The Annals of Statistics

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

Sanat K. Sarkar

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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 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 Šidák 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 Šidák 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 Šidák 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.

Article information

Ann. Statist., Volume 34, Number 1 (2006), 394-415.

First available in Project Euclid: 2 May 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62J15: Paired and multiple comparisons 62H15: Hypothesis testing
Secondary: 62H99: None of the above, but in this section

Modified Bonferroni and Šidák procedures mixture model positive false discovery rate positive false nondiscovery rate


Sarkar, Sanat K. False discovery and false nondiscovery rates in single-step multiple testing procedures. Ann. Statist. 34 (2006), no. 1, 394--415. doi:10.1214/009053605000000778.

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