  The logistic linear mixed model (LLMM) is one of the most widely used
statistical models. Generally, Markov chain Monte Carlo algorithms are used to
explore the posterior densities associated with the Bayesian LLMMs. Polson,
Scott and Windle's (2013) Polya-Gamma data augmentation (DA) technique can be
used to construct full Gibbs (FG) samplers for the LLMMs. Here, we develop
efficient block Gibbs (BG) samplers for Bayesian LLMMs using the Polya-Gamma DA
method. We compare the FG and BG samplers in the context of a real data
example, as the correlation between the fixed effects and the random effects
changes as well as when the dimensions of the design matrices vary. These
numerical examples demonstrate superior performance of the BG samplers over the
FG samplers. We also derive conditions guaranteeing geometric ergodicity of the
BG Markov chain when the popular improper uniform prior is assigned on the
regression coefficients, and proper or improper priors are placed on the
variance parameters of the random effects. This theoretical result has
important practical implications as it justifies the use of asymptotically
valid Monte Carlo standard errors for Markov chain based estimates of the
posterior quantities.
