Robins' causal inference theory assumes existence of treatment specific counterfactual variables so that the observed data augmented by the counterfactual data will satisfy a consistency and a randomization assumption. In this paper we provide an explicit function that maps the observed data into a counterfactual variable which satisfies the consistency and randomization assumptions. This offers a practically useful imputation method for counterfactuals. Gill & Robins [2001]'s construction of counterfactuals can be used as an imputation method in principle, but it is very hard to implement in practice. Robins [1987] shows that the counterfactual distribution can be identified from the observed data distribution by a G-computation formula under an additional identifiability assumption. He proves this for discrete variables. Gill & Robins [2001] prove the G-computation formula for continuous variables under some continuity assumptions and reformulation of the consistency and the randomization assumptions. We prove that if treatment is discrete (which deals with a less general case compared with Gill & Robins [2001], then Robins' G-computation formula holds under the original consistency, randomization assumptions and a generalized version of the identifiability assumption.


Statistical Methodology | Statistical Theory