The Importance of Scale for Spatial-Confounding Bias and Precision of Spatial Regression Estimators

TL;DR

This paper investigates how the spatial scale of residuals and covariates influences bias and precision in spatial regression models, highlighting that model choice should consider the spatial scales to mitigate bias from unmeasured confounders.

Contribution

It provides a simple analytic framework to understand the impact of spatial scales on bias and efficiency in spatial regression, especially under unmeasured confounding.

Findings

Bias depends on the relative spatial scales of covariate and residuals.

Fitting spatial models only when covariate variation is at a smaller scale than confounding reduces bias.

Controlling for large-scale spatial variation can reduce bias but increase uncertainty.

Abstract

Residuals in regression models are often spatially correlated. Prominent examples include studies in environmental epidemiology to understand the chronic health effects of pollutants. I consider the effects of residual spatial structure on the bias and precision of regression coefficients, developing a simple framework in which to understand the key issues and derive informative analytic results. When unmeasured confounding introduces spatial structure into the residuals, regression models with spatial random effects and closely-related models such as kriging and penalized splines are biased, even when the residual variance components are known. Analytic and simulation results show how the bias depends on the spatial scales of the covariate and the residual: one can reduce bias by fitting a spatial model only when there is variation in the covariate at a scale smaller than the scale of the unmeasured confounding. I also discuss how the scales of the residual and the covariate affect efficiency and uncertainty estimation when the residuals are independent of the covariate. In an application on the association between black carbon particulate matter air pollution and birth weight, controlling for large-scale spatial variation appears to reduce bias from unmeasured confounders, while increasing uncertainty in the estimated pollution effect.