## The Annals of Statistics

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

Sanat K. Sarkar

#### Abstract

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

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

Dates
First available in Project Euclid: 2 May 2006

https://projecteuclid.org/euclid.aos/1146576268

Digital Object Identifier
doi:10.1214/009053605000000778

Mathematical Reviews number (MathSciNet)
MR2275247

Zentralblatt MATH identifier
1091.62060

#### Citation

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. https://projecteuclid.org/euclid.aos/1146576268

#### References

• Benjamini, Y. and Hochberg, Y. (1995). Controlling the false discovery rate: A practical and powerful approach to multiple testing. J. Roy. Statist. Soc. Ser. B 57 289--300.
• Benjamini, Y. and Hochberg, Y. (2000). On the adaptive control of the false discovery rate in multiple testing with independent statistics. J. Educational and Behavioral Statistics 25 60--83.
• Benjamini, Y., Krieger, A. M. and Yekutieli, D. (2002). Adaptive linear step-up false discovery rate controlling procedures. Unpublished manuscript.
• Benjamini, Y. and Liu, W. (1999). A step-down multiple hypotheses testing procedure that controls the false discovery rate under independence. J. Statist. Plann. Inference 82 163--170.
• Benjamini, Y. and Yekutieli, D. (2001). The control of the false discovery rate in multiple testing under dependence. Ann. Statist. 29 1165--1188.
• Efron, B. (2003). Robbins, empirical Bayes and microarrays. Ann. Statist. 31 366--378.
• Efron, B., Tibshirani, R., Storey, J. D. and Tusher, V. (2001). Empirical Bayes analysis of a microarray experiment. J. Amer. Statist. Assoc. 96 1151--1160.
• Genovese, C. and Wasserman, L. (2002). Operating characteristics and extensions of the false discovery rate procedure. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 499--517.
• Genovese, C. and Wasserman, L. (2004). A stochastic process approach to false discovery control. Ann. Statist. 32 1035--1061.
• Hochberg, Y. and Benjamini, Y. (1990). More powerful procedures for multiple significance testing. Statistics in Medicine 9 811--818.
• Karlin, S. (1968). Total Positivity 1. Stanford Univ. Press.
• Lehmann, E. (1986). Testing Statistical Hypotheses, 2nd ed. Wiley, New York.
• Sarkar, S. K. (1998). Some probability inequalities for ordered MTP$_2$ random variables: A proof of the Simes conjecture. Ann. Statist. 26 494--504.
• Sarkar, S. K. (2002). Some results on false discovery rate in stepwise multiple testing procedures. Ann. Statist. 30 239--257.
• Sarkar, S. K. (2004). FDR-controlling stepwise procedures and their false negatives rates. J. Statist. Plann. Inference 125 119--137.
• Simes, R. J. (1986). An improved Bonferroni procedure for multiple tests of significance. Biometrika 73 751--754.
• Storey, J. D. (2002). A direct approach to false discovery rates. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 479--498.
• Storey, J. D. (2003). The positive false discovery rate: A Bayesian interpretation and the $q$-value. Ann. Statist. 31 2013--2035.
• Storey, J. D., Taylor, J. E. and Siegmund, D. (2004). Strong control, conservative point estimation and simultaneous conservative consistency of false discovery rates: A unified approach. J. R. Stat. Soc. Ser. B Stat. Methodol. 66 187--205.