  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.
