Randomization Tests for Weak Null Hypotheses
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The Fisher randomization test (FRT) is applicable for any test statistic, under a sharp null hypothesis that can recover all missing potential outcomes. However, it is often of interest 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 an appropriate test statistic. For a general weak null hypothesis, we propose an approach to imputing missing potential outcomes under a compatible sharp null hypothesis. With this imputation scheme, we advocate using 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 preserves correct large-sample type I errors under the weak null hypothesis of interest. Therefore, our FRT is agnostic to treatment effect heterogeneity. We establish a unified theory for general factorial experiments. We also extend it to stratified and clustered experiments.
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