Abstract
Thurstonian models have proven useful in a wide range of applications because they can describe the multidimensional nature of choice objects and the effects of similarity and comparability in choice situations. Special cases of Thurstonian ranking models are formulated that impose different constraints on the covariance matrix of the objects’ utilities. In addition, mixture models are developed to account for individual differences in rankings. Two estimation procedures, maximum likelihood and generalized least squares, are discussed. To illustrate the approach, data from three ranking experiments are analyzed.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arbuckle, J., & Nugent, J. H. A general procedure for parameter estimation for the law of comparative judgment. British Journal of Mathematical and Statistical Psychology, 26:240–260, 1973.
Bartholomew, D. J. Latent variable models and factor analysis. London: Oxford University Press, 1987.
Bock, R. D., & Aitkin, M. Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46:443–459, 1981.
Bock, R. D., Gibbons, R., & Muraki, E. Full-information item factor analysis. Applied Psychological Measurement, 12:261–280, 1988.
Bock, R. D., & Jones, L. V. The measurement and prediction of judgment and choice. San Francisco: Holden-Day, 1968.
Böckenholt, I., & Gaul, W. Analysis of choice behavior via probabilistic ideal point and vector models. Applied Stochastic Models and Data Analysis, 2:209–226, 1986.
Böckenholt, U. Thurstonian representation for partial ranking data. British Journal of Mathematical and Statistical Psychology, in press, 1992.
Brady, H. Factor and ideal point analysis for interpersonally incomparable data. Psychometrika, 54:181–202, 1989.
Cohen, A., & Mallows, C. L. Assessing goodness of fit of ranking models to data. The Statistician, 32:361–373, 1983.
Christoffersson, A. Factor analysis of dichotomized variables. Psychometrika, 40:5–32, 1975.
Critchlow, D. E. Metric methods for analyzing partially ranked data. New York: Springer, 1985.
Croon, M. Latent class models for the analysis of rankings. In G. DeSoete, H. Feger, & K. C. Klauer (Eds.) New developments in psychological choice modeling (pp. 99–121). Elsevier: Holland, 1989.
Daniels, H. E. Rank correlation and population models. Journal of the Royal Statistical Society B, 12:171–181 1950.
DeSoete, G., & Carroll, J. D. A maximum likelihood method for fitting the wandering vector model. Psychometrika, 48:553–566 1983.
DeSoete, G., Carroll, J. D., & DeSarbo, W. S. The wandering ideal point model: A probabilistic multidimensional unfolding model for paired comparison data. Journal of Mathematical Psychology, 30:28–41, 1986.
Fligner, M. A., & Verducci, J. S. Aspects of two group concordance. Communications in Statistics, Theory and Methods, 1479–1503 1987.
Hathaway, R. J. A constrained formulation of maximum-likelihood estimation for normal mixture distributions. Annals of Statistics, 13: 795–800, 1985.
Heiser, W., amp; De Leeuw, J. Multidimensional mapping of preference data. Mathematiques et Sciences humaines, 19:39–96, 1981.
Hollander, M., & Sethuraman, J. Testing for agreement between groups of judges. Biometrika, 65:403–411 1978.
Inglehart, R. The silent revolution. Princeton: Princeton University Press, 1977.
Luce, R. D. Individual choice behavior. New York: Wiley. 1989.
Lwin, T., & Martin, P. J. Probits of mixtures. Biometrics, 45:721–732, 1959.
Manski, C., & McFadden, D. Alternative estimators and sample designs for discrete choice analysis. In C. Manski, & D. McFadden (Ed.) Structural analysis of discrete data with econometric applications. MIT Press: Cambridge, 1981.
McLachlan, G. J., & Basford, K. E. Mixture models. New York: Marcel Dekker, 1989.
Muthén, B. Latent variable structural equation modeling with categorical data. Journal of Econometrics, 22: 43–65, 1983.
Pendergrass, P. N., & Bradley, R. A. Ranking in triple comparisons. In I. Olkin (Ed.), Contributions to probability and statistics (pp.331–351). Palo Alto, CA: Stanford University Press, 1960.
Pettitt, A. N. Parametric tests for agreement amongst groups of judges. Biometrika, 69:365–375, 1982.
Schervish, M. Algorithm AS 195. Multivariate normal probabilities with error bound. Applied Statistics, 33:81–94, 1984.
Stroud, A. H., & Sechrest, D. Gaussian quadrature formulas. New York: Prentice Hall, 1966.
Takane, Y. Maximum likelihood estimation in the generalized cases of Thurstone’s law of comparative judgment. Japanese Psychological Research, 22:188–196, 1980.
Takane, Y. Analysis of covariance structures and probabilistic binary choice data. Cognition and Communication, 20:45–62, 1987.
Teicher, H. Identifiability of finite mixtures. Annals of Mathematical Statistics, 34:1265–1269, 1963.
Thurstone, L. L. A law of comparative judgment. Psychological Review, 15:284–297, 1927.
Thurstone, L. L. Rank order as a psychophysical method. Journal of Experimental Psychology, 14:187–201, 1931.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer-Verlag New York, Inc.
About this paper
Cite this paper
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. Lecture Notes in Statistics, vol 80. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2738-0_9
Download citation
DOI: https://doi.org/10.1007/978-1-4612-2738-0_9
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-97920-5
Online ISBN: 978-1-4612-2738-0
eBook Packages: Springer Book Archive