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Some results on false discovery rate in stepwise multiple testing procedures. (English) Zbl 1101.62349

Summary: The concept of false discovery rate (FDR) has been receiving increasing attention by researchers in multiple hypotheses testing. This paper produces some theoretical results on the FDR in the context of stepwise multiple testing procedures with dependent test statistics. It was recently shown byY. Benjamini and D. Yekutieli [ibid. 29, No. 4, 1165–1188 (2001; Zbl 1041.62061)] that the Benjamini-Hochberg step-up procedure controls the FDR when the test statistics are positively dependent in a certain sense. This paper strengthens their work by showing that the critical values of that procedure can be used in a much more general stepwise procedure under similar positive dependency. It is also shown that the FDR-controlling the Y. Benjamini and W. Liu step-down procedure [J. Stat. Plann. Inference 82, No. 1–2, 163–170 (1999; Zbl 1063.62558)] originally developed for independent test statistics works even when the test statistics are positively dependent in some sense. An explicit expression for the FDR of a generalized stepwise procedure and an upper bound to the FDR of a step-down procedure are obtained in terms of probability distributions of ordered components of dependent random variables before establishing the main results.

MSC:

62H15 Hypothesis testing in multivariate analysis
62J15 Paired and multiple comparisons; multiple testing
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[1] ABRAMOVICH, F. and BENJAMINI, Y. (1996). Adaptive thresholding of wavelets coefficients. Comput. Statist. Data Anal. 22 351-361.
[2] ABRAMOVICH, F., BENJAMINI, Y., DONOHO, D. L. and JOHNSTONE, I. M. (2000). Adapting to unknown sparsity by controlling the false discovery rate. Technical Report 2000-19, Dept. Statistics, Stanford Univ. · Zbl 1092.62005
[3] BASFORD, K. E. and TUKEY, J. W. (1997). Graphical profiles as an aid to understanding plant breeding experiments. J. Statist. Plann. Inference 57 93-107. · Zbl 0900.62028
[4] 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. JSTOR: · Zbl 0809.62014
[5] BENJAMINI, Y. and LIU, W. (1999). A step-down multiple hypotheses testing procedures that controls the false discovery rate under independence. J. Statist. Plann. Inference 82 163- 170. · Zbl 1063.62558
[6] BENJAMINI, Y. and YEKUTIELI, D. (2001). The control of the false discovery rate in multiple testing under dependency. Ann. Statist. 29 1165-1188. · Zbl 1041.62061
[7] DRIGALENKO, E. I. and ELSTON, R. C. (1997). False discoveries in genome scanning. Gen. Epidem. 14 779-784.
[8] FINNER, H. (1999). Stepwise multiple test procedures and control of directional errors. Ann. Statist. 27 274-289. · Zbl 0978.62057
[9] FINNER, H. and ROTERS, M. (1998). Asymptotic comparison of step-down and step-up multiple test procedures based on exchangeable test statistics. Ann. Statist. 26 505-524. · Zbl 0934.62073
[10] FINNER, H. and ROTERS, M. (1999). Asymptotic comparisons of the critical values of step-down and step-up multiple comparison procedures. J. Statist. Plann. Inference 79 11-30. · Zbl 0951.62060
[11] KARLIN, S. and RINOTT, Y. (1980). Classes of orderings of measures and related correlation inequalities I. J. Multivariate Anal. 10 467-498. · Zbl 0469.60006
[12] LIU, W. (1996). Multiple tests of a non-hierarchical finite family of hypotheses. J. Roy. Statist. Soc. Ser. B 58 455-461. JSTOR: · Zbl 0853.62054
[13] SARKAR, S. K. (1998). Some probability inequalities for ordered MTP2 random variables: a proof of the Simes conjecture. Ann. Statist. 26 494-504. · Zbl 0929.62065
[14] SARKAR, S. K. and CHANG, C.-K. (1997). The Simes method for multiple hypothesis testing with positively dependent test statistics. J. Amer. Statist. Assoc. 92 1601-1608. JSTOR: · Zbl 0912.62079
[15] SIMES, R. J. (1986). An improved Bonferroni procedure for multiple tests of significance. Biometrika 73 751-754. JSTOR: · Zbl 0613.62067
[16] TAMHANE, A. C., LIU, W. and DUNNETT, C. W. (1998). A generalized step-up-down multiple test procedure. Canad. J. Statist. 26 353-363. JSTOR: · Zbl 0914.62013
[17] WILLIAMS, V. S. L., JONES, L. V. and TUKEY, J. W. (1999). Controlling error in multiple comparisons with examples from state-to-state differences in educational achievement. J. Edu. Behav. Statist. 24 42-69.
[18] PHILADELPHIA, PENNSYLVANIA 19122 E-MAIL: sanat@sbm.temple.edu
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