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Contributed by Donald B. Rubin, June 29, 2018 (sent for review May 17, 2018; reviewed by Robert J. Tibshirani and C. F. Jeff Wu)
Significance
Rerandomization refers to experimental designs that enforce covariate balance. This paper studies the asymptotic properties of the difference-in-means estimator under rerandomization, based on the randomness of the treatment assignment without imposing any parametric modeling assumptions on the covariates or outcome. The non-Gaussian asymptotic distribution allows for constructing large-sample confidence intervals for the average treatment effect and demonstrates the advantages of rerandomization over complete randomization.
Abstract
Although complete randomization ensures covariate balance on average, the chance of observing significant differences between treatment and control covariate distributions increases with many covariates. Rerandomization discards randomizations that do not satisfy a predetermined covariate balance criterion, generally resulting in better covariate balance and more precise estimates of causal effects. Previous theory has derived finite sample theory for rerandomization under the assumptions of equal treatment group sizes, Gaussian covariate and outcome distributions, or additive causal effects, but not for the general sampling distribution of the difference-in-means estimator for the average causal effect. We develop asymptotic theory for rerandomization without these assumptions, which reveals a non-Gaussian asymptotic distribution for this estimator, specifically a linear combination of a Gaussian random variable and truncated Gaussian random variables. This distribution follows because rerandomization affects only the projection of potential outcomes onto the covariate space but does not affect the corresponding orthogonal residuals. We demonstrate that, compared with complete randomization, rerandomization reduces the asymptotic quantile ranges of the difference-in-means estimator. Moreover, our work constructs accurate large-sample confidence intervals for the average causal effect.
Footnotes
- ↵1To whom correspondence should be addressed. Email: dbrubin{at}mac.com.
Author contributions: X.L., P.D., and D.B.R. designed research, performed research, contributed new reagents/analytic tools, analyzed data, and wrote the paper.
Reviewers: R.J.T., Stanford University; and C.F.J.W., Georgia Institute of Technology.
The authors declare no conflict of interest.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1808191115/-/DCSupplemental.
Published under the PNAS license.
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Asymptotic theory of rerandomization in treatment–control experiments
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