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Nonparametric Estimation of Mediation Effects with A General Treatment

Lukang Huang, Wei Huang, Oliver Linton, Zheng Zhang

Abstract

To investigate causal mechanisms, causal mediation analysis decomposes the total treatment effect into the natural direct and indirect effects. This paper examines the estimation of the direct and indirect effects in a general treatment effect model, where the treatment can be binary, multi-valued, continuous, or a mixture. We propose generalized weighting estimators with weights estimated by solving an expanding set of equations. Under some sufficient conditions, we show that the proposed estimators are consistent and asymptotically normal. Specifically, when the treatment is discrete, the proposed estimators attain the semiparametric efficiency bounds. Meanwhile, when the treatment is continuous, the convergence rates of the proposed estimators are slower than $N^{-1/2}$; however, they are still more efficient than that constructed from the true weighting function. A simulation study reveals that our estimators exhibit a satisfactory finite-sample performance, while an application shows their practical value

Nonparametric Estimation of Mediation Effects with A General Treatment

Abstract

To investigate causal mechanisms, causal mediation analysis decomposes the total treatment effect into the natural direct and indirect effects. This paper examines the estimation of the direct and indirect effects in a general treatment effect model, where the treatment can be binary, multi-valued, continuous, or a mixture. We propose generalized weighting estimators with weights estimated by solving an expanding set of equations. Under some sufficient conditions, we show that the proposed estimators are consistent and asymptotically normal. Specifically, when the treatment is discrete, the proposed estimators attain the semiparametric efficiency bounds. Meanwhile, when the treatment is continuous, the convergence rates of the proposed estimators are slower than ; however, they are still more efficient than that constructed from the true weighting function. A simulation study reveals that our estimators exhibit a satisfactory finite-sample performance, while an application shows their practical value
Paper Structure (17 sections, 7 theorems, 48 equations, 2 figures, 3 tables)

This paper contains 17 sections, 7 theorems, 48 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Basic Framework
  3. Estimation
  4. Unified Framework for Estimating the Weighting Functions
  5. Final Estimator of $\mu(t,t')$
  6. Large Sample Properties
  7. Variance Estimation
  8. Selecting the Smoothing Parameters
  9. Monte Carlo Simulation
  10. Continuous Treatment
  11. Binary Treatment
  12. Application
  13. Discussion and Conclusion
  14. Proof of \ref{["id:mutt'"]}
  15. Some Preliminary Results
  16. ...and 2 more sections

Key Result

Theorem 1

Under Assumptions as:SequentialIgnore -- as:ddmatrix and as:suppX -- as:u&v presented in Appendix appendix:preliminary,we have

Figures (2)

  • Figure 1: Estimated direct effects $\widehat{\mu}(t,t)-\widehat{\mu}(40,t)$ (top left) and $\widehat{\mu}(t,40)-\widehat{\mu}(40,40)$ (top right), and indirect effects $\widehat{\mu}(t,t)-\widehat{\mu}(t,40)$ (bottom left) and $\widehat{\mu}(40,t)-\widehat{\mu}(40,40)$ (bottom right) for $t\in\{100,200,\cdots,2000\}$, with the estimated $95\%$ confidence bands (dashed lines).
  • Figure 2: Estimated direct effects $\widehat{\mu}_h(t,t)-\widehat{\mu}_h(40,t)$ (top left) and $\widehat{\mu}_h(t,40)-\widehat{\mu}_h(40,40)$ (top right), and indirect effects $\widehat{\mu}_h(t,t)-\widehat{\mu}_h(t,40)$ (bottom left) and $\widehat{\mu}_h(40,t)-\widehat{\mu}_h(40,40)$ (bottom right) for $t\in\{100,200,\cdots,2000\}$, with the estimated $95\%$ confidence bands (dashed lines).

Theorems & Definitions (13)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Remark 5
  • Proposition 1
  • Proposition 2
  • ...and 3 more