Statistical inference for the probability of necessity for causal attribution

Abstract

To answer questions of "causes of effects", the probability of necessity was previously introduced for assessing whether an observed outcome was caused by an earlier treatment. However, statistical inference for the probability of necessity is understudied due to several difficulties, which hinder its application in practice. The evaluation of the probability of necessity involves the joint distribution of potential outcomes, and thus it is generally not point identified and one can at best obtain lower and upper bounds even in randomized experiments, unless fairly stringent monotonicity assumptions on potential outcomes are made. Moreover, these bounds are non-smooth functionals of the observed data distribution and standard estimation and inference methods cannot be directly applied. In this paper, we investigate the statistical inference for the probability of necessity in general situations where it may not be point identified. We introduce a mild margin condition to tackle the non-smoothness, under which the bounds become pathwise differentiable. We establish the semiparametric efficiency theory and propose novel asymptotically efficient estimators of the lower and upper bounds, and further construct confidence intervals for the probability of necessity based on the proposed bound estimators. The resultant confidence intervals can effectively utilize the observed covariates to reduce lengths. The proposed approach has potential application in biomedical, epidemiological, and legal studies where understanding causal attribution beyond traditional causal effects is essential.