October 2022 Bounds on the conditional and average treatment effect with unobserved confounding factors
Steve Yadlowsky, Hongseok Namkoong, Sanjay Basu, John Duchi, Lu Tian
Author Affiliations +
Ann. Statist. 50(5): 2587-2615 (October 2022). DOI: 10.1214/22-AOS2195


For observational studies, we study the sensitivity of causal inference when treatment assignments may depend on unobserved confounders. We develop a loss minimization approach for estimating bounds on the conditional average treatment effect (CATE) when unobserved confounders have a bounded effect on the odds ratio of treatment selection. Our approach is scalable and allows flexible use of model classes in estimation, including nonparametric and black-box machine learning methods. Based on these bounds for the CATE, we propose a sensitivity analysis for the average treatment effect (ATE). Our semiparametric estimator extends/bounds the augmented inverse propensity weighted (AIPW) estimator for the ATE under bounded unobserved confounding. By constructing a Neyman orthogonal score, our estimator of the bound for the ATE is a regular root-n estimator so long as the nuisance parameters are estimated at the op(n1/4) rate. We complement our methodology with optimality results showing that our proposed bounds are tight in certain cases. We demonstrate our method on simulated and real data examples, and show accurate coverage of our confidence intervals in practical finite sample regimes with rich covariate information.

Funding Statement

SY was partially supported by a Stanford Graduate Fellowship, and HN was partially supported by Samsung Fellowship. JCD was partially supported by NSF CAREER Award CCF-1553086 and the Office of Naval Research Young Investigator Award N00014-19-1-2288.
Research reported in this publication was supported by the National Heart, Lung, and Blood Institute; the National Institute Of Diabetes And Digestive And Kidney Diseases; and the National Institute On Minority Health And Health Disparities of the National Institutes of Health under Award Numbers R01HL144555, R01DK116852, R21MD012867, DP2MD010478 and U54MD010724.


We would like to acknowledge useful comments, discussions and feedback from Stefan Wager and Nigam Shah on this work. SY would also like to thank Mike Baiocchi for his discussions and encouragement to study unobserved confounding.

The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.


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Steve Yadlowsky. Hongseok Namkoong. Sanjay Basu. John Duchi. Lu Tian. "Bounds on the conditional and average treatment effect with unobserved confounding factors." Ann. Statist. 50 (5) 2587 - 2615, October 2022. https://doi.org/10.1214/22-AOS2195


Received: 1 June 2020; Revised: 1 February 2022; Published: October 2022
First available in Project Euclid: 27 October 2022

MathSciNet: MR4505372
zbMATH: 07628833
Digital Object Identifier: 10.1214/22-AOS2195

Primary: 62H12
Secondary: 62F03 , 62F30 , 62H15

Keywords: Causal estimation , omitted variable , semiparametric inference , sensitivity analysis

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 5 • October 2022
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