Variable elimination, graph reduction and efficient g-formula

TL;DR

This paper introduces graphical criteria to identify and eliminate uninformative variables in causal models, enabling more efficient estimation of interventional means without sacrificing statistical efficiency.

Contribution

It develops a complete set of graphical criteria for variable elimination and constructs a reduced graph to simplify estimation while maintaining efficiency.

Findings

Uninformative variables can be eliminated without losing estimation efficiency.

The reduced graph allows identification of the interventional mean via the g-formula.

The semiparametric variance bounds are preserved after variable elimination.

Abstract

We study efficient estimation of an interventional mean associated with a point exposure treatment under a causal graphical model represented by a directed acyclic graph without hidden variables. Under such a model, it may happen that a subset of the variables are uninformative in that failure to measure them neither precludes identification of the interventional mean nor changes the semiparametric variance bound for regular estimators of it. We develop a set of graphical criteria that are sound and complete for eliminating all the uninformative variables so that the cost of measuring them can be saved without sacrificing estimation efficiency, which could be useful when designing a planned observational or randomized study. Further, we construct a reduced directed acyclic graph on the set of informative variables only. We show that the interventional mean is identified from the marginal law by the g-formula (Robins, 1986) associated with the reduced graph, and the semiparametric variance bounds for estimating the interventional mean under the original and the reduced graphical model agree. This g-formula is an irreducible, efficient identifying formula in the sense that the nonparametric estimator of the formula, under regularity conditions, is asymptotically efficient under the original causal graphical model, and no formula with such property exists that only depends on a strict subset of the variables.