutting oil stability
Chapter 9 Chapter 9 Chapter 9 Chapter 9 Chapter 9
p. p. p. p. p.
166 166, 168, 171 167 172 175
p. 180
Chapter 10 Chapter 10 Chapter 10
p. 180 p. 180 p. 180
p. 190 10 10 10 11
p. 185 p. 190 p. 199 p. 230
11 11 11 11
p. 210 p. 211 p. 212 p. 213
Chapter 11 Chapter 11 Chapter 1 1 Chapter 11 Chapter 11
p. 220 p. 220 p. 220 p. 231 p. 236
Chapter Chapter Chapter Chapter p. 210
Chapter Chapter Chapter Chapter p. 220
p. 240
Chapter 12 Chapter 12 Chapter 12
p. 240 p. 240 p. 244
p. 258
Chapter 13 Chapter 13 Chapter 13 p. 268
p. 258 p. 257 p. 263
446
Complete two-level factorial design 22 Optimization Test of the model 19. Thickness of epitaxial deposits
Fractional design 284 Experimental design and quality Interpreting with several responses 20. Truck suspension spring leaves
Practical method of calculation Fractional design 24-' Fractional design 25-1 Experimental design and quality Interpreting with several responses Dispersion Decreasing experimental error 21. Propane remover optimizing
Complete two-level factorial design 2 3 Experimental designs and computer simulations Optimization 22. Strafluid optimization
Fractional design 2s-2 Experimental designs and computer simulations Optimization 23. Natural gas plant optimization
Complete two-level factorial design 2' Experimental designs and computer simulations Interpreting with several responses Optimization
Chapter 13 Chapter 13 Chapter 13
p. 268 p. 268 p. 270
p. 271
Chapter 13 Chapter 13 Chapter 13
p. 271 p. 271 p. 275
p. 284
Chapter 14 Chapter 14 Chapter 14 Chapter 14 Chapter 14 Chapter 14 Chapter 14
p. 283 p. 284 p. 284 p. 284 p. 287, 296 p. 292 p. 296
p. 335
Chapter 15 Chapter 15 Chapter 15
p. 335 p. 333 p. 337
p. 339
Chapter 15 Chapter 15 Chapter 15
p. 339 p. 339 p. 345
p. 347
Chapter 15 Chapter 15 Chapter 15 Chapter 15
p. 347 p. 347 p. 353 p. 361
447
SUBJECT INDEX Algebra of columns of signs Alias Alias generator Analysis of variance Anti-driR designs Arithmetic mean
p. 98 p. 93 p. 99 p. 220 p. 166, 168, 171,421 p. 51
Basic design Block error Blocking Blocking: one 25 design in four blocks Box notation
p. 101, 154 p. 161 p. 179, 180,428 p. 185 P 34
Calculation of contrasts Choosing the basic design Coded variables Complementary design Complementary trials Complete two-level factorial design 22 Complete two-level factorial design Z3 Complete two-level factorial design 2s Computing of the effects Computing of the interactions Concept of optimal design Concept of resolution Construction of fractional designs: one extra factor Construction of fractional designs: two extra factors Construction of fractional designs: p extra factors Construction of complete factorial design Continuous and discrete variables.
p. 95, 122, 130 p. 111 p. 17, 215, 373 p. 121, 134 p. 220,239 p. 10, 220, 268 p. 29, 166, 180, 335, 347 p. 38, 185 p. 15 p. 24 p. 144 p. 100 p. 104 p. 108 p. 37 p. 31
Decreasing experimental error Dispersion Domain of interest Drift Drift error Duplicated design
p. 296 p. 51, 292 p. 257 p. 166 p. 161, 162 p 231
Effect computing matrix Effect definition Effects matrix Effects matrix and mathematical matrix
p. 26 p. 11, 13 1 4 , 2 1 p. 26, 3 1, 153, 208 p. 44
448
Error Error distribution Error of the mean Error transmission Estimating error and significant effects Experimental design and standard method Experimental design and quality Experimental designs and computer simulations Experimental matrix Experimental quality Extra trials
p. 49, 52, 56, 161, 296, 377,420 p. 236,417 p. 420 p. 377 p. 49 p. 70 p. 271,284 p. 333,339,347 p. 12, 151 p. 388 p.240, 244, 246, 250
Fractional design
p. 116
Fractional design 2' Fractional design 24-1 Fractional design Z5-' Fractional design 25-2 Fractional design 214
p. 92
Fractional design 284
p. 144,271
Going from AGS to contrasts Going from contrasts to AGS Graphical representation of experimental designs
p. 110 p. 110
Interaction definition Interpretation of fractional designs Interpreting with several responses
p. 21 p. 94, 398 p. 140, 263, 275, 287,
Inverse matrix Isoresponse curve
296, 353 p. 46, 411 p. 212
Labelling of trials in a complete factorial design
p. 38
Mathematical modelling of factorial designs Matrix multiplication Matrix transposition Measurement units Modelling
p. 211 p. 45, 410
Non-duplicated design Notation of fractional designs
p. 220 p. 104
Optimal design Optimality criteria
p. 69 p. 76
p. 180,210,240, 284 p. 284 p. 116,339 p. 127,258
p. 36
p. 45, 407 p. 47 p. 21 1,376
449
Optimization Origin of the total error
p. 268,302,337,345,361 p. 53
Points misplaced Positioning experimental points for one factor Positioning experimental points for two factors Positioning experimental points for k factors Practical method of calculation Presentation of the results
p. 366, 372 p. 80 p. 85 p. 89 p. 283 p. 19, 63
Random error Random error of an effect Randomization Randomization and drift Reconstructing two 22 design from a 23-design Reconstructing two 23 designs from a 24design Reconstructing four 23 designs from a Z5 design: Reduced centred variables Residual analysis Resolution (concept of) Responses corrected from drift
p. 52 p. 56 p. 162, 166, 167 p. 175 p. 35 p. 252 p. 41, 199 p. 16 p. 230 p. 142, 144 p 172
Selection of a research direction Sign rule Steepest ascent Systematic error
p. 213 p. 25 P. p. 52
Table of effects Techniques for desaliasing Test of the model Two-level complete Factorial Design 2k Types of matrices
p. 32 p. 119, 134 p. 213,.270
Variance theorem
p. 419
p. 151, 158
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