Adjustment for Missing Confounders Using External Validation Data and Propensity Scores

Reducing bias from missing confounders is a challenging problem in the analysis of observational data. Information about missing variables is sometimes available from external validation data, such as surveys or secondary samples drawn from the same source population. In principle, the validation data permits us to recover information about the missing data, but the difficulty is in eliciting a valid model for nuisance distribution of the missing confounders. Motivated by a British study of the effects of trihalomethane exposure on risk of full-term low birthweight, we describe a flexible Bayesian procedure for adjusting for a vector of missing confounders using external validation data. We summarize the missing confounders with a scalar summary score using the propensity score methodology of Rosenbaum and Rubin. The score has the property that it induces conditional independence between the exposure and the missing confounders given the measured confounders. It balances the unmeasured confounders across exposure groups, within levels of measured covariates. To adjust for bias, we need only model and adjust for the summary score during Markov chain Monte Carlo simulation. Simulation results illustrate that the proposed method reduces bias from several missing confounders over a range of different sample sizes for the validation data.