  The marginal structure quantile model (MSQM) provides a unique lens to
understand the causal effect of a time-varying treatment on the full
distribution of potential outcomes. Under the semiparametric framework, we
derive the efficiency influence function for the MSQM, from which a new doubly
robust estimator is proposed for point estimation and inference. We show that
the doubly robust estimator is consistent if either of the models associated
with treatment assignment or the potential outcome distributions is correctly
specified, and is semiparametric efficient if both models are correct. To
implement the doubly robust MSQM estimator, we propose to solve a smoothed
estimating equation to facilitate efficient computation of the point and
variance estimates. In addition, we develop a confounding function approach to
investigate the sensitivity of several MSQM estimators when the sequential
ignorability assumption is violated. Extensive simulations are conducted to
examine the finite-sample performance characteristics of the proposed methods.
We apply the proposed methods to the Yale New Haven Health System Electronic
Health Record data to study the effect of antihypertensive medications to
patients with severe hypertension and assess the robustness of findings to
unmeasured baseline and time-varying confounding.
