Hierarchical copula regression models for areal data

Regression analysis for spatially aggregated data is common in a number of fields, including public health, ecology, and econometrics. Often, the goal of such an analysis is to quantify the relationship between an outcome of interest and one or more covariates. The mixed model with proper conditional autoregressive (CAR) spatial random effects is commonly used to model such data but suffers serious drawbacks. First, an analyst must interpret covariate effects conditionally although marginal effects may be of interest. Second, the dependence parameter of the proper CAR model has an intuitive conditional interpretation, but the parameter's marginal interpretation is complicated and counterintuitive; specifically, spatial units with a similar number of neighbors have different marginal correlations. To overcome these two drawbacks, we propose a copula-based hierarchical model with covariance selection. Our approach allows for unbiased estimation of marginal parameters and thus an intuitive marginal interpretation. The covariance-selection copula's single dependence parameter is the first-order correlation. This provides a dependence structure having intuitive conditional and marginal interpretations. We develop a computational framework that permits efficient frequentist inference for our model, even for large datasets. We evaluate the small- and large-sample performance of our method under simulated conditions, and apply our procedure to a widely studied Slovenia stomach cancer dataset.