Modern social and biomedical scientific publications require the reporting of covariate balance tables with not only covariate means by treatment group but also the associated $p$-values from significance tests of their differences. The practical need to avoid small $p$-values renders balance check and rerandomization by hypothesis testing standards an attractive tool for improving covariate balance in randomized experiments. Despite the intuitiveness of such practice and its arguably already widespread use in reality, the existing literature knows little about its implications on subsequent inference, subjecting many effectively rerandomized experiments to possibly inefficient analyses. To fill this gap, we examine a variety of potentially useful schemes for rerandomization based on $p$-values (ReP) from covariate balance tests, and demonstrate their impact on subsequent inference. Specifically, we focus on three estimators of the average treatment effect from the unadjusted, additive, and fully interacted linear regressions of the outcome on treatment, respectively, and derive their respective asymptotic sampling properties under ReP. The main findings are twofold. First, the estimator from the fully interacted regression is asymptotically the most efficient under all ReP schemes examined, and permits convenient regression-assisted inference identical to that under complete randomization. Second, ReP improves not only covariate balance but also the efficiency of the estimators from the unadjusted and additive regressions asymptotically. The standard regression analysis, in consequence, is still valid but can be overly conservative.

翻译：现代社会科学和生物医学科学出版物要求报告协变量平衡表，不仅需包含按处理组划分的协变量均值，还需报告其差异显著性检验的相应$p$值。为避免出现小$p$值的实际需求，使基于假设检验标准的平衡检验与再随机化成为随机实验中改善协变量平衡的有效工具。尽管此类方法直观且已在现实中广泛应用，但现有文献对其后续推断的影响知之甚少，导致许多实际采用再随机化的实验可能采用低效的分析方法。为填补这一空白，我们考察了多种基于协变量平衡检验$p$值（ReP）的再随机化潜在方案，并论证其对后续推断的影响。具体而言，我们聚焦于三种平均处理效应估计量——分别来自结果变量对处理变量的未调整线性回归、加法线性回归及完全交互线性回归，并推导了它们在ReP下的渐近抽样性质。主要发现有两方面：第一，在所有考察的ReP方案下，完全交互线性回归的估计量渐近有效，且允许与完全随机化下相同的便捷回归辅助推断；第二，ReP不仅能改善协变量平衡，还能渐近提升未调整回归与加法回归估计量的效率。因此，标准回归分析依然有效，但可能过于保守。