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Quadratic robust methods for causal mediation analysis

Zhen Qi, Yuqian Zhang

TL;DR

Quadratic robust methods for causal mediation analysis introduces a quadruply robust framework to identify natural direct and indirect effects under weaker model assumptions in high-dimensional contexts. It presents two estimation strategies: a nonparametric QR estimator compatible with machine-learning nuisance estimation and a parametric MQR estimator that uses specialized losses to achieve sqrt(N)-consistency under misspecification. Theoretical results establish consistency and asymptotic normality under broader model classes than existing triply robust methods, and simulations show favorable finite-sample performance with a real-data application (ACTG175) confirming substantive mediation through early immune recovery. Overall, the work advances robust causal mediation analysis by expanding identifiability conditions and providing practical estimators for high-dimensional mediation problems.

Abstract

Estimating natural effects is a core task in causal mediation analysis. Existing triply robust (TR) frameworks (Tchetgen Tchetgen & Shpitser 2012) and their extensions have been developed to estimate the natural effects. In this work, we introduce a new quadruply robust (QR) framework that enlarges the model class for unbiased identification. We study two modeling strategies. The first is a nonparametric modeling approach, under which we propose a general QR estimator that supports the use of machine learning methods for nuisance estimation. We also study high-dimensional settings, where the dimensions of covariates and mediators may both be large. In these settings, we adopt a parametric modeling strategy and develop a model quadruply robust (MQR) estimator to limit the impact of model misspecification. Simulation studies and a real data application demonstrate the finite-sample performance of the proposed methods.

Quadratic robust methods for causal mediation analysis

TL;DR

Quadratic robust methods for causal mediation analysis introduces a quadruply robust framework to identify natural direct and indirect effects under weaker model assumptions in high-dimensional contexts. It presents two estimation strategies: a nonparametric QR estimator compatible with machine-learning nuisance estimation and a parametric MQR estimator that uses specialized losses to achieve sqrt(N)-consistency under misspecification. Theoretical results establish consistency and asymptotic normality under broader model classes than existing triply robust methods, and simulations show favorable finite-sample performance with a real-data application (ACTG175) confirming substantive mediation through early immune recovery. Overall, the work advances robust causal mediation analysis by expanding identifiability conditions and providing practical estimators for high-dimensional mediation problems.

Abstract

Estimating natural effects is a core task in causal mediation analysis. Existing triply robust (TR) frameworks (Tchetgen Tchetgen & Shpitser 2012) and their extensions have been developed to estimate the natural effects. In this work, we introduce a new quadruply robust (QR) framework that enlarges the model class for unbiased identification. We study two modeling strategies. The first is a nonparametric modeling approach, under which we propose a general QR estimator that supports the use of machine learning methods for nuisance estimation. We also study high-dimensional settings, where the dimensions of covariates and mediators may both be large. In these settings, we adopt a parametric modeling strategy and develop a model quadruply robust (MQR) estimator to limit the impact of model misspecification. Simulation studies and a real data application demonstrate the finite-sample performance of the proposed methods.
Paper Structure (28 sections, 32 theorems, 421 equations, 1 figure, 4 tables, 2 algorithms)

This paper contains 28 sections, 32 theorems, 421 equations, 1 figure, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background and Related Works
  3. Quadruply Robust Estimation
  4. A quadruply robust estimator
  5. Theoretical properties of the QR estimator
  6. Model Quadruply Robust Estimation
  7. A model quadruply robust estimator
  8. Theoretical properties of the MQR estimator
  9. Numerical Experiments
  10. Simulation results
  11. Real data application
  12. Discussion
  13. Additional Discussions and Justifications
  14. An alternative DR estimator for in Section \ref{['sec:QR']}
  15. Uniqueness of in Section \ref{['sec:MQR']}
  16. ...and 13 more sections

Key Result

Theorem 3.1

Let Assumptions cond:basic hold. If either $\mu^*=\mu$ or $q^*=q$, then $\tau(\boldsymbol{X})=\tau_n(\boldsymbol{X})+\mathbb{E}[q^*(\boldsymbol{S})\{Y-\mu^*(\boldsymbol{S})\}\mid A=1,\boldsymbol{X}]$.

Figures (1)

  • Figure 1: A directed acyclic graph illustrating the causal relationships among treatment $A$, mediator $\boldsymbol{M}$, outcome $Y$, and confounder $\boldsymbol{X}$.

Theorems & Definitions (60)

  • Theorem 3.1: A DR representation for $\tau$
  • Theorem 3.2: A QR representation for $\theta$
  • Theorem 3.3: Consistency
  • Theorem 3.4: Asymptotic normality
  • Theorem 4.1: Improved convergence
  • Theorem 4.2: Robust inference
  • Lemma A.1
  • proof : Proof of Lemma \ref{['lem:r_identi']}
  • Example 1
  • proof : Proof of Theorem \ref{['thm:dr_tau']}
  • ...and 50 more