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The method of covariate adjustment is often used for estimation of populationaverage treatment effects in observational studies. Graphical rules fordetermining all valid covariate adjustment sets from an assumed causalgraphical model are well known. Restricting attention to causal linear models,a recent article derived two novel graphical criteria: one to compare theasymptotic variance of linear regression treatment effect estimators thatcontrol for certain distinct adjustment sets and another to identify theoptimal adjustment set that yields the least squares treatment effect estimatorwith the smallest asymptotic variance among consistent adjusted least squaresestimators. In this paper we show that the same graphical criteria can be usedin non-parametric causal graphical models when treatment effects are estimatedby contrasts involving non-parametrically adjusted estimators of theinterventional means. We also provide a graphical criterion for determining theoptimal adjustment set among the minimal adjustment sets, which is valid forboth linear and non-parametric estimators. We provide a new graphical criterionfor comparing time dependent adjustment sets, that is, sets comprised bycovariates that adjust for future treatments and that are themselves affectedby earlier treatments. We show by example that uniformly optimal time dependentadjustment sets do not always exist. In addition, for point interventions, weprovide a sound and complete graphical criterion for determining when anon-parametric optimally adjusted estimator of an interventional mean, or of acontrast of interventional means, is as efficient as an efficient estimator ofthe same parameter that exploits the information in the conditionalindependencies encoded in the non-parametric causal graphical model.

Efficient adjustment sets for population average treatment effect estimation in non-parametric causal graphical models