Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-01-08T08:56:14.633Z Has data issue: false hasContentIssue false
  • English
  • Français

Remarks on the Identifiability of Thurstonian Paired Comparison Models Under Multiple Judgment

Published online by Cambridge University Press:  01 January 2025

Rung-Ching Tsai
Rung-Ching Tsai*
Affiliation:
National Taiwan Normal University
*
Requests for reprints should be sent to Rung-Ching Tsai, Department of Mathematics, National Taiwan Normal University, No. 88, Sec. 4, Ting-Zhou Road, Taipei 116, Taiwan. E-Mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

It is well-known that the representations of the Thurstonian models for difference judgment data are not unique. It has been shown that equivalence classes can be formed to provide a more meaningful partition of the covariance structures of the Thurstonian ranking models. In this paper, we examine the equivalence relations between Thurstonian covariance structure models for paired comparison data obtained under multiple judgment and discuss their implications on the general identification constraints and methods to check for parameter identifiability in restricted models.

Keywords

Thurstonian paired comparison modelsidentifiabilitymultiple judgment
Type
Article
Information
Psychometrika , Volume 68 , Issue 3 , September 2003 , pp. 361 - 372
Copyright
Copyright © 2003 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The author is indebted to Ulf Böckenholt and Albert Maydeu-Olivares for their significant comments and suggestions which led to considerable improvement in this article.

References

Arbuckle, J., Nugent, J.H. (1973). A general procedure for parameter estimation for the law of comparative judgment. British Journal of Mathematical and Statistical Psychology, 26, 240–260.CrossRefGoogle Scholar
Bekker, P., Wansbeek, T. (2000). Matrix inequality application in econometrics. In Heijmans, R.D.H., Pollock, D.S.G., Satorra, A. (Eds.), Innovations in multivariate statistical analysis: A Festschrift for Heinz Neudecker (pp. 51–66). Dordrecht, The Netherlands: Kluwer Academia Publishers.CrossRefGoogle Scholar
Bock, R.D., Jones, L.V. (1968). The measurement and prediction of judgment and choice. San Francisco, CA: Holden-Day.Google Scholar
Böckenholt, U. (2001). Hierarchical modeling of paired comparison data. Psychological Methods, 6, 49–66.CrossRefGoogle ScholarPubMed
Böckenholt, U., Tsai, R. (2001). Individual differences in paired comparison data. British Journal of Mathematical and Statistical Psychology, 54, 265–277.CrossRefGoogle ScholarPubMed
Chan, W., Bentler, P.M. (1998). Covariance structure analysis of ordinal ipsative data. Psychometrika, 63, 369–399.CrossRefGoogle Scholar
Dansie, B.R. (1986). Normal order statistics as permutation probability models. Applied Statistics, 35, 269–275.CrossRefGoogle Scholar
Guttman, L. (1946). An approach for quantifying paired comparisons rank order. Annals of Mathematical Statistics, 17, 144–163.CrossRefGoogle Scholar
Harville, D.A. (1997). Matrix algebra from a statistician's perspective. New York, NY: Springer-Verlag.CrossRefGoogle Scholar
Maydeu-Olivares, A. (2001). Limited information estimation and testing of Thurstonian models for paired comparison data under multiple judgment sampling. Psychometrika, 66, 209–228.CrossRefGoogle Scholar
Maydeu-Olivares, A. (2003). Thurstonian covariance and correlation structure for multiple judgment paired comparison data. Madrid: Instituto de Empresa.Google Scholar
Mosteller, F. (1951). Remarks on the method of paired comparisons: I. The least squares solution assuming equal standard deviations and equal correlations. Psychometrika, 16, 3–11.CrossRefGoogle Scholar
Shuford, E.H., Jones, L.V., Bock, R.D. (1960). A rational origin obtained by the method of contingent paired comparisons. Psychometrika, 25, 343–356.CrossRefGoogle Scholar
Takane, Y. (1987). Analysis of covariance structures and probabilistic binary choice data. Cognition and Communication, 20, 45–62.Google Scholar
Thurstone, L.L. (1927). A law of comparative judgment. Psychological Review, 34, 273–286.CrossRefGoogle Scholar
Torgerson, W.S. (1958). Theory and methods of scaling. New York, NY: John Wiley & Sons.Google Scholar
Tsai, R. (2000). Remarks on the identifiability of Thurstonian ranking models: Case V, Case III, or neither?. Psychometrika, 65, 233–240.CrossRefGoogle Scholar
Tsai, R., Böckenholt, U. (2002). Two-level linear paired comparison models: Estimation and identifiability issues. Mathematical Social Sciences, 43, 429–449.CrossRefGoogle Scholar
Yao, G., Böckenholt, U. (1999). Bayesian estimation of Thurstonian ranking models based on the Gibbs sampler. British Journal of Mathematical and Statistical Psychology, 52, 79–92.CrossRefGoogle Scholar