Asymptotic Uncertainty of False Discovery Proportion for Dependent $t$-Tests

TL;DR

This paper investigates the asymptotic behavior and uncertainty of the false discovery proportion (FDP) in dependent $t$-tests, extending previous results from $z$-tests and providing methods to evaluate its variance.

Contribution

It extends the analysis of FDP asymptotics from $z$-tests to $t$-tests with unknown variances, and develops an efficient approximation method for its variance under dependence.

Findings

FDP converges to a fixed quantity regardless of dependence structure.

The asymptotic variance of FDP varies with dependence, affecting uncertainty estimates.

Simulation and real-data studies validate the theoretical results.

Abstract

Multiple testing is a fundamental problem in high-dimensional statistical inference. Although many methods have been proposed to control false discoveries, it is still a challenging task when the tests are correlated to each other. To overcome this challenge, various methods have been proposed to estimate the false discovery rate (FDR) and/or the false discovery proportion (FDP) under arbitrary covariance among the test statistics. An interesting finding of these works is that the estimation of FDP and FDR under weak dependence is identical to that under independence. However, Mei et al. (2021) pointed out that unlike FDR, the asymptotic variance of FDP can still differ drastically from that under independence, and the difference depends on the covariance structure among the test statistics. In this paper, we further extend this result from $z$-tests to $t$-tests when the marginal variances are unknown and need to be estimated. With weakly dependent $t$-tests, we show that FDP still converges to a fixed quantity unrelated to the dependence structure, and further derive the asymptotic expansion and uncertainty of FDP leading to similar results as in Mei et al. (2021). In addition, we develop an approximation method to efficiently evaluate the asymptotic variance of FDP for dependent $t$-tests. We examine how the asymptotic variance of FDP varies as well as the performance of its estimators under different dependence structures through simulations and a real-data study.