Multiple conditional randomization tests for lagged and spillover effects

We consider the problem of constructing multiple conditional randomization tests. They may test different causal hypotheses but always aim to be nearly independent, allowing the randomization p-values to be interpreted individually and combined using standard methods. We start with a simple, sequential construction of such tests, and then discuss its application to lagged treatment effect in stepped-wedge trials, and spillover effect in randomized trials with interference. Finally, we provide a sufficient condition for creating multiple nearly independent conditional randomization tests.