Web-based Supplementary Materials for. of the Null p-values

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1 Web-based Supplementary Materials for ocedures Controlling the -FDR Using Bivariate Distributions of the Null p-values Sanat K Sarar and Wenge Guo Temple University and National Institute of Environmental Health Sciences

2 Web Appendix A: oofs oof of Lemma 2 Let ˆP () ˆP (n0 ) be the ordered values of ˆP,, ˆP n0 Note that ˆP (j) P (n n0 +j) for all j n 0 Therefore, the original stepwise procedure rejects less number of null hypotheses, and hence has less number of falsely rejected null hypotheses, than the corresponding stepwise procedure where the p-values corresponding to the false null hypotheses are all very close to zero Thus, the lemma follows oof of Theorem 3 V -FDR E I (V ) E R r α i r n 0 i r r i r ˆPi α r, V, R r ( ) I ˆPi α r, V, R r P V, R r ˆPi α r (A) Now, for any r > and i n 0, V, R r ˆPi α r V, R r ˆPi α r V, R r ˆPi α r V, R r + ˆPi α r V, R r + ˆPi α r The inequality follows from the assumption (), since V, R r is a decreasing set for a 2

3 stepwise procedure Thus, we get -FDR α α α i α i α i V, R ˆPi α + i r+ i V, R r ˆPi α r V, R ˆPi α + V, R + ˆPi α V ˆP i α i V, ˆP i α (A2) Applying Lemma 2 to (A2), we get -FDR i P ˆRn0, ˆP i α ˆR n 0 Let denote the number of rejections in the corresponding stepwise procedure based on the ordered values ˆP () ˆP (n 0 ) of ˆP,, ˆP n0 \ ˆPi and the n 0 critical values α n n0 +2 α n Then, for any r n 0, we notice that, when our procedure is a stepup procedure ˆRn0 r, ˆP i α ˆP(r) α n n0 +r, ˆP (r+) > α n n0 +r+,, ˆP (n0 ) > α n, ˆPi α ˆP (r ) α n n 0 +r, ˆP (r) > α n n0 +r+,, ˆP (n 0 ) > α n, ˆPi α ˆR n 0 r, ˆP i α ; (A3) 3

4 whereas, for a stepdown procedure, we have ˆRn0 r, ˆP i α ˆP() α n n0 +,, ˆP (r) α n n0 +r, ˆP (r+) > α n n0 +r+, ˆPi α ˆP () α n n0 +2,, ˆP (r ) α n n 0 +r, ˆP (r) > α n n0 +r+, ˆPi α ˆR n 0 r, ˆP i α (A4) Thus, for a stepwise (stepup or stepdown) procedure, we get -FDR n 0 i r n 0 n 0 i j( i) r ˆR n 0 r, ˆP i α r ˆR n 0 r, ˆP i α, ˆP j α n n0 +r (A5) As explained in (A3) and (A4), ˆR n 0 r, ˆP i α, ˆP j α n n0 +r ˆR( i, j) n 0 2 r 2, ˆP i α, ˆP j α n n0 +r, for a stepup procedure, and ˆR n 0 r, ˆP i α, ˆP j α n n0 +r ˆR( i, j) n 0 2 r 2, ˆP i α, ˆP j α n n0 +r, ˆR ( i, j) for a stepdown procedure, where n 0 2 is the number of rejections in the corresponding stepwise procedure involving ˆP,, ˆP n0 \ ˆPi, ˆP j and critical values α n n0 +3 α n Thus, -FDR (n n 0 + )α 2 ( ) i j( i) r ˆR( i, j) n 0 2 r 2, ˆP i α ˆPj α n n0 +r 4

5 The inequality follows from the fact that n n 0 + r r n n 0 +, for all r n 0 Since for r >, ˆR( i, j) n 0 2 r 2, ˆP i α ˆP j α n n0 +r ˆR( i, j) n 0 2 r 2, ˆP i α ˆP j α n n0 +r ˆP i α ˆP j α n n0 +r ˆR( i, j) n 0 2 r 2, ˆP i α ˆP j α n n0 +r, ˆP i α ˆPj α n n0 +r ˆR( i, j) n 0 2 r, ˆR( i, j) n 0 2 r, ( i, j) with the inequality following due to the property () and the fact that ˆR n 0 2 r 2, ˆP i α is decreasing, we have r ˆR( i, j) n 0 2 r 2, ˆP i α ˆPj α n n0 +r ˆR( i, j) n 0 2 2, ˆP i α ˆPj α n n0 + r+ [ ˆR( i, j) n r 2, ˆP i α ˆP j α n n0 +r ˆR( i, j) n 0 2 r, ˆP ] i α ˆPj α n n0 +r ˆR( i, j) n 0 2 2, ˆP i α ˆPj α n n0 + + ˆR( i, j) n 0 2, ˆP i α ˆPj α n n0 + ˆR( i, j) n 0 2 2, ˆP i α ˆPj α n n0 + 5

6 Therefore, -FDR (n n 0 + )α 2 ( ) ˆP j α n n0 + (n n 0 + )α 2 ( ) ( ) i j( i) i j( i) i j( i) ˆR( i, j) n 0 2 2, ˆP i α ˆPi α ˆPj α n n0 + ˆPi α, ˆP j α n n0 + (A6) The theorem then follows by considering the maximum of the right-hand side in (A6) over the set of values of n 0 oof of (8) Since the p-values are independent and α i i β/n, i,, n, we see from the first line in (A5) that -FDR n 0β n r ˆRn0 r n 0β n ˆRn0, the desired inequality oof of Theorem 5 Define p ijr P ˆPi [α j, α j ], V, R r, given a set critical 6

7 values 0 α 0 < α < < α n Then, from (A) we have, -FDR r i n 0 r ˆPi α r, V, R r i j r j i j n 0 i j r p ijr i j j r j j ˆPi [α j, α j ], V α j α j j r r i j p ijr r p ijr Let α i (i )α /, for some fixed 0 < α < Then, -FDR n 0 α + j+ j (A7) The theorem then follows by considering α α/n + n j+ j in (A7) 7

8 Web Figure : rho 0 Average Power !FDR BH!FDR SD!FDR SD The number of false null hypotheses Figure : Power of two -FDR stepdown procedures in the case of independence with parameters n 200, 5, d 2 and α 005 8