We consider estimation of and inference for the mean outcome under the optimal dynamic two time-point treatment rule defined as the rule that maximizes the mean outcome under the dynamic treatment, where the candidate rules are restricted to depend only on a user-supplied subset of the baseline and intermediate covariates. This estimation problem is addressed in a statistical model for the data distribution that is nonparametric beyond possible knowledge about the treatment and censoring mechanism. This contrasts from the current literature that relies on parametric assumptions. We establish that the mean of the counterfactual outcome under the optimal dynamic treatment is a pathwise differentiable parameter under conditions, and develop a targeted minimum loss-based estimator (TMLE) of this target parameter. We establish asymptotic linearity and statistical inference for this estimator under specified conditions. In a sequentially randomized trial the statistical inference relies upon a second order difference between the estimator of the optimal dynamic treatment and the optimal dynamic treatment to be asymptotically negligible, which may be a problematic condition when the rule is based on multivariate time-dependent covariates. To avoid this condition, we also develop targeted minimum loss based estimators and statistical inference for data adaptive target parameters that are defined in terms of the mean outcome under the estimate of the optimal dynamic treatment. In particular, we develop a novel cross-validated TMLE approach that provides asymptotic inference under minimal conditions, avoiding the need for any empirical process conditions. We offer simulation results to support our theoretical findings. This work expands upon that of an earlier technical report (van der Laan, 2013; van der Laan and Luedtke, 2014) with new results and simulations, and is accompanied by a work which explores the estimation of the optimal rule (Luedtke and van der Laan, 2014).
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机译：我们考虑最佳动态两个时间点处理规则下对平均结果的估计和推断，该最佳动态两个时间点处理规则定义为在动态处理下最大化平均结果的规则，其中候选规则仅依赖于用户提供的基线和中间协变量。在估计数据的统计模型中解决了这一估计问题，该模型超出了关于处理和检查机制的可能知识，是非参数的。这与依赖于参数假设的当前文献形成对比。我们建立了在最佳动态处理下反事实结果的平均值在一定条件下是一个沿途可微分的参数，并开发了该目标参数的目标最小基于损失的估计量（TMLE）。我们在指定条件下为该估计量建立渐近线性和统计推断。在顺序随机试验中，统计推断依赖于最优动态处理的估计量与最优动态处理的估计量之间的二阶差异，渐近可忽略，当规则基于多变量时变协变量时，这可能是一个有问题的情况。为了避免这种情况，我们还针对数据自适应目标参数开发了基于目标最小损失的估计量和统计推断，这些数据根据最佳动态治疗方法下的平均结果定义。特别是，我们开发了一种新颖的交叉验证TMLE方法，该方法可在最小条件下提供渐近推断，而无需任何经验过程条件。我们提供仿真结果以支持我们的理论发现。这项工作是在早期技术报告（van der Laan，2013年； van der Laan和Luedtke，2014年）的基础上进行扩展的，并提供了新的结果和模拟，并伴随着探索最佳规则估计的工作（Luedtke和van der Laan，2014年）。
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