# Heterogeneous Indirect Effects for Multiple Mediators using Interventional Effect Models

## TL;DR

This paper introduces new estimation methods for heterogeneous interventional indirect effects with multiple mediators, accommodating unknown causal structures and unmeasured confounding, using interventional effect models and two estimation procedures.

## Contribution

It develops simplified, flexible estimation procedures for interventional indirect effects that allow effect modification by baseline covariates, applicable to continuous and noncontinuous mediators.

## Key findings

- Proposed inverse weighting and Monte Carlo integration methods for effect estimation.
- Methods accommodate unknown causal structures and unmeasured confounding.
- Effect modification by baseline covariates is readily incorporated.

## Abstract

Decomposing an exposure effect on an outcome into separate natural indirect effects through multiple mediators requires strict assumptions, such as correctly postulating the causal structure of the mediators, and no unmeasured confounding among the mediators. In contrast, interventional indirect effects for multiple mediators can be identified even when - as often - the mediators either have an unknown causal structure, or share unmeasured common causes, or both. Existing estimation methods for interventional indirect effects require calculating each distinct indirect effect in turn. This can quickly become unwieldy or unfeasible, especially when investigating indirect effect measures that may be modified by observed baseline characteristics. In this article, we introduce simplified estimation procedures for such heterogeneous interventional indirect effects using interventional effect models. Interventional effect models are a class of marginal structural models that encode the interventional indirect effects as causal model parameters, thus readily permitting effect modification by baseline covariates using (statistical) interaction terms. The mediators and outcome can be continuous or noncontinuous. We propose two estimation procedures: one using inverse weighting by the counterfactual mediator density or mass functions, and another using Monte Carlo integration. The former has the advantage of not requiring an outcome model, but is susceptible to finite sample biases due to highly variable weights. The latter has the advantage of consistent estimation under a correctly specified (parametric) outcome model, but is susceptible to biases due to extrapolation.