Causal mediation analysis with mediator-outcome confounders affected by exposure - on definition and identification of generalized natural indirect effect.

Causal mediation analysis aims to disentangle the pathways through which an exposure influences an outcome. In the presence of mediator-outcome confounders affected by exposure (intermediate confounders), the natural indirect effect (NIE) is not identifiable under nonparametric structural equation models (SEM) with independent errors. To address this challenge, we focus on the indirect pathway and introduce a novel class of indirect effect measures, referred to as generalized natural indirect effects, of which the NIE is a special case. In particular, we introduce a case of generalized NIE defined through a randomized intervention, which, under the nonparametric SEM with independent errors, coincides with the interventional indirect effect (IIE)-even though identifying the IIE generally does not rely on the cross-world assumptions implied by nonparametric SEM with independent errors. Furthermore, when an additional no-heterogeneity assumption is imposed, the NIE becomes equal to this generalized NIE and hence identifiable. Unlike prior approaches, we propose new indirect effect measures criteria that ensure valid mediation interpretation even in the presence of intermediate confounders. Under traditional identification assumptions alone, the IIE fails to satisfy these criteria. In contrast, all proposed generalized NIEs meet them, providing a wide range of options beyond the existing measures. Our findings highlight the generalized NIEs as a more pragmatic and reasonable alternative in settings where intermediate confounders are inevitable.