  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}.
