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
di�culty is in eliciting a valid model for nuisance distribution of the missing confounders. Motivated
by a British study of the e�ects of trihalomethane exposure on risk of full-term low
birthweight, we describe a
exible 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 di�erent sample sizes for the validation data.