  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 errors 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.
We also extend it to stratified and clustered experiments.
