Source: Electron. J. Statist.
We show that the control of the false discovery rate (FDR) for a multiple testing procedure is implied by two coupled simple sufficient conditions. The first one, which we call “self-consistency condition”, concerns the algorithm itself, and the second, called “dependency control condition” is related to the dependency assumptions on the p-value family. Many standard multiple testing procedures are self-consistent (e.g. step-up, step-down or step-up-down procedures), and we prove that the dependency control condition can be fulfilled when choosing correspondingly appropriate rejection functions, in three classical types of dependency: independence, positive dependency (PRDS) and unspecified dependency. As a consequence, we recover earlier results through simple and unifying proofs while extending their scope to several regards: weighted FDR, p-value reweighting, new family of step-up procedures under unspecified p-value dependency and adaptive step-up procedures. We give additional examples of other possible applications. This framework also allows for defining and studying FDR control for multiple testing procedures over a continuous, uncountable space of hypotheses.
Benjamini, Y. and Heller, R. (2007). False discovery rates for spatial signals., J. Amer. Statist. Assoc., 102(480):1272–1281.
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(1):289–300.
Benjamini, Y. and Hochberg, Y. (1997). Multiple hypotheses testing with weights., Scand. J. Statist., 24(3):407–418.
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.
Benjamini, Y. and Liu, W. (1999a). A distribution-free multiple test procedure that controls the false discovery rate. Technical report, Dept. of statistics, University of, Tel-Aviv.
Benjamini, Y. and Liu, W. (1999b). A step-down multiple hypotheses testing procedure that controls the false discovery rate under independence., J. Statist. Plann. Inference, 82(1-2):163–170.
Benjamini, Y. and Yekutieli, D. (2001). The control of the false discovery rate in multiple testing under dependency., Ann. Statist., 29(4):1165–1188.
Black, M. A. (2004). A note on the adaptive control of false discovery rates., J. R. Stat. Soc. Ser. B Stat. Methodol., 66(2):297–304.
Blanchard, G. and Fleuret, F. (2007). Occam’s hammer. In Bshouty, N. and Gentile, C., editors, Proceedings of the 20th. conference on learning theory (COLT 2007), volume 4539 of Springer Lecture Notes on Computer Science, pages 112–126.
Blanchard, G. and Roquain, E. (2008a). Adaptive FDR control under independence and dependence. ArXiV preprint, math.ST/0707.0536v2.
Blanchard, G. and Roquain, E. (2008b). Self-consistent multiple testing procedures. ArXiV preprint, math.ST/0802.1406v1.
Farcomeni, A. (2007). Some results on the control of the false discovery rate under dependence., Scandinavian Journal of Statistics, 34(2):275–297.
Finner, H., Dickhaus, T., and Roters, M. (2008). On the false discovery rate and an asymptotically optimal rejection curve., Ann. Statist. To appear.
Gavrilov, Y., Benjamini, Y., and Sarkar, S. K. (2008). An adaptive step-down procedure with proven FDR control under independence., Ann. Statist. To appear.
Genovese, C., Roeder, K., and Wasserman, L. (2006). False discovery control with, p-value weighting. Biometrika, 93(3):509–524.
Holm, S. (1979). A simple sequentially rejective multiple test procedure., Scand. J. Statist., 6(2):65–70.
Mathematical Reviews (MathSciNet): MR538597
Hommel, G. (1983). Tests of the overall hypothesis for arbitrary dependence structures., Biometrical J., 25(5):423–430.
Mathematical Reviews (MathSciNet): MR735888
Lehmann, E. L. (1966). Some concepts of dependence., Ann. Math. Statist., 37:1137–1153.
Mathematical Reviews (MathSciNet): MR202228
Lehmann, E. L. and Romano, J. P. (2005). Generalizations of the familywise error rate., The Annals of Statistics, 33:1138–1154.
Neuvial, P. (2008). Asymptotic properties of false discovery rate controlling procedures under independence. ArXiv preprint:, math.ST/0803.2111v1.
Perone Pacifico, M., Genovese, C., Verdinelli, I., and Wasserman, L. (2004). False discovery control for random fields., J. Amer. Statist. Assoc., 99(468):1002–1014.
Romano, J. P. and Shaikh, A. M. (2006a). On stepdown control of the false discovery proportion. In Rojo, J., editor, Optimality : the 2nd Lehmann Symposium, volume 49 of Lecture Notes-Monograph Series, pages 33–50. Institute of Mathematical statistics.
Romano, J. P. and Shaikh, A. M. (2006b). Stepup procedures for control of generalizations of the familywise error rate., Ann. Statist., 34(4):1850–1873.
Roquain, E. (2007)., Exceptional motifs in heterogeneous sequences. Contributions to theory and methodology of multiple testing. PhD thesis, Université Paris XI.
Roquain, E. and van de Wiel, M. (2008). Multi-weighting for fdr control. ArXiV preprint, math.ST/0807.4081.
Sarkar, S. K. (2002). Some results on false discovery rate in stepwise multiple testing procedures., Ann. Statist., 30(1):239–257.
Sarkar, S. K. (2008). Two-stage stepup procedures controlling FDR., Journal of Statistical Planning and Inference, 138(4):1072–1084.
Tamhane, A. C., Liu, W., and Dunnett, C. W. (1998). A generalized step-up-down multiple test procedure., Canad. J. Statist., 26(2):353–363.
Wasserman, L. and Roeder, K. (2006). Weighted hypothesis testing. Technical report, Dept. of statistics, Carnegie Mellon University. ArXiv preprint, math/0604172v1.