Statistics and Computing

, Volume 6, Issue 3, pp 251–262

A general maximum likelihood analysis of overdispersion in generalized linear models

  • Murray Aitkin
Papers

Abstract

This paper presents an EM algorithm for maximum likelihood estimation in generalized linear models with overdispersion. The algorithm is initially derived as a form of Gaussian quadrature assuming a normal mixing distribution, but with only slight variation it can be used for a completely unknown mixing distribution, giving a straightforward method for the fully non-parametric ML estimation of this distribution. This is of value because the ML estimates of the GLM parameters may be sensitive to the specification of a parametric form for the mixing distribution. A listing of a GLIM4 algorithm for fitting the overdispersed binomial logit model is given in an appendix.

A simple method is given for obtaining correct standard errors for parameter estimates when using the EM algorithm.

Several examples are discussed.

Keywords

Overdispersion random effects GLM EM algorithm mixture model 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abramowitz, M. and Stegun, I. A. (eds) (1964) Handbook of Mathematical Functions. National Bureau of Standards, Washington DC.Google Scholar
  2. Aitkin, M. (1995) Probability model choice in single samples from exponential families using Poisson log-linear modelling, and model comparison using Bayes and posterior Bayes factors. Statistics and Computing, 5, 113–20.Google Scholar
  3. Aitkin, M. (1996) A general maximum likelihood analysis of variance components in generalized linear models. Submitted.Google Scholar
  4. Aitkin, M. and Aitkin, I. (1996) A hybrid EM/Gauss-Newton algorithm for maximum likelihood in mixture distributions. Statistics and Computing (to appear).Google Scholar
  5. Aitkin, M., Anderson, D. A., Francis, B. J. and Hinde, J. P. (1989) Statistical Modelling in GLIM. Oxford University Press.Google Scholar
  6. Aitkin, M. and Francis, B. J. (1995) Fitting overdispersed generalized linear models by nonparametric maximum likelihood. GLIM Newsletter, 25, 37–45.Google Scholar
  7. Aitkin, M. and Tunnicliffe Wilson, G. T. (1980) Mixture models, outliers and the EM algorithm. Technometrics, 22, 325–31.Google Scholar
  8. Anderson, D. A. (1988) Some models for overdispersed binomial data. Aust. J. Statist., 30, 125–48.Google Scholar
  9. Anderson, D. A. and Aitkin, M. (1985) Variance component models with binary response: interviewer variability. J. Roy. Statist. Soc. B 47, 203–10.Google Scholar
  10. Anderson, D. A. and Hinde, J. P. (1988) Random effects in generalized linear models and the EM algorithm. Commun. Statist.-Theory Meth., 17, 3847–56.Google Scholar
  11. Barry, J. T., Francis, B. J. and Davies, R. B.(1989) SABRE: software for the analysis of binary recurrent events. In Statistical Modelling, Springer-Verlag, New York.Google Scholar
  12. Bock, R. D. and Aitkin, M. (1981) Marginal maximum likelihood estimation of item parameters: an application of an EM algorithm. Psychometrika, 46, 443–59.Google Scholar
  13. Böhning, D., Schlattman, P. and Lindsay, B. (1992) Computerassisted analysis of mixtures (C.A.MAN): statistical algorithms. Biometrics, 48, 285–303.Google Scholar
  14. Breslow, N. (1984) Extra-Poisson variation in log-linear models. Appl. Statist., 33, 38–44.Google Scholar
  15. Breslow, N. (1989) Score tests in overdispersed GLMs. In Statistical Modelling, Springer-Verlag, New York.Google Scholar
  16. Breslow, N. (1990) Tests of hypotheses in overdispersed Poisson regression and other quasi-likelihood models. J. Amer. Statist. Assoc., 85, 565–71.Google Scholar
  17. Brownlee, K. A. (1965) Statistical Theory and Methodology in Science and Engineering (2nd edn). Wiley, New York.Google Scholar
  18. Crouch, E. A. C. and Spiegelman, D. (1990) The evaluation of integrals of the form ∫-∞/+∞(t) exp(-t 2)dt: application to logistic-normal models. J. Amer. Statist. Assoc., 85, 464–9.Google Scholar
  19. Davies, R. B. (1987) Mass point methods for dealing with nuisance parameters in longitudinal studies. In: R. Crouchley, ed. Longitudinal Data Analysis. Avebury, Aldershot, Hants.Google Scholar
  20. Dean, C. B. (1992) Testing for overdispersion in Poisson and binomial regression models. J. Amer. Statist. Assoc., 87, 451–7.Google Scholar
  21. Dempster, A. P., Laird, N. M. and Rubin D. A. (1977) Maximum likelihood estimation from incomplete data via the EM algorithm (with Discussion). J. Roy. Statist. Soc. B, 39, 1–38.Google Scholar
  22. Dietz, E. (1992) Estimation of heterogeneity-a GLM approach. In Advances in GLIM and Statistical Modelling. Springer-Verlag, New York.Google Scholar
  23. Dietz, E. and Böhning, D. (1995) Statistical inference based on a general model of unobserved heterogeneity. In Statistical Modelling. Springer-Verlag, New York.Google Scholar
  24. Efron, B. (1986) Double exponential families and their use in generalized linear regression. J. Amer. Statist. Assoc., 81, 709–21.Google Scholar
  25. Ezzet, F. and Davies, R. B. (1988) A manual for MIXTURE. Centre for Applied Statistics, Lancaster, UK.Google Scholar
  26. Feigl, P. and Zelen, M. (1965) Estimation of exponential probabilities with concomitant information. Biometrics, 21, 826–38.Google Scholar
  27. Follman, D. A. and Lambert, D. (1989) Generalizing logistic regression by nonparametric mixing. J. Amer. Statist. Assoc., 84, 295–300.Google Scholar
  28. Francis, B. J., Green, M. and Payne, C. (eds) (1993) The GLIM System: Release 4 Manual. Clarendon Press, Oxford.Google Scholar
  29. Heckman, J. J. and Singer, B. (1984) A method for minimizing the impact of distributional assumptions in econometric models of duration. Econometrica, 52, 271–320.Google Scholar
  30. Hinde, J. P. (1982) Compound Poisson regression models. In R. Gilchrist, ed. GLIM 82 Springer-Verlag, New York.Google Scholar
  31. Hinde, J. P. and Wood, A. T. A. (1987) Binomial variance component models with a non-parametric assumption concerning random effects. In R. Crouchley, ed. Longitudinal Data Analysis. Avebury, Aldershot, Hants.Google Scholar
  32. Kiefer, J. and Wolfowitz, J. (1956) Consistency of the maximum likelihood estimator in the presence of infinitely many nuisance parameters. Ann. Math. Statist., 27, 887–906.Google Scholar
  33. Laird, N. M. (1978) Nonparametric maximum likelihood estimation of a mixing distribution. J. Amer. Statist. Assoc., 73, 805–11.Google Scholar
  34. Lesperance, M. L. and Kalbfleisch, J. D. (1992) An algorithm for computing the non-parametric MLE of a mixing distribution. J. Amer. Statist. Assoc., 87, 120–6.Google Scholar
  35. Lindsay, B. G. (1983) The geometry of mixture likelihoods, part I: a general theory. Ann. Statist., 11, 86–94.Google Scholar
  36. Louis, T. A. (1982) Finding the observed information matrix when using the EM algorithm. J. Roy. Statist. Soc., B, 44, 226–33.Google Scholar
  37. McCullagh, P. and Nelder, J. A. (1989) Generalized Linear Models. Chapman & Hall, London.Google Scholar
  38. Moore, D. F. (1987) Modelling the extraneous variance in the presence of extrabinomial variation. Appl. Statist., 36, 8–14.Google Scholar
  39. Nelder, J. A. (1985) Quasi-likelihood and GLIM. In R. Gilchrist, B. Francis and J. Whittaker, eds, Generalized Linear Models Springer-Verlag, Berlin.Google Scholar
  40. Williams, D. A. (1982) Extra-binomial variation in logistic linear models. Appl. Statist;., 31, 144–8.Google Scholar

Copyright information

© Chapman & Hall 1996

Authors and Affiliations

  • Murray Aitkin
    • 1
  1. 1.Department of MathematicsUniversity of Western AustraliaNedlandsAustralia

Personalised recommendations