Skip to main content
SpringerLink
Log in

Multilevel logistic regression for polytomous data and rankings

  • Articles
  • Published:
Psychometrika Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

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.

    Google Scholar 

  • Allison, P.D., & Christakis, N.A. (1994). Logit models for sets of ranked items. In P.V. Marsden (Ed.),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.

    Google Scholar 

  • Beggs, S., Cardell, S., & Hausman, J. (1981). Assessing the potential demand for electric cars.Journal of Econometrics, 16, 1–19.

    Google Scholar 

  • Block, H.D., & Marchak, J. (1960). Random orderings and stochastic theories of responses. In I. Olkin, W. Hoeffting, S. Ghurye, W.G. Madow & H.D. Mann (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.

    Google Scholar 

  • Bock, R.D. (1969). Estimating multinomial response relations. In R. Bose (Ed.),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.

    Google Scholar 

  • Böckenholt, U. (1992). Thurstonian representation for partial ranking data.British Journal of Mathematical and Statistical Psychology, 45, 31–39.

    Google Scholar 

  • Böckenholt, U. (1993). Application of Thurstonian models to ranking data. In M.A. Fligner & J.S. Verducci (Eds.),Probability models and statistical analysis for ranking data (pp. 157–172). New York, NY: Springer.

    Google Scholar 

  • Böckenholt, U. (2001a). Mixed-effects analyses of rank-ordered data.Psychometrika, 66, 45–62.

    Google Scholar 

  • Böckenholt, U. (2001b). Hierarchical modeling of paired comparison data.Psychological Methods, 6, 49–66.

    Google Scholar 

  • Brady, H.E. (1989). Factor and ideal point analysis for interpersonally incomparable data.Psychometrika, 54, 181–202.

    Google Scholar 

  • Breslow, N.E. (1974). Covariance analysis of censored survival data.Biometrics, 30, 89–100.

    Google Scholar 

  • Brody, R.A., & Page, B.I. (1972). Comment: The assessment of policy voting.American Political Science Review, 66, 450–458.

    Google Scholar 

  • Bunch, D.S. (1991). Estimability in the multinomial probit model.Transportation Research, Part B: Methodological, 25, 1–12.

    Google Scholar 

  • Carroll, J.D. (1980). Models and methods for multidimensional analysis of preference choice (or other dominance) data. In E.D. Lantermann & H. Feger (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.

    Google 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.

    Google Scholar 

  • Critchlow, D.E., Fligner, M.A., & Verducci, J.S. (1991). Probability models on rankings.Journal of Mathematical Psychology, 35, 294–318.

    Google Scholar 

  • Croon, M.A. (1989). Latent class models for the analysis of rankings. In G. De Soete & K.C. Klauer (Eds.),New developments in psychological choice modeling (pp. 99–121). Amsterdam: North-Holland.

    Google Scholar 

  • Daniels, M.J., & Gatsonis, C. (1997). Hierarchical polytomous regression models with applications to health services research.Statistics in Medicine, 16, 2311–2325.

    Google Scholar 

  • De Soete, G., & Carroll, J.D. (1983). A maximum likelihood method for fitting the wandering vector model.Psychometrika, 48, 553–566.

    Google 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.

    Google 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.

    Google Scholar 

  • Farewell, V.T., & Prentice, R.L. (1980). The approximation of partial likelihood with emphasis on case-control studies.Biometrika, 67, 273–278.

    Google 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.

    Google 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.

    Google Scholar 

  • Hajivassiliou, V., & Ruud, P. (1994). Classical estimation methods for LDV models using simulation. In R. Engle & D. McFadden (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.

    Google Scholar 

  • Hausman, J.A., & Ruud, P.A. (1987). Specifying and testing econometric models for rank-ordered data.Journal of Econometrics, 34, 83–103.

    Google 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.

    Google 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. (Eds.) (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 [Distributor].

    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.

    Google Scholar 

  • Hedeker, D. (1999). MIXNO: A computer program for mixed-effects logistic regression.Journal of Statistical Software, 4, 1–92.

    Google 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.

    Google 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.

    Google 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 R.D. Luce, R. Bush & E. Galanter (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.

    Google Scholar 

  • Marchak, J. (1960). Binary choice constraints on random utility indicators. In K.J. Arrow, S. Karlin & P. Suppes (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 P. Zarembka (Ed.),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.

    Google Scholar 

  • Morris, C. (1983). Parametric empirical Bayes inference, theory and applications.Journal of the American Statistical Association, 78, 47–65.

    Google Scholar 

  • Neuhaus, J.M. (1992). Statistical methods for longitudinal and clustered designs with binary responses.Statistical Methods in Medical Research, 1, 249–273.

    Google Scholar 

  • Plackett, R.L. (1975). The analysis of permutations.Applied Statistics, 24, 193–202.

    Google Scholar 

  • Rabe-Hesketh, S., & Skrondal, A. (2001). Parameterization of multivariate random effects models for categorical data.Biometrics, 57, 1256–1264.

    Google Scholar 

  • Rabe-Hesketh, S., Pickles, A., & Skrondal, A. (2001).GLLAMM Manual (Tech. Rep. 2001/01). London: King's College, University of London, Institute of Psychiatry, Department of Biostatistics and Computing. Computer software and documentation can be retreived at http://www.iop.kcl.ac.uk/iop/departments/biocomp/programs/gllamm.html

    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.

    Google 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.

  • Rabe-Hesketh, S., Pickles, A., & Skrondal, A. (in press). Correcting for covariate measurement error in logistic regression using nonparametric maximum likelihood estimation.Statistical Modelling.

  • Rabe-Hesketh, S., Skrondal, A. & Pickles, A. (in press). Generalized multilevel structural equation modeling.Psychometrika.

  • 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.

    Google Scholar 

  • Rivers, D. (1988). Heterogeneity in models of electoral choice.American Journal of Political Science, 32, 737–757.

    Google Scholar 

  • Skrondal, A. & Rabe-Hesketh, S. (2004).Generalized latent variable modeling: Multilevel, longitudinal and structural equation models. Boca Raton, FL: Chapman & Hall CRC.

    Google 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.

    Google 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 A.H. Leyland & H. Goldstein (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.

    Google 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, 109–44.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Biostatistics and Computing, Institute of Psychiatry, London

    Sophia Rabe-Hesketh

  2. Division of Epidemiology, Norwegian Institute of Public Health, P.O. Box 4404 Nydalen, N-0403, Oslo, Norway

    Anders Skrondal

Authors
  1. Anders Skrondal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sophia Rabe-Hesketh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anders Skrondal.

Additional information

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.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Skrondal, A., Rabe-Hesketh, S. Multilevel logistic regression for polytomous data and rankings. Psychometrika 68, 267–287 (2003). https://doi.org/10.1007/BF02294801

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02294801

Key words

Search

Navigation