Microarray data: s of hypotheses. Example: Using co-expressed gene clusters to hunt for cis-regulatory elements (Nelander 2005)

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2 Mcroarray data: 0.000s o hypotheses Example: Usng co-expressed gene clsters to hnt or cs-reglatory elements elander 2005 Many other examples! Let S be the sample space o possble realtes, and let S. Let H H,..., H : S {0,} where H 0 or means For every S, let T T,..., T T represents the collecton o "test statstcs", or more accrately the collecton o resltng p - vales, as we assme : be a ncton specyng "Hypotheses", that the hypothess s alse or tre, respectvely. be a stochastc varable on [0,] For any and any sch that H : T ~ UIFORM [0,] 2

3 Based on the test statstcs or p - vales T, we want to predct the vales o H. In other words : For a ncton rom T, varable dened, or gven and, by Err, V, Z z H :[0,] we stdy the error,.e., where v H T T {0,} V predctng vales or H we stdy the stochastc v and Z z where The Type I and Type II error rates are, or gven vales o and the expectatons o V and Z, respectvely :, E V E Z E v E z H H E T E T ote that, as sal, we cannot make comptatons or Type II errors whthot makng more assmptons abot the dstrbton o T 3

4 For a gven > 0, dene 0 Then E V < H :[0,] {0,} by The FWER s dened, or gven vales o and the probablty Pr V > 0, as It measres, or the whole "amly"o hypotheses, the probablty o one or more Type I errors. EXAMPLE: FWER Pr V For > 0 H Pr dened as above, we get Pr v T 0 H 4

5 or FWER at level ths sad to control correcton s The Bonerron / 0 Pr 0 Pr and / Then / / 0 by {0,} :[0,] dene 0, gven For a, > < > B H T H V FWER H V E FWER at level the Holm method controls Ths 0 Pr 0 Pr and reorderng ndces : We get by condtonng on or the rest. then set 0 as long as set,2,..., For -... Sort the ndces so that - by {0,} :[0,] dene the Holm method 0, gven For a 2, + > + < > j j j H T H V FWER T j H

6 Assme a ncton M, and F :[0,] Then F [0,] M, :[0,] F, where {0,} s some ncton. s called a p - vale adjstment. can be wrtten as s the ncton dened beore Wth adjsted p - vales, one can "reject" and "accept" Hypotheses jst as sal based on the adjsted p - vales, whle stll gettng or example control over FWER. The ncton F,..., mn,,...,mn, comptes adjsted p - vales or the Bonerron method. The ncton dened by rst orderng p - valesn ncreasng order and then comptng F max j k,..., j mn, k + comptes adjsted p - vales or the Holm method. k 6

7 Takemoto et.al.:large-scale dentcaton o genes mplcated n kdney glomerls development and ncton He et.al.:analyss o 5,000 mose glomerlar EST and dentcaton o novel glomerlar enrched genes "! $#% '&! *, %2 3! # # / 9!:4 ;2 3! < <>2.?6 $@%# A B C2.!!?D A C+E+ A BF 0HG%! % % C + 5@%6 JK#%2L& MJ C& %O I P 2 9 J%Q LCR L & MBJ% C&T %O S P 2 9 J% p-vales nadjsted Bonerron Holm mber o rejected hypotheses 7

8 The ncton dened by F comptes adjsted p - vales or the Sdák method. I we assme that the components o T T,..., T are ndependent, one can easly show that ths method controls FWER at level. Ths can also be proven even nder somewhat more general crcmstances. [ mn k, ] F mn k,..., + k 8

9 p-vales nadjsted Bonerron Holm Sdak Hochberg Hommel mber o rejected hypotheses In addton to the stochastc varablesv and Z dened above, dene a stochastc varable R T, and then dene Q as ollows: H T V when R > 0 Q R T 0 otherwse Then the FDR s dened as the expectaton o Q or xed and. As or FWER, we can dene adjsted p - vales controllng or FDR 9

10 The Benjamn and Hochberg adjstment : - Sort the ndces so that - Dene F mn - Sort the ndces so that - Dene F mn k,..., Ths always controls or FDR... mn, Ths controls or FDR nder some assmptons The Benjamn and Yektel adjstment : k,..., 2 2 k... mn , k k 2 k p-vales nadjsted Bonerron Holm Sdak Hochberg Hommel Benjamn and Hochberg Benjamn and Yektel p-vales mber o rejected hypotheses mber o rejected hypotheses 0

11 p.adjst

12 J $ B 2! B!?R <>2.?6 '! / & MBJ% % C2! % 9 2LB ' 2 %6 Q2 S &?6;2L%! 2

13 FWER Pr V > 0 / R R > 0 Pr > 0 FDR E V R / R > 0 pfdr E V R PCER E V / PFER E V Strong control verss weak control 3

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