Generalized Conditionally Autoregressive Models

Abstract

In many studies, lattice or area data are observed and spatial analysis is performed. A spatial process observed over a lattice or a set of irregular regions is usually modeled using a conditionally autoregressive (CAR) model. The neighborhoods within a CAR model are generally formed using only the inter-distances or boundaries between the regions. To accommodate the effect of directions, a new class of spatial models is developed using different weights given to neighbors in different directions. The proposed model generalizes the usual CAR model by accounting for spatial anisotropy. Maximum likelihood (ML) estimators are derived and shown to be consistent and asymptotically normal under some regularity conditions. Also, the posterior distribution of the parameters are derived using conjugate and non-informative priors. Efficient MCMC sampling algorithms are provided to generate samples from the marginal posterior distribution.

Simulation studies are presented to illustrate the finite sample performance of the new model as compared to CAR model. The method is demonstrated using a data set on the crime rates in Columbus, OH. Further generalization of the directional CAR model is proposed that adaptively chosen the neighborhoods based on a smooth function of the inter-distances and inter-angles between the regions.

The parameters of this generalized CAR are estimated using ML and Bayes estimators. A data set on the prevalence of elevated blood lead levels of children under the age of six years observed in the state of Virginia is used to illustrate the use of the generalized CAR models.