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

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1 Vol. 29, No. 2, (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 of Mathematical Siences and Infomation Engineering, Nanzan University In this paper, we introduce the definition of FDR (False Discovery Rate), which gets a lot of attention as a new concept for considering multiplicity effect, and expound properties of it. Furthermore, we enumerate multiple testing procedures that control FDR; Benjamini- Hochberg procedure (linear step-up procedure), Adaptive Benjamini-Hochberg procedure, Benjamini-Yekutieli procedure, Storey s procedure, two-stage linear step-up procedure, and Student-Newman-Kuels procedure. In addition, we review comparisons of them in order to consider which procedure is best to use. As a result, we show the conservativeness of Benjamini-Hochberg procedure and the availability of two-stage linear step-up procedure by Monte-Carlo simulation, in the case of dependent test statistics. Key words: false discovery rate; FWER (familywise error rate); multiple comparisons; linear step-up procedure; two-stage procedure; Monte-Carlo simulation. 1. FDR false discovery rate (1997) n Received August Revised September Accepted September 2008.

2 R. A. Fisher n x i ˆσ 2 2 x i x p j 2ˆσ2 /n Fisher 1 1 Fisher 2 t 2 protected LSD least significant difference J. W. Tukey 1 FWE familywise error FWE Hochberg and Tamhane (1987) (1997) FWE 3 FWE FWER familywise error rate 4 FWER 4 H 0 : µ 1 = µ 2 = µ 3 = µ 4 2 H 01 : µ 1 = µ 2 H 02 : µ 1 = µ 3 H 03 : µ 1 = µ 4, H 04 : µ 2 = µ 3 H 05 : µ 2 = µ 4 H 06 : µ 3 = µ 4 6 H 01 H 06 H 0 Fisher protected LSD FWER H 01 H 06 H 0 H 01 H 06

3 FDR 127 protected LSD 1 (1 α) 2 α = FWER 4. 1 FWE FWER 2 1 FDR FWER Benjamini and Hochberg (1995) Benjamini et al. (2005) 5. FDR Benjamini and Hochberg (1995) FDR 5.1 FDR m m 0 m 1 m = m 0 + m 1 R 1 R U, V, S, T FWER P (V 1) 1. m U V m 0 T S m m 0 m R R m R Q = V R R = 0 Q = 0 Benjamini and Hochberg (1995) Q Q e V Q e = E(Q) = E R

4 128 Q e FDR FDR FDR FDR FWER FWER FDR FWER FDR Benjamini and Hochberg (1995) 1 FDR FWER 2 m 0 < m FDR FWER FWER FDR FDR FWER FDR FDR 5.3 FDR FWER FDR Benjamini and Hochberg (1995) q FDR FWER 0.05 q q 6. FDR FDR 3 Benjamini et al. (2005) 1 p 2 p 3 1 Benjamini and Hochberg (1995) BH Benjamini and Yekutieli (2001) BY 2 BH BH 3 BH 2 1 BH BY 2 Adaptive BH Storey 3 2 SNK

5 6.1 BH FDR 129 H 1,H 2,,H m p P 1,P 2,,P m p P (1) P (2) P (m) P (i) H (i) BH Benjamini and Hochberg (1995) (2006) 1 i = m 2 P (i) i m q k = i 3 i 1 i 2 i = 1 3 H (i) ;i = 1,2,,k BH FDR BH p linear step-up procedure 6.2 BY BH BY Benjamini and Yekutieli (2001) (2006) mx 1 q = q 1 / j j=1 q q BH 2 i = m 3 P (i) i m q k = i 4 i 1 i 3 i = 1 4 H (i) ;i = 1,2,,k BY FDR 6.3 Adaptive BH BH Adaptive BH ABH Benjamini and Hochberg (2000) (2006) 1 q BH S i = 3 i = 2 1 P (i) Si (i = 1,2,,m) m + 1 i

6 130 4 S i S i 1 i + 1 i 4 5 S i < S i 1 S i S S i S i 1 S m S 6 ˆm 0 = min 1 S + 1,m [ ] 7 i = m 8 P (i) i ˆm 0 q k = i i 1 i 8 ˆm 0 m k 9 H (i) ;i = 1,2,,k ABH FDR 6.4 Storey Storey et al. (2004) Storey (2002) ST λ t q ([ FDRλ ) p 2 1 p t q ([ FDRλ=0 ) BH 2 p t q ([ FDRλ ) m ˆπ 0(λ)m BH 6 ST BH Storey et al. (2004) MST. Storey et al. (2004) λ = 0.5 ST BH q λ ST q Storey et al. (2004) λ q Storey Storey (2008) Storey 1 p P 1,P 2,,P m t (0 t 1) V (t) = (P i t ) S(t) = (P i t ) R(t) = V (t) + S(t)

7 FDR 131 FDR(t) = E V (t) max(r(t), 1) Storey (2002) λ FDR(t) [ FDRλ (t) [FDR λ (t) = ˆπ 0(λ)t max(r(t),1)/m ˆπ 0(λ) π 0 = m 0/m ˆπ 0(λ) = m R(λ) (1 λ)m [0,1] F t q (F ) t q (F ) = sup{0 t 1 : F (t) q } p p t q ([ FDRλ ) ST MST 2 1 ˆπ 0(λ) 1 [ FDRλ (t) [FDR λ(t) = ˆπ 0(λ) = m + 1 R(λ) (1 λ)m 8 >< >: ˆπ 0(λ)t max(r(t),1)/m (t λ ) 1 (t > λ ) BH 2 Two-stage linear step-up procedure TST Benjamini et al. (2005) 1 q = q /(1 + q ) BH r 1 r 1 = 0 2 r 1 = m 2 ˆm 0 = m r 1 3 q m/ ˆm 0 BH FDR

8 SNK Student-Newman-Kuels SNK FDR Oehlert (2000) BH Tukey-Welsch FWER FDR SNK Tukey- Welsch SNK α FWER F (1997) p p p 4 2 p FDR BH ABH λ = 0.5 ST TST q = BH p P (1) = P (2) = P (3) = P (4) = P (5) = P (6) = P (7) = P (8) = P (9) = P (10) = P (11) = P (12) = P (13) = P (14) = P (15) = P (16) = P (17) = P (18) = P (19) = P (20) =

9 FDR i = 20 2 P (20) = > q = 0.05 i = 19 2 P (19) = > q = i = 18 2 i = 13 P (12) = q = k = H (i) ;i = 1,2,, ABH 1 BH 2 S 1 = 1 P (1) = 0.05, S2 = 1 P (2) = , S3 = , S4 = , S 5 = , S 6 = , S 7 = , S 8 = , S 9 = , S 10 = , S 11 = , S 12 = , S 13 = , S 14 = , S 15 = , S 16 = , S 17 = , S 18 = , S 19 = , S 20 = i = 2 4 S 2 S 1 i = 3 4 i = 19 5 S 20 < S 19 S 20 S 6 ˆm 0 = min 7 i = 20 1 S + 1,20 = 11 8 P (20) = > 20 ˆm 0 q = i = 19 8 i = 15 i = 14 P (14) = k = H (i) ;i = 1,2,,14 BH = 0:5 ST 20 R(0.5) ˆπ 0(0.5) = (1 0.5) 20 = 1 = m ˆπ 0(0.5) 20 = 2 BH 1 i = 20 2 P (20) = > 20 2 q = 0.5 i = 19 2

10 134 P (19) = q = k = H (i) ;i = 1,2,,19 Storey ST ˆπ 0 = ST ˆπ 0 = TST 1 q = q /(1 + q ) = BH P (13) = > q = , P (12) = q = r 1 = ˆm 0 = 20 r 1 = 8 3 q 20 ˆm 0 = BH P (15) = > = , P (14) = = H (i) ;i = 1,2,,14 ABH BH p q ABH TST p FDR FDR p Bonferroni 0.05/20 = Holm 0.05/(m i + 1) 8 FDR λ = 0.5 ST λ ST 8. Benjamini et al. (2005) λ MST λ MST ABH FDR TST (2006) BH ABH BY FDR q = 0.05

11 FDR (2006) p BH FDR Benjamini and Hochberg (1995) FDR ABH FDR BY FDR FWER 3. FDR BH ABH BY TST TST (2006) 4 ABH FDR 3 TST FDR 8.2 FDR

12 TST FDR BH ABH BY TST 2: 0.5,, : 0.9,, : 0.5,, : 0.9,, : : BH 2. ABH 3. λ = 0.5 ST 4. λ ST 5. TST BH ABH BH q λ = 0.5 ST 2 q λ ST BH TST Benjamini et al. (2005) BH λ ST TST FDR 9. FDR FDR FDR FDR FDR

13 FDR Benjamini, Y. and 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. and 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), Hochberg, Y. and Tamhane, A.C. (1987). Multiple Comparison Procedures, John Wiley and Sons. (2006). FDR., 6, (URL: 006pdf/ pdf) (1997)... Oehlert, G. W. (2000). Student-Newman-Kuels controls the false discovery rate. Statistics and Probability Letters, 46, Storey, J. D. (2002). A direct approach to false discovery rates. J. R. Statist. Soc. ser.b, 64(3), Storey, J. D. (2008). Q-Value. 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),

14 138 1 F Tukey-Welsch (1997) 2 2 t 2 2 Welch 3 FWE FWE 1 FWE type I FWE (1997) 1. 4 FWE (1997) FWE FWER FDR FWER (1997) I FWE generalized type I FWE Hochberg and Tamhane (1987) FWE 5 FDR FDR Benjamini and Hochberg (1995) 5.1 FDR FDR. 6 BH q q /ˆπ 0(λ)

15 FDR ST MST 100 Storey λ 8 47 p 20 9 BH BH p p

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