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Limited Information Estimation and Testing of Thurstonian Models for Paired Comparison Data Under Multiple Judgment Sampling

Published online by Cambridge University Press:  01 January 2025

Albert Maydeu-Olivares
Albert Maydeu-Olivares*
Affiliation:
University of Illinois at Urbana-Champaign
*
Requests for reprints should be sent to Albert Maydeu-Olivares, Faculty of Psychology, University of Barcelona, Passeig Vall d'Hebron, 171, 08035 Barcelona SPAIN. E-Mail: amaydeu@psi.ub.es
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Abstract

We relate Thurstonian models for paired comparisons data to Thurstonian models for ranking data, which assign zero probabilities to all intransitive patterns. We also propose an intermediate model for paired comparisons data that assigns nonzero probabilities to all transitive patterns and to some but not all intransitive patterns.

There is a close correspondence between the multidimensional normal ogive model employed in educational testing and Thurstone's model for paired comparisons data under multiple judgment sampling with minimal identification restrictions. Alike the normal ogive model, Thurstonian models have two formulations, a factor analytic and an IRT formulation. We use the factor analytic formulation to estimate this model from the first and second order marginals of the contingency table using estimators proposed by Muthén. We also propose a statistic to assess the fit of these models to the first and second order marginals of the contingency table. This is important, as a model may reproduce well the estimated thresholds and tetrachoric correlations, yet fail to reproduce the marginals of the contingency table if the assumption of multivariate normality is incorrect.

A simulation study is performed to investigate the performance of three alternative limited information estimators which differ in the procedure used in their final stage: unweighted least squares (ULS), diagonally weighted least squares (DWLS), and full weighted least squares (WLS). Both the ULS and DWLS show a good performance with medium size problems and small samples, with a slight better performance of the ULS estimator.

Keywords

UMDEWMDWMDGLS estimationLISRELcategorical data analysispreference dataMPLUSbinary datagoodness of fit

Information

Type
Articles
Information
Psychometrika , Volume 66 , Issue 2 , June 2001, pp. 209 - 227
Copyright
Copyright © 2001 The Psychometric Society

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Footnotes

This paper is based on the author's doctoral dissertation; Ulf Böckenholt, advisor. The final stages of this research took place while the author was at the Department of Statistics and Econometrics, Universidad Carlos III de Madrid. The author is indebted to Adolfo Hernández for stimulating discussions that helped improve this paper, and to Ulf Böckenholt and the Associate Editor for a number of helpfulsuggestions to a previous draft.

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