  Hierarchical models for regionally aggregated disease incidence data commonly
involve region specific latent random effects that are modeled jointly as
having a multivariate Gaussian distribution. The covariance or precision matrix
incorporates the spatial dependence between the regions. Common choices for the
precision matrix include the widely used ICAR model, which is singular, and its
nonsingular extension which lacks interpretability. We propose a new parametric
model for the precision matrix based on a directed acyclic graph (DAG)
representation of the spatial dependence. Our model guarantees positive
definiteness and, hence, in addition to being a valid prior for regional
spatially correlated random effects, can also directly model the outcome from
dependent data like images and networks. Theoretical results establish a link
between the parameters in our model and the variance and covariances of the
random effects. Substantive simulation studies demonstrate that the improved
interpretability of our model reaps benefits in terms of accurately recovering
the latent spatial random effects as well as for inference on the spatial
covariance parameters. Under modest spatial correlation, our model far
outperforms the CAR models, while the performances are similar when the spatial
correlation is strong. We also assess sensitivity to the choice of the ordering
in the DAG construction using theoretical and empirical results which testify
to the robustness of our model. We also present a large-scale public health
application demonstrating the competitive performance of the model.
