IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v160y2020ics016771522030016x.html
   My bibliography  Save this article

Strong consistency and asymptotic normality for quantities related to the Benjamini–Hochberg false discovery rate procedure

Author

Listed:
  • Izmirlian, Grant

Abstract

The Benjamini–Hochberg (BH) false discovery rate (FDR) procedure enjoys widespread use in multiple testing scenarios when the cost of false positives is low and the number of simultaneous tests prohibits use of the Bonferroni procedure. The FDR and the average power are each expected values, of the false discovery proportion (FDP) and true positive proportion (TPP), respectively. Chi (2007), proved a law of the iterated logarithm (LIL) for the positive proportion (PP), FDP and TPP, and discusses a criticality phenomenon whereby a minimal FDR corresponding to the effect size is required for a nonzero rate of positive calls. While almost sure convergence follows as a corollary, no direct simple proof exists in the literature. We provide this result under the same set of conditions. Of greater consequence, we prove central limit results (CLT) for the PP, FPF and TPP under these weak conditions, providing full characterization of the limits.

Suggested Citation

  • Izmirlian, Grant, 2020. "Strong consistency and asymptotic normality for quantities related to the Benjamini–Hochberg false discovery rate procedure," Statistics & Probability Letters, Elsevier, vol. 160(C).
    • Handle: RePEc:eee:stapro:v:160:y:2020:i:c:s016771522030016x
    DOI: 10.1016/j.spl.2020.108713
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S016771522030016X
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spl.2020.108713?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Robert Lee Taylor & Ronald Frank Patterson, 1985. "Strong laws of large numbers for arrays of row-wise exchangeable random elements," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 8, pages 1-10, January.
    2. Sun, Wenguang & Cai, T. Tony, 2007. "Oracle and Adaptive Compound Decision Rules for False Discovery Rate Control," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 901-912, September.
    3. Ibrahim J. G. & Chen M-H. & Gray R. J., 2002. "Bayesian Models for Gene Expression With DNA Microarray Data," Journal of the American Statistical Association, American Statistical Association, vol. 97, pages 88-99, March.
    4. Benditkis, Julia & Heesen, Philipp & Janssen, Arnold, 2018. "The false discovery rate (FDR) of multiple tests in a class room lecture," Statistics & Probability Letters, Elsevier, vol. 134(C), pages 29-35.
    5. Efron B. & Tibshirani R. & Storey J.D. & Tusher V., 2001. "Empirical Bayes Analysis of a Microarray Experiment," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1151-1160, December.
    6. Won, Joong-Ho & Lim, Johan & Yu, Donghyeon & Kim, Byung Soo & Kim, Kyunga, 2014. "Monotone false discovery rate," Statistics & Probability Letters, Elsevier, vol. 87(C), pages 86-93.
    7. Christopher Genovese & Larry Wasserman, 2002. "Operating characteristics and extensions of the false discovery rate procedure," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(3), pages 499-517, August.
    8. John D. Storey, 2002. "A direct approach to false discovery rates," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(3), pages 479-498, August.
    9. Glueck Deborah H & Mandel Jan & Karimpour-Fard Anis & Hunter Lawrence & Muller Keith E, 2008. "Exact Calculations of Average Power for the Benjamini-Hochberg Procedure," The International Journal of Biostatistics, De Gruyter, vol. 4(1), pages 1-22, June.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Cipolli III, William & Hanson, Timothy & McLain, Alexander C., 2016. "Bayesian nonparametric multiple testing," Computational Statistics & Data Analysis, Elsevier, vol. 101(C), pages 64-79.
    2. T. Tony Cai & Wenguang Sun & Weinan Wang, 2019. "Covariate‐assisted ranking and screening for large‐scale two‐sample inference," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 81(2), pages 187-234, April.
    3. E. M. Conlon & B. L. Postier & B. A. Methe & K. P. Nevin & D. R. Lovley, 2009. "Hierarchical Bayesian meta-analysis models for cross-platform microarray studies," Journal of Applied Statistics, Taylor & Francis Journals, vol. 36(10), pages 1067-1085.
    4. Tingting Cui & Pengfei Wang & Wensheng Zhu, 2021. "Covariate-adjusted multiple testing in genome-wide association studies via factorial hidden Markov models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(3), pages 737-757, September.
    5. Wenguang Sun & T. Tony Cai, 2009. "Large‐scale multiple testing under dependence," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(2), pages 393-424, April.
    6. Joshua Habiger & Edsel Peña, 2011. "Randomised -values and nonparametric procedures in multiple testing," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 23(3), pages 583-604.
    7. Habiger, Joshua D. & Peña, Edsel A., 2014. "Compound p-value statistics for multiple testing procedures," Journal of Multivariate Analysis, Elsevier, vol. 126(C), pages 153-166.
    8. Ruth Heller & Saharon Rosset, 2021. "Optimal control of false discovery criteria in the two‐group model," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 83(1), pages 133-155, February.
    9. Xiaoquan Wen, 2017. "Robust Bayesian FDR Control Using Bayes Factors, with Applications to Multi-tissue eQTL Discovery," Statistics in Biosciences, Springer;International Chinese Statistical Association, vol. 9(1), pages 28-49, June.
    10. Alejandro Ochoa & John D Storey & Manuel Llinás & Mona Singh, 2015. "Beyond the E-Value: Stratified Statistics for Protein Domain Prediction," PLOS Computational Biology, Public Library of Science, vol. 11(11), pages 1-21, November.
    11. Zhigen Zhao, 2022. "Where to find needles in a haystack?," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(1), pages 148-174, March.
    12. T. Tony Cai & Wenguang Sun, 2017. "Optimal screening and discovery of sparse signals with applications to multistage high throughput studies," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(1), pages 197-223, January.
    13. Jiaying Gu & Roger Koenker, 2020. "Invidious Comparisons: Ranking and Selection as Compound Decisions," Papers 2012.12550, arXiv.org, revised Sep 2021.
    14. Debashis Ghosh & Wei Chen & Trivellore Raghuanthan, 2004. "The false discovery rate: a variable selection perspective," The University of Michigan Department of Biostatistics Working Paper Series 1040, Berkeley Electronic Press.
    15. Debashis Ghosh & Arul Chinnaiyan, 2004. "Covariate adjustment in the analysis of microarray data from clinical studies," The University of Michigan Department of Biostatistics Working Paper Series 1030, Berkeley Electronic Press.
    16. Cai, Qingyun, 2018. "A scoring criterion for rejection of clustered p-values," Computational Statistics & Data Analysis, Elsevier, vol. 121(C), pages 180-189.
    17. Jiaying Gu & Roger Koenker, 2023. "Invidious Comparisons: Ranking and Selection as Compound Decisions," Econometrica, Econometric Society, vol. 91(1), pages 1-41, January.
    18. Shiyun Chen & Ery Arias-Castro, 2021. "On the power of some sequential multiple testing procedures," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(2), pages 311-336, April.
    19. Wen Shi & Xi Chen & Jennifer Shang, 2019. "An Efficient Morris Method-Based Framework for Simulation Factor Screening," INFORMS Journal on Computing, INFORMS, vol. 31(4), pages 745-770, October.
    20. B. Moerkerke & E. Goetghebeur & J. De Riek & I. Roldán‐Ruiz, 2006. "Significance and impotence: towards a balanced view of the null and the alternative hypotheses in marker selection for plant breeding," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 169(1), pages 61-79, January.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:stapro:v:160:y:2020:i:c:s016771522030016x. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.