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The logistic linear mixed model (LLMM) is one of the most widely usedstatistical models. Generally, Markov chain Monte Carlo algorithms are used toexplore the posterior densities associated with the Bayesian LLMMs. Polson,Scott and Windle's (2013) Polya-Gamma data augmentation (DA) technique can beused to construct full Gibbs (FG) samplers for the LLMMs. Here, we developefficient block Gibbs (BG) samplers for Bayesian LLMMs using the Polya-Gamma DAmethod. We compare the FG and BG samplers in the context of a real dataexample, as the correlation between the fixed effects and the random effectschanges as well as when the dimensions of the design matrices vary. Thesenumerical examples demonstrate superior performance of the BG samplers over theFG samplers. We also derive conditions guaranteeing geometric ergodicity of theBG Markov chain when the popular improper uniform prior is assigned on theregression coefficients, and proper or improper priors are placed on thevariance parameters of the random effects. This theoretical result hasimportant practical implications as it justifies the use of asymptoticallyvalid Monte Carlo standard errors for Markov chain based estimates of theposterior quantities.

Block Gibbs samplers for logistic mixed models: convergence properties and a comparison with full Gibbs samplers