When to initiate treatment on patients is an important problem in many medical studies such as AIDS and cancer. In this article, we formulate the treatment initiation time problem for time-to-event data and propose an optimal individualized regime that determines the best treatment initiation time for individual patients based on their characteristics. Different from existing optimal treatment regimes where treatments are undertaken at a pre-specified time, here new challenges arise from the complicated missing mechanisms in treatment initiation time data and the continuous treatment rule in terms of initiation time. To tackle these challenges, we propose to use restricted mean residual lifetime as a value function to evaluate the performance of different treatment initiation regimes, and develop a nonparametric estimator for the value function, which is consistent even when treatment initiation times are not completely observable and their distribution is unknown. We also establish the asymptotic properties of the resulting estimator in the decision rule and its associated value function estimator. In particular, the asymptotic distribution of the estimated value function is nonstandard, which follows a weighted chi-squared distribution. The finite-sample performance of the proposed method is evaluated by simulation studies and is further illustrated with an application to a breast cancer data.

中文翻译：

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基于受限平均剩余寿命估计治疗开始时间的最佳方案

何时开始对患者进行治疗是艾滋病和癌症等许多医学研究中的一个重要问题。在这篇文章中，我们针对事件发生时间数据制定了治疗开始时间问题，并提出了一种最佳的个体化方案，根据患者的特征确定最佳治疗开始时间。与在预先指定的时间进行治疗的现有最佳治疗方案不同，这里的新挑战来自治疗开始时间数据中复杂的缺失机制和开始时间方面的连续治疗规则。为了应对这些挑战，我们建议使用受限平均剩余寿命作为价值函数来评估不同治疗启动方案的性能，并为价值函数开发一个非参数估计器，即使不能完全观察到治疗开始时间并且它们的分布未知，这也是一致的。我们还在决策规则及其关联的价值函数估计器中建立了结果估计器的渐近特性。特别是，估计值函数的渐近分布是非标准的，服从加权卡方分布。所提出方法的有限样本性能通过模拟研究进行评估，并通过对乳腺癌数据的应用进一步说明。估计值函数的渐近分布是非标准的，服从加权卡方分布。所提出方法的有限样本性能通过模拟研究进行评估，并通过对乳腺癌数据的应用进一步说明。估计值函数的渐近分布是非标准的，服从加权卡方分布。所提出方法的有限样本性能通过模拟研究进行评估，并通过对乳腺癌数据的应用进一步说明。