Rejoinder: Microarrays, Empirical Bayes and the Two-Groups Model

Abstract

The Fisher–Neyman–Pearson theory of hypothesis testing was a triumph of mathematical elegance and practical utility. It was never designed, though, to handle 10,000 tests at once, and one can see contemporary statisticians struggling to develop theories appropriate to our new scientific environment. This paper is part of that effort: starting from just the two-groups model (2.1), it aims to show Bayesian and frequentist ideas merging into a practical framework for largescale simultaneous testing. False discovery rates, Benjamini and Hochberg’s influential contribution to modern statistical theory, is the main methodology featured in the paper, but I really was not trying to sell any specific technology as the final word. In fact, the discussants offer an attractive menu of alternatives. It is still early in the large-scale hypothesis testing story, and I expect, and hope for, major developments in both theory and practice. The central issue, as Carl Morris makes clear, is the combination of information from a collection of more or less similar sources, for example from the expression levels of different genes in a microarray study. Crucial questions revolve around the comparability and relevance of the various sources, as well as the proper choice of a null distribution. Technical issues such as the exact control of Type I errors are important as well, but, in my opinion, have played too big a role in the microarray literature. The discussions today are an appealing mixture of technical facility and big-picture thinking. They are substantial essays in their own right, and I will be able to respond here to only a few of the issues raised. I once wrote, about the jackknife, that good simple ideas are our most precious intellectual commodity. False discovery rates fall into that elite category. The two-groups model is used here to unearth the Bayesian roots of Benjamini and Hochberg’s originally frequentist construction. In a Bayesian framework it is natural