# An Assumption-Free Exact Test For Fixed-Design Linear Models With Exchangeable Errors

## TL;DR

The paper introduces the Cyclic Permutation Test (CPT), an exact, assumption-free method for testing linear hypotheses in fixed-design linear models with exchangeable errors, valid in finite samples and applicable to arbitrary designs.

## Contribution

It presents a novel non-randomized test that guarantees exact Type I error control and extends existing methods by using cyclic permutations and solving a traveling salesman problem for enhanced power.

## Key findings

- CPT achieves comparable power to existing tests in simulations.
- The test provides exact confidence intervals for single coefficient contrasts.
- It is valid for arbitrary fixed designs and exchangeable errors in finite samples.

## Abstract

We propose the Cyclic Permutation Test (CPT) to test general linear hypotheses for linear models. This test is non-randomized 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 $n / p \ge 1 / \alpha - 1$. 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 non-standard cyclic permutation group under the null hypothesis.The power can be further enhanced by solving a secondary non-linear travelling 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, highlighting the novelty of our test.