Generalized linear mixed model

In statistics, a generalized linear mixed model (GLMM) is a particular type of mixed model with both random and fixed effects in the linear predictor. GLMMs provide a broad range of models for the analysis of grouped data; they extend the idea of linear mixed models to non-normal data.

Introduction

A GLMM is an extension to the generalized linear model in which the linear predictor contains random effects in addition to the usual fixed effects.

Fitting a model

The Akaike information criterion (AIC) is a common criterion for model selection. Estimates of AIC for GLMMs based on certain exponential family distributions have recently been obtained.

Fitting GLMMs via maximum likelihood (as AIC does) involves integrating over the random effects. In general, those integrals cannot be expressed in analytical form. Various approximate methods have been developed, but none has good properties for all possible models and data sets (e.g. ungrouped binary data are particularly problematic). For this reason, methods involving numerical quadrature or Markov chain Monte Carlo have increased in use, as increasing computing power and advances in methods have made them more practical.

See also

• Generalized estimating equation

References

• Berridge, D. M.; Crouchley, R. (2011), Multivariate Generalized Linear Mixed Models Using R, CRC Press.
• Breslow, N. E.; Clayton, D. G. (1993), "Approximate Inference in Generalized Linear Mixed Models", Journal of the American Statistical Association, 88 (421): 9–25, doi:10.2307/2290687, JSTOR 2290687.
• Jiang, J. (2007), Linear and Generalized Linear Mixed Models and Their Applications, Springer.
• Fitzmaurice, G. M.; Laird, N. M.; Ware, J. H. (2011), Applied Longitudinal Analysis (2nd ed.), John Wiley & Sons, ISBN 0-471-21487-6.
• Saefken, B.; Kneib, T.; van Waveren, C.-S.; Greven, S. (2014), "A unifying approach to the estimation of the conditional Akaike information in generalized linear mixed models", Electronic Journal of Statistics, 8: 201–225, doi:10.1214/14-EJS881.
• Stroup, W.W. (2012), Generalized Linear Mixed Models, CRC Press.