Normal Approximation for Bayesian Mixed Effects Binomial Regression Models

Bayesian inference for generalized linear mixed models implemented with Markov chain Monte Carlo (MCMC) sampling methods have been widely used. In this paper, we propose to substitute a large sample normal approxima-tion for the intractable full conditional distribution of the latent effects (of size k) in order to simplify the computation. In addition, we develop a second approxi-mation involving what we term a sufficient reduction (SR). We show that the full conditional distributions for the model parameters only depend on a small, say r << k, dimensional function of the latent effects, and also that this reduction is asymptotically normal under mild conditions. Thus we substitute the sampling of an r dimensional multivariate normal for sampling the k dimensional full con-ditional for the latent effects. Applications to oncology physician data, to cow abortion data and simulation studies confirm the reasonable performance of the proposed approximation method in terms of estimation accuracy and computa-tional speed.