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1 R BH 1 (2004) 1 (Familywise error rate) Tukey step-down Tukey-Welsch R (False Discovery Rate) () Benjamini and Hochberg(1995) BH adaptivebh BY R 2 (False Discovery Rate) m m 0 m 1 m = m 0 + m 1 R Benjamini and Hochberg(1995) 1 R U, V, S, T α α P (V 1) 1 α/m (Bonferroni ) V Q = (V + S) V + S =0 1 Q =0 m 0 Q q Benjamini and

2 1: m U V m 0 T S m m 0 m R R m Hochberg(1995) Q Q e ( ) ( ) V V Q e = E(Q) =E = E V + S R Q e (False Discovery Rate) Benjamini and Hochberg(1995) (a) (b) m 0 <m S BH H 1,H 2,,H m p P 1,P 2,,P m p P (1) P (2) P (m) P (i) H (i) q 0.05 BH Benjamini and Hochberg(1995) 1 i = m 2 P (i) i m q k = i i 1 i i =1 3 H (i) ; i =1, 2,,k

3 2 p BH The Linear Step-Up Procedure 3.2 Adaptive BH BH Adaptive BH ( ABH ) Benjamini and Hochberg(2000) 1 q BH 1 2 S i = 1 P (i) m +1 i S i (i =1, 2,,m) 3 i =2 4 S i S i 1 i +1 i 5 S i <S i 1 S i S S i S i 1 S m S ([ ] ) 1 6 ˆm 0 = min S +1,m [] 7 i = m 8 P (i) i q k = i i 1 ˆm 0 i ˆm 0 m k 9 H (i) ; i =1, 2,,k 3.3 BY Benjamini and Yekutieli BH Benjamini and Yekutieli ( BY ) Benjamini and Yekutieli(2001) m ( ) 1 1 q = q / j j=1 q q BH 2 i = m 3 P (i) i m q k = i i 1 i i =1 4 H (i) ; i =1, 2,,k

4 4 Benjamini and Hochberg(2000) BH ABH BH ABH Benjamini, Krieger and Yekutieli(2005) ( ) ABH Resampling Storey, Taylor and Siegmund(2004) 5 BH ABH BY q = P ,4 10 5,6 8,9,10 3,4, P

5 P [ ] 8 3 [ 5 ] ( 1) ( 2,3,4,8,9) ( 5,6,7,10) BH ABH Benjamini, Krieger and Yekutieli(2005) Benjamini, Krieger and Yekutieli(2005) BH ABH 2: BKY(2005) m 0 : m 1 BH ABH m 0 : m 1 BH ABH 0: : : : : : : : : : BY q = q / m ( 1 j ) q q BH q 0.05 BY j=1

6 (2004) BY ( ) 3: m 0:m 1 BH ABH BY BH ABH BY 5: : : : : : : : : m 0 :m 1 5:5, 15:5, 25:5, 10:10, 20:10, 5:15, 15:15, 10:20, 5: ABH BY BH 5 5 ABH ( )/( ) BH ABH 0.05 BY 0.05 BH ABH ( ) cor=0.1, 0.5, m 0 :m 1 5:5, 15:5, 25:5, 10:10, 20:10, 5:15, 15:15, 10:20, 5:

7 4: (0.1,0.5,0.9) [ ] m 0:m 1 cor BH ABH BY BH ABH BY 5: : : : : : : : : : : : : : : : : : : : : : : : : : : BH BY 0.05 ABH 1/ ABH BH BY 0.05 ABH 0.05 BH ( ) BH 0.05 BY 0.05 BY BH ABH 0.05

8 5: (0.1,0.5,0.9) [ ] m 0:m 1 cor BH ABH BY BH ABH BY 5: : : : : : : : : : : : : : : : : : : : : : : : : : : ,4 ( ) BH ABH ( ) 2 ABH (cor=0.5,0.9) 0.05 BH BH ABH 0.05

9 6: 1 [ 3] m 0:m 1 BH ABH BY BH ABH BY 5: : : : : : : : : : 1 [ 4] m 0:m 1 BH ABH BY BH ABH BY 5: : : : : : : : : BY ( ) 8 ABH 0.05 BH ABH 0.05 BH ( ) ABH BH BY BY 0.05

10 8: 1 ( ) m 0:m 1 BH ABH BY BH ABH BY 5: : : : : : : : : : 1 m 0:m 1 BH ABH BY BH ABH BY 5: : : : : : : : : ( ) BH ABH 5 BH ABH ABH BH 5 BH ABH 5: BY

11 10: ( ) a:b BH ABH BY BH ABH BY 5: : : : : : ( ) a:b 5:0, 4:1, 3:2, 2:3, 1:4, 0: ABH BH (5:0, 0:5) (3:2, 2:3) (3:2, 2:3) (5:0, 0:5) ABH BH 6.7 8,9,10 ( ) : 3 ( 8) m 0:m 1 BH ABH BY BH ABH BY 2 5: : : : : , 4, 8, 16, 20 3 BH ABH

12 12: 5 4 ( 9) m 0:m 1 BH ABH BY BH ABH BY 25: : : : 5 6 ( 10) m 0:m 1 BH ABH BY BH ABH BY 25: : : BH ABH BH ABH BH ABH 2 ( 8,9) BH 0.05 ABH 0.05 BY BH BY 0.05 BY BH (0.05 ) (25:5) (cor=0) (0:5)(5:0)

13 14: BH 0.05 ( ) m 0:m 1 cor BH 25: (5 ) 0: : BH 0.05 BH ( ) ABH ( ) BH ABH BH BY BH 8 BH ABH ABH BH

14 Benjamini, Y., Hochberg, Y. (1995): Controlling the False Discovery Rate: a Practical and Powerful Approach to Multiple Testing, J. R. Statist. Soc. ser.b, 57(1), Benjamini, Y., Hochberg, Y. (2000): On the Adaptive Control of the False Discovery Rate in Multiple Testing With Independent Statistics, J. Edu. Behavioral Statistics, 25(1), Benjamini, Y., Kriegery, A. M. and Yekutieli, D. (2005): Adaptive Linear Step-up Procedures that control the False Discovery Rate, Unpublished paper, ybenja/mypapers/bkymarch9.pdf. Benjamini, Y. and Yekutieli, D. (2001): The Control of the False Discovery Rate in Multiple Testing under Dependency, Annals of Statistics, 29(4), (2004): R,. (2004):,. 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. Statist. Soc. Ser.B, 66(1),

Vol. 29, No. 2, (2008) FDR Introduction of FDR and Comparisons of Multiple Testing Procedures that Control It Shin-ichi Matsuda Department of

Vol. 29, No. 2, (2008) FDR Introduction of FDR and Comparisons of Multiple Testing Procedures that Control It Shin-ichi Matsuda Department of Vol. 29, No. 2, 125 139 (2008) FDR Introduction of FDR and Comparisons of Multiple Testing Procedures that Control It Shin-ichi Matsuda Department of Information Systems and Mathematical Sciences, Faculty