Heterogeneous interventional indirect effects with multiple mediators: non-parametric and semi-parametric approaches

We propose semi- and non-parametric methods to estimate conditional interventional indirect effects in the setting of two discrete mediators whose causal ordering is unknown. Average interventional indirect effects have been shown to decompose an average treatment effect into a direct effect and interventional indirect effects that quantify effects of hypothetical interventions on mediator distributions. Yet these effects may be heterogeneous across the covariate distribution. We therefore consider the problem of estimating these effects at particular points. We first propose an influence-function based estimator of the projection of the conditional effects onto a working model, and show that under some conditions we can achieve root-n consistent and asymptotically normal estimates of this parameter. Second, we propose a fully non-parametric approach to estimation and show the conditions where this approach can achieve oracle rates of convergence. Finally, we propose a sensitivity analysis for the conditional effects in the presence of mediator-outcome confounding given a bounded outcome. We propose estimating bounds on the conditional effects using these same methods, and show that these results easily extend to allow for influence-function based estimates of the bounds on the average effects. We conclude by demonstrating our methods to examine heterogeneous mediated effects with respect to the effect of COVID-19 vaccinations on depression via social isolation and worries about health during February 2021.