Estimating weighted quantile treatment effects with missing outcome data by double sampling

Causal weighted quantile treatment effects (WQTEs) complement standard mean-focused causal contrasts when interest lies at the tails of the counterfactual distribution. However, existing methods for estimating and inferring causal WQTEs assume complete data on all relevant factors, which is often not the case in practice, particularly when the data are not collected for research purposes, such as electronic health records (EHRs) and disease registries. Furthermore, these data may be particularly susceptible to the outcome data being missing-not-at-random (MNAR). This paper proposes to use 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 estimating causal WQTEs. With the additional data, we present identifying conditions that do not require missingness assumptions 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 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. We illustrate the proposed method using EHR data examining the relative effects of 2 bariatric surgery procedures on BMI loss 3 years post-surgery.