A conformal test of linear models via permutation-augmented regressions

TL;DR

This paper introduces PALMRT, a conformal permutation-augmented regression test that reliably controls type I error and maintains power in linear models, outperforming existing permutation tests especially in biomedical research.

Contribution

The paper develops PALMRT, a novel conformal permutation test for linear models that guarantees error control and high power without strict assumptions or large sample requirements.

Findings

PALMRT controls type I error at no more than 2α across various settings.

PALMRT demonstrates competitive power compared to traditional methods.

In a long-Covid study, PALMRT validated key findings where other tests failed.

Abstract

Permutation tests are widely recognized as robust alternatives to tests based on normal theory. Random permutation tests have been frequently employed to assess the significance of variables in linear models. Despite their widespread use, existing random permutation tests lack finite-sample and assumption-free guarantees for controlling type I error in partial correlation tests. To address this ongoing challenge, we have developed a conformal test through permutation-augmented regressions, which we refer to as PALMRT. PALMRT not only achieves power competitive with conventional methods but also provides reliable control of type I errors at no more than $2\alpha$, given any targeted level $\alpha$, for arbitrary fixed designs and error distributions. We have confirmed this through extensive simulations. Compared to the cyclic permutation test (CPT) and residual permutation test (RPT), which also offer theoretical guarantees, PALMRT does not compromise as much on power or set stringent requirements on the sample size, making it suitable for diverse biomedical applications. We further illustrate the differences in a long-Covid study where PALMRT validated key findings previously identified using the t-test after multiple corrections, while both CPT and RPT suffered from a drastic loss of power and failed to identify any discoveries. We endorse PALMRT as a robust and practical hypothesis test in scientific research for its superior error control, power preservation, and simplicity. An R package for PALMRT is available at \url{https://github.com/LeyingGuan/PairedRegression}.