Construction of Counterfactuals and the G-computation Formula

Zhuo Yu, Department of Statistics, University of California, Berkeley
Mark J. van der Laan, Division of Biostatistics, School of Public Health, University of California, Berkeley

Abstract

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. Gill and Robins [2001] show that the consistency and randomization assumptions do not add any restrictions to the observed data distribution. In particular, they provide a construction of counterfactuals as a function of the observed data distribution. In this paper we provide a construction of counterfactuals as a function of the observed data itself. Our construction provides a new statistical tool for estimation of counterfactual distributions. Robins [1987b] 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 and Robins [2001] prove the G-computation formula for continuous variable under some additional conditions and modifications of the consistency and the randomization assumptions. We prove that if treatment is discrete, then Robins' G-computation formula holds under the original consistency, randomization assumptions and a generalized version of identifiability assumption.