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A Thurstonian Pairwise Choice Model with Univariate and Multivariate Spline Transformations

Published online by Cambridge University Press:  01 January 2025

Geert De Soete  and
Suzanne Winsberg
Geert De Soete*
Affiliation:
University of Ghent, Belgium
Suzanne Winsberg
Affiliation:
IRCAM, Paris, France
*
Requests for reprints should be sent to Geert De Soete, Department of Data Analysis, University of Ghent, Henri Dunantlaan 2, B-9000 Ghent, BELGIUM.
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Abstract

A probabilistic choice model is developed for paired comparisons data about psychophysical stimuli. The model is based on Thurstone's Law of Comparative Judgment Case V and assumes that each stimulus is measured on a small number of physical variables. The utility of a stimulus is related to its values on the physical variables either by means of an additive univariate spline model or by means of multivariate spline model. In the additive univariate spline model, a separate univariate spline transformation is estimated for each physical dimension and the utility of a stimulus is assumed to be an additive combination of these transformed values. In the multivariate spline model, the utility of a stimulus is assumed to be a general multivariate spline function in the physical variables. The use of B splines for estimating the transformation functions is discussed and it is shown how B splines can be generalized to the multivariate case by using as basis functions tensor products of the univariate basis functions. A maximum likelihood estimation procedure for the Thurstone Case V model with spline transformation is described and applied for illustrative purposes to various artificial and real data sets. Finally, the model is extended using a latent class approach to the case where there are unreplicated paired comparisons data from a relatively large number of subjects drawn from a heterogeneous population. An EM algorithm for estimating the parameters in this extended model is outlined and illustrated on some real data.

Keywords

paired comparisons dataThurstonian choice modelmultivariate splineslatent class analysisEM algorithmpsychophysical transformation
Type
Original Paper
Information
Psychometrika , Volume 58 , Issue 2 , June 1993 , pp. 233 - 256
Copyright
Copyright © 1993 The Psychometric Society

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Footnotes

The first author is supported as “Bevoegdverklaard Navorser” of the Belgian “Nationaal Fonds voor Wetenschappelijk Onderzoek”. The authors are indebted to Ulf Böckenholt and Yoshio Takane for useful comments on an earlier draft of this paper.

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