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Thurstonian Modeling of Ranking Data Via Mean and Covariance Structure Analysis

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-Otivares, Faculty of Psychology, University of Barcelona, P. Vail d'Hebron, 171, 08035 Barcelona, SPAIN. E-mail: [email protected]
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Abstract

Although Thurstonian models provide an attractive representation of choice behavior, they have not been extensively used in ranking applications since only recently efficient estimation methods for these models have been developed. These, however, require the use of special-purpose estimation programs, which limits their applicability. Here we introduce a formulation of Thurstonian ranking models that turns an idiosyncratic estimation problem into an estimation problem involving mean and covariance structures with dichotomous indicators. Well-known standard solutions for the latter can be readily applied to this specific problem, and as a result any Thurstonian model for ranking data can be fitted using existing general purpose software for mean and covariance structure analysis. Although the most popular programs for covariance structure analysis (e.g., LISREL and EQS) cannot be presently used to estimate Thurstonian ranking models, other programs such as MECOSA already exist that can be straightforwardly used to estimate these models.

Keywords

GMM, GLS, WLS estimationUMD, ULS, EWMD estimationpermutation datarandom utility modelsstructural equations models
Type
Original Paper
Information
Psychometrika , Volume 64 , Issue 3 , September 1999 , pp. 325 - 340
Copyright
Copyright © 1999 The Psychometric Society

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Footnotes

This paper is based on the author's doctoral dissertation. Ulf Böckenholt was my advisor. The author is indebted to Ulf Böckenholt for his comments on a previous version of this paper and to Gerhard Arminger for his extensive support on the use of MECOSA. The final stages of this research took place while the author was at the Department of Statistics and Econometrics, Universidad Carlos III de Madrid. Conversations with my colleague there, Adolfo Hernández, helped to greatly improve this paper.

References

Arminger, G., Wittenberg, J., & Schepers, A. (1996). MECOSA 3. User guide, Friedrichsdorf: Additive GmbH.Google Scholar
Bekker, P.A., Merckens, A., & Wansbeek, T.J. (1994). Identification, equivalent models and computer algebra, San Diego: Academic Press.Google Scholar
Bentler, P.M. (1995). EQS Structural Equations Program Manual, Encino, CA: Multivariate Software.Google Scholar
Bock, R.D. (1975). Multivariate statistical methods in behavioral research, New York: McGraw Hill.Google Scholar
Bock, R.D., & Jones, L.V. (1968). The measurement and prediction of judgment and choice, San Francisco: Holden-Day.Google Scholar
Böckenholt, U. (1992). Thurstonian representation for partial ranking data. British Journal of Mathematical and Statistical Psychology, 45, 31–49.CrossRefGoogle Scholar
Böckenholt, U. (1993). Applications of Thurstonian models to ranking data. In Fligner, M.A., & Verducci, J.S. (Eds.), Probability models and statistical analyses for ranking data, New York: Springer-Verlag.Google Scholar
Brady, H.E. (1989). Factor and ideal point analysis for interpersonally incomparable data. Psychometrika, 54, 181–202.CrossRefGoogle Scholar
Chan, W., & Bentler, P.M. (1998). Covariance structure analysis of ordinal ipsative data. Psychometrika, 63, 369–399.CrossRefGoogle Scholar
Christoffersson, A. (1975). Factor analysis of dichotomized variables. Psychometrika, 40, 5–32.CrossRefGoogle Scholar
Clark, T.E. (1996). Small-sample properties of estimators of nonlinear models of covariance structure. Journal of Bussiness and Economic Statistics, 14, 367–373.CrossRefGoogle Scholar
Dansie, B.R. (1986). Normal order statistics as permutation probability models. Applied Statistics, 3, 269–275.CrossRefGoogle Scholar
Jöreskog, K.G. (1994). On the estimation of polychoric correlations and their asymptotic covariance matrix. Psychometrika, 59, 381–389.CrossRefGoogle Scholar
Jöreskog, K.G., & Sörbom, D. (1993). LISREL 8. User's reference guide, Chicago, IL: Scientific Software.Google Scholar
Küsters, U.L. (1987). Hierarchische Mittelwert- und Kovarianztrukturmodelle mit nichtmetrischen endogenen Variablen, Heidelberg: Physica-Verlag [Hierarchical mean and covariance structure models on nonmetric endogenous variables]CrossRefGoogle Scholar
Lee, S.Y., Poon, W.Y., & Bentler, P.M. (1995). A two-stage estimation of structural equation models with continuous and polytomous variables. British Journal of Mathematical and Statistical Psychology, 48, 339–358.CrossRefGoogle ScholarPubMed
Maydeu-Olivares, A. (1995, July). Structural equation modeling of paired comparisons and ranking data. Paper presented at the 9th European Meeting of the Psychometric Society. Leiden, The Netherlands.Google Scholar
Muthén, B. (1978). Contributions to factor analysis of dichotomous variables. Psychometrika, 43, 551–560.CrossRefGoogle Scholar
Muthén, B. (1982). Some categorical response models with continuous latent variables. In Jöreskog, K.G., & Wold, H. (Eds.), Systems under indirect observation, Amsterdam: North Holland.Google Scholar
Muthén, B. (1984). A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators. Psychometrika, 49, 115–132.CrossRefGoogle Scholar
Muthén, B. (1987). LISCOMP: Analysis of linear structural equations using a comprehensive measurement model, Mooresville, IN: Scientific Software.Google Scholar
Muthén, B. (1993). Goodness of fit with categorical and other non normal variables. In Bollen, K.A., & Long, J.S. (Eds.), Testing structural equation models, Newbury Park, CA: Sage.Google Scholar
Muthén, B., & Satorra, A. (1995). Technical aspects of Muthén's LISCOMP approach to estimation of latent variable relations with a comprehensive measurement model. Psychometrika, 60, 489–503.CrossRefGoogle Scholar
Rao, C.R. (1973). Linear statistical inference and its applications 2nd ed,, New York: Wiley.CrossRefGoogle Scholar
Satorra, A. (1989). Alternative test criteria in covariance structure analysis: A unified approach. Psychometrika, 54, 131–151.CrossRefGoogle Scholar
Satorra, A., & Bentler, P.M. (1988). Scaling corrections for chi-square statistics in covariance structure analysis. ASA 1988 Proceedings of the Business and Statistics section, 308–313.Google Scholar
Takane, Y. (1987). Analysis of covariance structures and probabilistic binary choice data. Communication and Cognition, 20, 45–62.Google Scholar
Takane, Y. (1989). Analysis of covariance structures and probabilistic binary choice data. In Soete, G., & Feger, H., Klauer, K.C. (Eds.), New developments in psychological choice modeling (pp. 139–160). Amsterdam: Elsevier Science.CrossRefGoogle Scholar
Takane, Y., & de Leeuw, J. (1987). On the relationship between item response theory and factor analysis of discretized variables. Psychometrika, 52, 393–408.CrossRefGoogle Scholar
Thurstone, L.L. (1927). A law of comparative judgment. Psychological Review, 79, 281–299.Google Scholar
Thurstone, L.L. (1931). Rank order as a psychological method. Journal of Experimental Psychology, 14, 187–201.CrossRefGoogle Scholar
Yao, K.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