Mixed models for spatially correlated data using PC priors

Generalized linear mixed models (GLMM) represent a flexible tool to model environmental data which are characterized by various sources of heterogeneity, e.g. spatial/temporal correlation. The usual interpretation is that fixed effects ‘explain’ the response by measuring the effect of observed covariates, while random effects ‘account’ for heterogeneity due to unobserved factors. Most popular models for random effects are Gaussian conditional on some flexibility parameter (e.g. variance, correlation range), the prior specification and estimation of which represents a crucial issue in many applications. Often, random effects have a more predominant role in the analysis and are used for explanatory purposes rather than as tools to capture residual structure; for instance, in community ecology spatial random effects are associated to the presence of biotic interactions among species. We focus on cases where random effects reflect precise assumptions on the behavior of the phenomenon under study and propose mixed models with an intuitive prior specification. Based on the Penalized Complexity (PC) prior framework, we discuss solutions to build priors for variance parameters while achieving intuitive control on the flexibility of the random effects. We illustrate the use of the proposed priors using environmental case studies, with particular emphasis on spatially correlated data.