  Confounding by unmeasured spatial variables has received some attention in
the spatial statistics and causal inference literatures, but concepts and
approaches have remained largely separated. In this paper, we aim to bridge
these distinct strands of statistics by considering unmeasured spatial
confounding within a causal inference framework, and estimating effects using
outcome regression tools popular within the spatial literature. First, we show
how using spatially correlated random effects in the outcome model, an approach
common among spatial statisticians, does not necessarily mitigate bias due to
spatial confounding, a previously published but not universally known result.
Motivated by the bias term of commonly-used estimators, we propose an affine
estimator which addresses this deficiency. We discuss how unbiased estimation
of causal parameters in the presence of unmeasured spatial confounding can only
be achieved under an untestable set of assumptions which will often be
application-specific. We provide a set of assumptions which describe how the
exposure and outcome of interest relate to the unmeasured variables, and we
show that this set of assumptions is sufficient for identification of the
causal effect based on the observed data when spatial dependencies can be
represented by a ring graph. We implement our method using a fully Bayesian
approach applicable to any type of outcome variable. This work is motivated by
and used to estimate the effect of county-level limited access to supermarkets
on the rate of cardiovascular disease deaths in the elderly across the whole
continental United States. Even though standard approaches return null or
protective effects, our approach uncovers evidence of unobserved spatial
confounding, and indicates that limited supermarket access has a harmful effect
on cardiovascular mortality.
