Controlling for confounding in the presence of measurement error in hierarchical models: a Bayesian approach

A major concern in studies that address the health effects of air pollution is whether an observed association between concentrations of a pollutant and a health outcome is all, or in part, due to the correlation between that exposure and either a second pollutant or a confounder. The addition of exposure measurement error to such data complicates matters further. To account for measurement error when data come from a multi-city study, Schwartz and Coull (2003) proposed a two-stage estimator. These authors showed via both first principles and simulation that their approach yields unbiased estimates for the parameters of interest. However, these estimates have large variability. In this paper, we describe a fully Bayesian approach that yields estimators that are much more efficient than the existing two-stage measurement error correction yet still unbiased. The proposed approach can also incorporate additional exposures or confounders without requiring strict assumptions that are necessary in existing formulations of the model. We compare the properties of the Bayesian estimators to existing approaches via simulation.