Simultaneous high-probability bounds on the false discovery proportion in structured, regression, and online settings

TL;DR

This paper develops new finite-sample, high-probability bounds on the false discovery proportion for multiple testing, applicable in structured, regression, and online scenarios, enhancing flexibility and power in statistical inference.

Contribution

It introduces a novel class of simultaneous FDP bounds tailored for nested rejection sets, leveraging side information, knockoff statistics, and online data, connecting FDP bounds with FDR control methods.

Findings

Finite-sample bounds based on FDP estimates from FDR procedures

Bounds applicable in structured, regression, and online settings

Demonstrated utility on UK Biobank dataset

Abstract

While traditional multiple testing procedures prohibit adaptive analysis choices made by users, Goeman and Solari (2011) proposed a simultaneous inference framework that allows users such flexibility while preserving high-probability bounds on the false discovery proportion (FDP) of the chosen set. In this paper, we propose a new class of such simultaneous FDP bounds, tailored for nested sequences of rejection sets. While most existing simultaneous FDP bounds are based on closed testing using global null tests based on sorted p-values, we additionally consider the setting where side information can be leveraged to boost power, the variable selection setting where knockoff statistics can be used to order variables, and the online setting where decisions about rejections must be made as data arrives. Our finite-sample, closed form bounds are based on repurposing the FDP estimates from false discovery rate (FDR) controlling procedures designed for each of the above settings. These results establish a novel connection between the parallel literatures of simultaneous FDP bounds and FDR control methods, and use proof techniques employing martingales and filtrations that are new to both these literatures. We demonstrate the utility of our results by augmenting a recent knockoffs analysis of the UK Biobank dataset.