Effect decomposition through multiple causally nonordered mediators in the presence of exposure-induced mediator-outcome confounding

Avin et al (2005) showed that, in the presence of exposure-induced mediator-outcome confounding, decomposing the total causal effect (TCE) using standard conditional exchangeability assumptions is not possible even under a nonparametric structural equation model with all confounders observed. Subsequent research has investigated the assumptions required for such a decomposition to be identifiable and estimable from observed data. One approach was proposed by VanderWeele et al (2014). They decomposed the TCE under three different scenarios: (1) treating the mediator and the exposure-induced confounder as joint mediators; (2) generating path-specific effects albeit without distinguishing between multiple distinct paths through the exposure-induced confounder; and (3) using so-called randomised interventional analogues where sampling values from the distribution of the mediator within the levels of the exposure effectively marginalises over the exposure-induced confounder. In this paper, we extend their approach to the case where there are multiple mediators that do not influence each other directly but which are all influenced by an exposure-induced mediator-outcome confounder. We provide a motivating example and results from a simulation study based on from our work in dental epidemiology featuring the 1982 Pelotas Birth Cohort in Brazil.