Fisher’s randomization test (frt) delivers exact $p$-values under the strong null hypothesis of no treatment effect on any units whatsoever and allows for flexible covariate adjustment to improve the power. Of interest is whether the resulting covariate-adjusted procedure could also be valid for testing the weak null hypothesis of zero average treatment effect. To this end, we evaluate two general strategies for conducting covariate adjustment in frts: the pseudo-outcome strategy that uses the residuals from an outcome model with only the covariates as the pseudo, covariate-adjusted outcomes to form the test statistic, and the model-output strategy that directly uses the output from an outcome model with both the treatment and covariates as the covariate-adjusted test statistic. Based on theory and simulation, we recommend using the ordinary least squares (ols) fit of the observed outcome on the treatment, centered covariates, and their interactions for covariate adjustment, and conducting frt with the robust $t$-value of the treatment as the test statistic. The resulting frt is finite-sample exact for testing the strong null hypothesis, asymptotically valid for testing the weak null hypothesis, and more powerful than the unadjusted counterpart under alternatives, all irrespective of whether the linear model is correctly specified or not. We start with complete randomization, and then extend the theory to cluster randomization, stratified randomization, and rerandomization, respectively, giving a recommendation for the test procedure and test statistic under each design. Our theory is design-based, also known as randomization-based, in which we condition on the potential outcomes but average over the random treatment assignment.

Fisher 随机化检验 ( frt ) 提供准确的$磷$- 强零假设下的值对任何单位都没有治疗影响，并允许灵活的协变量调整以提高功效。有趣的是，由此产生的协变量调整程序是否也适用于测试零平均治疗效果的弱零假设。为此，我们评估了在frt s 中进行协变量调整的两种一般策略：伪结果策略，它使用结果模型中的残差，仅将协变量作为伪协变量调整结果来形成检验统计量，以及模型输出直接使用结果模型的输出，将处理和协变量作为协变量调整检验统计量的策略。基于理论和模拟，我们建议使用观察结果对治疗的普通最小二乘 ( ols ) 拟合、居中的协变量及其相互作用进行协变量调整，并使用鲁棒性进行frt$吨$- 作为检验统计量的处理值。得到的frt是有限样本精确的，用于测试强零假设，渐近有效地测试弱零假设，并且比替代方案下未调整的对应物更强大，所有这些都与是否正确指定线性模型无关。我们从完全随机化开始，然后将理论分别扩展到整群随机化、分层随机化和再随机化，对每种设计下的检验程序和检验统计量提出建议。我们的理论是基于设计的，也称为基于随机化的，我们以潜在结果为条件，但对随机治疗分配进行平均。