In clinical studies, patients take treatment over time according to pre-determined plans by their physicians. The effect of treatment duration is often of clinical interest in such studies. The treatment, however, may discontinue due to some undesired complications which compete with treatment assignment sequentially over time and the existence of such discontinuation complicates the data analysis. Thus, although the treatment length is specified by a physician \textit{a priori} the actual treatment length will be adapted based on patient's ongoing health status. In such settings,

treatment duration is considered as a continuous endogenous variable that is affected by a set of time-dependent confounders.  Existing methods often discretize the treatment duration or fail to adjust for time-varying confounders which can result in a biased estimate of the mean outcome under an adaptive treatment length strategy.

We propose an inverse probability risk-set weighted estimator that can be used with continuous treatment duration when treatment assignment and treatment discontinuation compete.

We show that the proposed estimator is asymptotically linear given the classical assumptions in causal inference 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 ESPRIT infusion trial data.

Keywords --- Adaptive treatment strategies, Causal inference,  Informative eligibility,  Treatment competing events,  Treatment discontinuation,  Survival analysis