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.

摘要

回归分析中空间随机效应和固定效应之间的空间混杂问题已被确定为统计文献中的一个关注点。多位作者提供了观点和潜在的解决方案。在这篇文章中，对于区域空间数据设置，我们展示了许多旨在减轻空间混杂的方法可以看作是一类通用模型的特例。我们将此类称为受限空间回归 (RSR) 模型，扩展了当前使用的术语。我们基于数学探索 RSR 方法对线性模型回归系数推断的影响。然后，我们使用模拟探索这些结果是否适用于计数数据的广义线性模型设置。我们表明，使用这些方法会产生违反直觉的后果，这违背了文献中的普遍预期。特别是，我们的结果和伴随的模拟表明 RSR 方法通常比非空间方法表现更差。这些结果对空间回归建模中的降维策略具有重要意义。具体来说，我们证明了 RSR 模型的问题无法通过选择“更好”的空间基向量或降维技术来解决。本文的补充材料可在线获取。这些结果对空间回归建模中的降维策略具有重要意义。具体来说，我们证明了 RSR 模型的问题无法通过选择“更好”的空间基向量或降维技术来解决。本文的补充材料可在线获取。这些结果对空间回归建模中的降维策略具有重要意义。具体来说，我们证明了 RSR 模型的问题无法通过选择“更好”的空间基向量或降维技术来解决。本文的补充材料可在线获取。