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

We propose semi- and non-parametric methods to estimate conditional interventional 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 consider the problem of estimating these effects at particular points. We propose an influence-function based estimator of the projection of the conditional effects onto a working model, and show under some conditions that we can achieve root-n consistent and asymptotically normal estimates. 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. 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 examining heterogeneous effects with respect to the effect of COVID-19 vaccinations on depression during February 2021.