  Missing data arise in most applied settings and are ubiquitous in electronic
health records (EHR). When data are missing not at random (MNAR) with respect
to measured covariates, sensitivity analyses are often considered. These
post-hoc solutions, however, are often unsatisfying in that they are not
guaranteed to yield concrete conclusions. Motivated by an EHR-based study of
long-term outcomes following bariatric surgery, we consider the use of double
sampling as a means to mitigate MNAR outcome data when the statistical goals
are estimation and inference regarding causal effects. We describe assumptions
that are sufficient for the identification of the joint distribution of
confounders, treatment, and outcome under this design. Additionally, we derive
efficient and robust estimators of the average causal treatment effect under a
nonparametric model and under a model assuming outcomes were, in fact,
initially missing at random (MAR). We compare these in simulations to an
approach that adaptively estimates based on evidence of violation of the MAR
assumption. Finally, we also show that the proposed double sampling design can
be extended to handle arbitrary coarsening mechanisms, and derive nonparametric
efficient estimators of any smooth full data functional.
