Bayesian inference for directional conditionally autoregressive models

Abstract

Counts or averages over arbitrary regions are often analyzed using conditionally autoregressive (CAR) models. The neighborhoods within CAR models are generally determined using only the inter-distances or boundaries between the sub-regions. To accommodate spatial variations that may depend on directions, a new class of models is developed using different weights given to neighbors in different directions. By accounting for such spatial anisotropy, the proposed model generalizes the usual CAR model that assigns equal weight to all directions. Within a fully hierarchical Bayesian framework, the posterior distributions of the parameters are derived using conjugate and non-informative priors. Efficient Markov chain Monte Carlo (MCMC) sampling algorithms are provided to generate samples from the marginal posterior distribution of the parameters. Simulation studies are presented to evaluate the performance of the estimators and are used to compare results with traditional CAR models. Finally the method is illustrated using data sets on local crime frequencies in Columbus, OH and on the elevated blood lead levels of children under the age of 72 months observed in Virginia counties for the year of 2000.