Multiple conditional randomization tests

We propose a general framework for (multiple) conditional randomization tests that incorporate several important ideas in the recent literature. We establish a general sufficient condition on the construction of multiple conditional randomization tests under which their p-values are "independent", in the sense that their joint distribution stochastically dominates the product of uniform distributions under the null. Conceptually, we argue that randomization should be understood as the mode of inference precisely based on randomization. We show that under a change of perspective, many existing statistical methods, including permutation tests for (conditional) independence and conformal prediction, are special cases of the general conditional randomization test. The versatility of our framework is further illustrated with an example concerning lagged treatment effects in stepped-wedge randomized trials.