# Restricted Spatial Regression Methods: Implications for Inference

## TL;DR

This paper critically examines Restricted Spatial Regression (RSR) methods, revealing that they often worsen inference in spatial data analysis and cannot be remedied by simple basis vector selection, challenging current practices.

## Contribution

The paper provides a unified mathematical framework for RSR methods, analyzes their impact on inference, and demonstrates their counterproductive effects through simulations in both linear and generalized linear models.

## Key findings

- RSR methods can impair inference for regression coefficients.
- Counterintuitive results show RSR often performs worse than non-spatial methods.
- Problems with RSR cannot be fixed by choosing better spatial basis vectors.

## 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 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.