Permutation p -value approximation via generalized Stolarsky invariance

It is common for genomic data analysis to use p-values from a large number of permutation tests. The multiplicity of tests may require very tiny p-values in order to reject any null hypotheses and the common practice of using randomly sampled permutations then becomes very expensive. We propose an inexpensive approximation to p-values for two sample linear test statistics, derived from Stolarsky.s invariance principle. The method creates a geometrically derived reference set of approximate p-values for each hypothesis. The average of that set is used as a point estimate p^ and our generalization of the invariance principle allows us to compute the variance of the p-values in that set. We find that in cases where the point estimate is small, the variance is a modest multiple of the square of that point estimate, yielding a relative error property similar to that of saddlepoint approximations. On a Parkinson.s disease data set, the new approximation is faster and more accurate than the saddlepoint approximation. We also obtain a simple probabilistic explanation of Stolarsky.s invariance principle.