Summary We study the selection of adjustment sets for estimating the interventional mean under a point exposure dynamic treatment regime, that is, a treatment rule that depends on the subject’s covariates. We assume a nonparametric causal graphical model with, possibly, hidden variables and at least one adjustment set comprised of observable variables. We provide the definition of a valid adjustment set for a point exposure dynamic treatment regime, which generalizes the existing definition for a static intervention. We show that there exists an adjustment set, referred to as optimal minimal, that yields the nonparametric estimator of the interventional mean with the smallest asymptotic variance among those that are based on observable minimal adjustment sets. An observable minimal adjustment set is a valid adjustment set such that all its variables are observable and the removal of any of its variables destroys its validity. We provide similar optimality results for the class of observable minimum adjustment sets, that is, valid observable adjustment sets of minimum cardinality among the observable adjustment sets. Moreover, we show that if either no variables are hidden or if all the observable variables are ancestors of either treatment, outcome or the variables that are used to decide treatment, a globally optimal adjustment set exists. We provide polynomial-time algorithms to compute the globally optimal, optimal minimal and optimal minimum adjustment sets. Because static interventions can be viewed as a special case of dynamic regimes, all our results also apply for static interventions.

中文翻译：

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具有隐藏变量的因果图模型中的有效调整集

总结 我们研究了在点暴露动态治疗方案（即依赖于受试者协变量的治疗规则）下估计介入平均值的调整集的选择。我们假设一个非参数因果图模型，可能包含隐藏变量和至少一个由可观察变量组成的调整集。我们为点暴露动态治疗方案提供了有效调整集的定义，它概括了静态干预的现有定义。我们表明存在一个调整集，称为最优最小值，它产生干预均值的非参数估计量，在基于可观察的最小调整集的那些中具有最小的渐近方差。一个可观察的最小调整集是一个有效的调整集，使得它的所有变量都是可观察的，并且删除它的任何变量都会破坏它的有效性。我们为可观察最小调整集的类别提供了类似的最优性结果，即在可观察调整集中具有最小基数的有效可观察调整集。此外，我们表明，如果没有隐藏变量，或者如果所有可观察变量都是治疗、结果或用于决定治疗的变量的祖先，则存在全局最优调整集。我们提供多项式时间算法来计算全局最优、最优最小和最优最小调整集。因为静态干预可以被视为动态制度的一个特例，我们所有的结果也适用于静态干预。