On empirical Bayes penalized quasi-likelihood inference in GLMMs and in Bayesian disease mapping and ecological modeling

Penalized quasi-likelihood(PQL) procedure for statistical inference in generalized linear mixed models (GLMMs) and in Bayesian disease mapping and ecological modeling are revisited. In GLMM framework, empirical Bayes PQL (EBPQL) procedure is discussed in the context of approximating posterior point and interval prediction of random effects. An in-depth Monte Carlo assessment on EBPQL point and interval estimation of random effects, fixed effects, and prior parameters in univariate and bivariate (shared component) disease mapping and ecological models is presented, with illustrative examples including spatial and ecological modeling of infant mortality rates (relative uncommon events), suicide hospitalization rates (rare events) and suicide mortality rates (extremely rare events), and associated ecological risk factors in local health areas in British Columbia Canada. In particular, EBPQL interval prediction of random effects is explored by prediction uncertainty attributions with respect to uncertainties associated with estimation of random effects, fixed effects, and prior parameters. Estimation of percent attributions of EBPQL random effects prediction errors to prior uncertainty is developed in the context of GLMMs and explored in Bayesian disease mapping and ecological models, suggesting evidence that uncertainty about prior parameter(s) may have minor and negligible influence on EBPQL interval prediction of random effects in Bayesian hierarchical disease mapping and ecological modeling of moderate Poisson observations. The EBPQL inference procedure may be judiciously and profitably utilized in Bayesian disease mapping and ecological model development.