Permutation p-values should never be zero: calculating exact p-values when permutations are randomly drawn

TL;DR

This paper highlights that permutation p-values are often incorrectly computed as zero in genomic studies, proposes a method for calculating exact p-values from random permutations, and emphasizes the importance of accurate p-value estimation in multiple testing scenarios.

Contribution

It introduces a computation strategy for exact p-values in permutation tests with random permutations, correcting common underestimation errors.

Findings

Permutation p-values are often understated by about 1/m.

A new method for calculating exact p-values from random permutations.

Recommendations for proper implementation of permutation tests.

Abstract

Permutation tests are amongst the most commonly used statistical tools in modern genomic research, a process by which p-values are attached to a test statistic by randomly permuting the sample or gene labels. Yet permutation p-values published in the genomic literature are often computed incorrectly, understated by about 1/m, where m is the number of permutations. The same is often true in the more general situation when Monte Carlo simulation is used to assign p-values. Although the p-value understatement is usually small in absolute terms, the implications can be serious in a multiple testing context. The understatement arises from the intuitive but mistaken idea of using permutation to estimate the tail probability of the test statistic. We argue instead that permutation should be viewed as generating an exact discrete null distribution. The relevant literature, some of which is likely to have been relatively inaccessible to the genomic community, is reviewed and summarized. A computation strategy is developed for exact p-values when permutations are randomly drawn. The strategy is valid for any number of permutations and samples. Some simple recommendations are made for the implementation of permutation tests in practice.