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Psychometrika

, Volume 58, Issue 3, pp 489–509 | Cite as

Estimating latent distributions in recurrent choice data

  • Ulf Böckenholt
Article

Abstract

This paper introduces a flexible class of stochastic mixture models for the analysis and interpretation of individual differences in recurrent choice and other types of count data. These choice models are derived by specifying elements of the choice process at the individual level. Probability distributions are introduced to describe variations in the choice process among individuals and to obtain a representation of the aggregate choice behavior. Due to the explicit consideration of random effect sources, the choice models are parsimonious and readily interpretable. An easy to implement EM algorithm is presented for parameter estimation. Two applications illustrate the proposed approach.

Key words

latent class models Poisson distribution gamma distribution Dirichlet distribution empirial Bayes estimation count data EM algorithm 

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References

  1. Abul-Libdeh, H., Turnbull, B. W., & Clark, L. C. (1990). Analysis of multi-type recurrent events in longitudinal studies; application to a skin cancer prevention trial.Biometrics, 46, 1017–1034.CrossRefGoogle Scholar
  2. Al-Hussaini, E. K., and Ahmad, K. E. (1981). On the identifiability of finite mixtures of distributions.IEEE Transactions on Information Theory, 27, 664–668.CrossRefGoogle Scholar
  3. Anscombe, F. J. (1950). Sampling theory of negative binomial and logarithmic distributions.Biometrika, 37, 358–382.PubMedGoogle Scholar
  4. Arbous, A. G., & Kerrich, J. E. (1951). Accident statistics and the concept of accident proneness.Biometrics, 7, 340–432.CrossRefGoogle Scholar
  5. Bates, G. E., & Neyman, J. (1952).Contributions to the theory of accident proneness, Parts I and II. Berkeley: University of California Press.Google Scholar
  6. Bozdogan, H. (1987). Model selection and Akaike's Information Criterion (AIC): the general theory and its analytical extensions.Psychometrika, 52, 345–370.CrossRefGoogle Scholar
  7. Chatfield, C., & Goodhardt, G. J. (1975). A consumer purchasing model with Erlang interpurchase times.Journal of the American Statistical Association, 68, 828–835.CrossRefGoogle Scholar
  8. Chernoff, H., & Lehmann, E. L. (1954). The use of maximum likelihood estimates inX 2-tests for goodness of fit.The Annals of Mathematical Statistics, 25, 579–586.CrossRefGoogle Scholar
  9. Devroye, L. (1986).Non-uniform random variate generation. New York: Springer-Verlag.Google Scholar
  10. Dillon, W. R., Madden, T. J., & Mullani, N. (1983). Scaling models for categorical variables: An application of latent structure models.The Journal of Consumer Research, 10, 209–224.CrossRefGoogle Scholar
  11. Dishon, M., & Weiss, G. H. (1980). Small sample comparison of estimation methods for the beta distribution.Journal of Statistical Computation and Simulation, 11, 1–11.CrossRefGoogle Scholar
  12. Ehrenberg, A. S. C. (1988).Repeat-buying. New York: Oxford University Press.Google Scholar
  13. Goodhardt, G. J., Ehrenberg, A. S. C., & Chatfield, C. (1984). The Dirichlet: A comprehensive model of buying behavior.Journal of the Royal Statistical Society, Series A, 147, 621–655.CrossRefGoogle Scholar
  14. Goodman, L. A. (1974). Exploratory latent structure models using both identifiable and unidentifiable models.Biometrika, 61, 215–231.CrossRefGoogle Scholar
  15. Hausman, J., Hall, B. H., & Griliches, Z. (1984). Econometric models for count data with an application to the patents-R&D relationship.Econometrica, 52, 909–938.CrossRefGoogle Scholar
  16. Johnson, N. L., & Kotz, S. (1969).Distribution in statistics: Discrete distributions. New York: Wiley.Google Scholar
  17. Lawless, J. F. (1987). Negative binomial and mixed Poisson regression.The Canadian Journal of Statistics, 15, 209–225.CrossRefGoogle Scholar
  18. Lazarsfeld, P. F. (1950). The logical and mathematical foundation of latent structure analysis. In S. A. Stouffer et al. (Eds.),Studies in social psychology in World War II, Vol. IV (pp. 362–412). Princeton: Princeton University Press.Google Scholar
  19. Lindsay, B., Clogg, C. C., & Grego, J. (1991). Semiparametric estimation in the Rasch model and related exponential response models, including a simple latent class model for item analysis.Journal of the American Statistical Association, 86, 96–107.CrossRefGoogle Scholar
  20. Luce, R. D. (1959).Individual choice behavior. New York: Wiley.Google Scholar
  21. Maritz, J. S., & Lwin, T. (1989).Empirical Bayes methods. London: Chapman and Hall.Google Scholar
  22. Mislevy, R. J. (1984). Estimating latent distributions.Psychometrika, 49, 359–381.CrossRefGoogle Scholar
  23. Moran, P. A. P. (1971). Maximum likelihood estimation in non-standard conditions.Proceedings of the Cambridge Philosophy Society, 70, 441–450.CrossRefGoogle Scholar
  24. Morrison, D. G., & Schmittlein, D. C. (1988). Generalizing the NBD model for customer purchases: What are the implications and is it worth the effort?Journal of Business and Economic Statistics, 6, 145–159.CrossRefGoogle Scholar
  25. Nelson, J. F. (1985). Multivariate gamma-Poisson models.Journal of the American Statistical Association, 80, 828–834.CrossRefGoogle Scholar
  26. Poulsen, C. S. (1983).Latent structure analysis with choice modeling applications. Unpublished dissertation. University of Pennsylvania.Google Scholar
  27. Pudney, S. (1989).Modeling individual choice: The econometrics of corners, kinks and holes. New York: Basil Blackwell.Google Scholar
  28. Redner, R. A., & Walker, H. F. (1984). Mixture densities, maximum likelihood and the EM algorithm.SIAM Review, 26, 195–240.CrossRefGoogle Scholar
  29. Shoemaker, R. W., Staelin, R., Kadane, J. B., & Shoaf, F. R. (1977). Relation of brand choice to purchase frequency.Journal of Marketing Research, 14, 458–468.CrossRefGoogle Scholar
  30. Teicher, H. (1961). Identifiability of mixtures.Annals of Mathematical Statistics, 32, 244–248.CrossRefGoogle Scholar
  31. Teicher, H. (1967). Identifiability of mixtures of product measures.Annals of Mathematical Statistics, 38, 1300–1302.CrossRefGoogle Scholar
  32. Thall, P. F. (1988). Mixed Poisson likelihood regression models for longitudinal interval count data.Biometrics, 44, 197–210.PubMedCrossRefGoogle Scholar
  33. Titterington, D. M., Smith, A. F. M., & Makov, U. E. (1985).Statistical analysis of finite mixture distributions. New York: Wiley.Google Scholar
  34. Wagner, U., & Taudes, A. (1986). A multivariate Polya model of brand choice and purchase incidence.Marketing Science, 5, 219–244.CrossRefGoogle Scholar
  35. Wasserman, S. (1983). Distinguishing between stochastic models of heterogeneity and contagion.Journal of Mathematical Psychology, 27, 201–215.CrossRefGoogle Scholar
  36. Yakowiz, S. J., and Spraggins, J. D. (1968). On the identifiability of finite mixtures.Annals of Mathematical Statistics, 39, 209–214.CrossRefGoogle Scholar

Copyright information

© The Psychometric Society 1993

Authors and Affiliations

  • Ulf Böckenholt
    • 1
  1. 1.Department of PsychologyUniversity of Illinois at Urbana-ChampaignChampaign

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