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Abstract:
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There has been an increased interest in the impact that spatial confounding between the spatial random effect and the fixed effects in regression analyses can have on inference. Multiple authors have offered perspectives on this issue and potential solutions. We offer a mathematically based exploration of how these solutions impact the inference on regression coefficients for the linear model. To do so, we illustrate that many of the methods designed to alleviate spatial confounding can be viewed as special cases of a general class of models, which we refer to as Restricted Spatial Regression (RSR) models (extending terminology currently in use). We show that RSR models 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. Furthermore, the problems with RSR models cannot be fixed with a selection of ``better" spatial basis vectors or dimension reduction techniques. A simulation study of count data indicates these results may extend to the generalized linear model setting.
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