Permutation $p$-value approximation via generalized Stolarsky invariance
Hera Y. He, Kinjal Basu, Qingyuan Zhao, Art B. Owen
The Annals of Statistics · 2019 · Volume 47 · Issue 1
Abstract
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 $\hat{p}$ 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.
Keywords
11K38, 62G09, 62G10, Discrepancy, gene sets, Hypothesis testing, quasi-Monte Carlo
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