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Mixed-Effects Analyses of Rank-Ordered Data

Published online by Cambridge University Press:  01 January 2025

Ulf Böckenholt
Ulf Böckenholt*
Affiliation:
University of Illinois at Urbana-Champaign
*
Requests for reprints should be sent to Ulf Böckenholt, Department of Psychology, University of Illinois, Champaign, IL 61820. E-Mail: ubockenh@s.psych.uiuc.edu
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Abstract

This paper presents a synthesis of Bock's (1972) nominal categories model and Luce's (1959) choice model for mixed-effects analyses of rank-ordered data. It is shown that the proposed ranking model is both parsimonious and flexible in accounting for preference heterogeneity as well as fixed and random effects of covariates. Relationships to other approaches, including Takane's (1987) ideal point discriminant model and Croon's (1989) latent-class version of Luce's ranking model, are also discussed. The application focuses on a ranking study of behavioral traits that parents find desirable in children.

Keywords

latent trait modelslatent class analysisprobabilistic choice modelsnominal categories modelLuce's choice model

Information

Type
Articles
Information
Psychometrika , Volume 66 , Issue 1 , March 2001 , pp. 45 - 62
Copyright
Copyright © 2001 The Psychometric Society

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Footnotes

The manuscript for this article was submitted and accepted during my tenure as the Editor of Psychometrika. — Willem Heiser

This research was partially supported by NSF grant SBR-9730197. The author is grateful to Rung-Ching Tsai and three anonymous reviewers for their helpful comments on this research.

References

Allison, P.D., Christakis, N.A. (1994). Logit models for sets of ranked items. In Marsden, P. (Eds.), Sociological Methodology (pp. 199–228). San Francisco: Jossey-Bass.Google Scholar
Alwin, D.F. (1990). Historical changes in parental orientations to children. Sociological Studies of Child Development, 3, 65–86.Google Scholar
Alwin, D.F., Jackson, D.J. (1982). The statistical analysis of Kohn's measures of parental values. In Jöreskog, K.G., Wold, H. (Eds.), Systems under indirect observations: Causality, structure, and prediction (pp. 197–223). Amsterdam: North-Holland.Google Scholar
Beggs, S.S., Cardell, S., Hausman, J. (1981). Assessing the potential demand for electric cars. Journal of Econometrics, 16, 1–19.CrossRefGoogle Scholar
Ben-Akiva, M., Lerman, S. (1985). Discrete choice analysis. Cambridge: MIT Press.Google Scholar
Ben-Akiva, M., Morikawa, T., Shiroishi, F. (1992). Analysis of the reliability of preference ranking data. Journal of Business Research, 24, 149–164.CrossRefGoogle Scholar
Bock, R.D. (1972). Estimating item parameters and latent ability when responses are scored in two or more nominal categories. Psychometrika, 37, 29–51.CrossRefGoogle Scholar
Bock, R.D., Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: An application of the EM algorithm. Psychometrika, 46, 443–459.CrossRefGoogle 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., Verducci, J. (Eds.), Probability models and statistical analyses for ranking data (pp. 157–172). New York: Springer.CrossRefGoogle Scholar
Chan, W., Bentler, P.M. (1998). Covariance structure analysis of ordinal ipsative data. Psychometrika, 63, 369–399.CrossRefGoogle Scholar
Chapman, R., Staelin, R. (1982). Exploiting rank ordered choice set data within the stochastic utility model. Journal of Marketing Research, 14, 288–301.CrossRefGoogle Scholar
Critchlow, D.E., Fligner, M.A. (1993). Ranking models with item covariates. In Fligner, M., Verducci, J. (Eds.), Probability models and statistical analyses for ranking data (pp. 1–19). New York: Springer.Google Scholar
Critchlow, D.E., Fligner, M., Verducci, J. (1991). Probability models in rankings. Journal of Mathematical Psychology, 35, 294–318.CrossRefGoogle Scholar
Croon, M.A. (1989). The analysis of partial rankings by log-linear and latent class model. In DeSoete, G., Feger, H., Klauer, K.C. (Eds.), New developments in psychological choice modeling (pp. 497–506). Amsterdam: Elsevier Science Publishers.Google Scholar
Dayton, C.M., MacReady, G.B. (1988). Concomitant variable latent class models. Journal of the American Statistical Association, 83, 173–178.CrossRefGoogle Scholar
Gill, P.E., Murray, W., Wright, M.H. (1981). Practical optimization. New York: Academic Press.Google Scholar
Goodman, L. (1979). Simple models for the analysis of association in cross-classifications having ordered categories. Journal of the American Statistical Association, 74, 537–552.CrossRefGoogle Scholar
Hausman, J.A., Ruud, P.A. (1987). Specifying and testing econometric models for rank-ordered data. Journal of Econometric, 34, 83–104.CrossRefGoogle Scholar
Hedeker, D. (1999). MIXNO: A computer program for mixed-effects nominal logistic regression. Journal of Statistical Software, 4, 1–92.CrossRefGoogle Scholar
Hedeker, D., Gibbons, R. (1994). A random-effects ordinal regression model for multilevel data. Biometrics, 50, 933–944.CrossRefGoogle Scholar
Hojo, H. (1997). A marginalization model for the multidimensional unfolding analysis of ranking data. Japanese Psychological Research, 39, 33–42.CrossRefGoogle Scholar
Kamakura, W.A., Mazzon, J.A. (1991). Value segmentation?: A model for the measurement of values and value systems. Journal of Consumer Research, 18, 208–218.CrossRefGoogle Scholar
Kamakura, W.A., Wedel, M., Agrawal, J. (1994). Concomitant variable latent class models for conjoint analysis. International Journal of Research in Marketing, 11, 451–464.CrossRefGoogle Scholar
Kerckhoff, A.C. (1972). Socialization and social class. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
Kohn, M. (1969). Class and conformity: A study in values. Homewood, IL: Irwin.Google Scholar
Kohn, M. (1976). Social class and parental values: Another confirmation of the relationship. American Sociological Review, 41, 538–545.CrossRefGoogle Scholar
Kohn, M.L., Slomczynski, K.M. (1990). Social structure and self-direction. Oxford: Basil Blackwell.Google Scholar
Luce, R.D. (1959). Individual choice behavior. New York: Wiley.Google Scholar
Lynd, R., Lynd, H. (1929). Middletown: A study in contemporary American culture. New York: Harcourt Brace.Google Scholar
Marden, J.I. (1995). Analyzing and modeling rank data. London: Chapman & Hall.Google Scholar
Maydeu-Olivares, A. (1999). Thurstonian modeling of ranking data via mean and covariance structure analysis. Psychometrika, 64, 325–340.CrossRefGoogle Scholar
McCullagh, P. (1993). Permutations and regression models. In Fligner, M., Verducci, J. (Eds.), Probability models and statistical analyses for ranking data (pp. 196–215). New York: Springer.CrossRefGoogle Scholar
McCullagh, P., Nelder, J.A. (1989). Generalized linear models. London: Chapman & Hall.CrossRefGoogle Scholar
National Opinion Research Center (1987). General social surveys, 1972–1987: Cumulative codebook. Chicago: Author.Google Scholar
Plackett, R.L. (1975). The analysis of permutations. Applied Statistics, 24, 193–202.CrossRefGoogle Scholar
Samejima, F. (1972). A general models for free-response data. Psychometric Monograph, No. 18.Google Scholar
Schwarz, G. (1978). Estimating the dimensions of a model. Annals of Statistics, 6, 461–464.CrossRefGoogle Scholar
Silverberg, A.R. (1984, August). Statistical models forq-permutations. Proceedings of the Biopharmaceutical Section, American Statistical Association, 107–112.Google Scholar
Stern, H. (1990). Models for distributions on permutations. Journal of the American Statistical Association, 85, 558–564.CrossRefGoogle Scholar
Stroud, A.H., Secrest, D. (1966). Gaussian quadrature formulas. New York: Prentice Hall.Google Scholar
Takane, Y. (1987). Analysis of contingency tables by ideal point discriminant analysis. Psychometrika, 52, 493–513.CrossRefGoogle Scholar
Takane, Y. (1996). Choice model analysis of the “pick any/n” type of binary data. Japanese Psychological Research, 40, 31–39.CrossRefGoogle Scholar
Takane, Y., Carroll, J.D. (1981). Nonmetric maximum likelihood multidimensional scaling from directional rankings of similarities. Psychometrika, 46, 389–405.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, 34, 273–286.CrossRefGoogle Scholar
Thurstone, L. L. (1931). Rank order as a psychophysical method. Journal of Experimental Psychology, 14, 187–201.CrossRefGoogle Scholar
Yao, 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
Train, K. (1986). Qualitative choice analysis. Cambridge: MIT Press.Google Scholar
Tversky, A., Russo, J.E. (1969). Substitutability and similarity in binary choices. Journal of Mathematical Psychology, 6, 1–12.CrossRefGoogle Scholar
Wright, J.D., Wright, S.A. (1976). Social class and parental values for children: A partial replication and extension of the Kohn thesis. American Sociological Review, 41, 527–537.CrossRefGoogle Scholar