Asymptotic theory of rerandomization in treatment–control experiments

Xinran Li, Peng Ding, Donald B. Rubin

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)9157-9162
Number of pages6
JournalProceedings of the National Academy of Sciences of the United States of America
Issue number37
StatePublished - Sep 11 2018
Externally publishedYes


  • Causal inference
  • Covariate balance
  • Geometry of rerandomization
  • Mahalanobis distance
  • Quantile range

ASJC Scopus subject areas

  • General


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