Abstract

Spatial confounding, where the inclusion of a spatial random effect introduces multicollinearity with spatially structured covariates, is a contentious and active area of research in spatial statistics. However, the majority of research into this topic has focused on the case of spatial mixed models. In this article, we demonstrate that spatial confounding can also arise in the setting of generalized estimating equations (GEEs). The phenomenon occurs when a spatially structured working correlation matrix is used, as it effectively induces a spatial effect which may exhibit collinearity with the covariates in the marginal mean. As a result, the GEE ends up estimating a so-called unpartitioned effect of the covariates. To overcome spatial confounding, we propose a restricted spatial working correlation matrix that leads the GEE to instead estimate a partitioned covariate effect, which additionally captures the portion of spatial variability in the response spanned by the column space of the covariates. We also examine the construction of sandwich-based standard errors, showing that the issue of efficiency is tied to whether the working correlation matrix aligns with the target effect of interest. We conclude by highlighting the need for practitioners to make clear the assumptions and target of interest when applying GEEs in a spatial setting, and not simply rely on the robustness property of GEEs to misspecification of the working correlation matrix.

摘要

空间混杂，其中包含空间随机效应引入了具有空间结构协变量的多重共线性，是空间统计研究中一个有争议且活跃的领域。然而，对该主题的大多数研究都集中在空间混合模型的情况下。在本文中，我们证明了在广义估计方程 (GEE) 的设置中也会出现空间混杂。当使用空间结构化的工作相关矩阵时，就会出现这种现象，因为它有效地引发了空间效应，该效应可能与边际均值中的协变量表现出共线性。结果，GEE 最终估计了协变量的所谓未分区效应。为了克服空间混淆，我们提出了一个受限的空间工作相关矩阵，该矩阵导致 GEE 改为估计分区协变量效应，该矩阵还捕获了协变量列空间所跨越的响应中的空间变异部分。我们还检查了基于三明治的标准误差的构造，表明效率问题与工作相关矩阵是否与感兴趣的目标效应一致。最后，我们强调从业者在空间环境中应用 GEE 时需要明确假设和感兴趣的目标，而不是简单地依赖 GEE 的稳健性来错误指定工作相关矩阵。它还捕获了由协变量的列空间跨越的响应中的空间可变性部分。我们还检查了基于三明治的标准误差的构造，表明效率问题与工作相关矩阵是否与感兴趣的目标效应一致。最后，我们强调从业者在空间环境中应用 GEE 时需要明确假设和感兴趣的目标，而不是简单地依赖 GEE 的稳健性来错误指定工作相关矩阵。它还捕获了由协变量的列空间跨越的响应中的空间可变性部分。我们还检查了基于三明治的标准误差的构造，表明效率问题与工作相关矩阵是否与感兴趣的目标效应一致。最后，我们强调从业者在空间环境中应用 GEE 时需要明确假设和感兴趣的目标，而不是简单地依赖 GEE 的稳健性来错误指定工作相关矩阵。