  Researchers in genetics and other life sciences commonly use permutation
tests to evaluate differences between groups. Permutation tests have desirable
properties, including exactness if data are exchangeable, and are applicable
even when the distribution of the test statistic is analytically intractable.
However, permutation tests can be computationally intensive. We propose both an
asymptotic approximation and a resampling algorithm for quickly estimating
small permutation p-values (e.g. $<10^{-6}$) for the difference and ratio of
means in two-sample tests. Our methods are based on the distribution of test
statistics within and across partitions of the permutations, which we define.
In this article, we present our methods and demonstrate their use through
simulations and an application to cancer genomic data. Through simulations, we
find that our resampling algorithm is more computationally efficient than
another leading alternative, particularly for extremely small p-values (e.g.
$<10^{-30}$). Through application to cancer genomic data, we find that our
methods can successfully identify up- and down-regulated genes. While we focus
on the difference and ratio of means, we speculate that our approaches may work
in other settings.
