  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.
