Efficient adjustment sets in causal graphical models with hidden variables

Abstract

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.