Some results on the control of the false discovery rate under dependence. (English) Zbl 1142.62048
Multiple hypothesis testing procedures are considered with control of false discovery rate: the Benjamini-Hochberg and plug-in algorithms. Asymptotics of these procedures are investigated when the number of tested hypotheses tends to infinity and the dependence between the test statistics for each hypothesis is weak (in the sense of \(\alpha\)-dependence, positive association or analogous measures of dependence). It is shown that these algorithms remain valid under weak dependence. Asymptotic normality of the false discovery proportion is demonstrated. The author proposes an iterative version of the plug-in algorithm in which the number of hypotheses rejected on the previous iteration is used for the estimation of the false null proportion on the next iteration. Simulation results are presented.
Reviewer: R. E. Maiboroda (Kyïv)
MSC:
| 62J15 | Paired and multiple comparisons; multiple testing |
| 62F05 | Asymptotic properties of parametric tests |
| 65C60 | Computational problems in statistics (MSC2010) |
Keywords:
mixture model; multiple testing; time course DNA array; \(\alpha\)-dependence; positive associationReferences:
| [1] | Alfo M., Econ. J. 7 pp 426– (2004) · Zbl 1064.62031 · doi:10.1111/j.1368-423X.2004.00138.x |
| [2] | Amaratunga D., Exploration and analysis of DNA microarray and protein array data (2004) |
| [3] | Arcones M. A., Chaos expansions, multiple Wiener-Ito integrals and their applications pp 205– (2000) |
| [4] | Benjamini Y., J. Roy. Statist. Soc. Ser. B 57 pp 289– (1995) |
| [5] | DOI: 10.1111/1467-9469.00072 · Zbl 1090.62548 · doi:10.1111/1467-9469.00072 |
| [6] | DOI: 10.2307/1165312 · doi:10.2307/1165312 |
| [7] | DOI: 10.1214/aos/1013699998 · Zbl 1041.62061 · doi:10.1214/aos/1013699998 |
| [8] | Benjamini Y., Two staged linear step up FDR controlling procedures. Technical report (2004) |
| [9] | Bickel D. R., On ?Strong control, conservative point estimation and simultaneous conservative consistency of false discovery rates?: does a large number of tests obviate confidence intervals of the FDR? (2004) |
| [10] | Billingsley P., Convergence of probability measures (1999) · Zbl 0944.60003 · doi:10.1002/9780470316962 |
| [11] | Bradley R. C., Ann. Probab. 21 pp 1921– (1993) |
| [12] | DOI: 10.1038/4462 · doi:10.1038/4462 |
| [13] | DOI: 10.1007/s004400050014 · doi:10.1007/s004400050014 |
| [14] | Doukan P., Mixing (1994) · doi:10.1007/978-1-4612-2642-0 |
| [15] | DOI: 10.1016/S0304-4149(99)00055-1 · Zbl 0996.60020 · doi:10.1016/S0304-4149(99)00055-1 |
| [16] | Dudoit S., Statist. Appl. Genet. Mol. Biol. 3 pp 1– (2004) · Zbl 1166.62338 · doi:10.2202/1544-6115.1040 |
| [17] | DOI: 10.1214/ss/1056397487 · Zbl 1048.62099 · doi:10.1214/ss/1056397487 |
| [18] | DOI: 10.1038/4434 · doi:10.1038/4434 |
| [19] | Dvoretzky A., Ann. Math. Statist. 33 pp 642– (1956) |
| [20] | Esary J. D., Ann. Math. Statist. 38 pp 1466– (1967) |
| [21] | DOI: 10.1214/aos/1015362191 · Zbl 1012.62020 · doi:10.1214/aos/1015362191 |
| [22] | DOI: 10.1111/1467-9868.00347 · Zbl 1090.62072 · doi:10.1111/1467-9868.00347 |
| [23] | Genovese C. R., Exceedance control of the false discovery proportion. Technical Report (2004) · Zbl 1171.62338 |
| [24] | DOI: 10.1214/009053604000000283 · Zbl 1092.62065 · doi:10.1214/009053604000000283 |
| [25] | Genovese C. R., Biometrika (2007) |
| [26] | Hochberg Y., Multiple comparisons procedures (1987) · doi:10.1002/9780470316672 |
| [27] | Kumar J. D., Ann. Statist. 11 pp 286– (1983) |
| [28] | Lehmann E. L., Ann. Math. Statist. 37 pp 1137– (1966) |
| [29] | DOI: 10.1214/009053605000000084 · Zbl 1072.62060 · doi:10.1214/009053605000000084 |
| [30] | DOI: 10.2307/2282907 · Zbl 0147.38106 · doi:10.2307/2282907 |
| [31] | Massart P., Ann. Probab. 18 pp 1269– (1990) |
| [32] | N. Meinshausen, and J. Rice (2005 ). , Seminar fur Statistik, Zurich. |
| [33] | Nze P. A., Ann. Statist. 30 pp 397– (2002) |
| [34] | Oliveira P., Wavelets and Statistics pp 331– (1995) · doi:10.1007/978-1-4612-2544-7_19 |
| [35] | Parmigiani G., The analysis of gene expression data: methods and software (2003) · Zbl 1012.00021 · doi:10.1007/b97411 |
| [36] | DOI: 10.1016/j.jspi.2003.07.019 · Zbl 1074.62009 · doi:10.1016/j.jspi.2003.07.019 |
| [37] | DOI: 10.1214/aos/1015362192 · Zbl 1101.62349 · doi:10.1214/aos/1015362192 |
| [38] | DOI: 10.1016/j.jspi.2003.06.019 · Zbl 1097.62062 · doi:10.1016/j.jspi.2003.06.019 |
| [39] | Sarkar S. K., Stepup procedures controlling generalized FWER and generalized FDR. Technical Report (2005) · Zbl 1129.62066 |
| [40] | DOI: 10.2307/2965431 · Zbl 0912.62079 · doi:10.2307/2965431 |
| [41] | Silvestrov D. S., Dokl. Akad. Nauk SSSR 199 pp 1251– (1971) |
| [42] | Silvestrov D. S., Teor. Veroyatn. Primen. 17 pp 707– (1972) |
| [43] | Silvestrov D. S., Limit theorems for randomly stopped stochastic processes (2004) · Zbl 1057.60021 · doi:10.1007/978-0-85729-390-9 |
| [44] | DOI: 10.1111/1467-9868.00346 · Zbl 1090.62073 · doi:10.1111/1467-9868.00346 |
| [45] | DOI: 10.1214/aos/1074290335 · Zbl 1042.62026 · doi:10.1214/aos/1074290335 |
| [46] | Storey J. D., A significance method for time course microarray experiments applied to two human studies (2004) |
| [47] | DOI: 10.1111/j.1467-9868.2004.00439.x · Zbl 1061.62110 · doi:10.1111/j.1467-9868.2004.00439.x |
| [48] | DOI: 10.1214/aos/1018031099 · Zbl 0937.62051 · doi:10.1214/aos/1018031099 |
| [49] | Tong Y. L., Probability inequalities in multivariate distributions (1980) · Zbl 0455.60003 |
| [50] | Van der Vaart A. W., Asymptotic statistics (1998) · Zbl 0910.62001 · doi:10.1017/CBO9780511802256 |
| [51] | DOI: 10.1091/mbc.02-02-0030. · doi:10.1091/mbc.02-02-0030 |
| [52] | DOI: 10.1093/bioinformatics/btg364 · doi:10.1093/bioinformatics/btg364 |
| [53] | DOI: 10.1214/aos/1018031114 · Zbl 0978.62039 · doi:10.1214/aos/1018031114 |
| [54] | Wu W. B., Empirical processes of dependent random variables (2004) |
| [55] | DOI: 10.1016/S0378-3758(99)00041-5 · Zbl 1063.62563 · doi:10.1016/S0378-3758(99)00041-5 |
| [56] | Yoshihara K., Yokohama Math. J. 23 pp 77– (1975) |
| [57] | DOI: 10.1007/BF01192169 · Zbl 0792.60018 · doi:10.1007/BF01192169 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
