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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.
Comments Published in at http://dx.doi.org/10.1214/10-STS326 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org)