Multiply Robust Causal Mediation Analysis with Continuous Treatments

In many applications, researchers are interested in the direct and indirect causal effects of an intervention on an outcome of interest. Mediation analysis offers a rigorous framework for the identification and estimation of such causal quantities. In the case of binary treatment, efficient estimators for the direct and indirect effects are derived by Tchetgen Tchetgen and Shpitser (2012). These estimators are based on influence functions and possess desirable multiple robustness properties. However, they are not readily applicable when treatments are continuous, which is the case in several settings, such as drug dosage in medical applications. In this work, we extend the influence function-based estimator of Tchetgen Tchetgen and Shpitser (2012) to deal with continuous treatments by utilizing a kernel smoothing approach. We first demonstrate that our proposed estimator preserves the multiple robustness property of the estimator in Tchetgen Tchetgen and Shpitser (2012). Then we show that under certain mild regularity conditions, our estimator is asymptotically normal. Our estimation scheme allows for high-dimensional nuisance parameters that can be estimated at slower rates than the target parameter. Additionally, we utilize cross-fitting, which allows for weaker smoothness requirements for the nuisance functions.