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Generalized linear mixed model with a penalized Gaussian mixture as a random effects distribution


  • Komárek, Arnost
  • Lesaffre, Emmanuel


Generalized linear mixed models are popular for regressing a discrete response when there is clustering, e.g. in longitudinal studies or in hierarchical data structures. It is standard to assume that the random effects have a normal distribution. Recently, it has been examined whether wrongly assuming a normal distribution for the random effects is important for the estimation of the fixed effects parameters. While it has been shown that misspecifying the distribution of the random effects has a minor effect in the context of linear mixed models, the conclusion for generalized mixed models is less clear. Some studies report a minor impact, while others report that the assumption of normality really matters especially when the variance of the random effect is relatively high. Since it is unclear whether the normality assumption is truly satisfied in practice, it is important that generalized mixed models are available which relax the normality assumption. A replacement of the normal distribution with a mixture of Gaussian distributions specified on a grid whereby only the weights of the mixture components are estimated using a penalized approach ensuring a smooth distribution for the random effects is proposed. The parameters of the model are estimated in a Bayesian context using MCMC techniques. The usefulness of the approach is illustrated on two longitudinal studies using R-functions.

Suggested Citation

  • Komárek, Arnost & Lesaffre, Emmanuel, 2008. "Generalized linear mixed model with a penalized Gaussian mixture as a random effects distribution," Computational Statistics & Data Analysis, Elsevier, vol. 52(7), pages 3441-3458, March.
  • Handle: RePEc:eee:csdana:v:52:y:2008:i:7:p:3441-3458

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    References listed on IDEAS

    1. Verbeke, Geert & Lesaffre, Emmanuel, 1997. "The effect of misspecifying the random-effects distribution in linear mixed models for longitudinal data," Computational Statistics & Data Analysis, Elsevier, vol. 23(4), pages 541-556, February.
    2. Jara, Alejandro & Jose Garcia-Zattera, Maria & Lesaffre, Emmanuel, 2007. "A Dirichlet process mixture model for the analysis of correlated binary responses," Computational Statistics & Data Analysis, Elsevier, vol. 51(11), pages 5402-5415, July.
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    6. 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.
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    Cited by:

    1. Dalla Valle, Luciana & De Giuli, Maria Elena & Tarantola, Claudia & Manelli, Claudio, 2016. "Default probability estimation via pair copula constructions," European Journal of Operational Research, Elsevier, vol. 249(1), pages 298-311.
    2. Bao, Junshu & Hanson, Timothy E., 2016. "A mean-constrained finite mixture of normals model," Statistics & Probability Letters, Elsevier, vol. 117(C), pages 93-99.
    3. Christian Schellhase & Göran Kauermann, 2012. "Density estimation and comparison with a penalized mixture approach," Computational Statistics, Springer, vol. 27(4), pages 757-777, December.
    4. Leonardo Grilli & Carla Rampichini, 2015. "Specification of random effects in multilevel models: a review," Quality & Quantity: International Journal of Methodology, Springer, vol. 49(3), pages 967-976, May.
    5. Xu, Peirong & Peng, Heng & Huang, Tao, 2018. "Unsupervised learning of mixture regression models for longitudinal data," Computational Statistics & Data Analysis, Elsevier, vol. 125(C), pages 44-56.
    6. Gwennaëlle Mabon, 2014. "Adaptive Estimation of Random-Effects Densities In Linear Mixed-Effects Model," Working Papers 2014-41, Center for Research in Economics and Statistics.
    7. Göran Kauermann & Renate Meyer, 2014. "Penalized marginal likelihood estimation of finite mixtures of Archimedean copulas," Computational Statistics, Springer, vol. 29(1), pages 283-306, February.
    8. Zhang, Tonglin, 2019. "General Gaussian estimation," Journal of Multivariate Analysis, Elsevier, vol. 169(C), pages 234-247.
    9. Broström, Göran & Holmberg, Henrik, 2011. "Generalized linear models with clustered data: Fixed and random effects models," Computational Statistics & Data Analysis, Elsevier, vol. 55(12), pages 3123-3134, December.
    10. 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.
    11. Jaspers, Stijn & Aerts, Marc & Verbeke, Geert & Beloeil, Pierre-Alexandre, 2014. "A new semi-parametric mixture model for interval censored data, with applications in the field of antimicrobial resistance," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 30-42.
    12. Oedekoven, C.S. & King, R. & Buckland, S.T. & Mackenzie, M.L. & Evans, K.O. & Burger, L.W., 2016. "Using hierarchical centering to facilitate a reversible jump MCMC algorithm for random effects models," Computational Statistics & Data Analysis, Elsevier, vol. 98(C), pages 79-90.
    13. Shun Yu & Xianzheng Huang, 2017. "Random-intercept misspecification in generalized linear mixed models for binary responses," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 26(3), pages 333-359, August.

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