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Asymptotic normality of posterior distributions for generalized linear mixed models

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  • Baghishani, Hossein
  • Mohammadzadeh, Mohsen

Abstract

Bayesian inference methods are used extensively in the analysis of Generalized Linear Mixed Models (GLMMs), but it may be difficult to handle the posterior distributions analytically. In this paper, we establish the asymptotic normality of the joint posterior distribution of the parameters and the random effects in a GLMM by using Stein’s Identity. We also show that while incorrect assumptions on the random effects can lead to substantial bias in the estimates of the parameters, the assumed model for the random effects, under some regularity conditions, does not affect the asymptotic normality of the joint posterior distribution. This motivates the use of the approximate normal distributions for sensitivity analysis of the random effects distribution. We additionally illustrate that the approximate normal distribution performs reasonably using both real and simulated data. This creates a primary alternative to Markov Chain Monte Carlo (MCMC) sampling and avoids a wide range of problems for MCMC algorithms in terms of convergence and computational time.

Suggested Citation

  • Baghishani, Hossein & Mohammadzadeh, Mohsen, 2012. "Asymptotic normality of posterior distributions for generalized linear mixed models," Journal of Multivariate Analysis, Elsevier, vol. 111(C), pages 66-77.
    • Handle: RePEc:eee:jmvana:v:111:y:2012:i:c:p:66-77
    DOI: 10.1016/j.jmva.2012.05.003
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    References listed on IDEAS

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    1. Saskia Litière & Ariel Alonso & Geert Molenberghs, 2007. "Type I and Type II Error Under Random-Effects Misspecification in Generalized Linear Mixed Models," Biometrics, The International Biometric Society, vol. 63(4), pages 1038-1044, December.
    2. Jo Eidsvik & Sara Martino & Håvard Rue, 2009. "Approximate Bayesian Inference in Spatial Generalized Linear Mixed Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(1), pages 1-22, March.
    3. Baghishani, Hossein & Mohammadzadeh, Mohsen, 2011. "A data cloning algorithm for computing maximum likelihood estimates in spatial generalized linear mixed models," Computational Statistics & Data Analysis, Elsevier, vol. 55(4), pages 1748-1759, April.
    4. Agresti, Alan & Caffo, Brian & Ohman-Strickland, Pamela, 2004. "Examples in which misspecification of a random effects distribution reduces efficiency, and possible remedies," Computational Statistics & Data Analysis, Elsevier, vol. 47(3), pages 639-653, October.
    5. Hosseini, Fatemeh & Eidsvik, Jo & Mohammadzadeh, Mohsen, 2011. "Approximate Bayesian inference in spatial GLMM with skew normal latent variables," Computational Statistics & Data Analysis, Elsevier, vol. 55(4), pages 1791-1806, April.
    6. Xianzheng Huang, 2009. "Diagnosis of Random-Effect Model Misspecification in Generalized Linear Mixed Models for Binary Response," Biometrics, The International Biometric Society, vol. 65(2), pages 361-368, June.
    7. Julie L. Yee, 2002. "Asymptotic approximations to posterior distributions via conditional moment equations," Biometrika, Biometrika Trust, vol. 89(4), pages 755-767, December.
    8. Su, Chun-Lung & Johnson, Wesley O., 2006. "Large-Sample Joint Posterior Approximations When Full Conditionals Are Approximately Normal: Application to Generalized Linear Mixed Models," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 795-811, June.
    9. Ole F. Christensen & Rasmus Waagepetersen, 2002. "Bayesian Prediction of Spatial Count Data Using Generalized Linear Mixed Models," Biometrics, The International Biometric Society, vol. 58(2), pages 280-286, June.
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    Cited by:

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    2. Murray, James & Philipson, Pete, 2022. "A fast approximate EM algorithm for joint models of survival and multivariate longitudinal data," Computational Statistics & Data Analysis, Elsevier, vol. 170(C).

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