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Multivariate Thurstonian Models

Published online by Cambridge University Press:  01 January 2025

Ulf Böckenholt
Ulf Böckenholt*
Affiliation:
University of Illinois, Champaign
*
Requests for reprints should be addressed to Ulf Böckenholt, Department of Psychology, University of Illinois, 603 East Daniel Street, Champaign, Illinois 61820.
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Abstract

The recent development of probabilistic Thurstonian choice models for analyzing preference data has been motivated by the need to describe both inter- and intra-individual difference, the multidimensional nature of choice objects, and the effects of similarity and comparability among choice objects. A common feature of these models is that they focus on a single preference judgment. It is customary, however, to ask subjects not only for an overall preference judgment but also for additional paired comparison responses regarding specific attributes. This paper proposes a generalization of Thurstonian probabilistic choice models for analyzing both multiple preference responses and their relationships. The approach is illustrated by modeling data from two multivariate preference experiments.

Keywords

paired comparisonBahadur's approximationThurstonian models
Type
Original Paper
Information
Psychometrika , Volume 55 , Issue 2 , June 1990 , pp. 391 - 403
Copyright
Copyright © 1990 The Psychometric Society

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Footnotes

I am grateful to James, Austin, Ingo Böckenholt, and anonymous referees for helpful comments on this research. This paper is based on research presented at the Meeting of the Society for the Multivariate Analysis in the Behavioral Sciences, Groningen, The Netherlands, December 1988.

References

Bahadur, R. R. (1961). A representation of the joint distribution of responses to n dichotomous items. In Solomon, H. (Eds.), Studies in item analysis and prediction (pp. 158–168). Sanford: Stanford University Press.Google Scholar
Beaver, R. J. (1983). Loglinear models for multivariate paired comparison experiments with ties. Journal of Statistical Planning and Inference, 7, 209–218.CrossRefGoogle Scholar
Böckenholt, I., & Gaul, W. (1986). Analysis of choice behavior via probabilistic ideal point and vector models. Applied Stochastic Models and Data Analysis, 2, 209–226.CrossRefGoogle Scholar
Böckenholt, U. (1988). A logistic representation of multivariate paired-comparison models. Journal of Mathematical Psychology, 32, 44–63.CrossRefGoogle Scholar
Bloxom, B. (1972). The simplex in pair comparisons. Psychometrika, 37, 119–136.CrossRefGoogle Scholar
Bradley, R. A. (1984). Paired comparisons: Some basic procedures and examples. In Krishnaiah, P. R., & Sen, P. K. (Eds.), Handbook of statistics (pp. 299–326). New York: North-Holland.Google Scholar
Bradley, R. A., & Terry, M. E. (1952). The rank analysis of incomplete block designs. I. The method of paired comparisons. Biometrika, 39, 324–345.Google 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
Christoffersson, A. (1975). Factor analysis of dichotomized variables. Psychometrika, 40, 5–32.CrossRefGoogle Scholar
Davidson, R. R., & Bradley, R. A. (1969). Multivariate paired comparisons: The extensions of a univariate model and associated estimation and test procedures. Biometrika, 56, 81–95.CrossRefGoogle Scholar
Davoodzadeh, J., & Beaver, R. J. (1983). Models for multivariate paired comparison with ties. Journal of Mathematical Psychology, 25, 269–281.CrossRefGoogle Scholar
De Soete, G. (1983). On the relation between two generalized cases of Thurstone's law of comparative judgment. Mathematiques et Sciences Humaines, 21, 45–57.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 mulktidimensional unfolding model for paired comparison data. Journal of Mathematical Psychology, 30, 28–41.CrossRefGoogle Scholar
Fienberg, S. E., & Larntz, K. (1976). Loglinear representation for paired and multiple comparison models. Biometrika, 63, 245–262.CrossRefGoogle Scholar
Gibbons, R. D., Bock, R. D., & Hedecker, D. (1987, June). Approximating multivariate normal orthant probabilities using the Clark alogrithm. Paper presented at the Joint Meeting of the Classification Society and Psychometric Society, Montreal, Canada.Google Scholar
Goodman, L. A. (1987). The analysis of a set of multidimensional contingency tables using loglinear models, latent class models, and correlation models: The Solomon data revistited. In Gelfand, A. E. (Eds.), Contributions to the theory and application of statistics, New York: Academic Press.Google Scholar
Halff, H. M. (1976). Choice theories for differentially comparable alternatives. Journal of Mathematical Psychology, 14, 244–246.CrossRefGoogle Scholar
Heiser, W., & de Leeuw, J. (1981). Multidimensional mapping of preference data. Mathematiques et Sciences Humaines, 19, 39–96.Google Scholar
Holland, P. W. (1987). The Dutch identity: A new tool for the study of item response theory models, Princeton: Educational Testing Service.Google Scholar
Johnson, N. L., & Kotz, S. (1975). On some generalized Farlie-Gumbel-Morgenstern distributions. Communications in Statistics, 4, 415–427.CrossRefGoogle Scholar
Johnson, N. L., & Kotz, S. (1977). On some generalized Farlie-Gumbel-Morgenstern distributions—II. Regression, correlation and further generalizations. Communications in Statistics, 6, 485–496.CrossRefGoogle Scholar
Langdon, M. G. (1984). Improved algorithms for estimating choice probabilities in the multinomial probit model. Transportation Science, 18, 267–299.CrossRefGoogle Scholar
Lord, F. (1962). Review of Studies in item analysis and prediction. Psychometrika, 27, 207–213.CrossRefGoogle Scholar
Lord, F., & Novick, M. R. (1968). Statistical theories of mental test scores, Reading: Addison-Wesley.Google Scholar
Luce, R. D. (1959). Individual choice behavior, New York: Wiley.Google Scholar
Luce, R. D. (1977). The choice axiom after twenty years. Jorunal of Mathematical Psychology, 15, 215–233.CrossRefGoogle Scholar
Muthén, B. (1978). Contributions to factor analysis dichotomous varialbes. Psychometrika, 43, 551–560.CrossRefGoogle Scholar
Muthén, B. (1983). Latent variable structural equation modeling with categorical data. Journal of Econometrics, 22, 43–65.CrossRefGoogle Scholar
Muthén, B. (1984). A general structural equation model with dichotomous, ordered categorical, and continous latent variable indicators. Psychometrika, 49, 115–132.CrossRefGoogle Scholar
Takane, Y. (1980). Maximum likelihood estimation int he generalized cases of Thurstone's 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
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. (1927). A law of comparative judgment. Psychological Review, 15, 284–297.Google Scholar
Schervish, M. (1984). Algorithm AS 195. Multivariate normal probabilities with error bound. Applied Statistics, 33, 81–94.CrossRefGoogle Scholar