Advanced Differential Expression Analysis

Transcription

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2 BIOINFORMATICS AND COMPUTATIONAL BIOLOGY SOLUTIONS USING R AND BIOCONDUCTOR Bostatstcs Rafael A. Irzarry Advanced Dfferental Exresson Analyss Outlne Revew of the basc deas Introducton to (Emrcal Bayesan Statstcs The multle comarson roblem SAM 1

3 Quantfyng Dfferentally Exresson Two questons Can we order genes by nterest? One goal s to assgn a one number summary and consder large values nterestng. We wll refer to ths number as a score How nterestng are the most nterestng genes? How do ther scores comare to the those of genes known not to be nterestng? Examle Consder a case were we have observed two genes wth fold changes of 2 Is ths worth reortng? Are they both as nterestng? Some journals requre statstcal sgnfcance. What does ths mean? * * 2

4 Reeated Exerment Reeated Exerment Revew of Statstcal Inference Let Y-X be our measurement reresentng dfferental exresson. What s the tycal null hyothess? P-value s Prob(Y-X as extreme under null and s a way to summarze how nterestng a gene s. Poular assumton: Under the null,y-x follows a normal dstrbuton wth mean 0 and standard devaton σ. Wthout σ we do not know the -value. We can estmate σ by takng a samle and usng the samle standard devaton s. Note: Dfferent genes have dfferent σ, 3

5 Samle Summares Observatons: X 1,K, X M Y 1,K,Y N Averages: SD 2 or varances: M X = 1 M M " =1 X Y = 1 N 1 s 2 X = #(X " X 2 s 2 M "1 Y = 1 N "1 =1 N # =1 N Y " =1 (Y "Y 2 The t-statstc t - statstc: Y " X 2 2 s Y N + s X M Proertes of t-statstc If the number of relcates s very large the t- statstc s normally dstrbuted wth mean 0 and and SD of 1 If the observed data,.e. Y-X, are normally dstrbuted then the t-statstc follows a t dstrbuton regardless of samle sze Wth one of these two we can comute - values wth one R command 4

6 Data Show Problems Data Show Problems Data Show Problems 5

7 Problems Problem 1: T-statstc bgger for genes wth smaller standard errors estmates Imlcaton: Rankng mght not be otmal Problem 2: T-statstc not t-dstrbuted. Imlcaton: -values/nference ncorrect Problem 1 Wth few relcates SD estmates are unstable Emrcal Bayes methodology and Sten estmators rovdes a statstcally rgorous way of mrovng ths estmate SAM, a more ad-hoc rocedure, works well n ractce Note: We won t talk about Sten estmators. See a aer by Gary Churchll for detals Problem 2 Even f we use a arametrc model to mrove standard error estmates, the assumtons mght not be good enough to rovde trust-worthy -values We wll descrbe non-arametrc aroaches for obtanng -values Note: We stll haven t dscussed the multle comarson roblem. That comes later. 6

8 Introducton to Emrcal Bayes Outlne General Introducton Models for relatve exresson Models for absolute exresson 7

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10 Borrowng Strength An advantage of havng tens of thousands of genes s that we can try to learn about tycal standard devatons by lookng at all genes Emrcal Bayes gves us a formal way of dong ths Modelng Relatve Exresson Courtesy of Gordon Smyth Herarchcal Model Normal Model Pror Rearametrzaton of Lönnstedt and Seed 2002 Normalty, ndeendence assumtons are wrong but convenent, resultng methods are useful 9

11 Posteror Statstcs Posteror varance estmators Moderated t-statstcs Elmnates large t-statstcs merely from very small s Margnal Dstrbutons The margnal dstrbutons of the samle varances and moderated t-statstcs are mutually ndeendent Degrees of freedom add Shrnkage of Standard Devatons The data decdes whether should be closer to t g,ooled or to t g 10

12 Posteror Odds Posteror robablty of dfferental exresson for any gene s Monotonc functon of for constant d Rearametrzaton of Lönnstedt and Seed 2002 Modelng the Absolute Exresson Courtesy of Chrstna Kendzorsk Herarchcal Model for Exresson Data (One condton y µ f µ ( g1, k g ~ 1 g1 y µ ( g2, k µ g ~ f 2 g2 y g µ ( g3, k µ ~ f g 3 3 µ g 1 µ µ g g 2 3 µ g ~ f (" µ g1 µ g2 µ g3 11

13 Herarchcal Model for Exresson Data (Two condtons Let x = [ x c1,x c 2 ] denote data (one gene n condtons C1 and C2. Two atterns of exresson: # P0 (EE : P1 (DE: ( f µ µ c1 = µ c 2 µ c1 " µ c 2 For P0, x ~ f x µ ( dµ " f 0 ( x For P1, x ~ " f ( x µ c1,µ c2 f ( µ c1,µ c2 dµ c1 dµ c 2 " # f ( x c1 µ c1 f ( µ c1 dµ c1 # f ( x c2 µ c 2 f ( µ c 2 dµ c 2 " f 1 ( x ( f 0 ( x c 2 f 0 x c1 Herarchcal Mxture Model for Exresson Data Two condtons: x ~ 0 f 0 ( x + 1 f 1 x Multle condtons: K ( => ( P1 x = x ~ " k f k ( x => Pk' x k=1 Parameter estmates va EM 1 f ( x P1 ( + 1 f ( x P1 0 f x P0 ( = k' f ( x Pk' # k f ( x Pk Bayes rule determnes threshold here; could target secfc FDR. k"k' For every transcrt, two condtons => two atterns (DE, EE EE: m 1 = m 2 DE: m 1 m 2 ( DE y ( EE y P f odds g = = P f ( yg DE P( DE ( y EE P( EE Emrcal Bayes methods make use all of the data to make gene secfc nferences. g 12

14 Odds lot: SCD knockout vs. SV129 (Atte lab EBarrays: Contour Plots of Odds Comments on Emrcal Bayes Aroach(EBarrays Herarchcal model s used to estmate osteror robabltes of atterns of exresson. The model accounts for the measurement error rocess and for fluctuatons n absolute exresson levels. Multle condtons are handled n the same way as two condtons (no extra work requred. Posteror robabltes of exresson atterns are calculated for every transcrt. Threshold can be adjusted to target a secfc FDR. In Boconductor 13

15 Emrcal Bayes for Mcroarrays (EBarrays On Dfferental Varablty of Exresson Ratos: Imrovng Statstcal Inference About Gene Exresson Changes from Mcroarray Data by M.A. Newton, C.M. Kendzorsk, C.S. Rchmond, F.R. Blattner, and K.W. Tsu Journal of Comutatonal Bology 8: 37-52, On Parametrc Emrcal Bayes Methods for Comarng Multle Grous Usng Relcated Gene Exresson Profles by C.M. Kendzorsk, M.A. Newton, H. Lan and M.N. Gould Statstcs n Medcne, to aear, Inference and the Multle Comarson Problem Many sldes courtesy of John Storey Hyothess testng Once you have a gven score for each gene, how do you decde on a cut-off? -values are oular. But how do we decde on a cut-off? Are 0.05 and 0.01 arorate? Are the -values correct? 14

16 P-values by ermutaton It s common for the assumtons used to derve the statstcs used to summarze nterest are not aroxmate enough to yeld useful -values An alternatve s to use ermutatons -values by ermutatons We focus on one gene only. For the bth teraton, b = 1,, B; 1. Permute the n data onts for the gene (x. The frst n 1 are referred to as treatments, the second n 2 as controls. 2. For each gene, calculate the corresondng two samle t-statstc, t b. After all the B ermutatons are done; 3. Put = #{b: t b t observed }/B ( lower f we use >. Multle Comarson Problem If we do have useful aroxmatons of our -values, we stll face the multle comarson roblem When erformng many ndeendent tests -values no longer have the same nterretaton 15

17 Hyothess Testng Test for each gene null hyothess: no dfferental exresson. Two tyes of errors can be commtted Tye I error or false ostve (say that a gene s dfferentally exressed when t s not,.e., reject a true null hyothess. Tye II error or false negatve (fal to dentfy a truly dfferentally exressed gene,.e.,fal to reject a false null hyothess Multle Hyothess Testng What haens f we call all genes sgnfcant wth -values 0.05, for examle? Null True Called Sgnfcant V Not Called Sgnfcant m 0 V Total m 0 Altern.True S m 1 S m 1 Total R m R m Null = Equvalent Exresson; Alternatve = Dfferental Exresson Other ways of thnkng of P-values A -value s defned to be the mnmum false ostve rate at whch an observed statstc can be called sgnfcant If the null hyothess s smle, then a null -value s unformly dstrbuted 16

18 Multle Hyothess Test Error Controllng Procedure Suose m hyotheses are tested wth - values 1, 2,, m A multle hyothess error controllng rocedure s a functon T(; α such that rejectng all nulls wth T(; α mles that Error α Error s a oulaton quantty (not random Weak and Strong Control If T(; α s such Error α only when m 0 = m, then the rocedure rovdes weak control of the error measure If T(; α s such Error α for any value of m 0, then the rocedure rovdes strong control of the error measure note that m 0 s not an argument of T(; α Error Rates Per comarson error rate (PCER: the exected value of the number of Tye I errors over the number of hyotheses PCER = E(V/m Per famly error rate (PFER: the exected number of Tye I errors PFER = E(V Famly-wse error rate: the robablty of at least one Tye I error FEWR = Pr(V 1 False dscovery rate (FDR rate that false dscoveres occur FDR = E(V/R; R>0 = E(V/R R>0Pr(R>0 Postve false dscovery rate (FDR: rate that dscoveres are false FDR = E(V/R R>0. 17

19 18 Bonferron Bonferron Procedure Procedure m m H m H m V m T m C " = # $ % & ' ( # $ % & ' ( * +, -. / 0 = 1 = Pr mn Pr 1 Pr( : max ; ( Provdes strong control.. Sdak Sdak Procedure Procedure { } ( ( = " " > " = " " # # $ " " # = % = (1 1 Pr 1 (1 1 mn Pr 1 Pr( (1 1 : max ; ( 1 0 1/ 0 1/ 1/ m m C m m H H V T Requres ndeendence for strong control Holm Procedure Holm Procedure ( { } 1 1/( ( ( ( ( ( (2 (1 1 1 : mn ; ( 1 : mn ; ( values Order the - + > = " # $ % & ' + > = ( ( ( m m T m T L Requres ndeendence for strong control

20 Hochberg Procedure % " ( T(;" = max& ( : ( # ' m $ +1*...the ste-u analogue of Holm Smes/BH Procedure & T (; = max% ( : ( $ ' # ( " m Weak controls the FWER (Smes 1986 Strongly controls FDR (Benjamn & Hochberg 1995 Both requre the null -values to be ndeendent False Dscovery Rate The false dscovery rate measures the roorton of false ostves among all genes called sgnfcant: #false ostves V V = = #called sgnfcant V + S R Ths s usually arorate because one wants to fnd as many truly dfferentally exressed genes as ossble wth relatvely few false ostves The false dscovery rate gves the rate at whch further bologcal verfcaton wll result n dead-ends 19

21 False Postve Rate versus False Dscovery Rate False ostve rate s the rate at whch truly null genes are called sgnfcant #false ostves V FPR = # truly null m False dscovery rate s the rate at whch sgnfcant genes are truly null FDR #false ostves #called sgnfcant = 0 V R False Postve Rate and P-values The -value s a measure of sgnfcance n terms of the false ostve rate (aka Tye I error rate P-value s defned to be the mnmum false ostve rate at whch the statstc can be called sgnfcant Can be descrbed as the robablty a truly null statstc s as or more extreme than the observed one False Dscovery Rate and Q-values The q-value s a measure of sgnfcance n terms of the false dscovery rate Q-value s defned to be the mnmum false dscovery rate at whch the statstc can be called sgnfcant Can be descrbed as the robablty a statstc as or more extreme s truly null 20

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23 Bayesan Connectons Ths allows Bayesans to estmate FDR as well: ( = Pr( = 0 X = x f ( x x #" FDR " H dx Ths motvates the name q-value drectly: " value( x = Pr q " value( x = Pr ( X x H = 0 ( H = 0 X x All the estmaton resented below can be vewed as an emrcal Bayes aroach Possble FDR Goals 1. For some re-chosen α, estmate a sgnfcance cut-off so that on average FDR α 2. For some re-chosen sgnfcance cut-off, estmate FDR so that E FD ˆ R [ ] " FDR 3. Estmate FDR so that t s smultaneously conservatve over all sgnfcance cut-offs 4. Estmate q-values for all genes that are smultaneously conservatve Unversal Goal 1. The q-value, an FDR-based measure of sgnfcance, s assocated wth each gene 2. The estmated q-values are conservatve over all genes smultaneously In dong so, all four otons wll be met 22

24 Estmate of FDR We begn by estmatng FDR when callng all genes sgnfcant wth - values t Heurstc motvaton: E FDR( t " E [ V ( t ] [ R( t ] [ t} ] E [#{ t} ] E #{null = mˆ " t Dˆ R( t = #{ t} F 0 =m 0 t Estmate of π 0 We frst estmate the more easly nterreted π 0 = m 0 /m, the roorton of truly null (non-dfferentally exressed genes: #{ > } " ˆ0( = m $ (1 # Then clearly m ˆ " m 0 = ˆ 0 #{ > } " ˆ0 ( = m $ (1 # ˆ0 23

25 Choose λ by Balancng Bas and Varance ˆ0 = 0.67 Overall FDR Estmate The overall estmate of FDR(t s ˆ m" t Dˆ R( t = # #{ t} F 0 The mlct estmate used n the orgnal FDR aer s a secal case of the above estmate wth π 0 estmated as 1. Numercal Examle Suose we call all genes sgnfcant wth -values 0.03 The estmate of the FDR s F Dˆ R = = = Could use any threshold 0 t 1 24

26 Q-value Estmate The mathematcal defnton of the q- value of gene s q - value( = mn t Snce FDR FDR, we estmate the q-value of gene by qˆ ( = mn t FDR( t FDˆ R( t Q-Plots Theoretcal Results Suose that the emrcal dstrbuton functons of the null statstcs and of the alternatve statstcs converge as the number of genes m gets large The FDR estmates are asymtotcally conservatve smultaneously over all sgnfcance regons The estmated q-values are smultaneously conservatve over all genes Ths s equvalent to controllng the FDR at all levels α smultaneously 25

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29 Smulaton Study Performed 3000 hyothess tests of H 0 :N(0,1 versus H 1 :N(2,1 The statstcs had correlaton 0.40 n blocks of 50 Two conclusons: 1. The true q-values under ths deendence structure are the same as those gven under the ndeendence model 2. The estmated q-values are smultaneously conservatve 28

30 3000 Deendent Tests of N(0,1 versus N(2,1 Power Comarson Power Comarson FDR Level # Sgnfcant BH # Sgnfcant PP ˆ0 =1 ˆ0 =

31 & ˆ0 FDˆ R( " = & ˆ ("' = 0 B b= 1 SAM Verson B 0b 0b 0b #{ d : d % l( " or d $ r( "}/ b= 1 #{ d : d % l( " or d $ r( "} # # { d : d > l( "' or d < r( "'} 0b 0b 0b { d : d > l( "' or d < r( "'}/ B & ˆ0 # avg no. nulls called sgnfcant = no. observed called sgnfcant B What should one look for n a multle testng rocedure? As we wll see, there s a bewlderng varety of multle testng rocedures. How can we choose whch to use? There s no smle answer here, but each can be judged accordng to a number of crtera: Interretaton: does the rocedure answer a relevant queston for you? Tye of control: strong or weak? Valdty: are the assumtons under whch the rocedure ales clear and defntely or lausbly true, or are they unclear and most robably not true? Comutablty: are the rocedure s calculatons straghtforward to calculate accurately, or s there ossbly numercal or smulaton uncertanty, or dscreteness? Selected references Westfall, PH and SS Young (1993 Resamlng-based multle testng: Examles and methods for -value adjustment, John Wley & Sons, Inc Benjamn, Y & Y Hochberg (1995 Controllng the false dscovery rate: a ractcal and owerful aroach to multle testng JRSS B 57: J Storey (2001: 3 aers (some wth other authors, www-stat.stanford.edu/~jstorey/ The ostve false dscovery rate: a Bayesan nterretaton and the q-value. A drect aroach to false dscovery rates Estmatng false dscovery rates under deendence, wth alcatons to mcroarrays Y Ge et al (2001 Fast algorthm for resamlng based -value adjustment for multle testng 30

32 Sgnfcance analyss of mcroarrays (SAM A clever adataton of the t-rato to borrow nformaton across genes In Boconductor, sggenes acakge s avalable SAM-statstc For gene d = y " x s + s 0 y = mean of Irradated samles x = mean of Unrradated samles = s = s 0 Standard devaton of resduals for gene a assumng same varance Exchangeablty factor estmated usng all genes The exchangeablty factor Chosen to make sgnal-to-nose ratos ndeendent of sgnal Comutaton Let s be the ercentle of the s values. Let d = r /( s s + Comute the 100 quantles of the values, denoted by q 1 < q2 < L < q100 " ( 0,0.05,0.10, K,1.0 For " Comute v j = mad( d s [ q j, q j+ 1, j = 1,2, K,99, where mad s the medan absolute devaton from the medan, dvded by 0.64 Comute cv( = coeffcent of varaton of the Choose ˆ = arg mn[ cv( ]. ˆ s ˆ = s. and v j 0 s 31

33 Scatter lots of relatve dfference Random fluctuatons n the data, measured by balanced ermutatons (for cell lne 1 and 2 Relatve dfference for a ermutaton of the data that was balanced betwe en cell lnes 1 and 2. The reference dstrbuton Order the values of (could be any stat d ( 1 d(2 L d( Permute the treatment labels, and comute a new set of ordered values * * * d ( 1 d(2 L d( Reeat ste 2 for, say, 100 ermutatons: *1 *1 *1 d d L d d (1 *2 (1 *100 *100 *100 d(1 d(2 L d( From these, comute the average largest, average second largest etc. d d (2 *2 (2 M L M d ( *2 ( Selected genes 32

34 exected d( Delta Ave # falsely sgnfcant # called sgnfcant False dscovery rate Delta s the half-wdth of the bar around the 45-degree lne. More general versons of SAM More than two grous Pared data Survval data, wth censored resonse 33

35 Lmtatons of SAM Solutons for s_0 are often at the extremes and senstve to the resoluton of the quantle grd. Permutaton analyss throws all genes n the same bag Requres a monotone sgnal-tonose relatonsh 34