Abstract

The Fisher randomization test (FRT) is appropriate for any test statistic, under a sharp null hypothesis that can recover all missing potential outcomes. However, it is often sought after to test a weak null hypothesis that the treatment does not affect the units on average. To use the FRT for a weak null hypothesis, we must address two issues. First, we need to impute the missing potential outcomes although the weak null hypothesis cannot determine all of them. Second, we need to choose a proper test statistic. For a general weak null hypothesis, we propose an approach to imputing missing potential outcomes under a compatible sharp null hypothesis. Building on this imputation scheme, we advocate a studentized statistic. The resulting FRT has multiple desirable features. First, it is model-free. Second, it is finite-sample exact under the sharp null hypothesis that we use to impute the potential outcomes. Third, it conservatively controls large-sample Type I error under the weak null hypothesis of interest. Therefore, our FRT is agnostic to the treatment effect heterogeneity. We establish a unified theory for general factorial experiments and extend it to stratified and clustered experiments. Supplementary materials for this article are available online.

摘要

Fisher 随机化检验 (FRT) 适用于任何检验统计量，在可以恢复所有缺失的潜在结果的尖锐零假设下。然而，人们经常寻求检验处理不影响平均单位的弱零假设。要将 FRT 用于弱零假设，我们必须解决两个问题。首先，我们需要估算缺失的潜在结果，尽管弱零假设无法确定所有这些结果。其次，我们需要选择一个合适的检验统计量。对于一般的弱零假设，我们提出了一种在兼容的锐零假设下估算缺失的潜在结果的方法。在此插补方案的基础上，我们提倡学生化统计。由此产生的 FRT 具有多个理想的特征。首先，它是无模型的。第二，在我们用来估算潜在结果的尖锐零假设下，它是有限样本精确的。第三，它在兴趣弱零假设下保守地控制了大样本 I 类错误。因此，我们的 FRT 与治疗效果的异质性无关。我们为一般析因实验建立了统一的理论，并将其扩展到分层和聚类实验。本文的补充材料可在线获取。