An assumption-free exact test for fixed-design linear models with exchangeable errors

Abstract

Summary We propose the cyclic permutation test to test general linear hypotheses for linear models. The test is nonrandomized and valid in finite samples with exact Type I error $\alpha$ for an arbitrary fixed design matrix and arbitrary exchangeable errors, whenever $1 / \alpha$ is an integer and $n / p \geqslant 1 / \alpha - 1$, where $n$ is the sample size and $p$ is the number of parameters. The test involves applying the marginal rank test to $1 / \alpha$ 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 travelling salesman problem, for which the genetic algorithm can find a reasonably good solution. Extensive simulation studies show that the cyclic permutation test 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.