Journal of Classification

, Volume 25, Issue 2, pp 225–247 | Cite as

Identifiability of Finite Mixtures of Multinomial Logit Models with Varying and Fixed Effects

  • Bettina Grün
  • Friedrich Leisch


Unique parametrizations of models are very important for parameter interpretation and consistency of estimators. In this paper we analyze the identifiability of a general class of finite mixtures of multinomial logits with varying and fixed effects, which includes the popular multinomial logit and conditional logit models. The application of the general identifiability conditions is demonstrated on several important special cases and relations to previously established results are discussed. The main results are illustrated with a simulation study using artificial data and a marketing dataset of brand choices.


Conditional logit Finite mixture Identifiability Multinomial logit Unobserved heterogeneity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. AGRESTI, A. (1990), Categorical Data Analysis (1st ed.), New York:Wiley.zbMATHGoogle Scholar
  2. AITKIN, M. (1999), “Meta-analysis by Random Effect Modelling in Generalized Linear Models”, Statistics in Medicine, 18(17–18), 2343–2351.CrossRefGoogle Scholar
  3. BLISCHKE, W.R. (1964), “Estimating the Parameters of Mixtures of Binomial Distributions”, Journal of the American Statistical Association, 59(306), 510–528.zbMATHCrossRefMathSciNetGoogle Scholar
  4. DAYTON, C.M., and MACREADY, G.B. (1988), “Concomitant-Variable Latent-Class Model”, Journal of the American Statistical Association, 83(401), 173–178.CrossRefMathSciNetGoogle Scholar
  5. ELMORE, R.T., and WANG, S. (2003), “Identifiability and Estimation in Finite Mixture Models with Multinomial Components”, Technical Report 03–04, Pennsylvania State University, Department of Statistics.Google Scholar
  6. FOLLMANN, D.A., and LAMBERT, D. (1991), “Identifiability of Finite Mixtures of Logistic Regression Models”, Journal of Statistical Planning and Inference, 27(3), 375–381.zbMATHCrossRefMathSciNetGoogle Scholar
  7. FRÜHWIRTH-SCHNATTER, S. (2006), Finite Mixture and Markov Switching Models, Springer Series in Statistics, New York: Springer.Google Scholar
  8. GREENE, W.H. (2002), LIMDEP Econometric Modeling Guide: Version 8.0, Plainview, NY: Econometric Software.Google Scholar
  9. GREENE,W.H., and HENSHER, D.A. (2003), “A Latent Class Model for Discrete Choice Analysis: Contrasts with Mixed Logit”, Transportation Research Part B, 37(8), 681–698.CrossRefGoogle Scholar
  10. GRÜN, B. (2002), Identifizierbarkeit von multinomialen Mischmodellen, Master’s thesis, Technische Universität Wien.Google Scholar
  11. GRÜN, B. (2006), Identification and Estimation of Finite Mixture Models, Ph.D. thesis, Technische Universität Wien.Google Scholar
  12. GRÜN, B., and LEISCH, F. (2004), “Bootstrapping Finite Mixture Models”, in Compstat 2004 — Proceedings in Computational Statistics, ed. J. Antoch, Heidelberg: Physica Verlag, pp. 1115–1122.Google Scholar
  13. GRÜN, B., and LEISCH, F. (2007), “Fitting Finite Mixtures of Generalized Linear Regressions in R”, Computational Statistics and Data Analysis, 51(11), 5247–5252.CrossRefMathSciNetGoogle Scholar
  14. HENNIG, C. (2000), “Identifiability of Models for Clusterwise Linear Regression”, Journal of Classification, 17(2), 273–296.zbMATHCrossRefMathSciNetGoogle Scholar
  15. HUBERT, L., and ARABIE, P. (1985), “Comparing Partitions”, Journal of Classification, 2(1), 193–218.CrossRefGoogle Scholar
  16. JAIN, D.C., VILCASSIM, N.J., and CHINTAGUNTA, P. K. (1994), “A Random-Coefficients Logit Brand-Choice Model Applied to Panel Data”, Journal of Business and Economic Statistics, 12(3), 317–328.CrossRefGoogle Scholar
  17. KAMAKURA,W.A., and RUSSELL, G.J. (1989), “A Probabilistic Choice Model for Market Segmentation and Elasticity Structure”, Journal of Marketing Research, 26(4), 379–390.CrossRefGoogle Scholar
  18. KAMAKURA, W.A., and WEDEL, M. (2004), “An Empirical Bayes Procedure for Improving Individual-Level Estimates and Predictions from Finite Mixtures of Multinomial Logit Models”, Journal of Business and Economic Statistics, 22(1), 121–125.CrossRefMathSciNetGoogle Scholar
  19. LEISCH, F. (2004), “FlexMix: A General Framework for Finite MixtureModels and Latent Class Regression in R”, Journal of Statistical Software, 11(8), 1–18.Google Scholar
  20. LINDSAY, B.G. (1995), Mixture Models: Theory, Geometry, and Applications, Hayward, CA: The Institute for Mathematical Statistics.Google Scholar
  21. MARIN, J.-M., MENGERSEN, K., and ROBERT, C. P. (2005), “Bayesian Modelling and Inference on Mixtures of Distributions, in Bayesian Thinking, Modeling and Computation, Handbook of Statistics, Vol. 25, Chap. 16, eds. D. Dey and C. Rao, Amsterdam: North–Holland, pp. 459–507.Google Scholar
  22. MCFADDEN, D. (1974), “Conditional Logit Analysis of Qualitative Choice Behavior”, in Frontiers in Econometrics, ed. P. Zarembka, New York: Academic Press, pp. 105–142.Google Scholar
  23. MEIJER, E., and YPMA, J.Y. (2008), “A Simple Identification Proof for a Mixture of Two Univariate Normal Distributions, Journal of Classification, 25(1), 113–123.CrossRefGoogle Scholar
  24. MUTHÉN, L.K., and MUTHÉN, B.O. (1998–2006), Mplus User’s Guide (4th ed.), Los Angeles, CA: Muthén & Muthén.Google Scholar
  25. R DEVELOPMENT CORE TEAM (2007), R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria.Google Scholar
  26. RABE-HESKETH, S., SKRONDAL, A., and PICKLES, A. (2004), “GLLAMM Manual”, Working Paper Series 160, U.C. Berkeley Division of Biostatistics.Google Scholar
  27. REVELT, D., and TRAIN, K. (1998), “Mixed Logit with Repeated Choices: Households’ Choices of Appliance Efficiency Level”, The Review of Economics and Statistics, 80(4), 647–657.CrossRefGoogle Scholar
  28. SOOFI, E.S. (1992), “A Generalizable Formulation of Conditional Logit with Diagnostics”, Journal of the American Statistical Association, 87(419), 812–816.CrossRefGoogle Scholar
  29. TEICHER, H. (1963), “Identifiability of Finite Mixtures”, The Annals of Mathematical Statistics, 34(4), 1265–1269.zbMATHCrossRefMathSciNetGoogle Scholar
  30. TITTERINGTON, D.M., SMITH, A. F.M., and MAKOV, U. E. (1985), Statistical Analysis of Finite Mixture Distributions, Chichester: Wiley.zbMATHGoogle Scholar
  31. VERMUNT, J.K., and MAGIDSON, J. (2003), Latent GOLD Choice User’s Guide, Statistical Innovations Inc., Boston.Google Scholar
  32. WEDEL, M. (2002), GLIMMIX—A Program for Estimation of Latent Class Mixture and Mixture Regression Models, Version 3.0, ProGAMMA bv, Groningen, The Netherlands.Google Scholar
  33. WEDEL, M., and DESARBO, W. S. (1995), “A Mixture Likelihood Approach for Generalized Linear Models”, Journal of Classification, 12(1), 21–55.zbMATHCrossRefGoogle Scholar
  34. WEDEL, M., and KAMAKURA, W.A. (2001), Market Segmentation — Conceptual and Methodological Foundations (2nd ed.), Boston: Kluwer Academic Publishers.Google Scholar
  35. WEGMAN, E.J. (1990), “Hyperdimensional Data Analysis Using Parallel Coordinates”, Journal of the American Statistical Association, 85(411), 664–675.CrossRefGoogle Scholar
  36. WU, C.F.J. (1983), “On the Converence Properties of the EM Algorithm”, The Annals of Statistics, 11(1), 95–103.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of Statistics and MathematicsWirtschaftsuniversit ät WienViennaAustria
  2. 2.Department of StatisticsLudwig-Maximilians-UniversitätMünchenGermany

Personalised recommendations