Hostname: page-component-745bb68f8f-5r2nc Total loading time: 0 Render date: 2025-01-08T16:50:18.706Z Has data issue: false hasContentIssue false
  • English
  • Français

Remarks on the Identifiability of Thurstonian Ranking Models: Case V, Case III, or Neither?

Published online by Cambridge University Press:  01 January 2025

Rung-Ching Tsai
Rung-Ching Tsai*
Affiliation:
University of Illinois, Champaign-Urbana
*
Requests for reprints should be sent to Rung-Ching Tsai, Department of Psychology, University of Illinois, 603 E. Daniel, Champaign, IL 61820. E-Mail: rtsai@s.psych.uiuc.edu.
Get access Rights & Permissions [Opens in a new window]

Abstract

It is well-known that the representations of the Thurstonian Case III and Case V models for paired comparison data are not unique. Similarly, when analyzing ranking data, other equivalent covariance structures can substitute for those given by Thurstone in these cases. That is, we may more broadly define the family of covariance structures satisfying Case III and Case V conditions. This paper introduces the notion of equivalence classes which defines a more meaningful partition of the covariance structures of the Thurstonian ranking models. In addition, the equivalence classes of Case V and Case III are completely characterized.

Keywords

Thurstonian modelsidentifiabilityranking data
Type
Original Paper
Information
Psychometrika , Volume 65 , Issue 2 , June 2000 , pp. 233 - 240
Copyright
Copyright © 2000 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 gratefully acknowledges the valuable feedback from Ulf Böckenholt and the Associate Editor. The author would also like to thank River Chiang and three reviewers for their helpful comments and suggestions.

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
Bock, R. D., Jones, L. V. (1968). The measurement and prediction of judgment and choice. San Francisco: Holden-Day.Google Scholar
Böckenholt, U. (1990). Multivariate Thurstonian models. Psychometrika, 55, 391–403.CrossRefGoogle Scholar
Böckenholt, U. (1992). Thurstonian representation for partial ranking data. British Journal of Mathematical and Statistical Psychology, 45, 31–49.CrossRefGoogle Scholar
Brady, H. (1989). Factor and ideal point analysis for interpersonally incomparable data. Psychometrika, 54, 181–202.CrossRefGoogle Scholar
Burros, R. H., Gibson, W. A. (1954). A solution for Case III of the law of comparative judgment. Psychometrika, 19, 57–64.CrossRefGoogle Scholar
Chan, W., Bentler, P. M. (1998). Covariance structure analysis of ordinal ipsative data. Psychometrika, 63, 369–399.CrossRefGoogle Scholar
Cohen, A., Mallows, C. L. (1983). Assessing goodness of fit of ranking models to data. The Statistician, 32, 361–373.CrossRefGoogle Scholar
Carroll, J. D. (1980). Models and methods for multidimensional analysis of preference choice (or other dominance) data. In Lantermann, E. D., Feger, H. (Eds.), Similarity and choice (pp. 234–289). Bern: Huber.Google Scholar
De Soete, G., Carroll, J. D. (1983). A maximum likelihood method for fitting the wandering vector model. Psychometrika, 48, 553–566.CrossRefGoogle Scholar
De Soete, G., Carroll, J. D., DeSarbo, W. S. (1986). The wandering ideal point model: A probabilistic multidimensional unfolding model for paired comparison data. Journal of Mathematical Psychology, 30, 28–41.CrossRefGoogle Scholar
Fligner, M. A., Verducci, J. S. (1986). Distance based ranking models. Journal of the Royal Statistical Association, Series B, 48, 359–369.CrossRefGoogle Scholar
Fligner, M. A., Verducci, J. S. (1988). Multistage ranking models. Journal of the American Statistical Association, 83, 892–901.CrossRefGoogle Scholar
Guttman, L. (1946). An approach for quantifying paired comparisons rank order. Annals of Mathematical Statistics, 17, 144–163.CrossRefGoogle Scholar
Hausman, J. A., Wise, D. A. (1978). A conditional probit model for qualitative choice: Discrete decision recognizing interdependence and heterogeneous preferences. Econometrica, 46, 403–426.CrossRefGoogle Scholar
Heiser, W., de Leeuw, J. (1981). Multidimensional mapping of preference data. Mathématiques et Sciences Humaines, 19, 39–96.Google Scholar
Iverson, G. J. (1987). Thurstonian psychophysics: Case III. Journal of Mathematical Psychology, 31, 219–247.CrossRefGoogle Scholar
Kamakura, W. A., Srivastava, R. K. (1984). Predicting choice shares under conditions of brand interdependence. Journal of Marketing Research, 21, 420–434.CrossRefGoogle Scholar
Luce, R. D. (1977). Thurstone's discriminal processes fifty years later. Psychometrika, 42, 461–490.CrossRefGoogle Scholar
MacKay, D. B., Chaiy, S. (1982). Parameter estimation for the Thurstone Case III model. Psychometrika, 47, 353–359.CrossRefGoogle Scholar
Marden, J. I. (1995). Analyzing and modeling rank data. London: Chapman & Hall.Google Scholar
Maydeu-Olivares, A. (1997). Structural equation modeling of binary preference data. Unpublished doctoral dissertation. University of Illinois at Urbana-Champaign.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
Takane, Y. (1980). Maximum likelihood estimation in the generalized cases of Thurstonian law of comparative judgment. Japanese Psychological Research, 22, 188–196.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: John Wiley & Sons.Google Scholar
Yao, K. G. (1995). Bayesian estimation of Thurstonian ranking models based on the Gibbs sampler. Unpublished doctoral dissertation. University of Illinois at Urbana-Champaign.Google Scholar