Causal Quantile Treatment Effects with missing data by double-sampling

Causal weighted quantile treatment effects (WQTE) are a useful complement to standard causal contrasts that focus on the mean when interest lies at the tails of the counterfactual distribution. To-date, however, methods for estimation and inference regarding causal WQTEs have assumed complete data on all relevant factors. In most practical settings, however, data will be missing or incomplete data, particularly when the data are not collected for research purposes, as is the case for electronic health records and disease registries. Furthermore, such data sources may be particularly susceptible to the outcome data being missing-not-at-random (MNAR). In this paper, we consider the use of double-sampling, through which the otherwise missing data are ascertained on a sub-sample of study units, as a strategy to mitigate bias due to MNAR data in the estimation of causal WQTEs. With the additional data in-hand, we present identifying conditions that do not require assumptions regarding missingness in the original data. We then propose a novel inverse-probability weighted estimator and derive its asymptotic properties, both pointwise at specific quantiles and uniformly across a range of quantiles over some compact subset of (0,1), allowing the propensity score and double-sampling probabilities to be estimated. For practical inference, we develop a bootstrap method that can be used for both pointwise and uniform inference. A simulation study is conducted to examine the finite sample performance of the proposed estimators. The proposed method is illustrated with data from an EHR-based study examining the relative effects of two bariatric surgery procedures on BMI loss at 3 years post-surgery.