Electronic Journal of Statistics

Inequalities for the false discovery rate (FDR) under dependence

Philipp Heesen and Arnold Janssen

Full-text: Open access

Enhanced PDF (478 KB)

Abstract

Inequalities are key tools to prove FDR control of a multiple test. The present paper studies upper and lower bounds for the FDR under various dependence structures of $p$-values, namely independence, reverse martingale dependence and positive regression dependence on the subset (PRDS) of true null hypotheses. The inequalities are based on exact finite sample formulas which are also of interest for independent uniformly distributed $p$-values under the null. As applications the asymptotic worst case FDR of step up and step down tests coming from an non-decreasing rejection curve is established. In addition, new step up tests are established and necessary conditions for the FDR control are discussed. The reverse martingale models yield sharper FDR results than the PRDS models. Already in certain multivariate normal dependence models the familywise error rate of the Benjamini Hochberg step up test can be different from the desired level $\alpha$. The second part of the paper is devoted to adaptive step up tests under dependence. The well-known Storey estimator is modified so that the corresponding step up test has finite sample control for various block wise dependent $p$-values. These results may be applied to dependent genome data. Within each chromosome the $p$-values may be reverse martingale dependent while the chromosomes are independent.

Article information

Source
Electron. J. Statist., Volume 9, Number 1 (2015), 679-716.

Dates
First available in Project Euclid: 2 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1427990069

Digital Object Identifier
doi:10.1214/15-EJS1016

Mathematical Reviews number (MathSciNet)
MR3331854

Zentralblatt MATH identifier
1309.62083

Subjects
Primary: 62G10: Hypothesis testing
Secondary: 62G20: Asymptotic properties

Keywords
False discovery rate (FDR) inequalities multiple hypotheses testing PRDS reverse martingale dependence blockwise dependence adaptive Benjamini Hochberg methods p-values Storey test

Citation

Heesen, Philipp; Janssen, Arnold. Inequalities for the false discovery rate (FDR) under dependence. Electron. J. Statist. 9 (2015), no. 1, 679--716. doi:10.1214/15-EJS1016. https://projecteuclid.org/euclid.ejs/1427990069


Export citation

References

  • [1] Benjamini, Y. and Hochberg, Y. (1995). Controlling the false discovery rate: A practical and powerful approach in multiple testing., J. Roy. Stat. Soc. B 57 289–300.
    Mathematical Reviews (MathSciNet): MR1325392
  • [2] Benjamini, Y. and Yekutieli, D. (2001). The control of the false discovery rate in multiple testing under dependency., Ann. Stat. 10 1165–1188.
    Mathematical Reviews (MathSciNet): MR1869245
    Zentralblatt MATH: 1041.62061
    Digital Object Identifier: doi:10.1214/aos/1013699998
    Project Euclid: euclid.aos/1013699998
  • [3] Benjamini, Y., Krieger, A. M. and Yekutieli, D. (2006). Adaptive linear step-up procedures that control the false discovery rate., Biometrika 93(3) 491–507.
    Mathematical Reviews (MathSciNet): MR2261438
    Zentralblatt MATH: 1032.62037
    Digital Object Identifier: doi:10.1093/biomet/93.3.491
  • [4] Blanchard, G. and Roquain, E. (2009). Adaptive false discovery rate control under independence and dependence., J. Mach. Learn. Res. 29 2837–2871.
    Mathematical Reviews (MathSciNet): MR2579914
    Zentralblatt MATH: 1235.62093
  • [5] Blanchard, G. and Roquain, E. (2008). Two simple sufficient conditions for FDR control., Electron. J. Stat. 2 963–992.
    Mathematical Reviews (MathSciNet): MR2448601
    Zentralblatt MATH: 06165722
    Digital Object Identifier: doi:10.1214/08-EJS180
    Project Euclid: euclid.ejs/1224078069
  • [6] Efron, B., Tibshirani, R. and Storey, D. (2001). Empirical Bayes analysis of a microarray experiment., J. Am. Stat. Soc. 96 1151–1160.
    Mathematical Reviews (MathSciNet): MR1946571
    Zentralblatt MATH: 1073.62511
    Digital Object Identifier: doi:10.1198/016214501753382129
  • [7] Finner, H., Gontscharuk, V., Dickhaus, T. (2012). False discovery rate control of step-up-down tests with special emphasis on the asymptotically optimal rejections curve., Scand. J. Stat. 39 382–397.
    Mathematical Reviews (MathSciNet): MR2927031
    Digital Object Identifier: doi:10.1111/j.1467-9469.2012.00791.x
  • [8] Finner, H. and Gontscharuk, V. (2009). Controlling the familywise error rate with plug-in estimator for the proportion of true null hypotheses., J. R. Stat. Soc. B 71 1031–1048.
    Mathematical Reviews (MathSciNet): MR2750256
    Digital Object Identifier: doi:10.1111/j.1467-9868.2009.00719.x
  • [9] Finner, H., Dickhaus, T. and Roters, M. (2009). On the false discovery rate and an asymptotically optimal rejection curve., Ann. Stat. 37 596–618.
    Mathematical Reviews (MathSciNet): MR2502644
    Zentralblatt MATH: 1162.62068
    Digital Object Identifier: doi:10.1214/07-AOS569
    Project Euclid: euclid.aos/1236693143
  • [10] Finner, H. and Roters, M. (2001). On the false discovery rate and expected type I errors., Biometrical J. 43 985–1005.
    Mathematical Reviews (MathSciNet): MR1878272
    Digital Object Identifier: doi:10.1002/1521-4036(200112)43:8<985::AID-BIMJ985>3.0.CO;2-4
  • [11] Gavrilov, Y., Benjamini, Y. and Sarkar, S. K. (2009). An adaptive step-down procedure with proven FDR control under independence., Ann. Stat. 37 619–629.
    Mathematical Reviews (MathSciNet): MR2502645
    Zentralblatt MATH: 1162.62069
    Digital Object Identifier: doi:10.1214/07-AOS586
    Project Euclid: euclid.aos/1236693144
  • [12] Gontscharuk, V. (2010). Asymptotic and exact results on FWER and FDR in multiple hypothesis testing. Diss. PhD thesis, Heinrich-Heine-Universität Düsseldorf., http://docserv.uni-duesseldorf.de/servlets/DocumentServlet?id=16990.
  • [13] Guo, W., Rao, M. B. (2008). On optimality of the Benjamini-Hochberg procedure for the false discovery rate., Statist. Probab. Lett. 78(14) 2024–2030.
    Mathematical Reviews (MathSciNet): MR2528629
  • [14] Guo, W. and Sarkar, S. K. (2013). Adaptive Controls of FWER and FDR Under Block Dependence., Unpublished manuscript. http://web.njit.edu/~wguo/Guo & Sarkar 2012.pdf.
  • [15] Heesen, P. and Janssen, A. (2014). Dynamic adaptive multiple tests with finite sample FDR control., arXiv:1410.6296v1, October 2014.
  • [16] Heesen, P. (2014). Adaptive step up tests for the false discovery rate (FDR) under independence and dependence. Ph.D. thesis, Heinrich-Heine-Universität Düsseldorf, Germany., http://docserv.uni-duesseldorf.de/servlets/DocumentServlet?id=33047.
  • [17] Liang, K. and Nettleton, D. (2012). Adaptive and dynamic adaptive procedures for false discovery rate control and estimation., J. Roy. Stat. Soc. B 74(1) 163–182.
    Mathematical Reviews (MathSciNet): MR2885844
    Digital Object Identifier: doi:10.1111/j.1467-9868.2011.01001.x
  • [18] Marshall, A. W. and Olkin, I. (1967). A multivariate exponential distribution., J. Amer. Stat. Assoc. 62(317) 30–44.
    Mathematical Reviews (MathSciNet): MR215400
    Zentralblatt MATH: 0147.38106
    Digital Object Identifier: doi:10.1080/01621459.1967.10482885
  • [19] Meskaldji, D. E., Thiran, J.-P. and Morgenthaler, S. (2013). A comprehensive error rate for multiple testing., arXiv:1112.4519v4, July 2013.
  • [20] Sarkar, S. K. (2002). Some results on false discovery rate in stepwise multiple testing procedures., Ann. Statist. 30 239–257.
    Mathematical Reviews (MathSciNet): MR1892663
    Zentralblatt MATH: 1101.62349
    Digital Object Identifier: doi:10.1214/aos/1015362192
    Project Euclid: euclid.aos/1015362192
  • [21] Sarkar, S. K. (2008). On methods controlling the false discovery rate., Sankhya Series A, 70 135–168.
    Mathematical Reviews (MathSciNet): MR2551809
  • [22] Scheer, M. (2013). Controlling the Number of False Rejections in Multiple Hypotheses Testing. Diss. PhD thesis, Heinrich- Heine-Universität Düsseldorf., http://docserv.uni-duesseldorf.de/servlets/DocumentServlet?id=23691.
  • [23] Storey, J. D. (2002). A direct approach to false discovery rates., J. Roy. Stat. Soc. B 64 479–498.
    Mathematical Reviews (MathSciNet): MR1924302
    Zentralblatt MATH: 1090.62073
    Digital Object Identifier: doi:10.1111/1467-9868.00346
  • [24] 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. Roy. Stat. Soc. B 66 187–205.
    Mathematical Reviews (MathSciNet): MR2035766
    Zentralblatt MATH: 1061.62110
    Digital Object Identifier: doi:10.1111/j.1467-9868.2004.00439.x
  • [25] Zeisel, A., Zuk, O. and Domany, E. (2011). FDR control with adaptive procedures and FDR monotonicity., Ann. Appl. Stat. 5(2A) 943–968.
    Mathematical Reviews (MathSciNet): MR2840182
    Zentralblatt MATH: 1232.62106
    Digital Object Identifier: doi:10.1214/10-AOAS399
    Project Euclid: euclid.aoas/1310562212

The Institute of Mathematical Statistics and the Bernoulli Society

Electronic Journal of Statistics

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more