Randomization Inference beyond the Sharp Null: Bounded Null Hypotheses and Quantiles of Individual Treatment Effects

Randomization (a.k.a. permutation) inference is typically interpreted as testing Fisher's "sharp" null hypothesis that all effects are exactly zero. This hypothesis is often criticized as uninteresting and implausible. We show, however, that many randomization tests are also valid for a "bounded" null hypothesis under which effects are all negative (or positive) for all units but otherwise heterogeneous. The bounded null is closely related to important concepts such as monotonicity and Pareto efficiency. Inverting tests of this hypothesis yields confidence intervals for the maximum (or minimum) individual treatment effect. We then extend randomization tests to infer other quantiles of individual effects, which can be used to infer the proportion of units with effects larger (or smaller) than any threshold. The proposed confidence intervals for all quantiles of individual effects are simultaneously valid, in the sense that no correction due to multiple analyses is needed. In sum, we provide a broader justification for Fisher randomization tests, and develop exact nonparametric inference for quantiles of heterogeneous individual effects. We illustrate our methods with simulations and applications, where we find that Stephenson rank statistics often provide the most informative results.