Rerandomization to Balance Tiers of Covariates.

When conducting a randomized experiment, if an allocation yields treatment groups that differ meaningfully with respect to relevant covariates, groups should be rerandomized. The process involves specifying an explicit criterion for whether an allocation is acceptable, based on a measure of covariate balance, and rerandomizing units until an acceptable allocation is obtained. Here we illustrate how rerandomization could have improved the design of an already conducted randomized experiment on vocabulary and mathematics training programs, then provide a rerandomization procedure for covariates that vary in importance, and finally offer other extensions for rerandomization, including methods addressing computational efficiency. When covariates vary in a priori importance, better balance should be required for more important covariates. Rerandomization based on Mahalanobis distance preserves the joint distribution of covariates, but balances all covariates equally. Here we propose rerandomizing based on Mahalanobis distance within tiers of covariate importance. Because balancing covariates in one tier will in general also partially balance covariates in other tiers, for each subsequent tier we explicitly balance only the components orthogonal to covariates in more important tiers.

在进行随机实验时，如果一种分配方式使得处理组在相关协变量方面存在有意义的差异，就应该重新进行随机分组。这个过程包括根据协变量平衡的度量明确指定一个分配是否可接受的标准，并对实验单位重新随机分组，直到获得可接受的分配为止。在此，我们举例说明重新随机分组如何能够改进一项已经实施的关于词汇和数学培训项目的随机实验的设计，然后针对重要性不同的协变量提供一种重新随机分组的程序，最后介绍重新随机分组的其他扩展内容，包括提高计算效率的方法。当协变量在先验重要性上存在差异时，对于更重要的协变量应要求更好的平衡。基于马氏距离的重新随机分组保留了协变量的联合分布，但对所有协变量进行了同等程度的平衡。在此，我们提出根据协变量重要性层级内的马氏距离进行重新随机分组。因为在一个层级中平衡协变量通常也会部分地平衡其他层级中的协变量，所以对于每个后续层级，我们只明确平衡与更重要层级中的协变量正交的成分。