Estimating the Null and the Proportion of Nonnull Effects in Large-Scale Multiple Comparisons

Abstract

SUMMARY In a recent paper [4], Efron pointed out that an important issue in large-scale multiple hypothesis testing is that the null distribution may be unknown and need to be estimated. Consider a Gaussian mixture model, where the null distribution is known to be normal but both null parameters-the mean and the variance-are unknown. We address the problem with a method based on Fourier transformation. The Fourier approach was first studied by Jin and Cai [9], which focuses on the scenario where any non-null effect has either the same or a larger variance than that of the null effects. In this paper, we review the main ideas in [9], and propose a generalized Fourier approach to tackle the problem under another scenario: any non-null effect has a larger mean than that of the null effects, but no constraint is imposed on the variance. This approach and that in [9] complement with each other: each approach is successful in a wide class of situations where the other fails. Also, we extend the Fourier approach to estimate the proportion of non-null effects. The proposed procedures perform well both in theory and on simulated data.

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

“…The resulting estimators are shown to be uniformly consistent over a wide class of parameters and outperform existing methods in simulations. The approach of Jin and Cai (2007) also yields a uniformly consistent estimator for the proportion of nonnull effects. In a two-component normal mixture setting, Cai, Jin and Low (2007) proposed an estimator of the proportion and developed a minimax theory for the estimation problem.…”

“…In the nonsparse case these methods of estimating the null densities do not perform well and it is not hard to show that the estimators are generally inconsistent. Jin and Cai (2007) introduced an alternative frequency domain approach for estimating the null parameters by using the empirical characteristic function and Fourier analysis. The approach demonstrates that the information about the null is well preserved in the high-frequency Fourier coefficients, where the distortion of the nonnull effects is asymptotically negligible.…”

“…Let us assume for the moment that we already have consistent estimatorsp 0 , f 0 andf . Such consistent estimators are provided, for example, in Jin and Cai (2007). Define the estimated Lfdr by Lfdr( Sun and Cai (2007) introduced the following adaptive step-up procedure:…”

Simultaneous Testing of Grouped Hypotheses: Finding Needles in Multiple Haystacks

“…Implementation of the Lfdr procedure requires the knowledge of population parameters such as the null density f 0 and proportion of nonnulls p, which may not be known in practice. Estimates of these unknown parameters for a normal mixture model have been developed in the literature; see Efron (2004) and Jin and Cai (2007). Letp,f 0 , andf be estimates of the unknown parameters and define the estimated Lfdr as Lfdr(x) =pf 0 (x)/f (x).…”

Penalized Multimarkervs.Single-Marker Regression Methods for Genome-Wide Association Studies of Quantitative Traits

“…Despite the correlation structure of the z-values, f 0 was fitted well by the N(0, 1) null distribution (see Supporting Information, Figure S1 as a typical representative of the data simulation replicates). We also used the consistent estimators of p 0 , f 0 , and f in equation 2 of Jin and Cai (2007). We refer to this procedure with the two sets of estimators for p 0 , f 0 , and f as LFDR1 and LFDR2.…”