Estimating mean potential outcome under adaptive treatment length strategies in continuous time

An adaptive treatment length strategy is a sequential stage‐wise treatment strategy where a subject's treatment begins at baseline and one chooses to stop or continue treatment at each stage provided the subject has been continuously treated. The effects of treatment are assumed to be cumulative and, therefore, the effect of treatment length on clinical endpoint, measured at the end of the study, is of primary scientific interest. At the same time, adverse treatment‐terminating events may occur during the course of treatment that require treatment be stopped immediately. Because the presence of a treatment‐terminating event may be strongly associated with the study outcome, the treatment‐terminating event is informative. In observational studies, decisions to stop or continue treatment depend on covariate history that confounds the relationship between treatment length on outcome. We propose a new risk‐set weighted estimator of the mean potential outcome under the condition that time‐dependent covariates update at a set of common landmarks. We show that our proposed estimator is asymptotically linear given mild assumptions and correctly specified working models. Specifically, we study the theoretical properties of our estimator when the nuisance parameters are modeled using either parametric or semiparametric methods. The finite sample performance and theoretical results of the proposed estimator are evaluated through simulation studies and demonstrated by application to the Enhanced Suppression of the Platelet Receptor IIb/IIIa with Integrilin Therapy (ESPRIT) infusion trial data.