Beyond the Sharp Null: Randomization Inference, Bounded Null Hypotheses, and Confidence Intervals for Maximum Effects

Abstract

Fisherian randomization inference is often dismissed as testing an uninteresting and implausible hypothesis: the sharp null of no effects whatsoever. We show that this view is overly narrow. Many randomization tests are also valid under a more general "bounded" null hypothesis under which all effects are weakly negative (or positive), thus accommodating heterogenous effects. By inverting such tests we can form one-sided confidence intervals for the maximum (or minimum) effect. These properties hold for all effect-increasing test statistics, which include both common statistics such as the mean difference and uncommon ones such as Stephenson rank statistics. The latter's sensitivity to extreme effects permits detection of positive effects even when the average effect is negative. We argue that bounded nulls are often of substantive or theoretical interest, and illustrate with two applications: testing monotonicity in an IV analysis and inferring effect sizes in a small randomized experiment.