Bayesian inference for generalized linear mixed models

Abstract

Generalized linear mixed models (GLMMs) continue to grow in popularity due to their ability to directly acknowledge multiple levels of dependency and model different data types. For small sample sizes especially, likelihood-based inference can be unreliable with variance components being particularly difficult to estimate. A Bayesian approach is appealing but has been hampered by the lack of a fast implementation, and the difficulty in specifying prior distributions with variance components again being particularly problematic. Here, we briefly review previous approaches to computation in Bayesian implementations of GLMMs and illustrate in detail, the use of integrated nested Laplace approximations in this context. We consider a number of examples, carefully specifying prior distributions on meaningful quantities in each case. The examples cover a wide range of data types including those requiring smoothing over time and a relatively complicated spline model for which we examine our prior specification in terms of the implied degrees of freedom. We conclude that Bayesian inference is now practically feasible for GLMMs and provides an attractive alternative to likelihood-based approaches such as penalized quasi-likelihood. As with likelihood-based approaches, great care is required in the analysis of clustered binary data since approximation strategies may be less accurate for such data.

Figures

Fig. 1.

Gamma prior for σ − 2 with parameters 0.5 and 0.005, (a) implied prior for σ, (b) implied prior for the effective degrees of freedom, and (c) effective degrees of freedom versus σ.

Fig. 2.

(a) Ten realizations (on the relative risk scale) from the random effects second-order random walk model in which the prior on the random-effects precision is Ga(0.5,0.001), (b) summaries of fitted models: the solid line corresponds to a log-linear model in birth cohort, the circles to birth cohort as a factor, and “+” to the Bayesian smoothing model.

Fig. 3.

SBMD versus age by ethnicity. Measurements on the same woman are joined with gray lines. The solid curve corresponds to the fitted spline and the dashed lines to the individual fits.

Fig. 4.

Prior summaries: (a) σ1, the standard deviation of the spline coefficients, (b) effective degrees of freedom associated with the prior for the spline coefficients, (c) effective degrees of freedom versus σ1, (d) σ2, the standard deviation of the between-individual random effects, (e) effective degrees of freedom associated with the individual random effects, and (f) effective degrees of freedom versus σ2. The vertical dashed lines on panels (a), (b), (d), and (e) correspond to the posterior medians.