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 $k$ false rejections, instead of at least one, for some fixed $k\ge 1$ can potentially increase the ability of a procedure to detect false null hypotheses. The $k$-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 $k$-FWER is provided in this article. In addition, an alternative and less conservative notion of error rate, the $k$-FDR, is introduced in the same spirit as the $k$-FWER by generalizing the usual false discovery rate (FDR). A $k$-FWER procedure is constructed given any set of increasing constants by utilizing the $k$th order joint null distributions of the $p$-values without assuming any specific form of dependence among all the $p$-values. Procedures controlling the $k$-FDR are also developed by using the $k$th order joint null distributions of the $p$-values, first assuming that the sets of null and nonnull $p$-values are mutually independent or they are jointly positively dependent in the sense of being multivariate totally positive of order two (MTP$_2$) and then discarding that assumption about the overall dependence among the $p$-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