Estimation of the False Discovery Proportion with Unknown Dependence

Abstract

Large-scale multiple testing with highly correlated test statistics arises frequently in many scientific research. Incorporating correlation information in estimating false discovery proportion has attracted increasing attention in recent years. When the covariance matrix of test statistics is known, Fan, Han & Gu (2012) provided a consistent estimate of False Discovery Proportion (FDP) under arbitrary dependence structure. However, the covariance matrix is often unknown in many applications and such dependence information has to be estimated before estimating FDP (Efron, 2010). The estimation accuracy can greatly affect the convergence result of FDP or even violate its consistency. In the current paper, we provide methodological modification and theoretical investigations for estimation of FDP with unknown covariance. First we develop requirements for estimates of eigenvalues and eigenvectors such that we can obtain a consistent estimate of FDP. Secondly we give conditions on the dependence structures such that the estimate of FDP is consistent. Such dependence structures include sparse covariance matrices, which have been popularly considered in the contemporary random matrix theory. When data are sampled from an approximate factor model, which encompasses most practical situations, we provide a consistent estimate of FDP via exploiting this specific dependence structure. The results are further demonstrated by simulation studies and some real data applications.

Asymptotics of empirical eigenstructure for high dimensional spiked covariance

“…Fan and Han (2013) proposed to employ the factor structure $$\mathrm{bold\Sigma }=\mathrm{boldB}\mathrm{boldB}\prime +\mathrm{boldA},$$ where $\mathrm{boldB}=false(\sqrt{{\lambda }_1}{\mathit{\xi }}_1,\dots ,\sqrt{{\lambda }_m}{\mathit{\xi }}_{m}false)$. λ j and ξ j are respectively the j th eigenvalue and eigenvector of Σ as before.…”

“…For simplicity, assume the maximal number of nonzero elements of each row of A is bounded. In Fan and Han (2013), they argued that the asymptotic upper bound $$\mathrm{normalF}\mathrm{normalD}{\mathrm{P}}_{A}false(tfalse)=\sum _{i=1}^{p}false[\mathrm{normal\Phi }false(a_{i}false(z_{t/2}+{\eta }_{i}false)false)+\mathrm{normal\Phi }false(a_{i}false(z_{t/2}-{\eta }_{i}false)false)false]/Rfalse(tfalse)$$ of FDP( t ) should be a realistic target to estimate for dependence tests, where z t /2 is the t /2-quantile of the standard normal distribution a i = (1 − ‖ b i ‖ 2 ) −1/2 , ${\eta }_i={\mathbf{b}}_i^{\prime }\mathrm{boldW}$ and ${\mathbf{b}}_i^{\prime }$ is the i th row of B .…”

The Empirical Distribution of a Large Number of Correlated Normal Variables

“…Proposition 3 and its proof imply that if there exist consistent estimates for the first K eigenvalues and eigenvectors, then the result of Theorem 4 (ii) still holds; see also Fan and Han (2014). A systematic study of the case that the correlation matrix is unknown is left to future research.…”

“…Fan et al (2012) and Fan and Han (2014) consider a related framework where some of the variables Z i may have a non-zero mean, and therefore use an estimate that minimizes the distance under sparsity assumptions.…”

Large-Scale Simultaneous Testing of Cross-Covariance Matrices with Applications to PheWAS

“…For example, Fan et al (2012) and Fan and Han (2013) considered multiple testing for normal means and required the covariance matrix to be known or well estimated. Efron (2004Efron ( , 2007 developed FDR controlling procedures for multiple t-tests.…”