Linear models of coregionalization for multivariate lattice data: a general framework for coregionalized multivariate CAR models.

We present a general coregionalization framework for developing coregionalized multivariate Gaussian conditional autoregressive (cMCAR) models for Bayesian analysis of multivariate lattice data in general and multivariate disease mapping data in particular. This framework is inclusive of cMCARs that facilitate flexible modelling of spatially structured symmetric or asymmetric cross-variable local interactions, allowing a wide range of separable or non-separable covariance structures, and symmetric or asymmetric cross-covariances, to be modelled. We present a brief overview of established univariate Gaussian conditional autoregressive (CAR) models for univariate lattice data and develop coregionalized multivariate extensions. Classes of cMCARs are presented by formulating precision structures. The resulting conditional properties of the multivariate spatial models are established, which cast new light on cMCARs with richly structured covariances and cross-covariances of different spatial ranges. The related methods are illustrated via an in-depth Bayesian analysis of a Minnesota county-level cancer data set. We also bring a new dimension to the traditional enterprize of Bayesian disease mapping: estimating and mapping covariances and cross-covariances of the underlying disease risks. Maps of covariances and cross-covariances bring to light spatial characterizations of the cMCARs and inform on spatial risk associations between areas and diseases. Copyright © 2016 John Wiley & Sons, Ltd.