Randomization test

A randomization test compares a statistic with values obtained by applying transformations that preserve the joint distribution under the null hypothesis.

A conditional randomization test resamples a tested variable from its known or estimated conditional distribution given covariates. Under conditional independence from the outcome, the observed and resampled test statistics are exchangeable conditional on those covariates and outcomes.

A permutation test uses relabelings of observations as its null-preserving transformations and computes a p-value from the resulting orbit of the test statistic.

A sign-flip randomization test independently multiplies observations by signs in $\{-1,1\}$. It is finite-sample exact for a zero-centered null when the joint distribution is invariant under those sign changes.