Bayesian inference for generalized linear mixed models: A comparison of different statistical software procedures

Bayesian inference for generalized linear mixed models (GLMM) is appealing, but its widespread use has been hampered by the lack of a fast implementation tool and the difficulty in specifying prior distributions. In this paper, we conduct an extensive simulation study to evaluate the performance of INLA for estimation of the hierarchical Poisson regression models with overdispersion in comparison with JAGS and Stan while assuming a variety of prior specifications for variance components. Further, we analysed the influence of different factors such as small number of observations per cluster, different values of the cluster variance and estimation from a misspecified model. A simulation study has shown that the approximation strategy employed by INLA is accurate in general and that all software leads to similar results for most of the cases considered. Estimation of the variance components, however, is difficult when their true value is small for all estimation methods and prior specifications. The estimates obtained for all software tend to be biased downward or upward depending on the assumed priors.