Abstract
We are concerned with the false discovery rate (FDR) of the linear step-up test $\varphi^{LSU}$ considered by Benjamini and Hochberg (1995). It is well known that $\varphi^{LSU}$ controls the FDR at level $m_{0}q/m$ if the joint distribution of $p$-values is multivariate totally positive of order $2$. In this, $m$ denotes the total number of hypotheses, $m_{0}$ the number of true null hypotheses, and $q$ the nominal FDR level. Under the assumption of an Archimedean $p$-value copula with completely monotone generator, we derive a sharper upper bound for the FDR of $\varphi^{LSU}$ as well as a non-trivial lower bound. Application of the sharper upper bound to parametric subclasses of Archimedean $p$-value copulae allows us to increase the power of $\varphi^{LSU}$ by pre-estimating the copula parameter and adjusting $q$. Based on the lower bound, a sufficient condition is obtained under which the FDR of $\varphi^{LSU}$ is exactly equal to $m_{0}q/m$, as in the case of stochastically independent $p$-values. Finally, we deal with high-dimensional multiple test problems with exchangeable test statistics by drawing a connection between infinite sequences of exchangeable $p$-values and Archimedean copulae with completely monotone generators. Our theoretical results are applied to important copula families, including Clayton copulae and Gumbel-Hougaard copulae.
Highlights
Control of the false discovery rate (FDR) has become a standard type I error criterion in large-scale multiple hypotheses testing
As shown by Benjamini and Yekutieli (2001) and Sarkar (2002), φLSU is generically FDRcontrolling over the class of models that lead to positive dependency among the random p-values P1, . . . , Pm in the sense of positive regression dependency on subsets (PRDS)
We extend these findings by deriving a sharper upper bound for the FDR of φLSU in the case that the dependency structure among P1, . . . , Pm can be expressed by an Archimedean copula
Summary
Control of the false discovery rate (FDR) has become a standard type I error criterion in large-scale multiple hypotheses testing. In an asymptotic setting (m → ∞), we show that the class of Archimedean p-value copulae with completely monotone generators includes certain models with p-values or test statistics, respectively, which are exchangeable under null hypotheses, H0-exchangeable for short. Such H0-exchangeable test statistics occur naturally in many multiple test problems, for instance in many-to-one comparisons or if test statistics are given by jointly Studentized means (cf Finner, Dickhaus and Roters (2007)).
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