Poster #90: Jason Liu

Extreme p-value estimation for large-scale permutation tests via the sub-Gaussian tail inequality

Liu, J. (Department of Statistics, Rutgers University), Xu, M., Ph.D. (Department of Statistics, Rutgers University), and Xing, J., Ph.D. (Department of Genetics, Rutgers University)

In a large-scale permutation test with high computational cost, the number of feasible permutations may be relatively small compared to the number of hypotheses. This presents a statistical difficulty because, after multiple testing adjustment, the empirical p-values often have no power at all as they cannot be smaller than 1/n for n permutations. We address this problem by computing p-values not from the empirical distribution but from the sub-Gaussian tail inequality, a result of probability theory that gives bounds on the tail probabilities of a distribution. The key technical challenge of this strategy is estimating a quantity called the sub-Gaussian parameter, which is defined through an optimization and has no closed form. We propose an estimator of the sub-Gaussian parameter and prove its rate of convergence, asymptotic normality, and minimax optimality. We demonstrate this on GO enrichment analysis of active module identification pipelines, where each permutation requires rerunning an expensive pipeline, making n necessarily small. Since enrichment scores are bounded, the null distribution is sub-Gaussian, and the estimator yields a valid tail bound. To ensure the p-value is based on the tail of the distribution, we employ a bulk- centering procedure that can only increase the power. Across various datasets and algorithms, the approach is competitive with existing methods. Overall, our sub-Gaussian parameter estimator provides a practical way for extreme p-value estimation when a large number of permutations is not available.