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Multiply Robust Causal Mediation Analysis with Continuous Treatments

Yizhen Xu, Numair Sani, AmirEmad Ghassami, Ilya Shpitser

TL;DR

This work extends the influence function-based estimator of Tchetgen Tchet Gen and Shpitser (2012) to deal with continuous treatments by utilizing a kernel smoothing approach and preserves the multiple robustness property of the estimator.

Abstract

In many applications, researchers are interested in the direct and indirect causal effects of a treatment or exposure on an outcome of interest. Mediation analysis offers a rigorous framework for identifying and estimating these causal effects. For binary treatments, efficient estimators for the direct and indirect effects are presented by Tchetgen Tchetgen and Shpitser (2012) based on the influence function of the parameter of interest. These estimators possess desirable properties such as multiple-robustness and asymptotic normality while allowing for slower than root-n rates of convergence for the nuisance parameters. However, in settings involving continuous treatments, these influence function-based estimators are not readily applicable without making strong parametric assumptions. In this work, utilizing a kernel-smoothing approach, we propose an estimator suitable for settings with continuous treatments inspired by the influence function-based estimator of Tchetgen Tchetgen and Shpitser (2012). Our proposed approach employs cross-fitting, relaxing the smoothness requirements on the nuisance functions and allowing them to be estimated at slower rates than the target parameter. Additionally, similar to influence function-based estimators, our proposed estimator is multiply robust and asymptotically normal, allowing for inference in settings where parametric assumptions may not be justified.

Multiply Robust Causal Mediation Analysis with Continuous Treatments

TL;DR

This work extends the influence function-based estimator of Tchetgen Tchet Gen and Shpitser (2012) to deal with continuous treatments by utilizing a kernel smoothing approach and preserves the multiple robustness property of the estimator.

Abstract

In many applications, researchers are interested in the direct and indirect causal effects of a treatment or exposure on an outcome of interest. Mediation analysis offers a rigorous framework for identifying and estimating these causal effects. For binary treatments, efficient estimators for the direct and indirect effects are presented by Tchetgen Tchetgen and Shpitser (2012) based on the influence function of the parameter of interest. These estimators possess desirable properties such as multiple-robustness and asymptotic normality while allowing for slower than root-n rates of convergence for the nuisance parameters. However, in settings involving continuous treatments, these influence function-based estimators are not readily applicable without making strong parametric assumptions. In this work, utilizing a kernel-smoothing approach, we propose an estimator suitable for settings with continuous treatments inspired by the influence function-based estimator of Tchetgen Tchetgen and Shpitser (2012). Our proposed approach employs cross-fitting, relaxing the smoothness requirements on the nuisance functions and allowing them to be estimated at slower rates than the target parameter. Additionally, similar to influence function-based estimators, our proposed estimator is multiply robust and asymptotically normal, allowing for inference in settings where parametric assumptions may not be justified.

Paper Structure

This paper contains 22 sections, 6 theorems, 143 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Mediation Analysis
  3. Estimating Natural Direct and Indirect Effects
  4. Continuous Mediation Analysis
  5. Asymptotic Analysis
  6. Simulation Study
  7. Application
  8. Conclusion
  9. Proof for Theorem 1
  10. Proof for Terms (CS1)-(CS6)
  11. Proof for Terms (E1)-(E8)
  12. Proof for Terms (TR1)-(TR5)
  13. Proof For Asymptotic Normality
  14. Calculation for $B(a, a')$ and $\mu_i$
  15. Calculation for $V(a, a')$ and $s^2_n$
  16. ...and 7 more sections

Key Result

Theorem 1

Suppose Assumptions assumption:id-assumption:DR hold. Then for any values of $a,a'\in\mathcal{A}$, Additionally, if Assumption assumption:lyapunov-conditions holds, then $\sqrt{nh^{d_A}}(\hat{\psi}^{MR}(a,a')-\psi_0(a,a')-h^2 B(a,a'))$ converges to the Gaussian distribution $\mathcal{N}(0,V(a,a'))$, where $B(a,a')$ and $V(a, a^\prime)$ are defined as and

Figures (3)

  • Figure 1: A graphical representation of the decomposition of total effect into direct and indirect effects. Part $(a)$ represents the indirect effect, part $(b)$ represents the direct effect, and part $(c)$ represents the total effect.
  • Figure 2: Direct effect $\hat{\psi}^{MR}(a, a'=40)$ for $a\in \{100, 200, \ldots, 2000\}$ under Silverman bandwidth and clipping of the Hajek propensity at 0.01.
  • Figure 3: (a) Contour plot of the estimated mean of the natural direct effect comparing treatments $a = 1500$ and $a' = 40$ using the proposed cross-fitting approach. (b) Contour plot of the estimated standard deviation of the natural direct effect comparing treatments $a = 1500$ and $a' = 40$ using the proposed cross-fitting approach.

Theorems & Definitions (8)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5