False discovery rate envelopes

TL;DR

This paper introduces FDR envelopes, a graphical tool based on resampling that controls false discovery rate in functional tests, allowing intuitive visualization of hypothesis rejection regions.

Contribution

It develops adaptive FDR-controlling envelopes for functional test statistics, extending global envelope testing to false discovery rate control with simple rejection rules.

Findings

The proposed methods effectively control FDR in simulations.

The envelopes provide clear visualization of hypothesis outcomes.

Applications to real data demonstrate practical utility.

Abstract

False discovery rate (FDR) is a common way to control the number of false discoveries in multiple testing. There are a number of approaches available for controlling FDR. However, for functional test statistics, which are discretized into $m$ highly correlated hypotheses, the methods must account for changes in distribution across the functional domain and correlation structure. Further, it is of great practical importance to visualize the test statistic together with its rejection or acceptance region. Therefore, the aim of this paper is to find, based on resampling principles, a graphical envelope that controls FDR and detects the outcomes of all individual hypotheses by a simple rule: the hypothesis is rejected if and only if the empirical test statistic is outside of the envelope. Such an envelope offers a straightforward interpretation of the test results, similarly as the recently developed global envelope testing which controls the family-wise error rate. Two different adaptive single threshold procedures are developed to fulfill this aim. Their performance is studied in an extensive simulation study. The new methods are illustrated by three real data examples.