University of Washington - Statistics
We present a Bayesian model for area-level count data that uses Gaussian random effects with a novel type of G-Wishart prior on the inverse variance-covariance matrix. The usual G-Wishart prior restricts off-diagonal elements of the precision matrix to 0 according to the neighborhood structure of the study region. This preserves conditional independence of non-neighboring regions but is more flexible than the traditional intrinsic autoregression prior. One drawback of the usual G-Wishart prior is that it allows for both positive and negative associations between neighboring areas; whereas, most spatial priors induce only positive pairwise associations between the relative risks of neighboring areas. In this work we introduce a new type of G-Wishart distribution, which we call the negative G-Wishart distribution. This distribution only puts support over precision matrices that lead to positive associations. We illustrate Markov chain Monte Carlo sampling algorithms for the negative G-Wishart prior in a disease mapping context and compare our results to Bayesian hierarchical models based on intrinsic autoregression priors. We show that using the negative G-Wishart prior improves over the intrinsic autoregressive priors when there are discontinuities in the disease risk surface.