Formulating Cross-World Mediation Estimands Through Single-World Mixtures

Abstract

In causal mediation analysis, natural direct and indirect effects are defined through nested counterfactuals that combine the outcome under one exposure level with the mediator value under another and are therefore inherently cross-world. Their canonical identification additionally relies on cross-world independence assumptions. Consequently, both the estimands and their identifying assumptions remain controversial. In this paper, we ask what additional single-world structure would be required to re-express cross-world estimands as single-world quantities, identifiable under single-world assumptions. We show that these quantities can be written as mixtures of controlled single-world effects under assumptions involving a susceptibility marker, a possibly latent baseline variable that encodes the "would-be" mediator value under a reference treatment arm. If observed, the marker would make several implications of the marker restrictions empirically testable. These results provide a transparent single-world formulation of cross-world mediation effects while making explicit the assumptions required. Although these conditions clarify how cross-world estimands can be interpreted within a single-world framework, their practical relevance depends on whether such susceptibility markers can be justified in specific applications. We discuss implications for principal stratification and show in the Supplementary Material how the formulation extends to ordered mediators and settings with exposure-induced mediator-outcome confounding.