  Despite the abundance of methods for variable selection and accommodating
spatial structure in regression models, there is little precedent for
incorporating spatial dependence in covariate inclusion probabilities for
regionally varying regression models. The lone existing approach is limited by
difficult computation and the requirement that the spatial dependence be
represented on a lattice, making this method inappropriate for areal models
with irregular structures that often arise in ecology, epidemiology, and the
social sciences. Here we present a novel method for spatial variable selection
in areal generalized linear models that can accommodate arbitrary spatial
structures and works with a broad subset of GLM likelihoods. The method uses a
latent probit model with a spatial dependence structure where the binary
response is taken as a covariate inclusion indicator for area-specific GLMs.
The covariate inclusion indicators arise via thresholding of latent standard
normals on which we place a conditionally autoregressive prior. We propose an
efficient MCMC algorithm for computation that is entirely conjugate in any
model with a conditionally Gaussian representation of the likelihood, thereby
encompassing logistic, probit, multinomial probit and logit, Gaussian, and
negative binomial regressions through the use of existing data augmentation
methods. We demonstrate superior parameter recovery and prediction in
simulation studies as well as in applications to geographic voting patterns and
population estimation. Though the method is very broadly applicable, we note in
particular that prior to this work, spatial population
estimation/capture-recapture models allowing for varying list dependence
structures has not been possible.
