Journal of the Iranian Statistical Society

A New Proof of FDR Control Based on Forward Filtration

Authors

1 Sharif University of Technology, Tehran, Iran.

2 University of Washington, USA.

10.29252/jirss.19.1.59

Abstract

For multiple testing problems, Benjamini and Hochberg (1995) proposed the false discovery rate (FDR) as an alternative to the family-wise error rate (FWER). Since then, researchers have provided many proofs to control the FDR under different assumptions. Storey et al. (2004) showed that the rejection threshold of a BH step-up procedure is a stopping time with respect to the reverse filtration generated by the p-values and proposed a new proof based on the martingale theory. Following this work, martingale methods have been widely used to establish FDR control in various settings, but have been primarily applied to reverse filtration only. However, forward filtration can be more amenable for generalized and adaptive FDR controlling procedures. In this paper, we present a new proof, based on forward filtration, for step-down FDR controlling procedures that start from small p-values and update the rejection regions as larger p-values are observed.

Keywords

  1. Benditkis, J., P. Heesen, and A. Janssen (2018). The false discovery rate (fdr) of multiple tests in a class room lecture. Statistics & Probability Letters, 134, 29-35. [DOI:10.1016/j.spl.2017.09.017]
  2. Benditkis, J. and A. Janssen (2017). Finite sample bounds for expected number of false rejections under martingale dependence with applications to FDR. Electronic Journal of Statistics, 11(1), 1827-1857. [DOI:10.1214/17-EJS1268]
  3. Benjamini, Y. and Y. Hochberg (1995). Controlling the false discovery rate: a practical and powerful approach to multiple testing. Journal of the Royal Statistical Society. Series B. Methodological, 57(1), 289-300. [DOI:10.1111/j.2517-6161.1995.tb02031.x]
  4. Benjamini, Y. and W. Liu (1999). A step-down multiple hypotheses testing procedure that controls the false discovery rate under independence. Journal of Statistical Planning and Inference, 82(1), 163-170. [DOI:10.1016/S0378-3758(99)00040-3]
  5. Benjamini, Y. and D. Yekutieli (2001). The control of the false discovery rate in multiple testing under dependency. The Annals of Statistics, 29(4), 1165-1188. [DOI:10.1214/aos/1013699998]
  6. Blanchard, G. and É. Roquain (2009). Adaptive false discovery rate control under independence and dependence. Journal of Machine Learning Research, 10(Dec), 2837-2871.
  7. Cai, T. T. and W. Sun (2009). Simultaneous testing of grouped hypotheses: Finding needles in multiple haystacks. Journal of the American Statistical Association, 104(488), 1467-1481. [DOI:10.1198/jasa.2009.tm08415]
  8. Heesen, P., A. Janssen, et al. (2015). Inequalities for the false discovery rate (fdr) under dependence. Electronic Journal of Statistics, 9(1), 679-716. [DOI:10.1214/15-EJS1016]
  9. Klenke, A. (2013). Probability Theory: A Comprehensive Course. Universitext. Springer London. [DOI:10.1007/978-1-4471-5361-0]
  10. Li, A. and R. F. Barber (2019). Multiple testing with the structure-adaptive benjamini- hochberg algorithm. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 81(1), 45-74. [DOI:10.1111/rssb.12298]
  11. Peña, E. A., J. D. Habiger, and W. Wu (2011). Power-enhanced multiple decision functions controlling family-wise error and false discovery rates. Annals of statistics, 39(1), 556. [DOI:10.1214/10-AOS844]
  12. Ploner, A., S. Calza, A. Gusnanto, and Y. Pawitan (2006). Multidimensional local false discovery rate for microarray studies. Bioinformatics, 22(5), 556-565. [DOI:10.1093/bioinformatics/btk013]
  13. Ramdas, A., J. Chen, M. J. Wainwright, and M. I. Jordan (2017). Dagger: A sequential algorithm for fdr control on dags. arXiv preprint arXiv:1709.10250.
  14. Romano, J. P., A. M. Shaikh, and M. Wolf (2008). Control of the false discovery rate under dependence using the bootstrap and subsampling. Test, 17(3), 417. [DOI:10.1007/s11749-008-0126-6]
  15. Sarkar, S. K. (2002). Some results on false discovery rate in stepwise multiple testing procedures. The Annals of Statistics, 30(1), 239-257. [DOI:10.1214/aos/1015362192]
  16. Storey, J. D., J. E. Taylor, and D. Siegmund (2004). Strong control, conservative point estimation and simultaneous conservative consistency of false discovery rates: a unified approach. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 66(1), 187-205. [DOI:10.1111/j.1467-9868.2004.00439.x]
  17. Sun, W. and T. T. Cai (2007). Oracle and adaptive compound decision rules for false discovery rate control. Journal of the American Statistical Association, 102(479), 901-912. [DOI:10.1198/016214507000000545]
Journal of the Iranian Statistical Society
Volume 19, Issue 1
June 2020
Pages 59-68
  • Receive Date: 23 July 2022
  • Revise Date: 20 May 2024
  • Accept Date: 23 July 2022