The positive false discovery rate: a Bayesian interpretation and the q-value

TLDR

Multiple hypothesis testing aims to control false positives, and the false discovery rate (FDR) is defined as the expected proportion of false positives among all significant results, making it suitable for exploratory analyses that seek several significant findings among many tests. This work introduces a modified FDR, the positive false discovery rate (pFDR), and examines its advantages, disadvantages, and statistical properties. Assuming a mixture distribution for test statistics, the authors show that pFDR equals a Bayesian posterior probability, link it to classification theory, and define the q‑value as the pFDR analogue of a p‑value. They demonstrate that these properties hold asymptotically under broad conditions, including certain dependent structures.

Abstract

Multiple hypothesis testing is concerned with controlling the rate of false positives when testing several hypotheses simultaneously. One multiple hypothesis testing error measure is the false discovery rate (FDR), which is loosely defined to be the expected proportion of false positives among all significant hypotheses. The FDR is especially appropriate for exploratory analyses in which one is interested in finding several significant results among many tests. In this work, we introduce a modified version of the FDR called the "positive false discoveryrate" (pFDR). We discuss the advantages and disadvantages of the pFDR and investigate its statistical properties. When assuming the test statistics follow a mixture distribution, we show that the pFDR can be written as a Bayesian posterior probability and can be connected to classification theory. These properties remain asymptotically true under fairly general conditions, even under certain forms of dependence. Also, a new quantity called the "$q$-value" is introduced and investigated, which is a natural "Bayesian posterior p-value," or rather the pFDR analogue of the p-value.