  The issue of spatial confounding between the spatial random effect and the
fixed effects in regression analyses has been identified as a concern in the
statistical literature. Multiple authors have offered perspectives and
potential solutions. In this paper, for the areal spatial data setting, we show
that many of the methods designed to alleviate spatial confounding can be
viewed as special cases of a general class of models. We refer to this class as
Restricted Spatial Regression (RSR) models, extending terminology currently in
use. We offer a mathematically based exploration of the impact that RSR methods
have on inference for regression coefficients for the linear model. We then
explore whether these results hold in the generalized linear model setting for
count data using simulations. We show that the use of these methods have
counterintuitive consequences which defy the general expectations in the
literature. In particular, our results and the accompanying simulations suggest
that RSR methods will typically perform worse than non-spatial methods. These
results have important implications for dimension reduction strategies in
spatial regression modeling. Specifically, we demonstrate that the problems
with RSR models cannot be fixed with a selection of "better" spatial basis
vectors or dimension reduction techniques.
