Estimating the causal effect of a time-dependent treatment on time to death is challenging. In this paper, we formulate the problem using the illness-death model and focus on a stochastic intervention that modifies the hazard governing the transition from no treatment to treatment initiation. Such an intervention can only be implemented at the level of the observed data, whereas the causally valid intervention is defined at the level of the true data-generating process. We provide conditions under which the practically feasible intervention corresponds to the desired causal intervention in the specific setting. We first consider an intervention in which treatment is initiated at a fixed time point, which may subsequently be varied across the relevant time span. However, the resulting estimand is not pathwise differentiable, preventing the development of assumption-lean inference. To address this, we instead consider a smoothed intervention that assigns treatment within a time window around the target time point, thereby yielding a parameter amenable to semiparametric analysis. We derive the corresponding efficient influence function and propose a debiased one-step estimator with desirable robustness properties. We investigate its finite-sample performance in a simulation study and apply the method to the classical Stanford Heart Transplant data, as well as to data on treatment delay among couples with unexplained subfertility seeking intrauterine insemination.

翻译：估计随时间变化治疗对生存时间的因果效应具有挑战性。本文利用疾病-死亡模型构建该问题，并聚焦于一种随机干预，该干预可改变控制从无治疗状态向治疗起始状态转化的风险函数。此类干预只能在观测数据层面实施，而因果有效的干预是在真实数据生成过程层面定义的。我们给出了在特定条件下，实际可行干预与期望因果干预相对应的条件。首先考虑治疗在固定时间点启动的干预，该时间点可在相关时间跨度内变化。但由此产生的估计量不具有路径可微性，阻碍了假设精简推断的发展。为应对这一问题，我们转而考虑一种平滑化干预，其在目标时间点周围的时间窗口内分配治疗，从而得到适用于半参数分析的参数。我们推导了相应的有效协函数，并提出了具有良好稳健性的去偏一步估计量。通过模拟研究考察其在有限样本下的表现，并将该方法应用于经典斯坦福心脏移植数据，以及寻求宫内人工授精的不明原因亚不孕夫妇的治疗延迟数据。