Proximal Causal Inference for Conditional Separable Effects

Scientists often pose questions about treatment effects on outcomes conditional on a post-treatment event. However, defining, identifying, and estimating causal effects conditional on post-treatment events requires care, even in perfectly executed randomized experiments. Recently, the conditional separable effect (CSE) was proposed as an interventionist estimand, corresponding to scientifically meaningful questions in these settings. However, while being a single-world estimand, which can be queried experimentally, existing identification results for the CSE require no unmeasured confounding between the outcome and post-treatment event. This assumption can be violated in many applications. In this work, we address this concern by developing new identification and estimation results for the CSE in the presence of unmeasured confounding. We establish nonparametric identification of the CSE in both observational and experimental settings when certain proxy variables are available for hidden common causes of the post-treatment event and outcome. We characterize the efficient influence function for the CSE under a semiparametric model of the observed data law in which nuisance functions are a priori unrestricted. Moreover, we develop a consistent, asymptotically linear, and locally semiparametric efficient estimator of the CSE using modern machine learning theory. We illustrate our framework with simulation studies and a real-world cancer therapy trial.