# Improving the Benjamini-Hochberg Procedure for Discrete Tests

## TL;DR

This paper introduces new modifications to the Benjamini-Hochberg procedure that account for data discreteness and signal strength, providing theoretical FDR control guarantees and demonstrating improved power in genomic data analysis.

## Contribution

It develops theoretically guaranteed, discrete-aware FDR procedures that adapt to signal proportion, filling a gap in existing methods.

## Key findings

- New FDR procedures control the false discovery rate under independence.
- The methods show increased power in simulations and real genomic datasets.
- The approach combines discreteness and signal proportion adaptively.

## Abstract

To find interesting items in genome-wide association studies or next generation sequencing data, a crucial point is to design powerful false discovery rate (FDR) controlling procedures that suitably combine discrete tests (typically binomial or Fisher tests). In particular, recent research has been striving for appropriate modifications of the classical Benjamini-Hochberg (BH) step-up procedure that accommodate discreteness. However, despite an important number of attempts, these procedures did not come with theoretical guarantees. The present paper contributes to fill the gap: it presents new modifications of the BH procedure that incorporate the discrete structure of the data and provably control the FDR for any fixed number of null hypotheses (under independence). Markedly, our FDR controlling methodology allows to incorporate simultaneously the discreteness and the quantity of signal of the data (corresponding therefore to a so-called $\pi\_0$-adaptive procedure). The power advantage of the new methods is demonstrated in a numerical experiment and for some appropriate real data sets.