Principal stratification with U-statistics under principal ignorability

Principal stratification is a popular framework for causal inference in the presence of an intermediate outcome. While the principal average treatment effects have traditionally been the default target of inference, it may not be sufficient when the interest lies in the relative favorability of one potential outcome over the other within the principal stratum. We thus introduce the principal generalized causal effect estimands, which extend the principal average causal effects to accommodate nonlinear contrast functions. Under principal ignorability, we expand the theoretical results in Jiang et. al. (2022) to a much wider class of causal estimands in the presence of a binary intermediate variable. We develop identification formulas and derive the efficient influence functions of the generalized estimands for principal stratification analyses. These efficient influence functions motivate a set of multiply robust estimators and lay the ground for obtaining efficient debiased machine learning estimators via cross-fitting based on $U$-statistics. The proposed methods are illustrated through simulations and the analysis of a data example.