Comment: Microarrays, Empirical Bayes and the Two-Group Model

Abstract

Professor Efron is to be congratulated for his innovative and valuable contributions to large-scale multiple testing. He has given us a very interesting article with much material for thought and exploration. The twogroup mixture model (2.1) provides a convenient and effective framework for multiple testing. The empirical Bayes approach leads naturally to the local false discovery rate (Lfdr) and gives the Lfdr a useful Bayesian interpretation. This and other recent papers of Efron raised several important issues in multiple testing such as theoretical null versus empirical null and the effects of correlation. Much research is needed to better understand these issues. Virtually all FDR controlling procedures in the literature are based on thresholding the ranked p-values. The difference among these methods is in the choice of the threshold. In multiple testing, typically one first uses a p-value based method such as the Benjamini– Hochberg procedure for global FDR control and then uses the Lfdr as a measure of significance for individual nonnull cases. See, for example, Efron (2004, 2005). In what follows I will first discuss the drawbacks of using p-value in large-scale multiple testing and demonstrate the fundamental role played by the Lfdr. I then discuss estimation of the null distribution and the proportion of the nonnulls. I will end with some comments about dealing with the dependency. In the discussion I shall use the notation given in Table 1 to summarize the outcomes of a multiple testing procedure. With the notation given in the table, the false discovery rate (FDR) is then defined as FDR = E(N10/R|R > 0)Pr(R > 0).