The False Discovery Rate (FDR) of Multiple Tests in a Class Room Lecture

TL;DR

This paper discusses methods for controlling the false discovery rate in multiple hypothesis testing, introduces a rapid approach to BH procedures, and examines FDR bounds under dependence, with implications for statistical testing.

Contribution

It presents a quick method for implementing BH step-up and step-down tests and explores sharp FDR inequalities for dependent p-values, including counter-examples.

Findings

Rapid approach to BH tests introduced

Sharp FDR inequalities for dependent p-values discussed

Bonferroni bound shown to be sharp under dependence

Abstract

Multiple tests are designed to test a whole collection of null hypotheses simultaneously. Their quality is often judged by the false discovery rate (FDR), i.e. the expectation of the quotient of the number of false rejections divided by the amount of all rejections. The widely cited Benjamini and Hochberg (BH) step up multiple test controls the FDR under various regularity assumptions. In this note we present a rapid approach to the BH step up and step down tests. Also sharp FDR inequalities are discussed for dependent p-values and examples and counter-examples are considered. In particular, the Bonferroni bound is sharp under dependence for control of the family-wise error rate.