×

False discovery rate control under Archimedean copula. (English) Zbl 1305.62269

Summary: We are concerned with the false discovery rate (FDR) of the linear step-up test \(\varphi^{\mathrm{LSU}}\) considered by Y. Benjamini and Y. Hochberg [J. R. Stat. Soc., Ser. B 57, No. 1, 289–300 (1995; Zbl 0809.62014)]. It is well known that \(\varphi^{\mathrm{LSU}}\) controls the FDR at level \(m_{0}q/m\) if the joint distribution of \(p\)-values is multivariate totally positive of order \(2\). In this, \(m\) denotes the total number of hypotheses, \(m_{0}\) the number of true null hypotheses, and \(q\) the nominal FDR level. Under the assumption of an Archimedean \(p\)-value copula with completely monotone generator, we derive a sharper upper bound for the FDR of \(\varphi^{\mathrm{LSU}}\) as well as a non-trivial lower bound. Application of the sharper upper bound to parametric subclasses of Archimedean \(p\)-value copulae allows us to increase the power of \(\varphi^{\mathrm{LSU}}\) by pre-estimating the copula parameter and adjusting \(q\). Based on the lower bound, a sufficient condition is obtained under which the FDR of \(\varphi^{\mathrm{LSU}}\) is exactly equal to \(m_{0}q/m\), as in the case of stochastically independent \(p\)-values. Finally, we deal with high-dimensional multiple test problems with exchangeable test statistics by drawing a connection between infinite sequences of exchangeable \(p\)-values and Archimedean copulae with completely monotone generators. Our theoretical results are applied to important copula families, including Clayton copulae and Gumbel-Hougaard copulae.

MSC:

62J15 Paired and multiple comparisons; multiple testing
62F05 Asymptotic properties of parametric tests
62F03 Parametric hypothesis testing

Citations:

Zbl 0809.62014

Software:

nacopula
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] Benjamini, Y. and Hochberg, Y. (1995). Controlling the false discovery rate: A practical and powerful approach to multiple testing., J. R. Stat. Soc. Ser. B Stat. Methodol. 57 289-300. · Zbl 0809.62014
[2] Benjamini, Y. and Liu, W. (1999). A step-down multiple hypotheses testing procedure that controls the false discovery rate under independence., J. Stat. Plann. Inference 82 163-170. · Zbl 1063.62558
[3] Benjamini, Y. and Yekutieli, D. (2001). The control of the false discovery rate in multiple testing under dependency., Ann. Stat. 29 1165-1188. · Zbl 1041.62061
[4] Blanchard, G. and Roquain, E. (2008). Two simple sufficient conditions for FDR control., Electron. J. Statist. 2 963-992. · Zbl 1320.62179
[5] Blanchard, G. and Roquain, E. (2009). Adaptive false discovery rate control under independence and dependence., Journal of Machine Learning Research 10 2837-2871. · Zbl 1235.62093
[6] Blanchard, G., Dickhaus, T., Roquain, E. and Villers, F. (2014). On least favorable configurations for step-up-down tests., Statistica Sinica 24 1-23. · Zbl 1416.62431
[7] Cai, T. T. and Jin, J. (2010). Optimal rates of convergence for estimating the null density and proportion of nonnull effects in large-scale multiple testing., Ann. Stat. 38 100-145. · Zbl 1181.62040
[8] Cerqueti, R., Costantini, M. and Lupi, C. (2012). A copula-based analysis of false discovery rate control under dependence assumptions. Economics & Statistics Discussion Paper No. 065/12, Università degli Studi del Molise, Dipartimento di Scienze Economiche, Gestionali e Sociali, (SEGeS).
[9] Chambers, J. M., Mallows, C. L. and Stuck, B. W. (1976). A method for simulating stable random variables., J. Am. Stat. Assoc. 71 340-344. · Zbl 0341.65003
[10] Delattre, S. and Roquain, E. (2011). On the false discovery proportion convergence under Gaussian equi-correlation., Stat. Probab. Lett. 81 111-115. · Zbl 1206.62132
[11] Dickhaus, T. (2013). Randomized \(p\)-values for multiple testing of composite null hypotheses., J. Stat. Plann. Inference 143 1968-1979. · Zbl 1279.62150
[12] Dickhaus, T. (2014)., Simultaneous Statistical Inference with Applications in the Life Sciences . Springer-Verlag Berlin Heidelberg. · Zbl 1296.62062
[13] Dickhaus, T. and Gierl, J. (2013). Simultaneous test procedures in terms of p-value copulae., Proceedings on the 2nd Annual International Conference on Computational Mathematics, Computational Geometry & Statistics (CMCGS 2013) 2 75-80. Global Science and Technology Forum (GSTF).
[14] Dudoit, S. and van der Laan, M. J. (2008)., Multiple Testing Procedures with Applications to Genomics. Springer Series in Statistics. New York, NY: Springer. · Zbl 1261.62014
[15] Fengler, M. R. and Okhrin, O. (2012). Realized Copula SFB 649 Discussion Paper No. 2012-034, Sonderforschungsbereich 649, Humboldt-Universität zu Berlin, Germany. available at, .
[16] Finner, H., Dickhaus, T. and Roters, M. (2007). Dependency and false discovery rate: Asymptotics., Ann. Stat. 35 1432-1455. · Zbl 1125.62076
[17] Finner, H., Dickhaus, T. and Roters, M. (2009). On the false discovery rate and an asymptotically optimal rejection curve., Ann. Stat. 37 596-618. · Zbl 1162.62068
[18] Finner, H., Gontscharuk, V. and Dickhaus, T. (2012). False discovery rate control of step-up-down tests with special emphasis on the asymptotically optimal rejection curve., Scandinavian Journal of Statistics 39 382-397. · Zbl 1246.62171
[19] Finner, H. and Roters, M. (1998). Asymptotic comparison of step-down and step-up multiple test procedures based on exchangeable test statistics., Ann. Stat. 26 505-524. · Zbl 0934.62073
[20] Gavrilov, Y., Benjamini, Y. and Sarkar, S. K. (2009). An adaptive step-down procedure with proven FDR control under independence., Ann. Stat. 37 619-629. · Zbl 1162.62069
[21] Genest, C., Nešlehová, J. and Ben Ghorbal, N. (2011). Estimators based on Kendall’s tau in multivariate copula models., Aust. N. Z. J. Stat. 53 157-177. · Zbl 1274.62367
[22] Genovese, C. and Wasserman, L. (2002). Operating characteristics and extensions of the false discovery rate procedure., J. R. Stat. Soc., Ser. B, Stat. Methodol. 64 499-517. · Zbl 1090.62072
[23] Genovese, C. and Wasserman, L. (2004). A stochastic process approach to false discovery control., Ann. Stat. 32 1035-1061. · Zbl 1092.62065
[24] Guo, W. and Rao, M. B. (2008). On control of the false discovery rate under no assumption of dependency., J. Stat. Plann. Inference 138 3176-3188. · Zbl 1140.62060
[25] Hofert, M., Mächler, M. and McNeil, A. J. (2012). Likelihood inference for Archimedean copulas in high dimensions under known margins., J. Multivariate Anal. 110 133-150. · Zbl 1244.62073
[26] Jin, J. and Cai, T. T. (2007). Estimating the null and the proportion of nonnull effects in large-scale multiple comparisons., J. Am. Stat. Assoc. 102 495-506. · Zbl 1172.62319
[27] Joe, H. (2005). Asymptotic efficiency of the two-stage estimation method for copula-based models., J. Multivariate Anal. 94 401-419. · Zbl 1066.62061
[28] Kanter, M. (1975). Stable densities under change of scale and total variation inequalities., Ann. Probab. 3 697-707. · Zbl 0323.60013
[29] Kingman, J. F. C. (1978). Uses of exchangeability., Ann. Probab. 6 183-197. · Zbl 0374.60064
[30] Marshall, A. W. and Olkin, I. (1988). Families of multivariate distributions., J. Am. Stat. Assoc. 83 834-841. · Zbl 0683.62029
[31] McNeil, A. J. and Nešlehová, J. (2009). Multivariate Archimedean copulas, \(d\)-monotone functions and \(\ell_1\)-norm symmetric distributions., Ann. Stat. 37 3059-3097. · Zbl 1173.62044
[32] Meinshausen, N., Maathuis, M. H. and Bühlmann, P. (2011). Asymptotic optimality of the Westfall-Young permutation procedure for multiple testing under dependence., Ann. Stat. 39 3369-3391. · Zbl 1246.62124
[33] Müller, A. and Scarsini, M. (2005). Archimedean copulae and positive dependence., J. Multivariate Anal. 93 434-445. · Zbl 1065.60018
[34] Nelsen, R. B. (2006)., An Introduction to Copulas. 2nd ed. Springer Series in Statistics. New York, NY: Springer. · Zbl 1152.62030
[35] Olshen, R. (1974). A note on exchangeable sequences., Z. Wahrscheinlichkeitstheor. Verw. Geb. 28 317-321. · Zbl 0265.60006
[36] Pollard, K. S. and van der Laan, M. J. (2004). Choice of a null distribution in resampling-based multiple testing., J. Stat. Plann. Inference 125 85-100. · Zbl 1074.62009
[37] Sarkar, S. K. (2002). Some results on false discovery rate in stepwise multiple testing procedures., Ann. Stat. 30 239-257. · Zbl 1101.62349
[38] Sarkar, S. K. (2006). False discovery and false nondiscovery rates in single-step multiple testing procedures., Ann. Stat. 34 394-415. · Zbl 1091.62060
[39] Sarkar, S. K. (2008a). Rejoinder: On methods controlling the false discovery rate., Sankhyā: The Indian Journal of Statistics Ser. A 70 183-185. · Zbl 1193.62102
[40] Sarkar, S. K. (2008b). On methods controlling the false discovery rate., Sankhyā: The Indian Journal of Statistics Ser. A 70 135-168. · Zbl 1193.62121
[41] Schweder, T. and Spjøtvoll, E. (1982). Plots of \(P\)-values to evaluate many tests simultaneously., Biometrika 69 493-502.
[42] Shorack, G. R. and Wellner, J. A. (1986)., Empirical Processes with Applications to Statistics . Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics . John Wiley & Sons Inc., New York. · Zbl 1170.62365
[43] Stange, J., Bodnar, T. and Dickhaus, T. (2013). Uncertainty quantification for the family-wise error rate in multivariate copula models. WIAS Preprint No. 1862, Weierstrass Institute for Applied Analysis and Stochastics Berlin. Available at, . · Zbl 1443.62140
[44] Storey, J. D. (2002). A direct approach to false discovery rates., J. R. Stat. Soc., Ser. B, Stat. Methodol. 64 479-498. · Zbl 1090.62073
[45] Storey, J. D., Taylor, J. E. and Siegmund, D. (2004). Strong control, conservative point estimation and simultaneous conservative consistency of false discovery rates: A unified approach., J. R. Stat. Soc., Ser. B, Stat. Methodol. 66 187-205. · Zbl 1061.62110
[46] Sun, W. and Cai, T. T. (2007). Oracle and adaptive compound decision rules for false discovery rate control., J. Am. Stat. Assoc. 102 901-912. · Zbl 1469.62318
[47] Tamhane, A. C., Liu, W. and Dunnett, C. W. (1998). A generalized step-up-down multiple test procedure., Can. J. Stat. 26 353-363. · Zbl 0914.62013
[48] Troendle, J. F. (2000). Stepwise normal theory multiple test procedures controlling the false discovery rate., Journal of Statistical Planning and Inference 84 139-158. · Zbl 1131.62310
[49] Westfall, P. H. and Young, S. S. (1993)., Resampling-Based Multiple Testing: Examples and Methods for p-Value Adjustment . Wiley Series in Probability and Mathematical Statistics. Applied Probability and Statistics . Wiley, New York. · Zbl 0850.62368
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.