Objective Bayesian Model Selection for Spatial Hierarchical Models with Intrinsic Conditional Autoregressive Priors

Abstract

We develop Bayesian model selection via fractional Bayes factors to simultaneously assess spatial dependence and select regressors in Gaussian hierarchical models with intrinsic conditional autoregressive (ICAR) spatial random effects. Selection of covariates and spatial model structure is difficult, as spatial confounding creates a tension between fixed and spatial random effects. Researchers have commonly performed selection separately for fixed and random effects in spatial hierarchical models. Simultaneous selection methods relieve the researcher from arbitrarily fixing one of these types of effects while selecting the other. Notably, Bayesian approaches to simultaneously select covariates and spatial model structure are limited. Our use of fractional Bayes factors allows for selection of fixed effects and spatial model structure under automatic reference priors for model parameters, which obviates the need to specify hyperparameters for priors. We also show the equivalence between two ICAR specifications and derive the minimal training size for the fractional Bayes factor applied to the ICAR model under the reference prior. We perform a simulation study to assess the performance of our approach and we compare results to the Deviance Information Criterion and Widely Applicable Information Criterion. We demonstrate that our fractional Bayes factor approach assigns low posterior model probability to spatial models when data is truly independent and reliably selects the correct covariate structure with highest probability within the model space. Finally, we demonstrate our Bayesian model selection approach with applications to county-level median household income in the contiguous United States and residential crime rates in the neighborhoods of Columbus, Ohio.