Spatial Confounding in Generalized Estimating Equations

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.

KEYWORDS:

• Marginal models
• Restricted spatial regression
• Sandwich covariance
• Spatial correlation
• Unconditional effect
• Working correlation

Supplementary Material

Additional Discussion:

Appendix A presents details for estimating the unrestricted and restricted spatial GEEs, Appendix B discusses the constant mean-variance function case in more detail, and Appendix F discusses large sample behaviour.

Additional Simulation and Application Results:

Appendices C and D presents further numerical results for the simulation study in Section 4, and Appendix E presents further exploratory plots for the application to the pelagics species richness data.

Funding

This work was supported by the Australian Research Council, under Grant DE200100435. Thanks to Thomas Lumley and Tao Zou for useful discussions.