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Results on the false discovery rate (FDR) and the false nondiscovery rate(FNR) are developed for single-step multiple testing procedures. In addition toverifying desirable properties of FDR and FNR as measures of error rates, theseresults extend previously known results, providing further insights,particularly under dependence, into the notions of FDR and FNR and relatedmeasures. First, considering fixed configurations of true and false nullhypotheses, inequalities are obtained to explain how an FDR- or FNR-controllingsingle-step procedure, such as a Bonferroni or \u{S}id\'{a}k procedure, canpotentially be improved. Two families of procedures are then constructed, onethat modifies the FDR-controlling and the other that modifies theFNR-controlling \u{S}id\'{a}k procedure. These are proved to control FDR or FNRunder independence less conservatively than the corresponding families thatmodify the FDR- or FNR-controlling Bonferroni procedure. Results of numericalinvestigations of the performance of the modified \u{S}id\'{a}k FDR procedureover its competitors are presented. Second, considering a mixture model wheredifferent configurations of true and false null hypotheses are assumed to havecertain probabilities, results are also derived that extend some of Storey'swork to the dependence case.

False discovery and false nondiscovery rates in single-step multiple testing procedures