Cyclic permutation test: A nonstandard exact test for linear models
We propose the Cyclic Permutation Test (CPT) to test general linear hypotheses for linear models. This test is nonrandomized and valid in finite samples with exact Type I error α for an arbitrary fixed design matrix and arbitrary exchangeable errors, whenever 1/α is an integer and n/p ≥ 1/α−1. The test applies the marginal rank test to 1/α linear statistics of the outcome vector, where the coefficient vectors are determined by solving a linear system such that the joint distribution of the linear statistics is invariant with respect to a nonstandard cyclic permutation group under the null hypothesis. The power can be further enhanced by solving a secondary nonlinear traveling salesman problem, for which the genetic algorithm can find a reasonably good solution. Extensive simulation studies show that the CPT has comparable power to existing tests. When testing for a single contrast of coefficients, an exact confidence interval can be obtained by inverting the test. Furthermore, we provide a selective yet extensive literature review of the century-long efforts on this problem from 1908 to 2018, highlighting the novelty of our test.