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.

Frequentists' inference often delivers point estimators associated with confidence intervals or sets for parameters of interest. Constructing the confidence intervals or sets requires understanding the sampling distributions of the point estimators, which, in many but not all cases, are related to asymptotic Normal distributions ensured by central limit theorems. Although previous literature has established various forms of central limit theorems for statistical inference in super population models, we still need general and convenient forms of central limit theorems for some randomization-based causal analysis of experimental data, where the parameters of interests are functions of a finite population and randomness comes solely from the treatment assignment. We use central limit theorems for sample surveys and rank statistics to establish general forms of the finite population central limit theorems that are particularly useful for proving asymptotic distributions of randomization tests under the sharp null hypothesis of zero individual causal effects, and for obtaining the asymptotic repeated sampling distributions of the causal effect estimators. The new central limit theorems hold for general experimental designs with multiple treatment levels and multiple treatment factors, and are immediately applicable for studying the asymptotic properties of many methods in causal inference, including instrumental variable, regression adjustment, rerandomization, clustered randomized experiments, and so on. Previously, the asymptotic properties of these problems are often based on heuristic arguments, which in fact rely on general forms of finite population central limit theorems that have not been established before. Our new theorems fill in this gap by providing more solid theoretical foundation for asymptotic randomization-based causal inference.

Summary Randomization is a basis for the statistical inference of treatment effects without strong assumptions on the outcome‐generating process. Appropriately using covariates further yields more precise estimators in randomized experiments. R. A. Fisher suggested blocking on discrete covariates in the design stage or conducting analysis of covariance in the analysis stage. We can embed blocking in a wider class of experimental design called rerandomization, and extend the classical analysis of covariance to more general regression adjustment. Rerandomization trumps complete randomization in the design stage, and regression adjustment trumps the simple difference‐in‐means estimator in the analysis stage. It is then intuitive to use both rerandomization and regression adjustment. Under the randomization inference framework, we establish a unified theory allowing the designer and analyser to have access to different sets of covariates. We find that asymptotically, for any given estimator with or without regression adjustment, rerandomization never hurts either the sampling precision or the estimated precision, and, for any given design with or without rerandomization, our regression‐adjusted estimator never hurts the estimated precision. Therefore, combining rerandomization and regression adjustment yields better coverage properties and thus improves statistical inference. To quantify these statements theoretically, we discuss optimal regression‐adjusted estimators in terms of the sampling precision and the estimated precision, and then measure the additional gains of the designer and the analyser. We finally suggest the use of rerandomization in the design and regression adjustment in the analysis followed by the Huber–White robust standard error.

There are two general views in causal analysis of experimental data: the super population view that the units are an independent sample from some hypothetical infinite populations, and the finite population view that the potential outcomes of the experimental units are fixed and the randomness comes solely from the physical randomization of the treatment assignment. These two views differs conceptually and mathematically, resulting in different sampling variances of the usual difference-in-means estimator of the average causal effect. Practically, however, these two views result in identical variance estimators. By recalling a variance decomposition and exploiting a completeness-type argument, we establish a connection between these two views in completely randomized experiments. This alternative formulation could serve as a template for bridging finite and super population causal inference in other scenarios.

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