## The Annals of Statistics

- Ann. Statist.
- Volume 47, Number 1 (2019), 583-611.

### Permutation $p$-value approximation via generalized Stolarsky invariance

Hera Y. He, Kinjal Basu, Qingyuan Zhao, and Art B. Owen

#### 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.

#### Article information

**Source**

Ann. Statist., Volume 47, Number 1 (2019), 583-611.

**Dates**

Received: March 2016

Revised: February 2018

First available in Project Euclid: 30 November 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1543568599

**Digital Object Identifier**

doi:10.1214/18-AOS1702

**Mathematical Reviews number (MathSciNet)**

MR3909943

**Zentralblatt MATH identifier**

07036212

**Subjects**

Primary: 62G10: Hypothesis testing

Secondary: 11K38: Irregularities of distribution, discrepancy [See also 11Nxx] 62G09: Resampling methods

**Keywords**

Discrepancy gene sets hypothesis testing quasi-Monte Carlo

#### Citation

He, Hera Y.; Basu, Kinjal; Zhao, Qingyuan; Owen, Art B. Permutation $p$-value approximation via generalized Stolarsky invariance. Ann. Statist. 47 (2019), no. 1, 583--611. doi:10.1214/18-AOS1702. https://projecteuclid.org/euclid.aos/1543568599

#### Supplemental materials

- Supplement to “Permutation $p$-value approximation via generalized Stolarsky invariance”. The supplement presents additional material, including lengthier proofs.Digital Object Identifier: doi:10.1214/18-AOS1702SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.