We establish a general sufficient condition on constructing multiple "nearly independent" conditional randomization tests, in the sense that the joint distribution of their p-values is almost uniform under the global null. This property implies that the tests are jointly valid and can be combined using standard methods. Our theory generalizes existing techniques in the literature that use independent treatments, sequential treatments, or post-randomization, to construct multiple randomization tests. In particular, it places no condition on the experimental design, allowing for arbitrary treatment variables, assignment mechanisms and unit interference. The flexibility of this framework is illustrated through developing conditional randomization tests for lagged treatment effects in stepped-wedge randomized controlled trials. A weighted Z-score test is further proposed to maximize the power when the tests are combined. We compare the efficiency and robustness of the commonly used mixed-effect models and the proposed conditional randomization tests using simulated experiments and real trial data.
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