Fast computation of p-values for the permutation test based on Pearson's correlation coefficient and other statistical tests

TL;DR

This paper introduces a fast Fourier transform-based algorithm for efficiently computing permutation test p-values, especially for Pearson's correlation, significantly reducing computational complexity compared to traditional sampling methods.

Contribution

The paper presents a novel, asymptotically faster sampling algorithm for permutation test p-values using FFT, applicable to Pearson's correlation and other tests.

Findings

Algorithm is practically faster than straightforward sampling.

Complexity is logarithmic in input size, versus linear for traditional methods.

Applicable to various statistical tests beyond Pearson's correlation.

Abstract

Permutation tests are among the simplest and most widely used statistical tools. Their p-values can be computed by a straightforward sampling of permutations. However, this way of computing p-values is often so slow that it is replaced by an approximation, which is accurate only for part of the interesting range of parameters. Moreover, the accuracy of the approximation can usually not be improved by increasing the computation time. We introduce a new sampling-based algorithm which uses the fast Fourier transform to compute p-values for the permutation test based on Pearson's correlation coefficient. The algorithm is practically and asymptotically faster than straightforward sampling. Typically, its complexity is logarithmic in the input size, while the complexity of straightforward sampling is linear. The idea behind the algorithm can also be used to accelerate the computation of p-values for many other common statistical tests. The algorithm is easy to implement, but its analysis involves results from the representation theory of the symmetric group.