Multivariate generalized linear mixed models with semi-nonparametric and smooth nonparametric random effects densities

Statistics and Computing volume 22, pages79–92(2012)Cite this article

Abstract

We extend the family of multivariate generalized linear mixed models to include random effects that are generated by smooth densities. We consider two such families of densities, the so-called semi-nonparametric (SNP) and smooth nonparametric (SMNP) densities. Maximum likelihood estimation, under either the SNP or the SMNP densities, is carried out using a Monte Carlo EM algorithm. This algorithm uses rejection sampling and automatically increases the MC sample size as it approaches convergence. In a simulation study we investigate the performance of these two densities in capturing the true underlying shape of the random effects distribution. We also examine the implications of misspecification of the random effects distribution on the estimation of the fixed effects and their standard errors. The impact of the assumed random effects density on the estimation of the random effects themselves is investigated in a simulation study and also in an application to a real data set.

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

References

  1. Agresti, A., Caffo, B., Ohman-Strickland, P.: Examples in which misspecification of a random effects distribution reduces efficiency, and possible remedies. Comput. Stat. Data Anal. 47(3), 639–653 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  2. Aitkin, M.: A general maximum likelihood analysis of variance components in generalized linear models. Biometrics 55(1), 117–128 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  3. Booth, J.G., Hobert, J.P.: Maximizing generalized linear mixed model likelihoods with an automated Monte Carlo EM algorithm. J. R. Stat. Soc., Ser. B Stat. Methodol. 61, 265–285 (1999)

    MATH  Article  Google Scholar 

  4. Breslow, N.E., Clayton, D.G.: Approximate inference in generalized linear mixed models. J. Am. Stat. Assoc. 88, 9–25 (1993)

    MATH  Article  Google Scholar 

  5. Carroll, R.J., Hall, P.: Optimal rates of convergence for deconvolving a density. J. Am. Stat. Assoc. 83, 1184–1186 (1988)

    MathSciNet  MATH  Article  Google Scholar 

  6. Chen, J., Zhang, D., Davidian, M.: A Monte Carlo EM algorithm for generalized linear mixed models with flexible random effects distribution. Biostatistics 3(3), 347–360 (2002)

    MATH  Article  Google Scholar 

  7. Davidian, M., Gallant, AR: The nonlinear mixed effects model with a smooth random effects density. Biometrika 80, 475–488 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  8. Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc., Ser. B, Methodol. 39, 1–22 (1977)

    MathSciNet  MATH  Google Scholar 

  9. Eilers, P.H.C., Marx, B.D.: Flexible smoothing with B-splines and penalties. Stat. Sci. 11(2), 89–121 (1996)

    MathSciNet  MATH  Article  Google Scholar 

  10. Fahrmeir, L., Kaufmann, H.: Consistency and asymptotic normality of the maximum likelihood estimator in generalized linear models. Ann. Stat. 13, 342–368 (1985)

    MathSciNet  MATH  Article  Google Scholar 

  11. Fahrmeir, L., Tutz, G.: Multivariate Statistical Modelling Based on Generalized Linear Models. Springer, Berlin (2001)

    Google Scholar 

  12. Follmann, D.A., Lambert, D.: Generalizing logistic regression by nonparametric mixing. J. Am. Stat. Assoc. 84, 295–300 (1989)

    Article  Google Scholar 

  13. Gallant, AR, Nychka, D.W.: Semi-nonparametric maximum likelihood estimation. Econometrica 55, 363–390 (1987)

    MathSciNet  MATH  Article  Google Scholar 

  14. Gallant, R., Tauchen, G.: A nonparametric approach to nonlinear time series analysis: estimation and simulation. In: Brillinger, D., Caines, P., Geweke, J., Parzen, E., Rosenblatt, M., Taqqu, M. (eds.) New Directions in Time Series Analysis, Part II, pp. 71–92. Springer, Berlin (1992)

    Google Scholar 

  15. Geweke, J.: Monte Carlo simulation and numerical integration. In: Amman, H.M., Kendrick, D.A., Rust, J. (eds.) Handbook of Computational Economics, pp. 731–800. Elsevier, Amsterdam (1996)

    Google Scholar 

  16. Ghidey, W., Lesaffre, E., Eilers, P.: Smooth random effects distribution in a linear mixed model. Biometrics 60(4), 945–953 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  17. Hartzel, J., Agresti, A., Caffo, B.: Multinomial logit random effects models. Stat. Model. 1(2), 81–102 (2001)

    MATH  Article  Google Scholar 

  18. Heagerty, P.J., Kurland, B.F.: Misspecified maximum likelihood estimates and generalised linear mixed models. Biometrika 88(4), 973–985 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  19. Hedeker, D., Gibbons, R.: Longitudinal Data Analysis. Wiley, Palo Alto (2006)

    Google Scholar 

  20. Laird, N.: Nonparametric maximum likelihood estimation of a mixing distribution. J. Am. Stat. Assoc. 73, 805–811 (1978)

    MathSciNet  MATH  Article  Google Scholar 

  21. Lesperance, M.L., Kalbfleisch, J.D.: An algorithm for computing the nonparametric MLE of a mixing distribution. J. Am. Stat. Assoc. 87, 120–126 (1992)

    MATH  Article  Google Scholar 

  22. Lindsay, B.G.: The geometry of mixture likelihoods, part II: the exponential family. Ann. Stat. 11, 783–792 (1983)

    MathSciNet  MATH  Article  Google Scholar 

  23. Litière, S., Alonso, A., Molenberghs, G.: The impact of a misspecified random-effects distribution on the estimation and the performance of inferential procedures in generalized linear mixed models. Stat. Med. 27(16), 3125–3144 (2008)

    MathSciNet  Article  Google Scholar 

  24. Magder, L.S., Zeger, S.L.: A smooth nonparametric estimate of a mixing distribution using mixtures of Gaussians. J. Am. Stat. Assoc. 91, 1141–1151 (1996)

    MathSciNet  MATH  Article  Google Scholar 

  25. McCulloch, C.E.: Maximum likelihood algorithms for generalized linear mixed models. J. Am. Stat. Assoc. 92, 162–170 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  26. Monahan, J.F.: An algorithm for generating chi random variables. ACM Trans. Math. Softw. 13, 168–172 (1987)

    MathSciNet  MATH  Article  Google Scholar 

  27. Neuhaus, J.M., Hauck, W.W., Kalbfleisch, J.D.: The effects of mixture distribution misspecification when fitting mixed-effects logistic models. Biometrika 79, 755–762 (1992)

    Article  Google Scholar 

  28. Tutz, G., Hennevogl, W.: Random effects in ordinal regression models. Comput. Stat. Data Anal. 22, 537–557 (1996)

    MATH  Article  Google Scholar 

  29. Verbeke, G., Lesaffre, E.: A linear mixed-effects model with heterogeneity in the random-effects population. J. Am. Stat. Assoc. 91, 217–221 (1996)

    MATH  Article  Google Scholar 

  30. Wei, G.C.G., Tanner, M.A.: A Monte Carlo implementation of the EM algorithm and the poor man’s data augmentation algorithms. J. Am. Stat. Assoc. 85, 699–704 (1990)

    Article  Google Scholar 

  31. Zhang, D., Davidian, M.: Linear mixed models with flexible distribution of random effects for longitudinal data. Biometrics 57(3), 795–802 (2001)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Author information

Affiliations

  1. School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

    Georgios Papageorgiou & John Hinde

Authors
  1. Georgios Papageorgiou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. John Hinde
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Georgios Papageorgiou.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Papageorgiou, G., Hinde, J. Multivariate generalized linear mixed models with semi-nonparametric and smooth nonparametric random effects densities. Stat Comput 22, 79–92 (2012). https://doi.org/10.1007/s11222-010-9207-y

Download citation

Keywords

  • Longitudinal data
  • Mixed models
  • Multinomial responses
  • Random effects
  • Semi-nonparametric densities
  • Smooth nonparametric densities