Abstract

Adjusting for several unmeasured confounders is a challenging problem in the analysis of observational data. Information about unmeasured confounders is sometimes available from external validation data, such as surveys or secondary samples drawn from the same source population. In principal, the validation permits us to recover information about the missing data, but the difficulty is in eliciting a valid model for nuisance distribution of the unmeasured confounders. Motivated by a British study of the effects of trihalomethane exposure on full-term low birthweight, we describe a flexible Bayesian procedure for adjusting for a vector of unmeasured confounders using external validation data. We summarize the unmeasured confounders with a scalar summary score using the propensity score methodology of Rosenbaum and Rubin. The score has the property that it breaks the dependence between the exposure and unmeasured confounders within levels of measured confounders. To adjust for unmeasured confounding in a Bayesian analysis, we need only update and adjust for the summary score during Markov chain Monte Carlo simulation. We demonstrate that trihalomethane exposure is associated with increased risk of full-term low birthweight, and this association persists even after adjusting for eight unmeasured confounders. Empirical results from simulation illustrate that our proposed method eliminates bias from several unmeasured confounders, even in small samples.