Gaussian prepivoting for finite population causal inference

In finite population causal inference exact randomization tests can be constructed for sharp null hypotheses, hypotheses which impute the missing potential outcomes. Oftentimes inference is instead desired for the weak null that the sample average of the treatment effects takes on a particular value while leaving the subject-specific treatment effects unspecified. Tests valid for sharp null hypotheses can be anti-conservative should only the weak null hold. We develop a general framework for unifying modes of inference for sharp and weak nulls, wherein a single procedure simultaneously delivers exact inference for sharp nulls and asymptotically valid inference for weak nulls. We employ randomization tests based upon prepivoted test statistics, wherein a test statistic is first transformed by a suitably constructed cumulative distribution function and its randomization distribution assuming the sharp null is then enumerated. For a large class of test statistics, we show that prepivoting may be accomplished by employing the push-forward of a sample-based Gaussian measure based upon a suitable covariance estimator. The approach enumerates the randomization distribution (assuming the sharp null) of a p-value for a large-sample test known to be valid under the weak null, and uses the resulting randomization distribution for inference. The versatility of the method is demonstrated through many examples, including rerandomized designs and regression-adjusted estimators in completely randomized designs.