62,188 research outputs found
False discovery and false nondiscovery rates in single-step multiple testing procedures
Results on the false discovery rate (FDR) and the false nondiscovery rate
(FNR) are developed for single-step multiple testing procedures. In addition to
verifying desirable properties of FDR and FNR as measures of error rates, these
results extend previously known results, providing further insights,
particularly under dependence, into the notions of FDR and FNR and related
measures. First, considering fixed configurations of true and false null
hypotheses, inequalities are obtained to explain how an FDR- or FNR-controlling
single-step procedure, such as a Bonferroni or \u{S}id\'{a}k procedure, can
potentially be improved. Two families of procedures are then constructed, one
that modifies the FDR-controlling and the other that modifies the
FNR-controlling \u{S}id\'{a}k procedure. These are proved to control FDR or FNR
under independence less conservatively than the corresponding families that
modify the FDR- or FNR-controlling Bonferroni procedure. Results of numerical
investigations of the performance of the modified \u{S}id\'{a}k FDR procedure
over its competitors are presented. Second, considering a mixture model where
different configurations of true and false null hypotheses are assumed to have
certain probabilities, results are also derived that extend some of Storey's
work to the dependence case.Comment: Published at http://dx.doi.org/10.1214/009053605000000778 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Stepup procedures controlling generalized FWER and generalized FDR
In many applications of multiple hypothesis testing where more than one false
rejection can be tolerated, procedures controlling error rates measuring at
least false rejections, instead of at least one, for some fixed
can potentially increase the ability of a procedure to detect false null
hypotheses. The -FWER, a generalized version of the usual familywise error
rate (FWER), is such an error rate that has recently been introduced in the
literature and procedures controlling it have been proposed. A further
generalization of a result on the -FWER is provided in this article. In
addition, an alternative and less conservative notion of error rate, the
-FDR, is introduced in the same spirit as the -FWER by generalizing the
usual false discovery rate (FDR). A -FWER procedure is constructed given any
set of increasing constants by utilizing the th order joint null
distributions of the -values without assuming any specific form of
dependence among all the -values. Procedures controlling the -FDR are
also developed by using the th order joint null distributions of the
-values, first assuming that the sets of null and nonnull -values are
mutually independent or they are jointly positively dependent in the sense of
being multivariate totally positive of order two (MTP) and then discarding
that assumption about the overall dependence among the -values.Comment: Published in at http://dx.doi.org/10.1214/009053607000000398 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Local False Discovery Rate Based Methods for Multiple Testing of One-Way Classified Hypotheses
This paper continues the line of research initiated in
\cite{Liu:Sarkar:Zhao:2016} on developing a novel framework for multiple
testing of hypotheses grouped in a one-way classified form using
hypothesis-specific local false discovery rates (Lfdr's). It is built on an
extension of the standard two-class mixture model from single to multiple
groups, defining hypothesis-specific Lfdr as a function of the conditional Lfdr
for the hypothesis given that it is within a significant group and the Lfdr for
the group itself and involving a new parameter that measures grouping effect.
This definition captures the underlying group structure for the hypotheses
belonging to a group more effectively than the standard two-class mixture
model. Two new Lfdr based methods, possessing meaningful optimalities, are
produced in their oracle forms. One, designed to control false discoveries
across the entire collection of hypotheses, is proposed as a powerful
alternative to simply pooling all the hypotheses into a single group and using
commonly used Lfdr based method under the standard single-group two-class
mixture model. The other is proposed as an Lfdr analog of the method of
\cite{Benjamini:Bogomolov:2014} for selective inference. It controls Lfdr based
measure of false discoveries associated with selecting groups concurrently with
controlling the average of within-group false discovery proportions across the
selected groups. Simulation studies and real-data application show that our
proposed methods are often more powerful than their relevant competitors.Comment: 26 pages, 17 figure
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