The Annals of Statistics

A robust and efficient approach to causal inference based on sparse sufficient dimension reduction

Shujie Ma, Liping Zhu, Zhiwei Zhang, Chih-Ling Tsai, and Raymond J. Carroll

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Abstract

A fundamental assumption used in causal inference with observational data is that treatment assignment is ignorable given measured confounding variables. This assumption of no missing confounders is plausible if a large number of baseline covariates are included in the analysis, as we often have no prior knowledge of which variables can be important confounders. Thus, estimation of treatment effects with a large number of covariates has received considerable attention in recent years. Most existing methods require specifying certain parametric models involving the outcome, treatment and confounding variables, and employ a variable selection procedure to identify confounders. However, selection of a proper set of confounders depends on correct specification of the working models. The bias due to model misspecification and incorrect selection of confounding variables can yield misleading results. We propose a robust and efficient approach for inference about the average treatment effect via a flexible modeling strategy incorporating penalized variable selection. Specifically, we consider an estimator constructed based on an efficient influence function that involves a propensity score and an outcome regression. We then propose a new sparse sufficient dimension reduction method to estimate these two functions without making restrictive parametric modeling assumptions. The proposed estimator of the average treatment effect is asymptotically normal and semiparametrically efficient without the need for variable selection consistency. The proposed methods are illustrated via simulation studies and a biomedical application.

Article information

Source
Ann. Statist., Volume 47, Number 3 (2019), 1505-1535.

Dates
Received: May 2017
Revised: February 2018
First available in Project Euclid: 13 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.aos/1550026847

Digital Object Identifier
doi:10.1214/18-AOS1722

Mathematical Reviews number (MathSciNet)
MR3911120

Zentralblatt MATH identifier
07053516

Subjects
Primary: 62G08: Nonparametric regression
Secondary: 62G10: Hypothesis testing 62G20: Asymptotic properties 62J07: Ridge regression; shrinkage estimators

Keywords
Average treatment effect dimension reduction high-dimensional data multiple-index model outcome regression semiparametric efficiency

Citation

Ma, Shujie; Zhu, Liping; Zhang, Zhiwei; Tsai, Chih-Ling; Carroll, Raymond J. A robust and efficient approach to causal inference based on sparse sufficient dimension reduction. Ann. Statist. 47 (2019), no. 3, 1505--1535. doi:10.1214/18-AOS1722. https://projecteuclid.org/euclid.aos/1550026847


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Supplemental materials

  • Supplement to “A robust and efficient approach to causal inference based on sparse sufficient dimension reduction”. The supplement contains the technical proof of Theorem 1, two lemmas that will be used in the proof of Theorem 2, and additional simulation studies.