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Multilevel Logistic Regression for Polytomous Data and Rankings

Published online by Cambridge University Press:  01 January 2025

Anders Skrondal  and
Sophia Rabe-Hesketh
Anders Skrondal*
Affiliation:
Division of Epidemiology, Norwegian Institute of Public Health, Oslo
Sophia Rabe-Hesketh
Affiliation:
Department of Biostatistics and Computing, Institute of Psychiatry, London
*
Requests for reprints should be sent to Anders Skrondal, Division of Epidemiology, Norwegian Institute of Public Health, P.O. Box 4404 Nydalen, N-0403 Oslo, NORWAY. E-Mail: anders.skrondal@fhi.no
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Abstract

We propose a unifying framework for multilevel modeling of polytomous data and rankings, accommodating dependence induced by factor and/or random coefficient structures at different levels. The framework subsumes a wide range of models proposed in disparate methodological literatures. Partial and tied rankings, alternative specific explanatory variables and alternative sets varying across units are handled. The problem of identification is addressed. We develop an estimation and prediction methodology for the model framework which is implemented in the generally available gllamm software. The methodology is applied to party choice and rankings from the 1987–1992 panel of the British Election Study. Three levels are considered: elections, voters and constituencies.

Keywords

multilevel modelsgeneralized linear latent and mixed modelsfactor modelsrandom coefficient modelspolytomous datarankingsfirst choicediscrete choicepermutationsnominal datagllamm

Information

Type
Articles
Information
Psychometrika , Volume 68 , Issue 2 , June 2003 , pp. 267 - 287
Copyright
Copyright © 2003 The Psychometric Society

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Footnotes

Parts of this work were completed while Anders Skrondal visited the Biostatistics Group at The University of Manchester, UK. gllamm and the script for the analyses in this article can be downloaded from: http://www.iop.kcl.ac.uk/IoP/Departments/BioComp/programs/gllamm.html.

References

Allenby, G.M., Lenk, P.J. (1994). Modelling household purchase behavior with logistic normal regression. Journal of the American Statistical Association, 89, 1218–1231.CrossRefGoogle Scholar
Allison, P.D., Christakis, N.A. (1994). Logit models for sets of ranked items. In Marsden, P.V. (Eds.), Sociological Methodology 1994 (pp. 199–228). Oxford, U.K.: Blackwell.Google Scholar
Arbuckle, J., Nugent, J.H. (1973). A general procedure for parameter estimation for the law of comparative judgment. British Journal of Mathematical and Statistical Psychology, 26, 240–260.CrossRefGoogle Scholar
Beggs, S., Cardell, S., Hausman, J. (1981). Assessing the potential demand for electric cars. Journal of Econometrics, 16, 1–19.CrossRefGoogle Scholar
Block, H.D., Marchak, J. (1960). Random orderings and stochastic theories of responses. In Olkin, I., Hoeffting, W., Ghurye, S., Madow, W.G., Mann, H.D. (Eds.), Contributions to probability and statistics: Essays in honor of Harold Hotelling (pp. 97–112). Stanford, CA: Stanford University Press.Google Scholar
Bloxom, B. (1972). The simplex in pair comparisons. Psychometrika, 37, 119–136.CrossRefGoogle Scholar
Bock, R.D. (1969). Estimating multinomial response relations. In Bose, R. (Eds.), Essays in probability and statistics (pp. 111–132). Chapel Hill, NC: University of North Carolina Press.Google 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
Böckenholt, U. (1992). Thurstonian representation for partial ranking data. British Journal of Mathematical and Statistical Psychology, 45, 31–39.CrossRefGoogle Scholar
Böckenholt, U. (1993). Application of Thurstonian models to ranking data. In Fligner, M.A., Verducci, J.S. (Eds.), Probability models and statistical analysis for ranking data (pp. 157–172). New York, NY: Springer.CrossRefGoogle Scholar
Böckenholt, U. (2001). Mixed-effects analyses of rank-ordered data. Psychometrika, 66, 45–62.CrossRefGoogle Scholar
Böckenholt, U. (2001). Hierarchical modeling of paired comparison data. Psychological Methods, 6, 49–66.CrossRefGoogle ScholarPubMed
Brady, H.E. (1989). Factor and ideal point analysis for interpersonally incomparable data. Psychometrika, 54, 181–202.CrossRefGoogle Scholar
Breslow, N.E. (1974). Covariance analysis of censored survival data. Biometrics, 30, 89–100.CrossRefGoogle ScholarPubMed
Brody, R.A., Page, B.I. (1972). Comment: The assessment of policy voting. American Political Science Review, 66, 450–458.CrossRefGoogle Scholar
Bunch, D.S. (1991). Estimability in the multinomial probit model. Transportation Research, Part B: Methodological, 25, 1–12.CrossRefGoogle 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
Chan, W., Bentler, P.M. (1998). Covariance structure analysis of ordinal ipsative data. Psychometrika, 63, 369–399.CrossRefGoogle Scholar
Chapman, R.G., 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., Verducci, J.S. (1991). Probability models on rankings. Journal of Mathematical Psychology, 35, 294–318.CrossRefGoogle Scholar
Croon, M.A. (1989). Latent class models for the analysis of rankings. In De Soete, G., Klauer, K.C. (Eds.), New developments in psychological choice modeling (pp. 99–121). Amsterdam: North-Holland.CrossRefGoogle Scholar
Daniels, M.J., Gatsonis, C. (1997). Hierarchical polytomous regression models with applications to health services research. Statistics in Medicine, 16, 2311–2325.3.0.CO;2-E>CrossRefGoogle ScholarPubMed
De Soete, G., Carroll, J.D. (1983). A maximum likelihood method for fitting the wandering vector model. Psychometrika, 48, 553–566.CrossRefGoogle Scholar
Durbin, J. (1951). Incomplete blocks in ranking experiments. British Journal of Psychology, 4, 85–90.Google Scholar
Elrod, T. (1988). Choice map: Inferring a product-market map from panel data. Marketing Science, 7, 21–40.CrossRefGoogle Scholar
Elrod, T., Keane, M.P. (1995). A factor analytic probit model for representing the market structure in panel data. Journal of Marketing Research, 32, 1–16.CrossRefGoogle Scholar
Farewell, V.T., Prentice, R.L. (1980). The approximation of partial likelihood with emphasis on case-control studies. Biometrika, 67, 273–278.CrossRefGoogle Scholar
Goldstein, H. (1995). Multilevel statistical models. London: Arnold.Google Scholar
Gurland, J., Lee, I., Dolan, P. (1960). Polychotomous quantal response in biological assay. Biometrics, 16, 382–398.CrossRefGoogle Scholar
Haaijer, M.E., Wedel, M., Vriens, M., Wansbeek, T.J. (1998). Utility covariances and context effects in conjoint MNP models. Marketing Science, 17, 236–252.CrossRefGoogle Scholar
Hajivassiliou, V., Ruud, P. (1994). Classical estimation methods for LDV models using simulation. In Engle, R., McFadden, D. (Eds.), Handbook of econometrics IV (pp. 2384–2441). Amsterdam: North-Holland.Google Scholar
Harris, K.M., Keane, M.P. (1999). A model for health plan choice: Inferring preferences and perceptions from a combination of revealed preference and attitudinal data. Journal of Econometrics, 89, 131–157.CrossRefGoogle Scholar
Hausman, J.A., Ruud, P.A. (1987). Specifying and testing econometric models for rank-ordered data. Journal of Econometrics, 34, 83–103.CrossRefGoogle Scholar
Hausman, J.A., Wise, D.A. (1978). A conditional probit model for qualitative choice: Discrete decisions recognizing interdependence and heterogeneous preferences. Econometrica, 46, 403–426.CrossRefGoogle Scholar
Heath, A., Curtice, J., Jowell, R., Evans, G., Fields, J., Witherspoon, S. (1991). Understanding political change: The British voter 1964–1987. Oxford, U.K.: Pergamon.Google Scholar
Heath, A., Jowell, R., Curtice, J. (1994). Labour's last chance? The 1992 election and beyond. Aldershot: Dartmouth.Google Scholar
Heath, A., Jowell, R., Curtice, J.K., Brand, J.A., Mitchell, J.C. (1993). SN2983—British Election Panel Study, 1987–1992 [Computer file]. Colchester, Essex: The Data Archive.Google Scholar
Heckman, J.J., Sedlacek, G. (1985). Heterogeneity, aggregation, and market wage functions: An empirical model of self-selection in the labor market. Journal of Political Economy, 93, 1077–1125.CrossRefGoogle Scholar
Hedeker, D. (1999). MIXNO: A computer program for mixed-effects logistic regression. Journal of Statistical Software, 4, 1–92.CrossRefGoogle Scholar
Hosmer, D.A., Lemeshow, S.A. (1989). Applied logistic regression. London: Wiley.Google Scholar
Hsiao, C. (1985). Benefits and limitations of panel data. Econometric Review, 4, 121–174.CrossRefGoogle Scholar
Johnson, N.L., Kotz, S., Balakrishnan, N. (1995). Continuous univariate distributions, Volume 2. New York, NY: Wiley.Google Scholar
Kalbfleisch, J.D., Prentice, R.L. (1980). The statistical analysis of failure time data. New York, NY: Wiley.Google Scholar
Keane, M.P. (1992). A note on identification in the multinomial probit model. Journal of Business and Economic Statistics, 10, 193–200.CrossRefGoogle Scholar
Luce, R.D. (1959). Individual choice behavior. New York, NY: Wiley.Google Scholar
Luce, R.D., Suppes, P. (1965). Preference, utility and subjective probability. In Luce, R.D., Bush, R., Galanter, E. (Eds.), Handbook of mathematical psychology III (pp. 249–410). New York, NY: Wiley.Google Scholar
Maddala, G.S. (1983). Limited dependent and qualitative variables in econometrics. Cambridge, MA: Cambridge University Press.CrossRefGoogle Scholar
Marchak, J. (1960). Binary choice constraints on random utility indicators. In Arrow, K.J., Karlin, S., Suppes, P. (Eds.), Mathematical methods in the social sciences, 1959 (pp. 312–329). Stanford, CA: Stanford University Press.Google Scholar
McFadden, D. (1973). Conditional logit analysis of qualitative choice behavior. In Zarembka, P. (Eds.), Frontiers in econometrics (pp. 105–142). New York, NY: Academic Press.Google Scholar
McFadden, D., Train, K. (2000). Mixed MLN models for discrete choice. Journal of Applied Econometrics, 15, 447–470.3.0.CO;2-1>CrossRefGoogle Scholar
Morris, C. (1983). Parametric empirical Bayes inference, theory and applications. Journal of the American Statistical Association, 78, 47–65.CrossRefGoogle Scholar
Neuhaus, J.M. (1992). Statistical methods for longitudinal and clustered designs with binary responses. Statistical Methods in Medical Research, 1, 249–273.CrossRefGoogle ScholarPubMed
Plackett, R.L. (1975). The analysis of permutations. Applied Statistics, 24, 193–202.CrossRefGoogle Scholar
Rabe-Hesketh, S., Skrondal, A. (2001). Parameterization of multivariate random effects models for categorical data. Biometrics, 57, 1256–1264.CrossRefGoogle ScholarPubMed
Rabe-Hesketh, S., Pickles, A., Skrondal, A. (2001). GLLAMM Manual. London: King's College, University of London, Institute of Psychiatry, Department of Biostatistics and Computing.Google Scholar
Rabe-Hesketh, S., Skrondal, A., Pickles, A. (2002). Reliable estimation of generalized linear mixed models using adaptive quadrature. The Stata Journal, 2, 1–21.CrossRefGoogle Scholar
Rabe-Hesketh, S., Skrondal, A. & Pickles, A. (2003). Maximum likelihood estimation of limited and discrete dependent variable models with nested random effects. Submitted for publication.Google Scholar
Rabe-Hesketh, S., Pickles, A., & Skrondal, A. (in press). Correcting for covariate measurement error in logistic regression using nonparametric maximum likelihood estimation. Statistical Modelling.Google Scholar
Rabe-Hesketh, S., Skrondal, A. & Pickles, A. (in press). Generalized multilevel structural equation modeling. Psychometrika.Google Scholar
Raudenbush, S.W., Bryk, A.S. (2002). Hierarchical linear models: Applications and data analysis methods. Thousand Oaks, CA: Sage.Google Scholar
Revelt, D., Train, K. (1998). Mixed logit with repeated choices: household's choices of appliance efficiency level. Review of Economics and Statistics, 80, 647–657.CrossRefGoogle Scholar
Rivers, D. (1988). Heterogeneity in models of electoral choice. American Journal of Political Science, 32, 737–757.CrossRefGoogle Scholar
Skrondal, A., Rabe-Hesketh, S. (2004). Generalized latent variable modeling: Multilevel, longitudinal and structural equation models. Boca Raton, FL: Chapman & Hall CRC.CrossRefGoogle Scholar
Takane, Y. (1987). Analysis of covariance structures and probabilistic binary data. Communication & Cognition, 20, 45–62.Google Scholar
Theil, H. (1969). A multinomial extension of the linear logit model. International Economic Review, 10, 251–259.CrossRefGoogle Scholar
Train, K. (1986). Qualitative Choice Analysis. Cambridge, MA: MIT Press.Google Scholar
Yang, M. (1997). Multilevel models for multiple category responses—A simulation. Multilevel Modelling Newsletter, 9, 10–16.Google Scholar
Yang, M. (2001). Multinomial regression. In Leyland, A.H., Goldstein, H. (Eds.), Multilevel modelling of health statistics (pp. 107–123). Chichester, U.K.: John Wiley & Sons.Google 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
Yellott, J. (1977). The relationship between Luce's choice axiom, Thurstone's theory of comparative judgement, and the double exponential distribution. Journal of Mathematical Psychology, 15, 109–144.CrossRefGoogle Scholar