Conditionally Affinely Invariant Rerandomization and its Admissible Complete Class

Rerandomization utilizes modern computing ability to improve covariate balance while adhering to the randomization principle originally advocated by RA Fisher. Affinely invariant rerandomization has the ``Equal Percent Variance Reducing'' (EPVR) property. When dealing with covariates of varying importance and/or mixed types, the conditionally EPVR property is often more desired. We discuss a general class of conditionally affinely invariant rerandomization methods and obtain their conditionally EPVR property. In addition, we set up a decision-theoretical framework to evaluate balance criteria for rerandomization. Popular rerandomization methods, such as the covariate balance table check, are found to be inadmissible. We suggest an admissible complete class of conditionally affinely invariant balance criteria, which can be applied to experimental designs involving tiers of covariates, stratification, and multiple treatment arms.