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 on this issue 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. Extending terminology currently in use, we refer to this class as Restricted Spatial Regression (RSR) models. Using this insight, 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.
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