A Comparison of Probabilistic Unfolding Theories for Paired Comparisons Data

ISBN-13: 9783540524915 / Angielski / Miękka / 1990 / 186 str.

Patrick Bossuyt

Opis

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.

Spis treści

1 Approaches to probabilistic unfolding.- 1.1 An overview of probabilistic unfolding theories.- 1.1.1 Random configuration theories.- 1.1.2 Random response theories.- 1.2 Probabilistic unfolding models.- 1.2.1 Models of the Coombs-Zinnes-Griggs theory.- 1.2.2 Random distance models.- 1.2.3 Strong unfolding models.- 1.2.4 Midpoint unfolding models.- 1.2.5 Summary.- 2 Properties of probabilistic choice behavior.- 2.1 Properties.- 2.1.1 Stochastic transitivity.- 2.1.2 The ideal point.- 2.1.3 Unidimensional unfolding.- 2.2 Proofs.- 2.2.1 Stochastic transitivity.- 2.2.2 The ideal point.- 2.2.3 Unidimensional unfolding.- 3 Evaluating probabilistic unfolding theories.- 3.1 Testing probabilistic choice theories and models.- 3.1.1 Models of theory.- 3.1.2 Models of data.- 3.1.3 Theory versus data: evaluating goodness of fit.- 3.2 Sampling schemes in paired comparison tasks.- 3.2.1 Data collection designs based on specified subjects.- 3.2.2 Data collection designs based on sampled subjects.- 3.2.3 Conclusion.- 4 Evaluating properties of probabilistic choice behavior.- 4.1 Preliminaries.- 4.2 Ordinal restrictions and rankings.- 4.2.1 Ideal point conditions.- 4.2.2 Unidimensional unfolding.- 4.2.3 Stochastic transitivity.- 4.3 Maximum likelihood estimates of binomial probabilities under ordinal restrictions.- 4.3.1 Basic principles.- 4.3.2 The algorithm.- 4.3.3 Gebhardt’s algorithm.- 4.3.4 Application.- 4.4 A branch search strategy.- 4.4.1 The branch search principle.- 4.4.2 The algorithm.- 4.4.3 Implementation.- 4.4.4 Application.- 4.4.5 Extensions.- 4.5 Testing ordinal restrictions on binomial parameters.- 4.5.1 A generalized likelihood ratio test.- 4.5.2 A Monte Carlo approach.- 4.6 Examples.- 4.6.1 Stochastic acyclicity.- 4.6.2 Strong stochastic transitivity.- 4.6.3 Characteristic monotonicity.- 5 An experimental evaluation of probabilistic unfolding theories.- 5.1 Experiment 1.- 5.1.1 Method.- 5.1.2 Results.- 5.1.3 Discussion.- 5.2 Experiment 2.- 5.2.1 Method.- 5.2.2 Results.- 5.2.3 Discussion.- 5.3 Conclusions.- References.- Appendix A — PSTRIX.- Appendix B — Choice proportions.- Appendix C — Distributions.- List of Symbols.- Author Index.