A geometric interpretation of the permutation $p$-value and its application in eQTL studies

TL;DR

This paper introduces a geometric approach to efficiently estimate permutation p-values in genetic studies, significantly reducing computational costs while maintaining accuracy, especially in large-scale eQTL analyses.

Contribution

It presents a novel geometric interpretation of permutation p-values and develops a constant-time estimation method applicable to regression models with binary predictors.

Findings

Method estimates permutation p-values with less than 5% of the computational time of direct permutations.

Provides reliable permutation p-value estimates across a wide p-value range.

Offers insights into the relationship between nominal and permutation p-values and the effective number of independent tests.

Abstract

Permutation $p$-values have been widely used to assess the significance of linkage or association in genetic studies. However, the application in large-scale studies is hindered by a heavy computational burden. We propose a geometric interpretation of permutation $p$-values, and based on this geometric interpretation, we develop an efficient permutation $p$-value estimation method in the context of regression with binary predictors. An application to a study of gene expression quantitative trait loci (eQTL) shows that our method provides reliable estimates of permutation $p$-values while requiring less than 5% of the computational time compared with direct permutations. In fact, our method takes a constant time to estimate permutation $p$-values, no matter how small the $p$-value. Our method enables a study of the relationship between nominal $p$-values and permutation $p$-values in a wide range, and provides a geometric perspective on the effective number of independent tests.