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Bias-Targeted Nonparametric Balancing for Stable Causal Mediation Analysis

Chang Liu, AmirEmad Ghassami

TL;DR

This paper tackles the instability of influence-function-based causal mediation estimators in the presence of continuous mediators by introducing a bias-targeted nuisance estimation framework. It reparametrizes the likelihood to avoid direct mediator-density estimation and employs nonparametric minimax balancing within RKHS to stabilize nuisance components, achieving consistency, multiple robustness, and asymptotic normality. The authors develop a two-stage nuisance estimation procedure for the single-mediator case and extend it to multiple mediators, including kernel-based implementations and theory for bias control. Through extensive simulations and an NHANES application, they demonstrate reduced bias and variance, improved inference, and substantive insights into obesity-related mediation of coronary heart disease via HbA1c.

Abstract

Influence function (IF)-based estimators are widely used in mediation analysis due to their modeling flexibility, but standard implementations require direct estimation of the distribution functions of the mediator and treatment variables. Since these functions appear in the denominator of IF-based estimators, they can induce significant instability, particularly with continuous mediators. In this work, we propose an alternative implementation of IF-based estimators for both single- and multiple-mediator settings, based on reparametrizations of the likelihood. The key idea is to estimate the involved nuisance functions according to their role in the bias structure of the IF-based estimators. In our approach, key nuisance functions that are potential sources of instability are estimated using a novel nonparametric weighted balancing method-which can be viewed as a nonparametric extension of covariate balancing generalized to mediation analysis-fully stabilizing the estimators. We establish consistency and multiple robustness under suitable regularity conditions, and asymptotic normality. Simulation studies demonstrate substantial reductions in bias and variance relative to existing methods for continuous mediators. We further illustrate the approach using NHANES 2013-2014 data to estimate the effect of obesity on coronary heart disease mediated by Glycohemoglobin.

Bias-Targeted Nonparametric Balancing for Stable Causal Mediation Analysis

TL;DR

This paper tackles the instability of influence-function-based causal mediation estimators in the presence of continuous mediators by introducing a bias-targeted nuisance estimation framework. It reparametrizes the likelihood to avoid direct mediator-density estimation and employs nonparametric minimax balancing within RKHS to stabilize nuisance components, achieving consistency, multiple robustness, and asymptotic normality. The authors develop a two-stage nuisance estimation procedure for the single-mediator case and extend it to multiple mediators, including kernel-based implementations and theory for bias control. Through extensive simulations and an NHANES application, they demonstrate reduced bias and variance, improved inference, and substantive insights into obesity-related mediation of coronary heart disease via HbA1c.

Abstract

Influence function (IF)-based estimators are widely used in mediation analysis due to their modeling flexibility, but standard implementations require direct estimation of the distribution functions of the mediator and treatment variables. Since these functions appear in the denominator of IF-based estimators, they can induce significant instability, particularly with continuous mediators. In this work, we propose an alternative implementation of IF-based estimators for both single- and multiple-mediator settings, based on reparametrizations of the likelihood. The key idea is to estimate the involved nuisance functions according to their role in the bias structure of the IF-based estimators. In our approach, key nuisance functions that are potential sources of instability are estimated using a novel nonparametric weighted balancing method-which can be viewed as a nonparametric extension of covariate balancing generalized to mediation analysis-fully stabilizing the estimators. We establish consistency and multiple robustness under suitable regularity conditions, and asymptotic normality. Simulation studies demonstrate substantial reductions in bias and variance relative to existing methods for continuous mediators. We further illustrate the approach using NHANES 2013-2014 data to estimate the effect of obesity on coronary heart disease mediated by Glycohemoglobin.
Paper Structure (23 sections, 16 theorems, 134 equations, 2 figures, 3 tables)

This paper contains 23 sections, 16 theorems, 134 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model Description
  3. Single Mediator
  4. Multiple Mediators
  5. Single-Mediator Case
  6. Influence Function-Based Estimator
  7. Proposed Estimation Approach
  8. Two-Stage Nuisance Function Estimation
  9. A Kernel Method-Based Implementation
  10. Theoretical Analysis
  11. Bias Analysis and Robustness Properties
  12. Asymptotic Normality
  13. Multiple-Mediator Case
  14. Influence Function-Based Estimator
  15. Proposed Estimation Approach
  16. ...and 8 more sections

Key Result

Proposition 1

$\hat{\psi}^{\text{TTS}}$ is unbiased if at least one of the following pairs of nuisance function estimators is correctly specified.

Figures (2)

  • Figure 1: A graphical causal model for the mediation model.
  • Figure 2: Nuisance functions estimated in each stage of the approach. The arrow from $\hat{\pi}_1$ to $\hat{\pi}_2$ indicates that the estimator of $\pi_1$ is used in estimating $\pi_2$. The arrow from $\hat{\mu}_1$ to $\hat{\mu}_2$ is interpreted similarly. Correct specification of any of the pairs of functions in an ellipse suffices unbiasedness of the final estimator of the parameter of interest.

Theorems & Definitions (33)

  • Proposition 1
  • Proposition 2
  • Remark 1
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Theorem 1
  • Proposition 6
  • Theorem 2
  • Proposition 7
  • ...and 23 more