  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 (ANCOVA) in the analysis
stage. We can embed blocking into a wider class of experimental design called
rerandomization, and extend the classical ANCOVA 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 analyzer to have access to
different sets of covariates. We find that asymptotically (a) for any given
estimator with or without regression adjustment, rerandomization never hurts
either the sampling precision or the estimated precision, and (b) 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 theoretically quantify these statements, 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 analyzer. We finally suggest using rerandomization in the design and
regression adjustment in the analysis followed by the Huber--White robust
standard error.
