Restricted Spatial Regression Methods: Implications for Inference

Abstract

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 article, 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 nonspatial 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. Supplementary materials for this article are available online.

KEYWORDS:

• Bayesian
• Confounding
• Dimension reduction
• Spatial statistics

Acknowledgments

Gabriel Khan is credited for suggesting the clever use of Hölder’s inequality in Theorem 1, which resulted in a relatively cleaner proof.

Additional information

Funding

The authors were partially supported by NIH funding: NIH NICHD R01-HD088545. They also acknowledge the support of The Ohio State University’s Mathematical Biosciences Institute (NSF DMS-1440386) and the Institute for Population Research (NIH funding: NIH NICHD P2C-HD058484-10).