Title: Spatial disease mapping using Directed Acyclic Graph Auto-Regressive (DAGAR) models,
Abstract: Hierarchical models for regionally aggregated disease incidence data commonly
involve region specific latent random effects that are modelled 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 intrinsic conditional autoregressive
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 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 and
empirical results demonstrate the interpretability of parameters in our model.
Our precision matrix is sparse and the model is highly scalable for large
datasets. We also derive a novel order-free version which remedies the
dependence of directed acyclic graphs on the ordering of the regions by
averaging over all possible orderings. The resulting precision matrix is
available in closed form. We demonstrate the superior performance of our models
over competing models using simulation experiments and a public health
application.