Table Of ContentRecent Research in Psychology
Springer-Verlag
Geschattsbibliothek - Heidelberg
Patrick Bossuyt
A Comparison of
Probabilistic Unfolding Theories
for Paired Comparisons Data
Springer-Verlag
Berlin Heidelberg New York
London Paris Tokyo Hong Kong
Author
Patrick Bossuyt
Center for Clinical Decision Making
Medical Faculty, Erasmus University Rotterdam
P.O. Box 1738,3000 DR Rotterdam, The Netherlands
ISBN-13 :978-3-540-52491-5 e-ISBN-13:978-3-642-84172-9
DOl: 10.1007/978-3-642-84172-9
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Springer-Verlag Berlin Heidelberg 1990
2126/3140-543210 - Printed on acid-free paper
to Hannah
Preface
Some data-analytic methods excel by their sheer elegance. Their basic
principles seem to have a particular attraction, based on a intricate combination
of simplicity, deliberation, and power. They usually balance on the verge of
two disciplines, data-analysis and foundational measurement, or statistics and
psychology.
To me, unfolding has always been one of them. The theory and the original
methodology were created by Clyde Coombs (1912-1988) to describe and
analyze preferential choice data. The fundamental assumptions are truly psy
chological; Unfolding is based on the notion of a single peaked preference
function over a psychological similarity space, or, in an alternative but
equivalent expression, on the assumption of implicit comparisons with an ideal
alternative.
Unfolding has proved to be a very constructive data-analytic principle, and
a source of inspiration for many theories on choice behavior. Yet the number
of applications has not lived up to the acclaim the theory has received among
mathematical psychologists. One of the reasons is that it requires far more
consistency in human choice behavior than can be expected.
Several authors have tried to attenuate these requirements by turning the
deterministic unfolding theory into a probabilistic one. Since Coombs first put
forth a probabilistic version of his theory, a number of competing proposals
have been presented in the literature over the past thirty years.
This monograph contains a summary and a comparison of unfolding theories
for paired comparisons data, and an evaluation strategy designed to assess the
validity of these theories in empirical choice tasks.
Chapter 1 contains a classification of the existing probabilistic unfolding
theories in which a distinction is made between random configuration theories
and random response theories.
Chapter 2 presents an organized description of possible properties of pro
babilistic choice behavior. Among these are the familiar probabilistic versions
of the transitivity property, as well as properties tied to Coombs' definition of
the ideal alternative, and properties that apply in unidimensional unfolding
representations only. For each probabilistic unfolding theory reintroduced in
Chapter 1, and for each property, proofs are presented on whether or not the
theory expects that property to hold.
VIII
In Chapter 3, the kernel of the evaluation strategy is presented. To assess the
validity of a particular probabilistic unfolding theory, one should look at the
goodness-of-fit between models of data and models of critical properties of
probabilistic choice behavior.
The strategy itself is discussed in greater detail in Chapter 4. It is based on
regarding properties of probabilistic choice behavior as conditional ordinal
restraints in estimating choice probabilities, and uses isotonic regression, a
branch search strategy and the generalized likelihood ratio principle. The
technique is applied to some models of data presented in the literature.
Chapter 5 contains a report on a collection of new paired comparisons tasks
in which the evaluation strategy presented in the earlier chapters was used.
After completing my psychology study at the University of Ghent, a grant
from the Netherlands Foundation for Scientific Research NWO (1983-1987, no.
40-30) enabled me to undertake this study at the Mathematical Psychology
Group of the University of Nijmegen, presently integrated in the Nijmegen
Institute of Cognition Research and Information Technology NIel.
I would like to record my gratitude for Geert De Soete, who has put me on
the tracks, for Professor Edward Roskam, for his inspiring supervision, for
Math Candel, who has been an arduous beta-tester of PSTRIX and has pointed
out several flaws in earlier versions of the package and the manuscript. I am
indebted to Frans Gremmen and his colleagues from the Nijmegen GRO for
assistance with statistical and programming problems. I want to thank the
students that participated in the experiments and Guillaume Vuist for his
assistance in running them. Professor Thorn Bezembinder, Professor Ivo
Molenaar, and Professor Luc Delbeke read the manuscript and helped to
improve it.
The Nijmegen Mathematical Psychology Group and the Rotterdam Center
for Clinical Decision Making have been stimulating environments to work and
live in. To all colleagues, former and present, go my thanks.
Last but certainly not least, lowe a very special debt to all that helped me
without even mentioning "probabilistic unfolding", my family and friends in
Belgium and the Netherlands.
Rotterdam P.M.B.
February 1990
Contents
Preface vii
1 Approaches to probabilistic unfolding 1
1.1 An overview of probabilistic unfolding theories 3
1.1.1 Random configuration theories 4
1.1.2 Random response theories 8
1.2 Probabilistic unfolding models 9
1.2.1 Models of the Coombs-Zinnes-Griggs theory 10
1.2.2 Random distance models 15
1.2.3 Strong unfolding models 17
1.2.4 Midpoint unfolding models 21
1.2.5 Summary 22
2 Properties of probabilistic choice behavior 25
2.1 Properties 26
2.1.1 Stochastic transitivity 26
2.1.2 The ideal point 29
2.1.3 Unidimensional unfolding 36
2.2 Proofs 41
2.2.1 Stochastic transitivity 41
2.2.2 The ideal point 46
2.2.3 Unidimensional unfolding 51
x
3 Evaluating probabilistic unfolding theories 53
3.1 Testing probabilistic choice theories and models 54
3.1.1 Models of theory 55
3.1.2 Models of data 59
3.1.3 Theory versus data: evaluating goodness of fit 61
3.2 Sampling schemes in paired comparison tasks 65
3.2.1 Data collection designs based on specified subjects 65
3.2.2 Data collection designs based on sampled subjects 67
3.2.3 Conclusion 70
4 Evaluating properties of probabilistic choice behavior 73
4.1 Preliminaries 74
4.2 Ordinal restrictions and rankings 75
4.2.1 Ideal point conditions 76
4.2.2 Unidimensional unfolding 76
4.2.3 Stochastic transitivity 78
4.3 Maximum likelihood estimates of binomial probabilities
under ordinal restrictions 82
4.3.1 Basic principles 82
4.3.2 The algorithm 86
4.3.3 Gebhardt's algorithm 93
4.3.4 Application 95
4.4 A branch search strategy 98
4.4.1 The branch search principle 99
4.4.2 The algorithm 100
4.4.3 Implementation 101
4.4.4 Application 103
4.4.5 Extensions 106
4.5 Testing ordinal restrictions on binomial parameters 107
4.5.1 A generalized likelihood ratio test 107
4.5.2 A Monte Carlo approach 110
4.6 Examples 113
4.6.1 Stochastic acyclicity 114
4.6.2 Strong stochastic transitivity 116
4.6.3 Characteristic mono tonicity 118
XI
5 An experimental evaluation of
probabilistic unfolding theories 121
5.1 Experiment 1 121
5.1.1 Method 122
5.1.2 Results 123
5.1.3 Discussion 126
5.2 Experiment 2 128
5.2.1 Method 129
5.2.2 Results 132
5.2.3 Discussion 136
5.3 Conclusions 139
References 143
Appendix A - PSTRIX 151
Appendix B - Choice proportions 169
Appendix C - Distributions 175
List of Symbols 179
Author Index 181
Subject Index 183
1 Approaches to
probabilistic unfolding
The method of paired comparisons is a technique for the collection of data
used in a variety of fields such as acoustics, animal ecology, choice behavior,
dentistry, economics, epidemiology, food science, marketing, optics, personnel
testing, preference testing, psychometrics, sensory testing, sports, taste testing
and others (David, 1988). Alternatives are presented in pairs to one or more
subjects, who are asked to pick one of them. Making such a binary choice is
probably the simplest of all choice tasks. For this reason, this method is primarily
used in cases where judgements are necessarily subjective. The method is
essentially due to Fechner, and several parametric and nonparametric methods
for analyzing such data have been proposed ever since. Bradley (1976) and
David (1988) have summarized most of these methodological approaches.
Some forty years ago Clyde Coombs (1950, 1964) provided the analysis of
choice data with a powerful paradigm. Coombs assumed that subjects, when
making a choice, are actually comparing the available alternatives with an ideal
alternative, which is the alternative they actually want, actually need or actually
should choose, depending on the choice situation. Asked to choose from two
alternatives, neither of which may be the ideal, the subject will choose the
alternative least dissimilar from the ideal. Coombs conjectured that subjects
differ in the choices they make because they do not necessarily agree on the
ideal alternative, but that even subjects who disagree in choice will share the
same underlying cognitive pattern of dissimilarities between alternatives.
Assuming subjective differences in the definition of the ideal, but intersub
jectivity in the perception of alternatives, Coombs proposed his "unfolding"
theory. The unfolding theory is closely related to the data-representation
technique from which it received its name. With this technique, the latent
unfolding structure of dissimilarities between ideals and alternatives is repre
sented as a set of distances in a psychological space, and a smallest-space
representation is aimed at.
In the past decades, the unfolding paradigm has never been absent from the
literature on psychological choice, and the amount of methodological con
tributions based on this paradigm makes it impossible to review them here. The
opportunity in Coombs' original unfolding technique of mapping the
dissimilarities in a unidimensional space has been complemented with options
involving general Euclidean spaces (Bennett & Hays, 1960; Hays & Bennett,
