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 article 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 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 those constructed from the true weighting function. A simulation study reveals that our estimators exhibit satisfactory finite-sample performance, while an application shows their practical value.

Keywords:

• Covariate balancing
• direct and indirect effects
• general treatment
• semiparametric efficiency

Acknowledgments

The authors sincerely thank the editor Esfandiar Maasoumi and the referees for their constructive suggestions and comments. Wei Huang’s research is supported by the Professor Maurice H. Belz Fund of the University of Melbourne.

Supplementary Material

Supplementary Material for “Nonparametric Estimation of Mediation Effects with A General Treatment”: The supplementary material is only for online publication (pdf file). It contains the assumptions required to derive the asymptotic properties of ${\hat{\pi }}_{\mathit{Z}}$ and detailed discussions on the assumptions, the asymptotic results of ${\hat{\pi }}_{\mathit{Z}}$ and the proofs of Theorems 1, 2 and Corollary 1.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1. Our causal mediation framework and the proposed estimation methods can also be adapted to the multiple dimensions of treatment. However, this article focuses on the univariate treatment variable for simplicity of presentation.

Additional information

Funding

Lukang Huang is supported by the Fundamental Research Funds for the Central Universities, Nankai University (No.63231181) and the Natural Science Funds of Tianjin (No.22JCQNJC01660). Zheng Zhang is supported by the fund from the Natural Science Foundation of Beijing, China [grant number 1222007] and the fund for building world-class universities (disciplines) of Renmin University of China [project number KYGJC2023011].