The generalized g-formula can be used to estimate the probability of survival under a sustained treatment strategy. When treatment strategies are deterministic, estimators derived from the so-called efficient influence function (EIF) for the g-formula will be doubly robust to model misspecification. In recent years, several practical applications have motivated estimation of the g-formula under non-deterministic treatment strategies where treatment assignment at each time point depends on the observed treatment process. In this case, EIF-based estimators may or may not be doubly robust. In this paper, we provide sufficient conditions to ensure existence of doubly robust estimators for intervention treatment distributions that depend on the observed treatment process for point treatment interventions, and give a class of intervention treatment distributions dependent on the observed treatment process that guarantee model doubly and multiply robust estimators in longitudinal settings. Motivated by an application to pre-exposure prophylaxis (PrEP) initiation studies, we propose a new treatment intervention dependent on the observed treatment process. We show there exist 1) estimators that are doubly and multiply robust to model misspecification, and 2) estimators that when used with machine learning algorithms can attain fast convergence rates for our proposed intervention. Theoretical results are confirmed via simulation studies.

翻译：通用 g 公式可以用来估计在持续治疗战略下存活的概率。当治疗战略具有确定性时,根据所谓的g 公式有效影响功能(EIF)得出的估计值将加倍加强,以模拟偏差。近年来,一些实际应用促使在非确定性治疗战略下根据每个时间点的治疗分配取决于观察的治疗过程对g 公式进行估计。在这种情况下,基于 EIF的估计值可能或可能不会加倍强大。在本文中,我们提供了充分的条件,以确保存在对干预治疗分布的双重强力估计值,这种估计值取决于点治疗干预措施的观察治疗过程,并给根据观察的治疗过程提供一类干预治疗分布值。近年来,一些实际应用促使在非确定性治疗战略下对g 公式进行估计,因为每次时间的治疗分配取决于观察到的治疗过程。在这种情况下,基于EF的估算值的估算值可能或可能不会加倍强大。在观察的治疗过程之后,我们提出了一种新的治疗干预干预干预干预干预干预干预干预措施。我们指出,有充足的估计值是双重和倍增倍的,因为根据观察性测算结果,通过模拟学习模型得出结果时,可以实现快速的模拟结果。