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# Interval Estimation for the Mean of the Selected Populations

## Material Information

Title:
Interval Estimation for the Mean of the Selected Populations
Physical Description:
1 online resource (60 p.)
Language:
english
Creator:
Fuentes,Claudio
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

## Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Statistics
Committee Chair:
Casella, George
Committee Members:
Ghosh, Malay
Daniels, Michael J
Peter, Gary F

## Subjects

Subjects / Keywords:
ci -- confidence -- coverage -- estimation -- inference -- interval -- largest -- mean -- normal -- population -- probability -- selected -- simultaneous
Statistics -- Dissertations, Academic -- UF
Genre:
Statistics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

## Notes

Abstract:
Consider an experiment in which p independent populations pi_i, with corresponding unknown means theta_i are available and suppose that for every 1<= i <= p, we can obtain a sample X_i1,...,X_in from pi_i. In this context, researchers are sometimes interested in selecting the populations that give the largest sample means as a result of the experiment, and to estimate the corresponding population means theta_i. In this dissertation, we present a frequentist approach to the problem, based on the minimization of the coverage probability, and discuss how to construct confidence intervals for the mean of k>=1 selected populations, assuming the populations pi_i are normal and have a common variance sigma^2. Finally, we extend the results for the case when the value of k is randomly chosen and discuss the potential connection of the procedure with false discovery rate analysis. We include numerical studies and a real application example that corroborate this new approach produces confidence intervals that maintain the nominal coverage probability while taking into account the selection procedure.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Claudio Fuentes.
Thesis:
Thesis (Ph.D.)--University of Florida, 2011.
Local:
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2012-08-31

## Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2011
System ID:
UFE0043284:00001

Full Text

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page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 6 LISTOFFIGURES ..................................... 7 ABSTRACT ......................................... 8 CHAPTER 1INTRODUCTION ................................... 9 1.1TwoFormulationsoftheProblem ....................... 9 1.2InferenceontheSelectedMean ....................... 10 2INTERVALESTIMATIONFOLLOWINGTHESELECTIONOFONEPOPULATION .................................... 13 2.1TheKnownVarianceCase .......................... 14 2.2TheUnknownVarianceCase ......................... 21 2.3NumericalStudies ............................... 22 2.4TablesandFigures ............................... 24 3CONFIDENCEINTERVALSFOLLOWINGTHESELECTIONOFk1POPULATIONS .................................... 29 3.1AnAlternativeApproach ............................ 35 3.2NumericalStudies ............................... 42 3.3TablesandFigures ............................... 44 4INTERVALESTIMATIONFOLLOWINGTHESELECTIONOFARANDOMNUMBEROFPOPULATIONS ........................... 46 4.1ConnectiontoFDR ............................... 48 4.2TablesandFigures ............................... 50 5APPLICATIONEXAMPLE .............................. 53 5.1FixedSelection ................................. 53 5.2RandomSelection ............................... 54 5.3TablesandFigures ............................... 55 6CONCLUSIONS ................................... 56 LISTOFREFERENCES .................................. 58 BIOGRAPHICALSKETCH ................................ 60 5

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Table page 2-1Congurationofthenewparameterizationforthecoverageprobability ..... 24 2-2Congurationofthenewparameterizationforthecasep=3 24 2-3Representationoftheparametersi,jwhenp=k+1 24 2-4Coverageprobabilityof95%CIfortheselectedmeanwhenp=4 25 3-1Structureofthe'sforthecasep=4,k=2 44 3-2Coverageprobabilitiesforthenumberofpopulationmeansvsthenumberofselectedpopulations ................................. 44 3-3Observedcondencecoefcientfor95%CIwhenp=6 44 3-4Cutoffpointsfor95%CIusingthenewmethod .................. 45 5-1Condenceintervalsforxedtoplog-scoredifferences .............. 55 5-2Condenceintervalsforrandomtoplog-scoredifferences ............ 55 6

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Figure page 2-1Coverageprobabilityasafunctionof21and32whenp=3 25 2-2Plotof@h=@21whenp=3 26 2-3Plotsofthersttwotermsof@h=@21 26 2-4Condencecoefcientvsthenumberofpopulationsfortheiidcaseand=0.05 27 2-5Cutoffpointversusnumberofpopulationsfortheiidcaseand=0.05 28 3-1Coverageprobabilitiesasafunctionofwhenp=6 45 4-1IndividualcomponentsforthecoverageprobabilityforrandomK 50 4-2LowerboundforrandomKvaryingtheprobabilityselection ........... 51 4-3CoverageprobabilitiesforrandomKfordifferentvaluesofp 52 7

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Bechhofer ( 1954 ), GuptaandSobel ( 1957 ).Inhispaper,Bechhoferpresentsasinglesamplemultipledecisionprocedureforrankingmeansofnormalpopulations.Assumingthevariancesofthepopulationsareknown,heisabletoobtainclosedformexpressionsfortheprobabilitiesofacorrectrankingindifferentscenarios.Thisapproachismoreconcernedwithselectionofthepopulationwiththelargestmeanratherthanestimationofthatmean.Guptaandco-authorshavepioneeredthesubsetselectionapproach,inwhichasubsetofpopulationsisselectedwithaminimumprobabilityguaranteeofcontainingthelargestmeanwithcertainprobabilityP(see GuptaandPanchapakesan ( 2002 ));whileBechhoferusesanindifferentzone.Thatis,thereisaminimumguaranteedprobabilityofselectingthepopulationwiththelargestmean,aslongasthatmeanisseparatedfromthesecondlargestbyaspecieddistance(see Bechhoferetal. ( 1995 )). 1. Selectthepopulationthathasthelargestparameter,maxf1,...,pg,andestimateitsvalue. 9

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Selectthepopulationwiththelargestsamplemean,andestimatethecorrespondingi.Therstoftheseproblemshasbeenwidelydiscussedintheliterature.Forexample, BlumenthalandCohen ( 1968 )considerestimatingthelargermeanfromtwonormalpopulationsandcomparedifferentestimators,buttheydonotdiscusshowtomaketheselection.Inthisdirection, GuttmanandTiao ( 1964 )proposeaBayesianprocedureconsistinginthemaximizationoftheexpectedposteriorutilityforacertainutilityfunctionU(i).Inthesamedirection,butfromafrequentistperspective, SaxenaandTong ( 1969 ), Saxena ( 1976 ),and ChenandDudewicz ( 1976 )considerpointandintervalestimationofthelargestmean. PutterandRubinstein ( 1968 )).Thisissuebecomesclearifweconsiderallthepopulationstobeidenticallydistributed,forwewillbeestimatingthepopulationmeanbyanextremevalue. Dahiya ( 1974 )addressesthisproblemforthecaseoftwonormalpopulationsandproposedestimatorsthatperformbetterintermsoftheMSE.Progresswasmadeby CohenandSackrowitz ( 1982 ), CohenandSackrowitz ( 1986 )and GuptaandMiescke ( 1990 ),whereBayesandgeneralizedBayesruleswereobtainedandstudied.However,performancetheoremsarescarce.Oneexceptionis Hwang ( 1993 ),whoproposesanempiricalBayesestimatorandshowsthatitperformsbetterintermsoftheBayesriskwithrespecttoanynormalprior.Anotherexceptionis SackrowitzandSamuel-Cahn ( 1984 )who,inthecaseofthenegativeexponentialdistribution,ndUMVUEandminimaxestimatorsofthemeanoftheselectedpopulation.Theproblemofimprovingtheintuitiveestimatoristechnicallydifcult.Inaddition,despitetheobviousbiasproblem,ithasbeendifculttoestablishitsoptimality 10

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Berger ( 1976 ), Brown ( 1979 )and Lele ( 1993 )arenotstraightforward.Inthisdirection, Stein ( 1964 )establishedtheminimaxityandadmissibilityofthenaiveestimatorfork=2.Minimaxityforthegeneralcase,wasestablishedlaterby SackrowitzandSamuel-Cahn ( 1986 ),weretheydiscussedthecasenormalcasefork3.Admissibility,forthegeneralcase,appearstobestillopen.Similarly,intervalestimationisanequallychallengingandagain,littlecanbefoundintheliterature.Typically,condenceintervalsareconstructedintheusualway,usingthestandardnormaldistributionasareferencetoattainthedesiredcoverageprobability.Howevertheseintervalsdonotmaintainthenominalcoverageprobability,asthenumberofpopulationsincrease. QiuandHwang ( 2007 )proposeanempiricalBayesapproachtoconstructsimultaneouscondenceintervalsforKselectedmeans,butwearenotawareofanyotherattemptstosolvethisproblem.Intheirpaper,QiuandHwangconsideranormal-normalmodelforthemeanoftheselectedpopulation,whichassumesthateachpopulationmeanifollowsanormaldistribution.UndertheseassumptionstheyareabletoconstructsimultaneouscondenceintervalsthatmaintainthenominalcoverageprobabilityandaresubstantiallyshorterthantheintervalsconstructedusingtheBonferroni'sbounds.Howeverthecondenceintervalstheyproposeareasymptoticallyoptimal,andsincetheircoverageprobabilitiesareobtainedaveragingoverbothsamplespaceandprior,theydonotgiveavalidfrequentistinterval.Wearenotawareofanyotherattemptstosolvethisproblem.Recently,amodernvariationofthisproblemhasbecomeverypopular,withamajorreasonbeingtheexplosionofgenomicdata,callingforthedevelopmentofnewmethodologies.Forinstance,ingenomicstudies,lookingeitherfordifferentialexpressionorgenomewideassociation,thousandsofgenesarescreened,butonlyasmallernumberareselectedforfurtherstudy.Consequently,theassessmentof 11

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2 )canbeexplicitlydeterminedusingthejointdistributionof(X1,...,Xp).Forexample,wheni=1(thersttermofthesum),wehave 2 ),assumingthepopulationvariance2isknown,andpresentanewapproachtoobtainthedesiredcondenceintervals. 14

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2 )intermsof21...p1,andobtainP(12X1c,X1X2,...,X1Xp)=P(jzjc,!221,...,!pp1)=1 (2)p=2Zcc(pYj=2Z1j1e1 2(!jz)2d!j)e1 2z2dz.Noticethatforxedz,theintegralswithinthecurlybracketsfgareessentiallythetailprobabilityofanormaldistributioncenteredatz.Therefore,wecanwriteP(jzjc,!221,...,!pp1)=Zcc(pYj=2(zj1))(z)dz,where()denotesthepdfofthestandardnormaldistribution.Ofcourse,thesameargumentisvalidfortheremainingtermsofthesumin( 2 ).ItfollowsthatwecanfullydescribetheprobabilityP((1)2X(1)c)intermsofanewsetofparametersij's,whereij=ijfor1i,jp.Underthisrepresentation,foreveryc>0,thevalueofthecoverageprobabilityP((1)2X(1)c)isdeterminedbytherelativedistancesbetweenthepopulationmeansi,i=1,...,p.Inotherwords,wecoverageprobabilitydenesafunctionhc()=P((1)2X(1)c),where=(11,12,...,pp)isthevectorofpossiblecongurationsoftherelativedistancesij's.Inthiscontext,wecanobtaincondenceintervalsfor(1),thathave(atleast)therightnominallevel,byminimizingrstthefunctionhc.Specically,given01,wecandeterminethevalueofc>0thatsatises 15

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1. 2. 3. Forj>k,jk=j,j1+j1,j2+...+k+1,k.Thesepropertiesrevealacertainunderlyingsymmetryinthestructureoftheproblem.ThissymmetryisportrayedinTable 2-1 whereeveryentryijcorrespondstothedifferencebetweenthevaluesofiandjlocatedinrowiandcolumnjrespectively.Inaddition,Property3indicatesthatweonlyneedtoconsiderp1parametersinordertodeterminethevalueofP((1)2X(1)c).Infact,foranygivenorderingoftheparametersi's,wecanalwayschoosearepresentationoftheprobabilityin( 2 )basedonp1parametersij.Asaresult,wehavethatthetrueorderingofthepopulationmeansi'sisnotparticularlyrelevantinthisapproach,andhence,wewillassume(withoutanylossofgenerality)that12...p.Althoughtheintroductionofthenewparameterizationseemstoreduce(inasense)thecomplexityoftheproblem,theminimizationofhcisstilldifcult.First,becauseofthedelicatebalanceexistingbetweentheij'sinthefullexpression(seeTable 2-1 )andsecond,becausetheformulaofthecoverageprobabilityissomehowinvolved.Toillustratetheseproblems,letusdiscussthecasep=2.WehaveP((1)2X(1)c)=Zcc(z12)(z)dz+Zcc(z+12)(z)dz=Zcc[(z12)+(z+12)](z)dz,where12>0.Sinceonlythequantityinbrackets[]dependson21and(z)>0,itseemsreasonabletothinkthathc(12)=P((1)2X(1)c)isminimizedatthesamepointwheregz(12)=(z12)+(z+12)ndsitsminimum.However,differentiatinggz

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2-2 .Weobtain 2z2dz 2z2dz+1 2z2dz,where12,230and()denotesthecdfofthestandardnormaldistribution.Preliminarystudiessuggestthattheglobalminimumofhc(12,23)=P((1)2X(1)c)islocatedattheorigin(seeFigure 2-1 ),butaformalproofisrequired.Tothisend,itissufcienttoshowthat@hc=@23>0and@hc=@12>0. 17

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2Zcc(z+23)e1 2(23+12+z)21 2z2dz 2Zcc(z12)e1 2(23+12z)21 2z2dz+1 2Zcc(z23)e1 2(12+z)21 2z2dz1 2Zcc(z2312)e1 2(12z)21 2z2dz.Sincethepartialderivativedependsonboth12and23,thebehaviorofitssignisnotobvious,butdifferentnumericalstudiessupporttheideathatthederivativeisnon-negative.Figure 2-2 showstheplotoftheintegrandof@hc=@12forxedvaluesof12and23.Noticethatifwegroupthersttwotermsandthelasttwotermsof( 2 ),wecanlookatthepartialderivativeasthesumoftwodifferences.InFigure 2-3 weobserve(inseparateplots)theintegrandsofthersttwotermsofthepartialderivative@hc=@12,forxedvaluesof12and23.Theplotsuggestthattheintegrandsdifferonlybyalocationparameter.Infact,changingvariables,wecanrewritetheexpressionin( 2 )as 2Z23+12+c23+12cZcc(z12)e1 2(23+12z)21 2z2dz,D2=1 2Z12+c12cZcc(z2312)e1 2(12z)21 2z2dz.Recallthat12>0,thenlookingatD2,wehavetwopossibilitiesfortheintervalsofintegration: 1.

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2 )weobtaintheratioisgreaterthan1(regardlessthecase)whichiscompellingtoconcludethatD2>0.Noticethattheargumentstillholdsifwereplacethecdf()byanynon-decreasingfunctionorifwechangetheinterval(c,c)for(c1,c2),wherec1,c2>0.Thisway,weobtainthefollowingmoregeneralresult: 2(1z)21 2z2dz0,wheretheinequalityisstrictwheneverthefunctionfismonotonicallyincreasinginz.

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2.1 isthatD1>0.Asaresult,weobtainthat@h=@12>0.Asimilarargumentshowsthat@h=@23>0,completingtheproof.ItfollowsthatcoverageprobabilityP((1)2X(1)c)isminimizedat12=23=0,thatis,whenever1=2=3.ObservethatProposition 2.1 givesastraightforwardproofforthecasep=2.Ineffect,forhc(12)=P((1)2X(1)c),wehavedhc 2.1 withf=1=2,weobtainthath0c(12)0.Itimmediatelyfollowsthatthecoverageprobabilityisminimizedat12=0,orequivalently,when1=2.Forthegeneralcase(p>3),weobservethatwhenmovingfromthecasep=ktothecasep=k+1,weonlyneedtoincludetheextraparameterk+1,kinordertodescribetheproblem(seeTable 2-3 ).Then,usingProposition 2.1 andmathematicalinductionweobtainthefollowingresult:

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whereZi=(Xii)=andij=(ij)=for1i,j3.Noticethattakingt=s=wecanrewriteeachterminthesum( 2 )asamixture.WeobtainP((1)2(X(1)sc)=Z10P(jZ1jct,Z1Z2+21,Z1Z3+31jt)'(t)dt+Z10P(Z2Z1+21,jZ2jct,Z2Z3+32jt)'(t)dt+Z10P(Z3Z1+13,Z3Z2+32,jZ3jctjt)'(t)dt,

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2-4 showstheresultofsimulationsconsideringupto30populationswiththesamemeanandsetting=0.05.Thesolidbluelinerepresentsthecondencecoefcientobtainedusingourproposedcondenceintervalsandthedashedredlinedepictsthebehaviorofthecondencecoefcientobtainedusingthestandardcondenceintervals.Observethatthesolidlineisconstantatthenominallevel95%.Intuitively,inordertomaintainthecoverageprobabilityconstant,thecondenceintervalsneedtogetwider.However,thisincrementisnotdramaticandslowdownasthenumberofpopulationsincrease.Forinstance,ifweconsider10000populations,thevalueofthecutoffpointisonlyabout4.41.Infact,fromtheinequalityinTheorem 2.1 itcanbedeterminedthatthebehaviorofthecutoffvaluecp 2-5 showsthebehaviorofthecutoffpointc,asthenumberofpopulationsincreaseforthecase=0.05.Thesolidlinecorrespondtothevalueofthestandardcutoffpointfora95%condenceinterval(z=2=1.96).Thedashed/dottedlinerepresentsthevalueofcforthenewcondenceintervalsandthedashedlinecorrespondtothecutoffvaluesfortheBonferroniintervals.Inanappliedsituation,thepopulationmeansi(1ip)willberarelyidentical.Henceweneedtocomparetheperformanceofthecondenceintervalswhenthepopulationsmeansaredifferent.Table 2-4 summarizesomeresultsobtainedby 23

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2-4 ),thetraditionalintervalsmayperformpoorly. CongurationofthenewparameterizationfortheprobabilityP((1)2X(1)c).Inthetableij=ij. ............p Table2-2. Congurationofthenewparameterizationforthecasep=3,when12and23arethefreeparameters.Inthetableij=ij. Representationoftheparametersi,jforthecasep=k+1. ...............k

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Observedcoverageprobabilityof95%CIforthemeanoftheselectedpopulationoutoffourpopulationsusingthetraditionalandthenewmethod.Thereportedvaluescorrespondtotheaverageaftertenreplicationsandthenumberinparenthesisisthecorrespondingstandarderror. (0,0,0,0)0.9040.952(0.0016)(0.0012) (0,0.25,0.5,1)0.9070.952(0.0020)(0.0011) (0,5,10,15)0.9500.974(0.0014)(0.0009) (0,0,0,2)0.9280.9584(0.0042)(0.0027) (0,0,0,5)0.9520.973(0.0031)(0.0028) Figure2-1. Coverageprobabilityasafunctionof21and32whenp=3. 25

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Plotof@h=@21forpredeterminedvaluesof21and32. Figure2-3. Plotsofthersttwotermsof@h=@21forpredeterminedvaluesof21and32. 26

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3 )asXj16=...6=jkP((1)2X(1)c,...,(k)2X(k)c,X(1)=Xj1,...,X(k)=Xjk),wherethesumhaspkterms.Letusconsiderrst,thecasep=4andk=2.Then,theprobabilityofinterestis 29

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3 )canbewrittenP(12X1c,22X2c,X1X2,X2X3,X2X4)=P(jZ1jc,jZ2jc,Z2Z1+12,Z3Z2+23,Z4Z2+24)andmakinguseofthenormalityassumptions,wecanexplicitlywriteP(12X1c,22X2c,X1X2,X2X3,X2X4)=ZccZmin(c,z121)c(z232)(z243g)(z1)(z2)dz2dz1+ZccZmin(c,z212)c(z131)(z141g)(z1)(z2)dz1dz2Ofcourse,thesameargumentisvalidfortheothertermsinthesum.Thisway,consideringallthe12possiblecongurationsfortheorderoftherandomvariablesX1, 30

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3-1 .Thispatternisparticularlyimportantsinceitsuggeststogeneralizetheexpressionforanyvaluesofpandk.Inordertodeterminethecongurationof'sthatminimizetheexpressionin( 3 ),weassume(withoutlossofgenerality)that1234,thiswayij0foranyij.Also,weconsider12,23and34asfreeparameters.Basedonourpreviousresults,itisreasonabletobelievethattheminimumof( 3 )isreachedattheorigin.Inordertoprovethisclaimwehavestudiedthebehaviorofthe 33

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whereIj=fj1,...,jkg,thesetofindicesforthetopkvariablesinthej-tharrangementandIcj=fjk+1,...,jpg,thesetofindicesforthebottompkvariablesinthej-tharrangement.Noticethatifk=1wearebackinthecasediscussedinChapter2andthecasek=pcorrespondtosimultaneouscondenceintervals.Letustakeacloserlookatthisformulaandconsiderrstthecasep=6andk=3.Insuchcase,thesumin( 3 )willhave63=20termsdeterminedbythecongurations 36

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3 )canbewrittenasZZR1(z1+12)(z1)(z3)dz1dz3+ZZR2(z3+32)(z1)(z3)dz1dz3.Similarly,theintegralin( 3 )canbewrittenasZZR1(z1+12)(z1)(z3)dz1dz3+ZZR2(z1+12)(z1)(z3)dz1dz3

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wheretheequalityisattainedasymptoticallyaspapproachesinnity.Integrating( 3 )withrespecttozp,weobtain((c)(c))24(p1k1)Xj=1ZccZccYm2Icjmin`2Ijfpgfz`+`gmY`2Ijfpg(z`)dz`35wherethequantityinbrackets[]isexactlythecoverageprobabilityforselectingk1outofp1.Repeatingtheargument,butnowlettingp1"1,weobtainthelowerbound((c)(c))224(p2k2)Xj=1ZccZccYm2Icjmin`2Ijfpgfz`+`gmY`2Ijfp,p2g(z`)dz`35.

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3-2 showstheresultofasimulationstudyinwhichweconsideredsixpopulationsandwevariedthenumberofoftheselectedones.Intherstcolumnwecanseethenumberofpopulationmeanssetequaltozero(the 42

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3-3 summarizestheresultsfortheobservedcoverageprobabilitiesconsidering6populationsobtainedinanumericalstudy.Thenominallevelis95%.Inthetable,therstcolumnshowsdifferentcongurationsforthepopulationmeansandtherstrowsindicatethenumberofselectedpopulations.Weobservedthatforeverycongurationtheobservedcoverageprobabilityisneverbelowthenominallevel.Theseresultremainvalidforeveryothercongurationwehaveconsidered(includingchangingthenumberofpopulations)whichvalidatesthereliabilityoftheprocedure.Finally,westudiedthebehaviorofthelengthoftheintervals.InChapter2weobservedthatthecondenceintervalsincreaseinlengthasthenumberofpopulationsincreases.Thisbehaviorisalsoexpectedwhenweareselectingk>1populations,howeverititisimportanttodeterminehowthevalueofkaffectsthelengthoftheintervals.Table 3-4 showstheresultsofanumericalstudyinwhichweconsidereddifferentvaluesofp(totalnumberofpopulations)andk(numberofselectedpopulations).Inthetable,therstcolumnsshowsthenumberofpopulations,andtherstrowthenumberofselectedpopulations.Inthebodyweobservethevaluesofthecutoffpointsfora95%condenceintervalsforthecorrespondingconguration,andthelastcolumnshowsthecutoffvaluesfor95%simultaneouscondenceintervalsusingBonferroni.WenoticethattheproposedintervalsarealwaysshortertoBonferroni,evenwhenweselectalltheavailablepopulations(p=k).Thisdifferenceincreasesasthenumberofpopulationsincreases. 43

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Structureofthe'sforthecasep=4,k=2(see 3 ).Eachrowrepresentaterminthesum. Top +13+23+14+24(X1,X3) +1223+14+34(X1,X4) +1224+1334(X2,X3) Coverageprobabilitiesforthenumberofpopulationmeansequalto0(rstcolumn)vsthenumberofselectedpopulation(rstrow). #ofi=0 0.7400.7400.7390.7380.7140.5315 0.8980.6980.6980.6970.6820.5314 0.9040.8130.6620.6620.6540.5313 0.8610.8530.7300.6260.6230.5312 0.8190.8180.8050.6580.5920.5311 0.7770.7770.7760.7570.5900.5310 0.7400.7400.7390.7380.7140.531 Table3-3. Observedcoverageprobabilityfor95%CIforthemeanoftheselectedpopulationswhenp=6usingthenewmethod. (0,0,0,0,0,0) 44

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Cutoffpointsfor95%CIfordifferentvaluesofpandkusingthenewmethod. NumPop 1 1.960 1.9602 1.9602.236 2.2413 2.1212.2362.388 2.3944 2.2342.3192.3882.491 2.4985 2.3192.3872.4432.4912.569 2.576 Figure3-1. Coverageprobabilitiesasafunctionofwhenp=6.Theplotssuggesttheminimumisnotreachedattheorigin. 45

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Fromourpreviousresults,wenoticethatforeveryterminthesumP((1)2X(1)c,...,(j)c)((c)(c))j1pj+1(c)pj+1(c),

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4 )andobtain where()isthecdfofthestandardnormaldistribution.Sincetheinequalityaboveisnotobtainedbydirectminimizationofthecoverageprobabilityin( 4 ),anysolutionbasedon( 4 )islikelytobetooconservative.Therefore,itisimportanttoassesstheperformanceoftheproposedboundintermsofotsproximitytothecoverageprobability.Therstthingtodetermine,isthebehaviorofthelowerboundsatthecomponentlevel(K=j).Figure 4-1 showstheresultsofanumericalstudyconsideringthecomponentsK=1,...,K=6ofthecoverageprobabilitywhenp=6.Thedashedbluelineshowsthebehavioroftherespectivecomponentasthenormof=(1,...,6)increasesandtheredsolidlineshowsthecorrespondinglowerbound.Weobservethelowerbound(fortheindividualterms)isnotextremelyconservative.Ontheotherhand,theprobabilitythatK=jisgivenby whereP(Xidj)=1(dji)istheprobabilityofselectionforpopulationi.Noticethattheexpressionin( 4 )resemblesabinomialdistribution.Infact,taking1=...=p=,wehave(pj)Xi=1Yi2Ii[1(dji)]Yi2Ici(dji)=pj[1(dj)]j[(dj)]pj,

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4-2 showstheresultsofanumericalstudyinwhichwetaked1=...=dp=d,andusethequantityqasatuningparameter.Weseethatbychangingthevalueoftheprobabilityofselectionwecanmovethepositionlowerbound(redsolidline)andproducesomeimprovementintheapproximationofthethecoverageprobability.Basedonthepreviousobservations,wecanobtainanapproximatesolutiontotheproblemanddeterminec>0usingtheequation1=pXj=1pj((c)(c))j1pj+1(c)pj+1(c)[1(qj)]j[(qj)]pj,forany0<<1.Numericalstudiessuggesttheresultsbasedontheexpressionabovearenotextremelyconservative.Inaddition,theresultssuggesttheperformanceofthemethodgreatlyimprovesasthenumberofpopulationincreases(seeFigure 4-3 ). BenjaminiandHochberg ( 1995 )andisatechniquecommonlyusedbypractitionersinthecontextofmultipletesting.Themainideaistocontroltheproportionoferrorscommittedbyfalselyrejectingnullhypotheses.Insimpleterms,theprocedureworksinthefollowingway:supposethatweneedtotestH1,...,Hmhypothesesandwearenotwillingtoacceptaproportionoffalsediscoveriesgreaterthanq.WerstranktheP-values(andcorrespondinghypothesis)resultingfromallthetestsfromsmallertolargestanddenethesequenceq1,q2,...,qpaccordingtoqi=(i=m)qfori=1...p).Then,wedenektobethelargestisuchthatP-valuei
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IndividualcomponentsandcorrespondingboundsforthetermsofthecoverageprobabilityforrandomKwhenp=6.Thebluedashedlinecorrespondtothecoverageprobabilityandtheredsolidlineisthelowerbound. 50

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BehaviorofthelowerboundforrandomKwhenp=6astheprobabilityofselectionvaries. 51

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BehaviorofthecoverageprobabilitiesandrespectivelowerboundsforrandomKasthepopulationsizepvaries. 52

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5-1 showsthemean,standarddeviationandcondenceintervalsforthetop5andbottom5selectedgenes.Thetableshowthat,althoughthevalueofthecutoffpointcisseeminglylarge,theactualcondenceintervalsarenarrowenoughtodrawpracticalconclusions. 5-2 showsthemean,standarddeviation,P-valueandcondenceintervalsforthemeandifferenceofthe25populationsselectedusingtheFDRcriteria.Again,weobservetheintervalsarenarrowenoughtocarryoutmeaningfulinference.Infacttheresultsofalltheintervalsagreewiththeconclusionsofthetests. 54

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Condenceintervalsbasedontheselectionofthetop100log-scoredifferences RankingMeanStDev95%CI 14.760.262(4.247,5.268)24.380.303(3.790,4.969)33.930.203(3.534,4.325)43.790.519(2.782,4.804)53.520.600(2.351,4.685)961.350.930(-0.457,3.163)971.350.680(0.029,2.675)981.351.459(-1.488,4.189)991.350.911(-0.428,3.118)1001.340.915(-0.445,3.117) Table5-2. Condenceintervalsbasedontheselectionofthetoplog-scoredifferences,randomlychosenusingFDR. MeanStDevP-value95%CI 4.380.3012.67e-08(3.778,4.981)3.930.2033.93e-08(3.526,4.333)4.760.2622.13e-07(4.237,5.279)2.850.6621.60e-06(1.532,4.161)0.950.2362.00e-05(0.483,1.420)3.030.5882.82e-05(1.859,4.194)0.990.3384.84e-05(0.320,1.662)0.480.0866.44e-05(0.311,0.652)1.830.3516.50e-05(1.130,2.526)1.240.4496.64e-05(0.345,2.129)0.880.2326.95e-05(0.417,1.337)1.250.4577.87e-05(0.344,2.159)2.611.1778.81e-05(0.271,4.943)1.320.4839.02e-05(0.357,2.277)0.980.1739.16e-05(0.638,1.324)3.520.6009.57e-05(2.328,4.709)2.740.7391.04e-04(1.277,4.212)1.420.4501.15e-04(0.530,2.319)1.000.4311.19e-04(0.139,1.851)1.120.5101.25e-04(0.103,2.127)3.500.5821.29e-04(2.343,4.654) 55

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