# Optimal control of false discovery criteria in the two-group model

## TL;DR

This paper develops optimal multiple testing procedures for controlling FDR and pFDR in the two-group model, even with dependent test statistics, by thresholding the local false discovery rate with data-dependent thresholds.

## Contribution

It derives the first optimal policies for FDR and pFDR control in the two-group model allowing dependence, with an efficient algorithm for large-scale problems.

## Key findings

- Optimal policies threshold locFDR with data-dependent thresholds.
- Algorithms efficiently handle thousands of hypotheses.
- Procedures demonstrated on gene expression data.

## Abstract

The highly influential two-group model in testing a large number of statistical hypotheses assumes that the test statistics are drawn independently from a mixture of a high probability null distribution and a low probability alternative. Optimal control of the marginal false discovery rate (mFDR), in the sense that it provides maximal power (expected true discoveries) subject to mFDR control, is known to be achieved by thresholding the local false discovery rate (locFDR), i.e., the probability of the hypothesis being null given the set of test statistics, with a fixed threshold. We address the challenge of controlling optimally the popular false discovery rate (FDR) or positive FDR (pFDR) rather than mFDR in the general two-group model, which also allows for dependence between the test statistics. These criteria are less conservative than the mFDR criterion, so they make more rejections in expectation. We derive their optimal multiple testing (OMT) policies, which turn out to be thresholding the locFDR with a threshold that is a function of the entire set of statistics. We develop an efficient algorithm for finding these policies, and use it for problems with thousands of hypotheses. We illustrate these procedures on gene expression studies.