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.
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References
Arminger, G., Wittenberg, J., & Schepers, A. (1996).MECOSA 3. User guide. Friedrichsdorf: Additive GmbH.
Bekker, P.A., Merckens, A., & Wansbeek, T.J. (1994).Identification, equivalent models and computer algebra. San Diego: Academic Press.
Bentler, P.M. (1995).EQS Structural Equations Program Manual. Encino, CA: Multivariate Software.
Bock, R.D. (1975).Multivariate statistical methods in behavioral research. New York: McGraw Hill.
Bock, R.D., & Jones, L.V. (1968).The measurement and prediction of judgment and choice. San Francisco: Holden-Day.
Böckenholt, U. (1992). Thurstonian representation for partial ranking data.British Journal of Mathematical and Statistical Psychology, 45, 31–49.
Böckenholt, U. (1993). Applications of Thurstonian models to ranking data. In M.A. Fligner & J.S. Verducci (Eds).Probability models and statistical analyses for ranking data. New York: Springer-Verlag.
Brady, H.E. (1989). Factor and ideal point analysis for interpersonally incomparable data.Psychometrika, 54, 181–202.
Chan, W., & Bentler, P.M. (1998). Covariance structure analysis of ordinal ipsative data.Psychometrika, 63, 369–399.
Christoffersson, A. (1975). Factor analysis of dichotomized variables.Psychometrika, 40, 5–32.
Clark, T.E. (1996). Small-sample properties of estimators of nonlinear models of covariance structure.Journal of Bussiness and Economic Statistics, 14, 367–373.
Dansie, B.R. (1986). Normal order statistics as permutation probability models.Applied Statistics, 3, 269–275.
Jöreskog, K.G. (1994). On the estimation of polychoric correlations and their asymptotic covariance matrix.Psychometrika, 59, 381–389.
Jöreskog, K.G., & Sörbom, D. (1993).LISREL 8. User's reference guide. Chicago, IL: Scientific Software.
Küsters, U.L. (1987).Hierarchische Mittelwert- und Kovarianztrukturmodelle mit nichtmetrischen endogenen Variablen [Hierarchical mean and covariance structure models on nonmetric endogenous variables]. Heidelberg: Physica-Verlag.
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.
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.
Muthén, B. (1978). Contributions to factor analysis of dichotomous variables.Psychometrika, 43, 551–560.
Muthén, B. (1982). Some categorical response models with continuous latent variables. In K.G. Jöreskog & H. Wold (Eds.).Systems under indirect observation. (Vol 1). Amsterdam: North Holland.
Muthén, B. (1984). A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators.Psychometrika, 49, 115–132.
Muthén, B. (1987).LISCOMP: Analysis of linear structural equations using a comprehensive measurement model. Mooresville, IN: Scientific Software.
Muthén, B. (1993). Goodness of fit with categorical and other non normal variables. In K.A. Bollen & J.S. Long (Eds.)Testing structural equation models. Newbury Park, CA: Sage.
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.
Rao, C.R. (1973).Linear statistical inference and its applications (2nd ed). New York: Wiley.
Satorra, A. (1989). Alternative test criteria in covariance structure analysis: A unified approach.Psychometrika, 54, 131–151.
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.
Takane, Y. (1987). Analysis of covariance structures and probabilistic binary choice data.Communication and Cognition, 20, 45–62.
Takane, Y. (1989). Analysis of covariance structures and probabilistic binary choice data. In G. Soete, H. Feger & K.C. Klauer (Eds.),New developments in psychological choice modeling (pp. 139–160). Amsterdam: Elsevier Science.
Takane, Y., & de Leeuw, J. (1987). On the relationship between item response theory and factor analysis of discretized variables.Psychometrika, 52, 393–408.
Thurstone, L.L. (1927). A law of comparative judgment.Psychological Review, 79, 281–299.
Thurstone, L.L. (1931). Rank order as a psychological method.Journal of Experimental Psychology, 14, 187–201.
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.
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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.
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Maydeu-Olivares, A. Thurstonian modeling of ranking data via mean and covariance structure analysis. Psychometrika 64, 325–340 (1999). https://doi.org/10.1007/BF02294299
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DOI: https://doi.org/10.1007/BF02294299