The positive false discovery rate: A Bayesian interpretation and the \(q\)-value. (English) Zbl 1042.62026

Summary: Multiple hypothesis testing is concerned with controlling the rate of false positives when testing several hypotheses simultaneously. One multiple hypothesis testing error measure is the false discovery rate (FDR), which is loosely defined to be the expected proportion of false positives among all significant hypotheses. The FDR is especially appropriate for exploratory analyses in which one is interested in finding several significant results among many tests.
We introduce a modified version of the FDR called the “positive false discovery rate” (pFDR). We discuss the advantages and disadvantages of the pFDR and investigate its statistical properties. When assuming the test statistics follow a mixture distribution, we show that the pFDR can be written as a Bayesian posterior probability and can be connected to classification theory. These properties remain asymptotically true under fairly general conditions, even under certain forms of dependence. Also, a new quantity called the “\(q\)-value” is introduced and investigated, which is a natural “Bayesian posterior \(p\)-value,” or rather the pFDR analogue of the \(p\)-value.


62F15 Bayesian inference
62J15 Paired and multiple comparisons; multiple testing
62F03 Parametric hypothesis testing
Full Text: DOI


[1] 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. · Zbl 0809.62014
[2] Benjamini, Y. and Hochberg, Y. (2000). On the adaptive control of the false discovery rate in multiple testing with independent statistics. J. Educational and Behavioral Statistics 25 60–83.
[3] 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
[4] Billingsley, P. (1968). Convergence of Probability Measures . Wiley, New York. · Zbl 0172.21201
[5] Brown, P. O. and Botstein, D. (1999). Exploring the new world of the genome with DNA microarrays. Nature Genetics 21 33–37.
[6] Cherkassky, V. S. and Mulier, F. M. (1998). Learning from Data : Concepts , Theory and Methods . Wiley, New York. · Zbl 0960.62002
[7] Efron, B., Tibshirani, R., Storey, J. D. and Tusher, V. (2001). Empirical Bayes analysis of a microarray experiment. J. Amer. Statist. Assoc. 96 1151–1160. · Zbl 1073.62511
[8] Genovese, C. and Wasserman, L. (2002a). Operating characteristics and extensions of the procedure. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 499–517. · Zbl 1090.62072
[9] Genovese, C. and Wasserman, L. (2002b). False discovery rates. Technical report, Dept. Statistics, Carnegie Mellon Univ. · Zbl 1090.62072
[10] Lehmann, E. L. (1986). Testing Statistical Hypotheses , 2nd ed. Wiley, New York. · Zbl 0608.62020
[11] Morton, N. E. (1955). Sequential tests for the detection of linkage. Amer. J. Human Genetics 7 277–318.
[12] Sarkar, S. K. (2002). Some results on false discovery rate in stepwise multiple testing procedures. Ann. Statist. 30 239–257. · Zbl 1101.62349
[13] Shaffer, J. (1995). Multiple hypothesis testing: A review. Annual Review of Psychology 46 561–584.
[14] Simes, R. J. (1986). An improved Bonferroni procedure for multiple tests of significance. Biometrika 73 751–754. · Zbl 0613.62067
[15] Storey, J. D. (2001). The positive false discovery rate: A Bayesian interpretation and the \(q\)-value. Technical Report 2001-12, Dept. Statistics, Stanford Univ.
[16] Storey, J. D. (2002a). A direct approach to false discovery rates. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 479–498. · Zbl 1090.62073
[17] Storey, J. D. (2002b). False discovery rates: Theory and applications to DNA microarrays. Ph.D. dissertation, Dept. Statistics, Stanford Univ.
[18] 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
[19] Weller, J. I., Song, J. Z., Heyen, D. W., Lewin, H. A. and Ron, M. (1998). A new approach to the problem of multiple comparisons in the genetic dissection of complex traits. Genetics 150 1699–1706.
[20] Zaykin, D. V., Young, S. S. and Westfall, P. H. (2000). Using the false discovery rate approach in the genetic dissection of complex traits: A response to Weller et al. Genetics 154 1917–1918.
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