  The method of covariate adjustment is often used for estimation of population
average treatment effects in observational studies. Graphical rules for
determining all valid covariate adjustment sets from an assumed causal
graphical model are well known. Restricting attention to causal linear models,
a recent article derived two novel graphical criteria: one to compare the
asymptotic variance of linear regression treatment effect estimators that
control for certain distinct adjustment sets and another to identify the
optimal adjustment set that yields the least squares treatment effect estimator
with the smallest asymptotic variance among consistent adjusted least squares
estimators. In this paper we show that the same graphical criteria can be used
in non-parametric causal graphical models when treatment effects are estimated
by contrasts involving non-parametrically adjusted estimators of the
interventional means. We also provide a graphical criterion for determining the
optimal adjustment set among the minimal adjustment sets, which is valid for
both linear and non-parametric estimators. We provide a new graphical criterion
for comparing time dependent adjustment sets, that is, sets comprised by
covariates that adjust for future treatments and that are themselves affected
by earlier treatments. We show by example that uniformly optimal time dependent
adjustment sets do not always exist. In addition, for point interventions, we
provide a sound and complete graphical criterion for determining when a
non-parametric optimally adjusted estimator of an interventional mean, or of a
contrast of interventional means, is as efficient as an efficient estimator of
the same parameter that exploits the information in the conditional
independencies encoded in the non-parametric causal graphical model.
