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Nonparametric Functional Mapping of Quantitative Trait Loci

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First,Iwouldliketothankmyparents,MingfaYangandQiyunLi,mysisters,JingYang,HuiYangandXiaominYang,foralltheirloveandsupportshownatallstagesofmylife.Iowethegreatestdebtofgratitudetomyhusband,SongWu,whohasbeenatmysidewithhistirelesslovesincemydayoneintheUSA.TheyaregreatcompanionsduringthesunnyandcloudydaystowardsmyPh.D.AndIwanttothankmyson,David,forbringussomuchjoyandpride.Iexpressmywarmestgratitudetomyacademicadvisorandcommitteechair,Dr.GeorgeCasella.Learningfromhisintuitionandhisclearunderstandingofmathematicalstatisticshasalwaysbeenapleasure.Ashisstudent,Ifeelmotivatedandenergizedaftereachresearchmeetingwithhim,althoughattimesImaynotknowexactlywhattodobutIworkmoreecientlyandammorecreativefollowingameetingwithhim.MyheartfeltthanksgotoDr.RonglingWuforhisgenerouscounsel,helpandencouragementduringallstagesofmyPh.D.studyhere.Ienjoyeverytalkwithhim.Iwouldalsoliketoextendmythankstotheothercommitteemembers:Dr.RamonLittell,Dr.RonaldH.RandlesandDr.MartaL.Wayne.IalsothankallotherfacultyandstumembersoftheDepartmentofStatistics,UniversityofFlorida;wheneverIneededadviceorfeedback,Ifeltwelcomeintheiroces.EquallyimportanthavebeenmyfriendsattheUniversityofFloridawiththeirhumor,comradeship,andunwaveringsupport.Iwillalwaysrecallwithfondnessthegoodtimesspenthereduringthegraduatework. iv

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page ACKNOWLEDGMENTS ............................. iv LISTOFTABLES ................................. vii LISTOFFIGURES ................................ ix ABSTRACT .................................... xiii CHAPTER 1INTRODUCTION .............................. 1 1.1GeneticBasics .............................. 2 1.2IntervalMapping ............................ 5 1.3FunctionalMapping .......................... 8 1.4Splines .................................. 12 1.5MixedModels .............................. 15 1.6MotivationandOutlineoftheDissertation .............. 17 2DENSE-MAPCASE:WHENTHEQTLISONAMARKER ...... 19 2.1DataSetting ............................... 19 2.2TheLikelihoodRatioTestatOneMarker .............. 21 2.3TheLikelihoodRatioTestoverAllMarkers ............. 22 2.4Examples ................................ 25 2.4.1Simulation ............................ 25 2.4.2PoplarData ........................... 27 2.5TechnicalDetails:DerivationoftheLikelihoodRatioTest ..... 30 3NONPARAMETRICFUNCTIONALINTERVALMAPPING ...... 32 3.1DataSetting ............................... 32 3.2ApproximateDistributionofTestStatisticsatEachQunderH0 35 3.3PrecisionoftheLinearApproximationtoLRTStatisticsunderH0 41 3.4AlternativeApproachtoCalculateP-value:Simulation ....... 49 3.5EstimationofConsistentVariance-CovarianceMatrixwithSmallSampleSize ............................... 56 3.6ApplicationtoPoplarData-Revisited ................ 61 3.6.1AChromosome-wiseAnalysisonLinkageGroup10 ..... 61 3.6.2AGenome-wiseAnalysisUsingREMLEstimate ....... 63 v

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. 64 3.8FurtherEvaluationofEmpiricalBayesEstimateofCovarianceMa-trix .................................... 69 3.9Discussion ................................ 71 3.10Appendix:PerformanceoftheApproximationGtoActualLikeli-hoodRatioTestStatistics ....................... 78 3.10.1JointdistributionofGateachQunderH0 78 3.10.2Examples ............................. 80 4MISSINGMARKERINFORMATION ................... 85 4.1SimpleMethods ............................. 85 4.2ModellingIncompletenessinMarkers ................. 86 4.3LikelihoodRatioTestwhenMissingMarkerisPresent ....... 89 4.4SimulationProceduretoGetThresholdPoint ............ 93 4.5Application ............................... 97 4.6AnotherWaytoHandleAllMissingMarkerCases{GibbsSampler 100 4.6.1Simulation ............................ 110 4.6.2Application ........................... 111 4.7Discussion ................................ 116 5FUTURESTUDY .............................. 122 REFERENCES ................................... 125 BIOGRAPHICALSKETCH ............................ 131 vi

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Table page 1{1ConditionalprobabilitiesofgenotypeataQTLbracketedbymarkersMandNinabackcrosspopulation ...................... 5 1{2Distributionofthephenotypicvaluesfordierentgenotypegroupsinabackcrosspopulation;1=r1r2 6 1{3ConditionalprobabilitiesofgenotypeataQTLbracketedbymarkersMandNinabackcrosspopulation,whenr1isrelativelysmall ....... 6 3{1AnalysisofPoplardatausingnonparametricfunctionalintervalmap-ping(NPFIM)andparametricfunctionalintervalmapping(PFIM).\EB",theempiricalBayesestimator.\REML",theREMLestimatorcorre-spondingtostructure2J+Autoregressive(2;): 64 3{2P-valuesanditsstandarddeviationofnonparametricfunctionalintervalmapping(NPFIM)usingdierentvariance-covarianceestimatorsunderdierenttrues.\EB"meanstheshrinkageestimatorwhichisguaran-teedtobeaconsistentestimator.\REMLw"estimatoristheREMLes-timatorcorrespondingtoawrongstructure.\REML"estimatoristheoneselectedfromSASProcMIXEDusingBICandassumingeachsub-jecthasdierentunderlyinggrowthcurve.\True"estimatorisofcoursethematrixweactuallyusedtogeneratedata. ............... 66 3{3P-valuesanditsstandarddeviationofnonparametricfunctionalinter-valmapping(NPFIM)usingdierentvariance-covarianceestimatorsfordierentsamplesizeand=3:\EB"meanstheshrinkageestimatorwhichisguaranteedtobeaconsistentestimator.\REMLw"estimatoristheREMLestimatorcorrespondingtoawrongstructure.\REML"estimatoristheoneselectedfromSASProcMIXEDusingBICandas-sumingeachsubjecthasdierentunderlyinggrowthcurve.\True"esti-matorisofcoursethematrixweactuallyusedtogeneratedata. ..... 68 3{4P-valuesandstandarddeviationofnonparametricfunctionalintervalmapping(NPFIM)fromHIVdynamicsdataandgrowthdatafordif-ferentcombinationofvariance-covarianceestimatorsandsamplesize.\EB"meanstheshrinkageestimatorwhichisguaranteedtobeaconsis-tentestimator.\REML"estimatorisobtainedfromSASProcMIXEDassumingeachsubjecthasadierentunderlyingmeancurve.\True"es-timatoris,ofcourse,thematrixweactuallyusedtogeneratedata. ... 72 vii

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............. 97 4{2Comparisonofcriticalvaluesfromdierentanalyzingmethods ...... 98 4{3Distributionofmissingmarkerconditionalontheotherobservedmarkerforkthmarkerintervalforsubjecti. .................... 103 4{4Conditionaldistributionofmissingmarkersofkthmarkerintervalforsubjecti. ................................... 103 4{5Distributionofmissingmarkerconditionalontheotherobservedmarkerforkthmarkerintervalforsubjecti. .................... 107 4{6Conditionaldistributionofmissingmarkersofkthmarkerintervalforsubjecti. ................................... 108 viii

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Figure page 1{1IllustrationsofoneisolatedB-splineandseveraloverlappingoneswithdegree1and2,respectively. ......................... 13 2{1Graphofthepvalueandits95%condenceintervalfordatafromtwounderlyingatlineswithdierentdistances. ................ 26 2{2Graphofpvalueandits95%condenceintervalfordatafromtwoun-derlyingcurves:oneisatlinetheotherisinquadraticform. ...... 27 2{3Originalpoplardataproles. ........................ 28 2{4ThegrowthcurverepresentingtwogroupsofgenotypesatmarkerCA/CCC-640Ronlinkagegroup10inthePopulusdeltoidesparentmap. ..... 29 3{1TheplotofPn3i=n1+1Siandmaxn1+1
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........................ 55 3{10Fittedcurvesfromusingtwodierentvariance-covarianceestimator. .. 63 3{11Theproleofthelikelihoodratioteststatisticsbetweenthefullandre-duced(noQTL)subject-specicmodelforthediametergrowthtrajec-toriesacrossthewholePopulusdeltoidesparentmap.Thegenomicpo-sitionscorrespondingtothepeakofthecurvearetheMLEsoftheQTLlocalization. .................................. 65 3{12Heritabilitycurvesforthefoursimulateddatasetsusingdierents. .. 67 3{13Theleftplotshowsthe10heritabilitycurvesofsimulatedgrowthdatasets.TherightplotshowsthetrendofP-valueswhenheritabilityonyear4increases.\EB"and\REML"meanstheEmpiricalBayesestimateandREMLestimateofcovariancematrix,respectively.\True"repre-sentsthecovariancematrixusedtogeneratedata. ............. 70 3{14TheHIVdynamicscurvesusedasthemeancurves. ............ 72 3{15TheheritabilitycurveacrossallobservationtimepointsintheHIVsim-ulationstudy. ................................. 73 3{16TheempiricaldistributionoftheteststatisticsinthesimulationstudyofChapter3.Dataset1-4correspondtothefourmatrixesinTable 3{2 .\EB"meanstheEmpiricalBayesestimate.\True"representsthecovariancematrixusedtogeneratethedata. ................ 76 3{17TheleftplotindicatestheempiricaldistributionoftheteststatisticsinthesimulatedgrowthdataofChapter3withcovariancematrix1,61subjectswith11observationpoints.TherightoneistheonefromthesimulatedHIVdynamicsdata,61subjectswith20observationpoints. .. 77 3{18Graphofthepvalueandits95%condenceintervalfordatafromtwounderlyingatlineswithdierentdistancesusingnonparametricfunc-tionalintervalmapping. ........................... 81 3{19Graphofpvalueandits95%condenceintervalfordatafromtwoun-derlyingcurves:oneisatlineandtheotherisinquadraticformusingnonparametricfunctionalintervalmapping. ................ 82 3{20Graphofthelikelihoodratioteststatisticsalongthelinkagegroup10usingnonparametricfunctionalintervalmapping. ............. 83 4{1HistogramsofLRTstatisticsfromdierentcombinationofmethodshan-dlingmissingdataandmethodstoestimateunderH0: 98 x

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................. 99 4{3Thisgureshowsthehistogramofgivenk=7fromthesimulateddatasetwhereQTLisassumedininterval7.TheREMLestimatefrominterval-wisedeleteddataisusedincalculation.\n"isthenumberoflastdrawnsamplesusedtoplotthehistogram. .............. 112 4{4ThehistogramsoftheintervalindexkandconditionalonkfromthesimulateddatasetunderH0:Thetruecovariancematrixisusedincalcu-lation. ..................................... 113 4{5Thehistogramsoftheintervalindexkfromthelast25,000iterations(left)andthelast10,000iterations(right).TheREMLestimatefrominterval-wisedeleteddataisused.DatasetissimulatedunderH0: 114 4{6Thehistogramsoffromthelast10,000iterations.TheREMLesti-matefrominterval-wisedeleteddataisused.Datasetissimulatedun-derH0: 115 4{7Thehistogramsof(k;)fromGibbssampler.TheleftgureshowsthehistogramofthepossibleintervalwhereQTLfallsin.Therightgureshowsthehistogramofgivenk=1: 116 4{8TheunderlyingfourdierentphenotypiccurvescontrolledbytwoQTLstogether. .................................... 118 4{9Theplotshowsthehistogramofgivenk=2:Medianofis:849whilethetrueis:917:TheREMLestimatefrominterval-wisedeleteddataisusedincalculation.Heritabilityvalueattheeightobservationpointissettobe.4.TwoQTLsareassumedtobeininterval2andinterval8,respectively.Startvalueforkis4.Thelast10,000samplesfrom50,000iterationsareusedtomaketheplots. .................... 119 4{10Thehistogramof(k;)fromthelast10,000samplesof50,000iterations.TwoQTLsareassumedtobeininterval2andinterval8,respectively.Theleftplotshowsthehistogramofintervalindexwhichgives^k=2:Therightplotshowsthehistogramofgivenk=2:Medianofis:749whilethetrueis:917:TheREMLestimatefrominterval-wisedeleteddataisusedincalculation.Heritabilityvalueattheeightobservationpointissettobe.1.Startvalueforkis6. ................. 120 xi

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1.2 ,abriefintroductiontointervalmapping,rstproposedbyLanderandBotstein(1989)isgiven.Typicallythesemethodstreattraitsasdiscrete,eventhoughthetraitisaninnite-dimensionalcharacteristic(KirkpatrickandHeckman1989)orafunctional-valuedtrait(PletcherandGeyer1999),whichmaybeexpressedasasmoothfunctionofonecontinuousvariable.Inreallife,asaconsequenceofnaturalselection,aninnite-dimensionalcharacteristicmaychangeitsphenotypetobemorecompetitive.Therefore,embeddingtheunderlyingbiologicalmechanismandprocessesintoaQTLmappingstrategyisareasonableapproach.Functionalmapping,developedbyMa,CasellaandWu(2002),implementsaparametric 1

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function,usuallyderivedfromauniversalbiologicallaw,todescribeaphenotypiccurve.Forexample,alogisticgrowthcurvecanbedescribedbyamathematicalformg(t)=a 1.3 providesanintroductionofthisfunctionalmappingframework.AshortdescriptionofthecomparisonbetweentraditionaldiscretemappingmethodsandfunctionalmappingmethodsisalsoincludedinSection 1.3 .Inthisdissertation,weextendedparametricfunctionalmappingtonon-parametricfunctionalmappingbecauseweproposetousesplinestoestimatetheunderlyingfunctionandusethespiritofmixedmodelstocharacterizethecomplex-ityofourbeautifulnature.InSection 1.4 andSection 1.5 webrieyexplainaboutsplinesandmixedmodels. 1. 2.

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Manytraitsofagronomicandhorticulturalinterestarecontrolledbyasinglegeneandfallintoafewdistinctphenotypicclasses.Theseclassescanbeusedtopredictthegenotypesoftheindividuals.Andifweknowthegenotypewecouldpredictthephenotype.Thesetypeofphenotypesarecalleddiscretetraits.Othertraitsmayhaveadistributionthatresemblesthebell-shapedcurveforanormaldistributionsuchashumanIQ,cropyield,weightgaininanimals,fatcontentofmeatetc.Thesetypesoftraitsarecalledcontinuoustraits.Becausecontinuoustraitsareoftengivenaquantitativevalue,theyareoftenreferredtoasquantitativetraitsandthereforethelocicontrollingthesetraitsarecalledquantitativetraitlociorQTL.ThepurposeofQTLmappingistolinkquantitativephenotypictraitswithregionsonachromosome.Twoquestionsareraised:(1)AretheresignicantQTLeects?and(2)wherearetheQTLsiftheydoexist?FormappingaQTL,acrossisarrangedbetweentwoinbredlines(whichmeanstheyarehomozygoteseverywhere)whichdiersubstantiallyinthequantitativetraitofconcern.AssumetherearetwoparentshavingallelesQQandqqatacertain(unknown)place(namedlocus)onthechromosome.TheospringoftheseparentsarecalledtheF1generationwhichhavethealleleQq:Iftheospringismatedbacktooneoftheparents,sayQQ;thenthenewgenerationproducedhasallelesQQandQq.SuchpopulationstructureiscalledabackcrossandthenewgenerationiscalledBC1:SinceourmappingmethodistondtheQTLQorq;ofcourse,thatlocusisunknown.Allweknowaremarkersalongthechromosome,sayMorm:Recom-binationoccurswhenallelescrossovertoanotherchromosomeandrecombinationratercanbeusedasameasureofclosenessbetweentwogenesorbetweengeneandmarker.SowecanusemarkerstolocateQTL.IftherecombinationraterbetweenmarkerandQTLissmall,withzerobeingthelimitingcaseofnorecom-bination,thenmarkerMiscloselylinkedtoQTLQ;thatis,whenweseeMwe

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hopeQisalsothere.Ifrislargewithvalue1=2beingthelimitingcase,thentheremaybenolinkagebetweenMandQ:Thebackcrossisthesimplestdesigninwhichwegetenoughinformationtoestimater:Asimplestatisticalmodeldirectlyfollows.AssumeYisarandomvariablethatfollowsanormaldistribution.ThevarianceofYis2andthemeanofYde-pendsontheallelesunderconsideration,QQ;Qqorqq:Clearlyinthebackcross,parentsareN(QQ;2)(orN(qq;2))andN(Qq;2)andgenerationBC1isamixtureofnormalswithmeansQqandQQ(orqq).IftheQTLQisknown,themixingfactorwouldbe1=2;butwecanonlyobservethemarkergenotypesMorm:Forexample,theparentsofBC1haveallelesMQjMQandallelesmqjMQ.WhenthereisnorecombinationbetweenMandQ;wewillseeallelesMQjMQandmqjMQingenerationBC1:ButwhenrecombinationbetweenMandQoc-curs,wewillexpectfourpossiblegenotypesMQjMQ;MqjMQ;mQjMQandmqjMQ:NowthedistributionofgenotypesinthisbackcrosspopulationisYjMM8><>:N(QQ;2);withprobability1rN(Qq;2);withprobabilityrandYjMm8><>:N(QQ;2);withprobabilityrN(Qq;2);withprobability1r:ThedierenceinmeansofthepopulationswhencategorizedbythemarkersisMMMm=(12r)(QQQq):AssumingQQQq6=0;atestofH0:MMMm=0isequivalenttotestingH0:r=1=2;whichmeansnolinkagebetweenMandQ:Thistestcanbecarriedoutwithsomethingassimpleasat-testorthemorepopularusedlikelihoodmethods.Usingthismixturemodelinlikelihoodanalysis,wecannotonlytestforlinkagebutalsocanestimater:

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Table1{1: ConditionalprobabilitiesofgenotypeataQTLbracketedbymarkersMandNinabackcrosspopulation MarkerQTL Qqqq 1rr1r2 1r

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Table1{2: Distributionofthephenotypicvaluesfordierentgenotypegroupsinabackcrosspopulation;1=r1r2 Table1{3: ConditionalprobabilitiesofgenotypeataQTLbracketedbymarkersMandNinabackcrosspopulation,whenr1isrelativelysmall MarkerQTL Qqqq TheconditionalprobabilityofaQTLgenotype,sayQq;giveneachofthefourtwo-markergenotypesisgiveninTable 1{1 .Let1=r1r2 1{2 ,whereQq;qq;2aretheunknownparameterscontainedinthemixturemodelsandcanbeestimatedusinglikelihoodprinciple.When=r1=rissmall,wecanusethefollowingapproximatedconditionalprobabilitieslistedinTable 1{3 .Inthedissertation,weusetheseapproximateconditionalprobabilities.SupposethatN1;N2;N3andN4arethesamplesizesofthefourmarkergroups( 1{1 ).Ifwedenen1=N1;n2=n1+N2;n3=n2+N3andn=n3+N4;theloglikelihoodofthephenotypedataconditionalonthemarker

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informationcanbewrittenas`(Qq;qq;2)=n 22(yiQq)2+n2Xi=n1+1log[(1)exp(1 22(yiQq)2)+exp(1 22(yiqq)2)]+n3Xi=n2+1log[exp(1 22(yiQq)2)+(1)exp(1 22(yiqq)2)]nXi=n3+11 22(yiqq)2:InLanderandBotstein's(1989)modelforintervalmappingofQTL,theproleoflikelihoodratiotest(LRT)statisticsisconstructedoverthegridofpossibleQTLlocations,whichisimpliedby;inalinkagegrouporanentiregenome.(Thegenomeisthecompletesetoflinkagegroupsorchromosomes.)ThemaximumoftheLRTstatisticsisusedasaglobalteststatistic.AtagivenpositionoftheQTL,thatis,whenisxed,theLRTstatisticisasymptotically2distributedwithdegreesoffreedomequaltonumberofassociatedQTLeectsunderthenullhypothesisthatnoQTLexistsatthis.However,underH0:noQTL,theQTLpositionisunidentiedandtherefore,thenalglobalmaximumLRTstatisticdoesnotfollowthestandard2-distributionasymptotically.ThisisexactlywhythereisnoclosedformtodecidethecriticalthresholdtodeclarethestatisticalsignicanceofaQTL.ButseveralauthorsderivedapproximateformulastodeterminecriticalthresholdsforaparticulardesignbasedontheresultsofDavies(1977,1987)(Rebaietal.(1994);DoergeandRebai(1996);Piepho(2001)).Toovercomethelimitationsduetothefailureoftheteststatistictofollowastandardstatisticaldistribution,onecanalsousesimulationstudies(LanderandBotstein(1989);VanOoijen(1992);ChenandChen(2005))andpermutationtests(seeChurchillandDoerge(1994);DoergeandChurchill(1996))tocalculatethe

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thresholdvaluethroughoutagenome.LanderandBotstein(1989)establishedthenite-dimensionalconvergenceofLODscoreprocess,whicharisesfromtheLRT,totheOrenstein-Uhlenbeckprocess.ChenandChen(2005)establishedanoveltheoremfortheasymptoticdistributionoftheLRTstatisticcalculatedfromonemarkerintervalforbackcrossmodel.Thenadistribution-freesimulationapproachcanbeusedtocalculatecriticalvalues.Eitherthesimulation-basedorpermutation-basedapproachisahighlycomputational-demandingapproach.Forpermutationtest,oneneedperformatleast10,000permutationsforthesamedatasettoobtainareasonablyaccurateestimateofacriticalthresholdatagenome-widetypeIerrorrateof0.01(DoergeandRehai(1996)).LanderandBotstein's(1989)intervalmappingmethodisbasedonasimpliedsituation,thatis,thesegregationpatternofallmarkersstrictlyagreeswiththeMendalianlawsandthereisonlyoneQTLonachromosomecontrollingatraitunderstudy.Manyauthorsextendedandimprovedthisworkbyincludingmarkersfromotherintervalsascovariatestocontroltheoverallgeneticbackground(JansenandStam(1994);Zeng(1994);Haleyetal.(1994);Xu(1996)).Kaoetal.(1999)usedmultiplemarkerintervalssimultaneouslytomapmultipleQTLofepistaticinteractionthroughoutalinkagemap.

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implementtheinformationaboutthecovariancesofdierenttraitsorthesametraitmeasuredatdierentagesorenvironments.Multitraitmappingtakesaccountofcovariancesamongsimultaneouslydierenttraitsorthesametraitmeasuredatdierentagesorenvironments.Butitisdiculttoproducepreciseestimatewhenthenumberoftraitsincreases(JiangandZeng(1995);Koroletal.(1995);Roninetal.(1995);Eavesetal.(1996);KnottandHaley(2000)).Tocircumventsuchdiculties,someauthorsattemptedtotransformtheinitialtraitspaceintoaspaceofalowerdimensionbasedonprincipalcomponentanalysis(Manginetal.(1998);Koroletal.(2001)).However,theystilltreatinnite-dimensionalcharactersasdiscretetraitsoreigenvaluesanddonotconsiderthephysiologicalmechanismspredisposingforthephenotypicvariationoffunctional-valuedcharacter.ThefunctionalmappingmethodrstintroducedbyMaetal.(2002),incorpo-ratestheunderlyingphysiologicalordevelopmentalmechanismsoftraitvariationintostatisticalanalysisofQTLmappingframework,thusismorelikelytoproducemoreaccurateresultsintermsofbiologicalreality.Intheirpaper,anapplicationtomapthegrowthofforesttreesisusedtoillustratetheirmethod.Sowealsousethatexampletobrieyintroducethefunctionalmappingmethod.Thesimplestbackcrossdesignisassumedsothatthereareonlytwogroupsofgenotypesatalocus.Ofcourse,thegenotypesofQTLalsoonlyhavetwopossibilities.Inpractice,insteadofacontinuumweonlyobserveanitesetofobservations,yi(1);:::;yi(m)foreachtreei;i=1:::n:Supposethisnitesetofdatacanbemodelledasthelogisticgrowthcurve,whichhasamathematicalformg(t)=a

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determinewhetherandhowtheQTLaectsgrowthtrajectoriesissimplytotestthedierenceorsomefunctionsoftheseparameters.Similartointervalmapping,thephenotypesofthetraitatalltimepointsforeachQTLgenotypegroupfollowsamultivariatenormaldistribution:fj(y)=1 (2)m=2jj1=2exp[(ygi)01(ygi)=2];wheregjisthevectoroftheexpectedphenotypicvaluecorrespondingtoQTLgenotypegroupjattmeasurementtimesandmodelsthedependencestructurebetweenobservationsfromsamesubject.Whenweuselogisticgrowthcurvetomodelythengj=[gj(t)]1m=aj =20BBBBBBB@1m11m2m1m211CCCCCCCA:(1{3)Thelikelihoodfunctionofthebackcrossprogenywithm-dimensionalmeasure-mentscanbeexpressedasamultivariatemixturemodelL(aj;bj;rj;;;)=nYi=1"2Xj=1pijfj(yi)#;wherepij;afunctionofparameterasinTable 1{3 ,istheconditionalproba-bilityofsubjectihavingQTLgenotypejgivenobservedankingmarkers.Themaximum-likelihood-basedmethodimplementedwiththeEMalgorithmisusedtoestimateallunknownparameters.Alotofbiologicallymeaningfulhypothesescanbetestedusingsuchgeneticmodels.Forexample,thehypothesistotesttheexistenceofaQTLaectingan

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overallgrowthcurvecanbestatedasH0:a1=a2;b1=b2;r1=r2againstH1:atleastoneoftheequalitiesabovedoesnothold:TotestifthedetectedQTLstartstoexertorceasesaneectongrowthcurvesatparticulartimetisessentiallytotestH0:g1(t)=g2(t)againstH1:g1(t)6=g2(t):Theteststatisticfortestingsuchhypothesesiscalculatedasthelog-likelihoodratioofthefullmodeloverthereducedmodel.Maetal.(2002)used1000per-mutationteststoobtainthechromosome-wideempiricalestimateofthecriticalvalue.ThisfunctionalmappingmethodallowsthemtosuccessfullydetectoneQTLunderlyingpoplarstemgrowthlocatingonlinkagegroup10whiletraditionalintervalmapping(LanderandBostein(1989))andcompositeintervalmapping(Zeng(1994))failedtodoso,whichsuggeststhatbyincorporatinglogisticgrowthcurvesfunctionalmappingprocedurehasgreaterpowertodetectasignicantQTLthanthecurrentmethods.Theincreaseddetectionpoweroffunctionalmappingresultsfromthesimultaneoususeofrepeatedmeasurementsthatarecorrelatedtoeachotherduetobiologicalor/andenvironmentalreasons.Anotheradvantageoffunctionalmappingisthatbytreatingphenotypicvaluesasafunctionofsomevariablesalargenumber(theoreticallyunlimitednumber)ofmeasurementscanbeanalyzed.Aninitiallyhigh-dimensionalmappingmodelbecomesmoretractableandtheprecisionoftheestimatesofQTLparameters

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getsimproved.Also,asmallsamplesizecanhaveenoughpowerofQTLdetectionbecausethefunctionmodelledfromrepeatedmeasurementsextractsmaximuminformationaboutQTLeectsandpositions. 1{1 :oneB-splineofdegree1.Itconsistsoftwolinearpieces:oneisfromx=1tox=2;theotherisfromx=2tox=3:Theknotsarex=1;2;3:Totheleftofx=1andtotherightofx=3thevalueofthisB-splineiszero.Intheright

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Figure1{1: IllustrationsofoneisolatedB-splineandseveraloverlappingoneswithdegree1and2,respectively.

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partofthisuppergureinFigure 1{1 ,threemoreB-splinesofdegree1areshownwitheachonebasedonthreeknots.AlsointheFigure 1{1 ,thelowergureillustratesB-splineswithdegree2.WecanseethattheleftmostB-splineconsistsofthreequadraticpiecesjoinedattwoknotsx=2andx=3:Atthejointknotsthevaluesofthesequadraticpiecesmatchandsodotherstderivatives(butnottheirsecondderivatives).Intherightpart,threeoverlappingB-splineswithdegree2areshown.Notearst-degreeB-splinecanoverlapwithonlyatmosttwoneighborsandasecond-degreeB-splineoverlapswithfourneighbors.Theleftmostandrightmostsplineshavelessoverlap.ThesesimpleexamplesshedlightonthegeneralpropertiesofaB-splineofdegreep:

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factorsisacomplextask.Someapproachesstartwithafreechoiceofknotlocationsandaratherdensesetofknotsandtheneliminateunneededknotsbyanalgorithmicproceduresimilartovariableselectiontechniquesusedinmultipleregression(see,forexample,FriedmanandSilverman(1989)).O'Sullivan(1986,1988)proposedtousearelativelylargenumberofknotsbutputapenaltyonthesecondderivativetorestricttheexibilityofcurvetting.EilersandMarx(1996)followedthisideaandusedasimpledierencepenaltyonthecoecientsthemselvesofadjacentB-splines,whichtheycallP-splines.P-splinescanbeappliedinanycontextwhereregressiononB-splinesisuseful.

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1. Norequirementforbalanceindata; 2. Explicitlymodellingandanalysisofwithin-andbetween-individualvariation; 3. Theindividualparametershaveanaturalexplanationandthereforefacilitatetheexploratoryanalysis.UnlessinaBayesianframework,theinferenceisbasedonleastsquaresandmaximumlikelihoodprinciplesfromthemarginaldistributionofyi:FromabovemodellingmarginallyyiMVN(Xi;ZiDZ0i+Ri):However,ifso,thehierarchicalstructureoftheoriginalmodel( 1{4 )isthennottakenintoaccount.Indeed,themarginalmodelisnotequivalenttotheoriginalhierarchicalmodel.Inferencesbasedonmarginalmodelsdonotexplicitlyassumethepresenceofrandomeectsrepresentingtheheterogeneitybetweensubjects.ThepurposeofourproposedmethodistondthesignicantstatisticalevidenceofexistingQTLwhichisreectedbythedierenceofaveragephenotypiccurves.Estimatesforrandomeectsshowthesubject-specicproleswhich

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deviatefromtheoverallaverageproleandareneededforpredictionofsubject-specicevolutions.Soinferencefromthemarginaldistributionisenoughforourstudy.Theselectionofvariance-covariancestructureisanontrivialstepinthemodelselectionprocess,whichwillbefurtheraddressedinChapter3.Severalstatisticalpackagesareavailableforestimationandinferenceonallparametersinthemarginalmodel.Amongthese,themostexiblecommerciallyusedoneistheSASprocedurePROCMIXED.Littelletal.(2000)providedaniceexampletohowtoapplyfrequentlyusedstatementsandoptionsinPROCMIXED.MoredetaileddescriptionsofallstatementsandoptionscanbefoundinSASmanuals. 1.3 ,weknowthebasicideaoffunctionalmappingisusingaparametricformtomodelthephenotypiccurve.However,insomecasestherearemanydierentfunctionsthatdescribesamephenotypictrajectory,forexample,therearefunctionsin3categoriestodescribeagrowthtrajectory:exponential,saturatingandsigmoidal(VonBertalany(1957),Niklas(1994)).Thus,itmaynotbeclearwhichoneshouldbeused,especiallywhentherearenotenoughobservationsforeachsubjecttoshowobviouscharacteristics.Moreover,inmanysituations,therearenoobviousfunctionalforms.Nonparametricfunctionalmappingavoidssuchproblemsoftheoriginalfunctionalmappingmethodwhileinheritingallitsadvantages.Inthisdissertationweproposeanonparametricfunctionalmappingprocedurefordierentsituations.InChapter2weconsiderthedense-mapsituationwhichassumesthattheQTLisonsomemarker.Weusenonparametricfunctionestimation,typicallyim-plementedwithB-splines,toestimatetheunderlyingfunctionalformofphenotypictrajectories,andthenconstructanonparametrictesttondevidenceofexistingquantitativetraitloci.Usingtherepresentationofanonparametricregressionasa

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mixedmodel,wecaneasilyderivealikelihoodratiotest(LRT)statistic.ThejointdistributionofLRTstatisticsateachputativelocus(eachmarker)isderived,sothatP-valuescanbeexactlycalculatedusinganimportancesamplingmethod.Intotal,onerealpoplardatasetandtwosimulateddatasetsareusedforapplication.InChapter3wedevelopanonparametricfunctionalintervalmappingproce-dureforthesituationwheretheactualgenesresponsiblefordierentunderlyingphenotypictrajectoriesmightnotjustbeonamarkerlocus,whichismorereal-istic.LRTstatisticsarecalculatedfromanonparametricmixture-mixedmodel.ThejointdistributionofallcalculatedLRTstatisticsalongagenomeorlinkagegroupcanbeapproximatelyobtained,buttheresultingP-valueisalowerboundoftheexactP-value.SoasimulationprocedureisproposedtocalculatetheexactP-value.Analysisfromsimulateddatasetsisperformedtoshowtheoperatingcharacteristicsofthenonparametricfunctionalintervalmappingprocedure.Twowaystoestimatethecovariancematrixareexaminedthroughsimulationstudiesalso.InChapter4weextendthenonparametricfunctionalintervalmappingproceduretothemissinggenotypicdatasituation.Theteststatisticisobtainedfrommaximumlikelihoodprincipleanditcriticalvaluecanalsobedeterminedbyasimulationapproach.Inthischapter,wealsoproposeaBayesianmappingproceduretotakeintoaccountofmissinggenotypicdata.InChapter5wesuggesttoextendthemappingprocedureintroducedinChapter4tohandleboththemissinggenotypicandphenotypicdatasituation.

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19

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Weconsiderasubject-specicmodel,foreachsubjecti;i=1;:::;N; 2{1 )isequivalenttothehierarchicalmodelyi(~t)=i(~t)+"ii(~t)=mi(~t)+i1T:ortheunivariatemodel, 2{2 )canalsobeviewedasarandomcoecientmodelthatassumestherandomdeviationofasinglecurvewithrespecttopopulationcurveatdierenttimepointsmaynotbethesame.Thischaracteristicwillbedeterminedbythecovariancematrix.ItisstraightforwardtoseethatthetestforadierencebetweentwophenotypiccurvesisatestofthehypothesisH0:m1=m2ateachmarkervsH1:m16=m2atsomemarker.Wederiveamaximumlikelihoodtestforthishypothesis.WeassumethatwecanuselB-splinebasestoestimatetheunderlyingfunctional,wherelT(Thisassuresthattheexpressionwillbeunique).Sowecanwritem=B;whereBisthebasismatrix,andisthesamefordierentfunctions.

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2l1Xi=1(yiB1)01(yiB1)1 2NXi=l1+1(yiB2)01(yiB2)g:Toconstructthelikelihood-ratiotestforH00:m1=m2ononeparticularmarker,werstneedtondtheMLEsoftheunknownparameters.DenotethecommoncoecientvectorunderH00by:Aftertakingtherstderivativeofthelog(L)wegetthelikelihoodequationsfor:NXi=11(yiB)=0:NowitisstraightforwardtoseethatthemaximumlikelihoodestimateforunderH00is^=(B01B)1B01y;wherey=1 2PNi=1(yiB^)01(yiB^)g 2Pl1i=1(yiB^1)01(yiB^1)1 2Pl1+l2i=l1+1(yiB^2)01(yiB^2)g:

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TheLRTrejectsH00whenissmall,whichisequivalenttorejectingwhen0=NXi=1(yiB^)01(yiB^)l1Xi=1(yiB^1)01(yiB^1)l1+l2Xi=l1+1(yiB^2)01(yiB^2) (2{3)islarge.Astraightforwardcalculation(Section 2.5 )showsthatthisisfurtherequivalenttorejectingH0when 2(m1m2)0A0(m1m2);where2nisanoncentralchisquaredrandomvariablewithdegreesoffreedomn=rank(A0)=rank((B01B)1(B01B))=rank(B)andnoncentralityparameter(1=2)(m1m2)0A0(m1m2):UnderH0;G2n,acentralchisquaredrandomvariable.

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where 2{5 )correspondingtokthmarkerislabelledasAk:Thus,ateachmarkeri,thecorrespondingteststatisticisY0kAkYk=Y01P0kAkPkY1:Noticethat,var(PkY1)=Pk~P0k=~because~isblockdiagonalwithblocksizeTTandPkjustinterchangestherowsof~.SounderH0westillhaveY0kAkYk=Y01P0kAkPkY12n:RecallthatunderH0;m(~t)=m1(~t)=m2(~t);Y1MVN(1Nm(~t);IN)MVN(~;~)andwecanwriteY0kAkYk=Y0k~1 2~1 2Ak~1 2~1 2Yk=Y0k~1 2WkW0k~1 2Yk=Y01P0k~1 2WkW0k~1 2PkY1Z0kZk;

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whereZk=W0k~1 2PkY1dMVN(W0k~1 2Pk~;W0kWk)andWkW0karethespectraldecompositionsof~1 2Ak~1 2.ThematrixWkiscomposedoftheeigenvectorscorrespondingtothennon-zeroeigenvalues,andithasnorthonormalcolumnsandisoforderNTn:Also,underH0thestructureofAkresultsin(W0k~1 2Pk~)0W0k~1 2Pk~=~0Ak~=~0(Uk1N)(A0m(~t))=~00(A0m(~t))=0:andthus,underH0;Zkn1=W0k~1 2PkY1MVN(0;In):TheentirevectorZ=(Z01;Z02;:::;Z0m)0hasdistributionZMVN(Z;)whereZi=W0i~1 2~and=0BBBBBBBBBB@W01~1 2W02~1 2P2...W0m~1 2Pm1CCCCCCCCCCA~~1 2W1P02~1 2W2P0m~1 2Wm;andunderH0,Z=0.IfweletBxdenotethen-dimensionballwithradiusequaltox,thenP0(max1kmY0kAkYkx)=P0(max1kmZ0kZkx)=P0(Z01Z1x;;Z0mZmx)=Z:::ZfZi2Bxg1 2exp(1 2XiZ0i1Zi)dZ1dZm:

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Thisprobability,whichisoneminusthep-valueforH0;canbedirectlycalculatedbysimulatingZMVN(0;)manytimesandcountinghowmanyZi'sfallinthen-dimensionalballBx:Alltheabovederivationsaremadeundertheassumptionthatweknow;whichistypicallyuntrueinpractice.WesuggestsubstitutingaREMLestimateofthevariance-covariancematrix^insteadof,and,inthatcase,theaboveformulaearecorrectasymptotically(Wolnger1993,Littellet.al2000).(PleaserefertoChapter3formoredetails.)

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Figure2{1: Graphofthepvalueandits95%condenceintervalfordatafromtwounderlyingatlineswithdierentdistances. asameasureofdistance.Thepvaluetrendwithrespecttodierentdistancesbetweentwounderlyingatlines,andits95%condenceinterval,isgiveninFigure 2{1 .Itisclearthatthepvaluedecreasesasthecurvemovesapart,whichisthebehaviorwewouldexpect.Theseconddatasetassumesonephenotypiccurveisat(m1(t)=1)andtheotherisquadratic(m2(t)=ax2+bx+c).Inthiscase,usingeightmarkers,50subjectsaregeneratedat12observationpointsusingthevariance-covariancematrixforeachyi=:1J12+Autoregressive(2=:05;rho=:5);whereJisamatrixwithallonesandthestructureofmatrixAutoregressive(2;rho)isshownas( 1{3 ).Basedonthesamemarkerinformation,100datasetsaregener-atedandanalyzed.Tomeasuredistance,weusetheareabetweenthetwocurves

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Figure2{2: Graphofpvalueandits95%condenceintervalfordatafromtwounderlyingcurves:oneisatlinetheotherisinquadraticform. overtheobservedtimeinterval.Figure 2{2 showsthepvalueprolefordierentdistancesandits95%condenceinterval.Again,thebehavioriswhatwedesire,withthep-valuedecreasingasthecurvesgetfurtherapart. 2{3 :Theleftgureshowstheoriginaldiametergrowthtrendforeverytree.Therightoneisthegrowthcurveaftertakinglogs.).Ageneticlinkagemapwasconstructedwhichcomprisesthe19largestlinkagegroupsforeachparentalmap,

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Figure2{3: Originalpoplardataproles. andrepresentsroughly19pairsofchromosomes.WeusedourmethodtodetectQTLaectingdiametergrowthonlinkagegroup10fromtheP.deltoidesparentmap.Becauseourmodelassumesthatalltobservationshaveequalvariance,wersttransformedtheoriginalobservationsbytakingthelog.Usingourmethodwiththevariance-covariancestructureautoregressive+simple(estimatedbyPROCMIXEDas^=0:001236J11+Autoregressive(2=0:06155;=0:8945)),wefoundstrongevidencethatthereisaQTLatamarkeronlinkagegroup10whichcontrolsthegrowthtrajectoryofstemdiameterintheinterspecichybridsofpoplar(p=:037,se=4:6e4).Thebiggestlikelihood-ratioteststatisticsGappearsatMarkerCA/CCC-640R.Figure 2{4 showstheestimatedgrowthcurvesusingB-splinesforeachgroupindexedbythismarker.ThisndingisconsistentwiththeresultinMa,CasellaandWu(2002).

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Figure2{4: ThegrowthcurverepresentingtwogroupsofgenotypesatmarkerCA/CCC-640Ronlinkagegroup10inthePopulusdeltoidesparentmap.

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2{3 )toG( 2{6 ).Rewrite0as0=l1Xi=1(yiB^)01(yiB^)l1Xi=1(yiB^1)01(yiB^1)+NXi=l1+1(yiB^)01(yiB^)NXi=l1+1(yiB^2)01(yiB^2)01+02:Noticethatl1Xi=1(yiB^)01(yiB^)=l1Xi=1(yiy1)01(yiy1)+l1Xi=1(y1B^1)1(y1B^1)+l1Xi=1(B^1B^)01(B^1B^)+2l1Xi=1(y1B^1)1(B^1B^):withl1Xi=1(y1B^1)1(B^1B^)=0:Thusl1Xi=1(yiB^)01(yiB^)=l1Xi=1(yiy1)01(yiy1)+l1Xi=1(y1B^1)1(y1B^1)+l1Xi=1(B^1B^)01(B^1B^):

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Similarly,wecangetl1Xi=1(yiB^1)01(yiB^1)=l1Xi=1(yiy1)01(yiy1)+l1Xi=1(y1B^1)1(y1B^1):Therefore,01=l1Xi=1(B^1B^)01(B^1B^)=l1(B^1B^)01(B^1B^);and02=l1+l2Xi=l1+1(B^2B^)01(B^2B^)=l2(B^2B^)01(B^2B^):Wethanhave,0=01+02=l1(B^1B^)01(B^1B^)+l2(B^2B^)01(B^2B^)=l1(y1y)0A0(y1y)+l2(y2y)0A0(y2y)=l1(y1l1y1+l2y2

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32

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1{3 .Bythecombinationoftwooutsidemarkers,arrangetheobservationsy0ssuchthaty1;;yn1isfrommarkerclassMm=Nn;yn1+1;;yn2isfrommarkerclassMm=nn;yn2+1;;yn3isfrommarkerclassmm=Nn;andyn3+1;;ynisfrommarkerclassmm=nn:Againweconsiderasubject-specicmodel,foreachsubjecti;i=1;:::;n @1=n1Xi=1B01(yiB1)+n2Xi=n1+1(1)(yi;B1;)B01(yiB1) (1)(yi;B1;)+(yi;B2;)+n3Xi=n2+1(yi;B1;)B01(yiB1) (1)(yi;B2;)+(yi;B1;):DenoteP(y;t)=(1t)(y;B1;) (1t)(y;B1;)+t(y;B2;):

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Thenthelikelihoodequationfor1canberewrittenas@logL @1=n1Xi=1B01(yiB1)+n2Xi=n1+1P(yi;)B01(yiB1)+n3Xi=n2+1P(yi;1)B01(yiB1):Similarly,thelikelihoodequationfor2canberewrittenas@logL @2=nXi=n3+1B01(yiB2)+n2Xi=n1+1(1P(yi;))B01(yiB2)+n3Xi=n2+1(1P(yi;1))B01(yiB2):Thus,itisstraightforwardtohavethefollowingEMalgorithmtonumericallyndtheMLEof1and2:EMAlgorithm:Forxedandknown;repeatthefollowingstepsuntiltheconvergencecriterionissatised.

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teststatisticsateachxedis2log=2logmaxH0L(jy) maxL(jy)=nXi=1kyiB^0k2n1Xi=1kyiB^1k2+2n2Xi=n1+1logf(1)exp(kyiB^1k2

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When2logisbigenough,H0canberejected.ByJensen'sInequality,wecanshowthatlogf1+exp(x)gx:(AsimpleproofcanbefoundinSection??).Letf(x)=logf1+exp(x)gx;thenasimplecalculationshowsthatf(x)increaseswhenx<0anddecreaseswhenx>0withmaximumvalue0atx=0:Sowhenxisintheneighborhoodof0;itisreasonabletoapproximatelogf1+exp(x)gbyx:UnderH0:1=2;weexpect^1^2sothatkyiB^1k2kyiB^2k2isaround0:Therefore,wecanapproximatethelikelihoodratioteststatistics2logwithitslowerbound2logn2Xi=1n(^1^0)0B01B(^1^0)+2(^1^0)0B01(yiB^1)o+nXi=n2+1n(^2^0)0B01B(^2^0)+2(^2^0)0B01(yiB^2)o+2n2Xi=n1+1kyiB^1k2

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UnderH0;GturnstobeaquadraticformY0AY;whereYisanT1vectorwithy1;:::;ynstackingtogetheranddetailsaboutAcanbefoundinSection??.Furthermore,thematrixAhasrank(B)=lnonzeroequaleigenvalues.UnderH0;YisdistributedasMVN(1nB;In)MVN(!;e);itisstraightforwardtoshowthatG2l(detailsinSection??).Thusthedistributionof2logunderH0isapproximatedbythismultipleofacentralChi-squaredistribution.TechnicalDetails:1.Proofoflogf1+exp(x)gx:Proof:LetadiscreterandomvariableYwithP(Y=x)=1P(Y=0)=;thenE(eY)=1+exp(x):Becauselog(:)isconcave,directlyfollowingJensen'sInequalitywecangetlogfE(eY)gEflog(eY)g=E(Y),thatis,logf1+exp(x)gx:2.DistributionofGunderH0RewriteGasG=n2(^1^0)0B01B(^1^0)+(nn2)(^2^0)0B01B(^2^0)+(n2n1)(^2^1)0B01B(^2^1)+(n3n2)(^1^2)0B01B(^1^2)+2n2Xi=1(^1^0)0B01(yiB^1)+2nXi=n2+1(^2^0)0B01(yiB^2)+2n2Xi=n1+1(^2^1)0B01(yiB^2)+2n3Xi=n2+1(^1^2)0B01(yiB^1):

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UnderH0;theMLEforunderlyingcoecientvector^0=(B01B)1B01Pni=1yi

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NowwecanrewriteGasG=n2Y0(W1W0)01B(B01B)1B01(W1W0)Y+(nn2)Y0(W2W0)01B(B01B)1B01(W2W0)Y+(n2n1)Y0(W2W1)01B(B01B)1B01(W2W1)Y+(n3n2)Y0(W1W2)01B(B01B)1B01(W1W2)Y+2Y0(W1W0)01B(B01B)1B01fV1n2W1gY+2Y0(W2W0)01B(B01B)1B01fV2(nn2)W2gY+2Y0(W2W1)01B(B01B)1B01fV3(n2n1)W2gY+2Y0(W1W2)01B(B01B)1B01fV4(n3n2)W1gY;WhereV1=1n20nn2ITv1ITV2=0n21nn2ITv2ITV3=0n11n2n10nn2ITv3ITV4=0n21n3n20nn3ITv4ITObviously,GcanbeexpressedasquadraticformY0AY;whereunderH0;YisdistributedasMVN(1TB;In)MVN(!;e)andA=(A1+A01)=2withA1=n2(W1W0)01B(B01B)1B01(W1W0)+(nn2)(W2W0)01B(B01B)1B01(W2W0)+(n2n1)(W2W1)01B(B01B)1B01(W2W1)+(n3n2)(W1W2)01B(B01B)1B01(W1W2)+2(W1W0)01B(B01B)1B01fV1n2W1g

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+2(W2W0)01B(B01B)1B01fV2(nn2)W2g+2(W2W1)01B(B01B)1B01fV3(n2n1)W2g+2(W1W2)01B(B01B)1B01fV4(n3n2)W1g=[n2(w1w0)0(w1w0)+(nn2)(w2w0)0(w2w0)+(n2n1)(w2w1)0(w2w1)+(n3n2)(w1w2)0(w1w2)+2(w1w0)0(v1n2w1)+2(w2w0)0fv2(nn2)w2g+2(w2w1)0fv3(n2n1)w2g+2(w1w2)0fv4(n3n2)w1g]1B(B01B)1B01A21B(B01B)1B01.SinceYMVN(!;e);thequadraticformY0AYcanbeexpressedasY0AY=P(k)i=1iWi,whereWi02mi(i):i;i=1;:::;karethosenonzerodistincteigenvaluesofAe(ore1 2Ae1 2),wheremultiplicityforeacheigenvalueiismi:NoticethatAe=1 2(A2+A02)1B(B01B)1B01=1 2(A2+A02)1B(B01B)1B0:Because1B(B01B)1B0isidempotentwithranklandrank(1 2(A2+A02))=rank(A2)=1duetothestructureofw0andw1;Aehaslnonzeroequaleigenvalues.Thusk=1:Also,isthenoncentralityparameter=1 2!0e1 2PP0e1 2!;wherePconsistsofleigenvectorscorrespondingtosatisfyingPP0=e1 2Ae1 2:Notethat=1 2!0e1 2(PP0)e1 2!=1 2!0A!=0ThusG2lfollows.Furthermore,>0becauseforallnonzerovectorsz;thequadraticformz0e1 2Ae1 2zcannallybewrittenassummationofnonnegativevaluez0izisinceispositivedenite.

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2:Proof:Thep

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Obviously,foranywith0<<1 2;P(nj^kkj=n1 2j^kkj 2>M 2)<:Obviously,whenn!1;M 2!0:So^=op(n);with0<<1 2.4Lemma2:UnderH0;eachterminthedierencebetweenthetrueLRTstatisticsanditsapproximatecontributedbysubjecti;logf(1)+exp(kyiB^1k2

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wherejx2j
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simplesimulationshowstheanswerisheartbreaking\NO",whichisimpliedinthefollowingtwogures:Figure 3{1 andFigure 3{2 .IntheupperpartofFigure 3{1 showsthewigglypatternofthesummationofallSiwhenthesamplesizeincreases,thatis,PiSidoesNOTgotozeroeventhoughitisnotbig.ThewigglypatternagainintheupperpartofFigure 3{2 resultingfrombiggersamplesizesthanthoseinFigure 3{1 conrmsthis.ThelowerpartsinbothFigure 3{1 andFigure 3{2 suggestSidoesconvergetozerowhensamplesizegoestoinnity.SohowaboutabetterapproximationinthesenseofincludinghigherordertermswhenweuseTaylorapproximationstoactualLRTstatistics2logk?Thatis,howaboutapproximatingf1(x)=2log(1+(exp(x)1))byf2(x)=2(x+x2=2)2(x+x2=2)2+23(x+x2=2)3=3:Weusethesimplestsettingthatalldataarefromstandardnormaldistributionswheretherecombinationratebetweentwomarkersis.2and=:5:.Figure 3{3 plotsthedierencebetweenf1(x)andf2(x):Wecanseethatwhenxisintheneighborof0;thedierencebetweentwofunctionsisalsoaround0:Figure 3{4 andFigure 3{5 showasimplesimulationresultaboutthesummationofthedierences,whichisthegapbetweenactualLRTanditsfollowingapproximation,G.Nowthegapissmallenoughwithexpression:2logG=n3Xi=n1+1f2log[(1)+exp(xi)]2(xi+x2i=2)+2(xi+x2i=2)223(xi+x2i=2)3=3gwherexi=kyiB^1k2 3{6 showsthebehavioroftheotherapproximation,whichalsomakesthegapgoesto0visuallywhenn!1:Butitishardtondthejointdistribution

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Figure3{1: TheplotofPn3i=n1+1Siandmaxn1+1
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Figure3{2: TheplotofPn3i=n1+1Siandmaxn1+1
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Figure3{3: Theplotoffunctionf(x)=f1(x)f2(x):

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Figure3{4: TheplotofPn3i=n1+1Siandmaxn1+1
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Figure3{6: TheplotofPn3i=n1+1Siandmaxn1+1
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^1=(B01B)1B01W1(Y)Yand^2=(B01B)1B01W2(Y)Y;whereW1(Y)=1n1110nn3IT 1. Generate1000dierentdatasetsfromH0;thatis,havingsamevariance-covariancestructureandbasedonsameknownmarkerinformation.WecaneasilygenerateyiB^0;yiB^1;yiB^2,i=1;:::;n;foreachxed: Foreachintervalcalculatethelikelihoodratioteststatisticsatthesexedbasedonsimulateddata;Obtain1000dierentmaximumlikelihoodratioteststatisticsthroughthewholegenome(linkagegroup).

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Figure3{7: Twosimulateddistributionsoflikelihoodratioteststatisticbasedonsame1000datasets. 3. GetthecutopointvaluefromthesimulateddistributionofmaximumofLRTstatistics.Tocheckthevalidityofthissimulationprocedure,wegetthedistributionofmaximumofLRTstatisticscalculatedusingMLEsfromEMalgorithm.Figure 3{7 showsthetwohistogramsresultedfromeachprocedure,whichconrmsthevalidityofthissimulationprocedure.Inthatgure,underH0;theunderlyingphenotypiccurveisaatline.100subjectsaremeasuredat4dierentpoints.Markerintervalis25CMandscanningunitis2CM.Tobesimple,weonlyassumethislinkagegrouphas3markers,thatis,twomarkerintervals.BytreatingasunknownparameterandndingitsMLE,weneedonlycalculateonelikelihoodratioteststatisticforeachmarkerinterval,whichisalready

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themaximumLRTstatisticforthatmarkerinterval.Thelog-likelihoodfunctionfortheunknowncoecientvectors1;2andislogL=n1Xi=1kyiB1k2 (1t)(y;B1;)+t(y;B2;);wecaneasilywriteouttheEMalgorithmfornding^1and^2whenisxed.Nowwhenisalsoanunknownparameter,itslikelihoodequationis@logL @=n2Xi=n1+1(yi;B2;)(yi;B1;) (1)(yi;B1;)+(yi;B2;)+n3Xi=n2+1(yi;B1;)(yi;B2;) (1)(yi;B2;)+(yi;B1;)=n2Xi=n1+11P(yi;) 1+n3Xi=n2+1P(yi;1) 1:Thus,itisstraightforwardtohavethisfollowingEMalgorithmtonumericallyndtheMLEof1;2andsimultaneously:EMAlgorithm2:Supposeisknown.Repeatthefollowingstepsuntiltheconvergencecriterionissatised.

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1. Generate1000dierentdatasetsfromH0: Foreachdatasetandeachinterval,ndtheMLEsofandhencecalculatethemaximumofLRTstatisticamongthisintervalbyeasilygeneratingyiB^0;yiB^1;yiB^2,i=1;:::;n: Foreachdataset,movetoeachoftheleftintervalsandrepeatstep2.Obtain1000dierentmaximumLRTstatisticsthroughthewholegenome(linkagegroup). 4. GetthecutopointvaluefromthesimulateddistributionofmaximumofLRTstatistics.Figure 3{8 showstwoempiricaldistributionsofthegenome-wisemaximumLRTstatistic.UnderH0;theunderlyingphenotypiccurveisaatline.100subjectsaremeasuredat4dierentpoints.Markerintervalis25CM.Tobesimple,weonlyassumethislinkagegrouphas3markers,thatis,twomarkerintervals.TheupperhistogramisfromtheEMalgorithmwithxedandthelowerhistogramisfromthesimulationprocedurewith^fromthesecondEMalgorithm.The95%and99%cutopointsfromtheupperprocedureare7.986

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Figure3{8: Twosimulateddistributionsbasedonsame1000datasetswhentreat-ingasunknownparameter.

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Figure3{9: Twosimulateddistributionsbasedondierent1000datasetswhiletreatingasunknownparameter. and11.818,respectively.Andthecorrespondingvaluesfromthelowerprocedureare7.944and11.802.Thesesimilarnumberswiththesimilardistributionshapessuggestthecorrectnessofabovesimulationprocedure.Remark: Whenwegenerate1000datasetsfromH0;theunderlyingphenotypiccurveunderH0canberandomlypickedincluding0=0:Figure 3{9 showstwoempiricaldistributionsfromgeneratingdatasetsfromtwodierentphenotypiccurves. 2. WhenH0istrue,theEMalgorithmforndingMLEofconvergesslow.ButwhenH0isnottrue,theconvergenceisveryfast.SoIthinkforthesimulationprocedurewecanjustusexedwhichiscomputationallylesstime-consuming.

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Theonlydierencebetweenthistwolog-likelihoodfunctionisthetermlogfjX01Xjg:Also,itisstraightforwardtoseetheformofthedesignmatrixXmustbecorrectlyspecied,otherwisewemaynotevengetconsistentestimatorsfor:SoDiggle,LiangandZeger(1994)recommendedtouseasaturatingmodelforthemeanresponseprolestogetaguaranteedconsistentestimateofvariance-covariancematrix.Forexperimentwhereobservedvaluesfromdierentsubjectsaremeasuredatsamesetofobservationpoints,arobustestimateforVcanbeobtainedusingtheREMLprinciple.Supposeforeachofmiexperimentunitsinithofgexperimentaltreatmentunits,measurementsaremadeateachofTobservation-pointstk:Thecompletesetofmeasurementsareyijk;i=1,...,g;j=1,...,mi;k=1,...,T:ThesaturatedmodelformeanresponseisE(yijk)=ik:Thus,wecouldusetheordinaryleast-squaresttogetaconsistentestimateofV:^V=gXi=1mig!1gXi=1miXj=1(yij^i)(yij^i)0;with^i=(m1iPmij=1yij1;::::;m1iPmij=1yijT)0:ThisapproachcanbeextendedtothesituationwhenmeasurementstimesarenotcommontoallunitsbyestimatingeachViintheblockdiagonalvariance-covariancematrix:Thisapproachfailsfortheextremecasewheresetsofmeasurementstimesareessentiallyuniqueforeachunit.However,explicitmodellingofthecovariancestructurehasbeenpopularlyconsideredforthesefollowingreasons.Firstly,whenthetruecovariancematrixhasmanyfewerparametersthantheunconstrainedvariancematrix,theestimatecanbemademoreaccurately.Secondly,theobjectiontoestimating1 2T(T+1)param-etersinthecovariancematrixgainsforcewhenT;thenumberofobservationsper

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experimentalunitislarge.Thirdly,thisrobustapproachusesreplicationsacrossexperimentunitstoestimatethecovariancematrixnon-parametrically.Whentherearealotofmissingvaluesorsetsofmeasurementstimesareessentiallyuniqueforallunits,aparametricmodellingofcovariancestructurecanavoidtheseproblems.Manystatisticalsoftwaresprovidesgenerallinearmodelttingwithavarietyofparameterizedcovariancestructures.Littellet.al(2000)providedatutorialofusingtheMIXEDprocedureinSAS,whichcanmodelarichselectionofcovariancestructuresthroughtheRANDOMandREPEATEDstatements.OnecanchoosethebestttingoneusingAkaike'sInformationCriterion(AIC)orBayesianInformationCriterion(BIC).TheformulaeforthesetwocriteriaareAIC=L()qBIC=L()(q=2)log(N)whereL()isthemaximizedlog-likelihoodorrestrictedlog-likelihood,qisthenumberofparametersinthecovariancematrix,pisthenumberofxedparametersandNisthenumberofsubjects:NforMLandNpforREML.ModelswithlargerAICorBICyieldabettert.ThedierencebetweenthesetwocriterionisthatBICputsmorepenaltyonthenumberofparametersusedtotcovariancematrix.Whenthecovariancestationarityassumptionislikelytofail,onecanusesometransformationtostabilizethedependencestructure.ZimmermanandNu~nez-Anton(2001)proposedastructuredantedependence(SAD)modeltoestimatesuchanonstationarycovariancematrixdirectlyandshowedsuchamodeldisplaysmanyfavorableproperties.Thevariancesoneachobservationpointarenotassumedtobeconstantinthismodel,andthecorrelationbetweenmeasurementsequidistantintimearenotassumedequal.

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Eitherasaguidetotheformulationofaparametricmodelornotdependingonparametricassumption,non-parametricestimationofcovariancestructurehasalsobeenproposedbyalotofauthors.Forexample,DiggleandVerbyla(1998)proposedtousekernelweightedlocallinearregressionsmoothingofsamplevariogramordinatesandofsquaredresidualstogetanonparametricestimatorforthecovariancestructure.Forournonparametricfunctionalmappingprocedure,weneedndaconsis-tentestimatorfortheunderlyingcovariancestructurethatbehaveswellwhenthesamplesizeissmall.Itisknownthatthesamplecovariancematrixisguaranteedtoaconsistentestimator,howeveritcanbeveryunstablewithsmallestestimatedeigenvaluesbeingtoosmallwhilelargestestimatedeigenvaluesbeingtoolarge,especiallyforsmallsamplesizes.Obviouslystandardestimators,likeREMLorMLcangainsomestabilityinestimatingthematrixinsmallsamplesbyassumingsomeparametriccovariancestructurethatinvolvesestimationoffewerparameters.Butsuchestimatorsareonlyconsistentwhenthehypothesizedstructuresarecorrect.Noasymptoticalpropertiesofnonparametricestimatorsarestudiedbyanyauthors.DanielsandKass(1999)providedabetterwaytoestimatethecovariancematrixbyspecifyinganappropriatepriorforthecovariancematrixandchoosinganestimatorbasedonaparticularlossfunction.Theirmethodrstgenericallystabilizesanunstructuredestimateandthenshrinkssuchestimatestowardaparsimonious,structuredformofthematrix.Howmuchshrinkagerequiredisdecidedbythedata.DanielsandKass(2001)extendedthismethodtomoreeasilycalculatedestimatorswithoutusingafullyMCMCapproach.Theycalledsuchestimatorsempirical(orapproximate)Bayesestimators.Thenalestimatorincludesacombinationofshrinkingtheeigenvaluesandthenshrinkingtowardstructure.Thedatadecidestheamountofshrinkage.Theseestimatorsinduce

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stabilityovertheunstructuredestimatorofthecovariancematrixwhileprovidingrobustnesstomisspecicationofthestructure.Theseestimatorsareconsistent.Fromtheirsimulationstudytoevaluatetheriskinestimatingacovariancematrix,DanielsandKassrecommendedrstshrinkingtheeigenvaluesoftheunstructuredestimatorbyreplacingitwiththeSteinestimatorandthenshrinkingtheSteinestimatortowardastructureusingastructuredlogeigenvalueorcorrelationshrinkageestimator.Thisisbecausewhenthestructureisfarfromcorrect,thecorrelationshrinkageestimatorisworsethanthestructuredlogeigenvalueestimator.Sowewilladoptsuchanestimatingprocedureintoourfunctionalmappingapproach: ^j^i

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3.6.1AChromosome-wiseAnalysisonLinkageGroup10WeapplythismethodtothesamePoplarDatasetasbefore,where61plantsallhaveinformationfrom8markersoflinkagegroup10and11observationsfromthesamesetofmeasurementtime.TocomparewithpreviousanalysisusingnonparametricfunctionalintervalmappingwithlinearapproximationofLRTstatistics,werstlog-transformedtheoriginalobservations.Weuseorder3B-splinesbecauseweneednottohavesmoothsecondderivativefunctionsfortheunderlyingphenotypiccurves.Andthemeasurementtimeisateachendofcontinuous11years.Soweuseequidistantinnerknotsat(1;3;5;7;9;11)totthegrowthcurve.AsweknowinSection 1.4 ,weneedtwomoreknotsateachendoftheinnerknotsvector.Thus,weactuallyuse7splinestotthegrowthcurve.Proceduretondtheconsistentestimateof:Wewillrstgetaconsistentvariance-covariancematrixfromasaturatingmodel,thenusingthises-timatetoproceedfollowingfunctionalintervalmappingapproach.Thisestimatingprocedureof^isadoptedfromDanielsandKass(2001).

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Step1:Findtheunstructuredcovariancematrixestimator^unfromasaturat-ingmodel,yi=k+"i;i=1;;nandvar("i)=:k=1;;4accordingto4possiblecombinationsfromthersttwomarkers.^unhas11distincteigenvalues:0:328;0:086;0:057;0:011;0:005;0:001;0:0009;0:0006;0:0002;0:0001;6:6e05:Step2:Shrink^untoitsStein'sestimator^st,whoseeigenvaluesare0:276;0:072;0:055;0:010;0:005;0:0011;0:0010;0:0007;0:0002;0:00017;8:3e05:Step3:Fittingthesaturatingmodelassumingsomestructure.Weusedthepreviousestimatorhavingstructureautoregressive+simplechosenbySASproce-durePROCMIXED,thatis,^s=0:001236J11+Autoregressive(2=0:06155;=0:8945):Thisestimatorhaseigenvalues:0:485;0:107;0:037;0:018;0:011;0:008;0:006;0:005;0:004;0:0037;0:0035:Step4:Determinetheamountofshrinkage2=6:533using^stand^s:Step5:Shrink^stusing2togetnalestimate^sh;whoseeigenvaluesnoware0:272;0:071;0:054;0:010;0:0054;0:0011;0:0010;0:0008;0:00024;0:00018;8:6e05:Fromtheeigenvaluesofeachvariance-covarianceestimatorwecanseethenalestimatorisclosertotheunstructuredvariance-covarianceestimator.UsingthisempiricalBayesvariance-covarianceestimator,wendthemaximumofLRTappearsontherstintervalwith^=:123(about3cMfromtherstmarkerCA/CCC-640R).Themaximumvalueof2logis15:487withpvalue=:143:Thecutopointfor-levelat:1is16:84:(18.99for=:05and23.10for=:01:)Therefore,wecouldnotmakeaconclusionthataQTLgoverningstemgrowthexistsinlinkagegroup10ifweusedEBestimator.IfweusetheREMLestimateoftheunderlyingvariance-covariancewiththesamestructureinSection 3.10.2 ,andusethesimulationproceduretondthecutopointsinsteadofusingtheapproximatejointdistributionofallLRTstatistics,thep-valueis0withmaximumLRTvalue62:79at^=:32(about8.5cM

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Figure3{10: Fittedcurvesfromusingtwodierentvariance-covarianceestimator. fromtherstmarkerCA/CCC-640R).The:01-levelcutopointfromsimulationis23:72:(17.26for=:1and19:26for=:05:)Thisndingareconsistentwithpreviousanalysis(pvalue=5e4withstandarddeviation5e4).InFigure 3{10 theblacklinesarefrom^shwhilethedashedlinesarefrom^s:Figure 3{10 showsthatthettedcurvedoesnotchangemuchduetodierent^butthesignicanceofthedierencebetweentwocurvescausedbytheputativeQTLdependsonhowtoestimate^:

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Table3{1: AnalysisofPoplardatausingnonparametricfunctionalintervalmap-ping(NPFIM)andparametricfunctionalintervalmapping(PFIM).\EB",theempiricalBayesestimator.\REML",theREMLestimatorcorrespondingtostruc-ture2J+Autoregressive(2;): stemdiameterstostabilizetheage-dependentvarianceheteroscedasticity(Wuet.al2004).Figure 3{11 showsthelikelihoodmapsresultingfromournonparametricfunctionalmappingapproach.Theempiricalestimateofthecriticalvalueisobtainedfrom1000simulationtestsandwendthethresholdvaluefordeclaringthegenome-wiseexistenceofaQTLis32.01atthesignicancelevelP=0.01.TheQTLcandidatepositionsarethepositionscorrespondingtothepeaksofcurveshigherthanthecriticalvalue.FromFigure2,thereissignicantevidenceshowingthatseveralQTLsexistinlinkagegroup1,2,4,7,10,14&18. 3{1 showstheanalysisresultwhenusingnonparametricfunctionalintervalmapping(NPFIM)andparametricfunctionalintervalmapping(PFIM)withtwodierentvariance-covarianceestimators,respectively.TherstnumberineachcellistheP-valueandthenumberinparenthesisisthemaximumLRTstatisticalongthelinkagegroup10.Sincethetrueunderlyingvariance-covariancestructureforthePoplardatasetisunknownandtheQTLlocationisalsounknown,itisinconclusiveregardingthebehaviorofNPFIMandthebehavioroftheconsistentestimatorusingproceduresintroducedinSection 3.5 .SoweusethefollowingsimulateddatasetstoexplorethecharacteristicsofourNPFIMprocedure.Weusethemarkerinformationoflinkagegroup10inthePoplarexperimentandthetrueQTLexistsintherstmarkerinterval,8cMawayfromtherstmarker.Growthprolesof61subjectsarefrom

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Figure3{11: Theproleofthelikelihoodratioteststatisticsbetweenthefullandreduced(noQTL)subject-specicmodelforthediametergrowthtrajectoriesacrossthewholePopulusdeltoidesparentmap.ThegenomicpositionscorrespondingtothepeakofthecurvearetheMLEsoftheQTLlocalization.

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Table3{2: P-valuesanditsstandarddeviationofnonparametricfunctionalintervalmapping(NPFIM)usingdierentvariance-covarianceestimatorsunderdierenttrues.\EB"meanstheshrinkageestimatorwhichisguaranteedtobeaconsis-tentestimator.\REMLw"estimatoristheREMLestimatorcorrespondingtoawrongstructure.\REML"estimatoristheoneselectedfromSASProcMIXEDus-ingBICandassumingeachsubjecthasdierentunderlyinggrowthcurve.\True"estimatorisofcoursethematrixweactuallyusedtogeneratedata. VarianceEstimateNPFIM 1234 1{3 ).Heritabilitycurvescorrespondingtothesefourcovariancestruc-turesareplottedinFigure 3{12 .Table 3{2 showstheanalysisoutcomeofNPFIMusingdierentvariance-covariancestructures.Fromthistable,wecansee 3.5 merelyshrinkstothetruestructurealittlebit,thatis,itisnotverystablebecauseofmoresimilaritytounstructuredestimators.

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Figure3{12: Heritabilitycurvesforthefoursimulateddatasetsusingdierents.

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Table3{3: P-valuesanditsstandarddeviationofnonparametricfunctionalinter-valmapping(NPFIM)usingdierentvariance-covarianceestimatorsfordierentsamplesizeand=3:\EB"meanstheshrinkageestimatorwhichisguaranteedtobeaconsistentestimator.\REMLw"estimatoristheREMLestimatorcorre-spondingtoawrongstructure.\REML"estimatoristheoneselectedfromSASProcMIXEDusingBICandassumingeachsubjecthasdierentunderlyinggrowthcurve.\True"estimatorisofcoursethematrixweactuallyusedtogeneratedata. VarianceEstimateNPFIM 3{2 .Butwhenthevariationisbig,\REMLw"estimatorworksbetterthan\REML"estimateintermsofsmallerP-value.Thismayduetothesmallsamplesizeandthevariance-covariancestructurewechosetosimulatethedataset.WealsodothesameNPFIManalysisfordatafromunderlyingvariance-covariancematrix3butthesamplesizehasbeenincreasedto200.Table 3{3 liststhechangeofP-valueswhenthesamplesizeincreases.Whenthesamplesizeincreases,becausetheunderlyingphenotypiccurvesareindeeddierent,thesmallerP-valueisexactlywhatweexpected.Withlargersamplesizes,theEBestimateworksne.WecanalsoseethatwhenthesamplesizeisbigtheREMLestimateoftruestructureoutperformstheREMLestimateofthewrongstructureintermsofsmallerP-values.

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3.7 ,werecommendtousetheREMLestimateselectedusingAICorBICfromasaturatingmodelinsteadoftheem-piricalBayes(EB)estimateeventhoughitisaguaranteedconsistentestimate.BecauseempiricalBayesestimateisessentiallyobtainedfromshrinkinganunstruc-turedREMLestimatetowardsomestructure,itisratherconservativeinthesensethatlargesamplesizesarerequiredtomaketheEBestimatebehaveslikeatruecovariancematrix.ThefollowingsimulationstudiesareconductedtofurtherstudytheperformanceoftheEmpiricalBayesestimate.Therstsimulationstudyusesthe61subjects'markerinformationoflinkagegroup10inthePoplardataset.Theunderlyingfunctionsaretwologisticgrowthcurves:20 1+20e:6tand30 1+27e:9t;wheret=1;:::;11:Autoregressivecorrelationisassumedforanytwoobservations.Thecovariancematrixisdeterminedbylettingtheheritabilityonyear4(Thegeneticvarianceisthebiggestoneinthisyear.)equalto(:15;:2;:25;:3;:35;:4;:45;:5;:55;:6);respectively.Theheritabilitycurvesacrossall11yearsareshownintheleftpartofFigure 3{13 .100datasetsforeachheritabilityvaluearegeneratedtoperformthenonparametricfunctionalintervalmapping(NPFIM)procedure.TheaverageP-valuesareshownintherightpartofFigure 3{13 correspondingtoeachheritabilityvalue.ThisguretellsusthatEBperformsbetterwhenheritabilityisbigger.But,inreality,wehavenoinformationabouthowbigtheheritabilityis.Thesecondsimulationstudyalsousesthe61subjects'markerinformationoflinkagegroup10inthePoplardatasetasgenotypicdata.ThetwounderlyingbiologicaltrajectoriesarefromtheHIVdynamicsmechanism,whichhavedoubleexponentialformsexp(12:7t)+exp(7:5:05t)andexp(11:4t)+exp(5:03t).ThetwomeancurvesareillustratedinFigure 3{14 .Assumethereare20

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Figure3{13: Theleftplotshowsthe10heritabilitycurvesofsimulatedgrowthdatasets.TherightplotshowsthetrendofP-valueswhenheritabilityonyear4increases.\EB"and\REML"meanstheEmpiricalBayesestimateandREMLes-timateofcovariancematrix,respectively.\True"representsthecovariancematrixusedtogeneratedata.

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observationpoint.Thecovariancematrixisrandomlygeneratedwithoutputtinganyknownstructure.Figure 3{15 plotstheheritabilityvalueoneachtimepoints.100datasetswith200subjectsaregeneratedtodotheNPFIManalysis.NPFIManalysisisalsoconductedforasub-datasetcontaining61subjectsrandomlyselectedfromeachdatasets.ThebeststructurepickedbySASProcMixedisautoregressivemovingaveragestructure,ARMA(1,1).TheresultisinTable 3{4 .ForeasycomparisonTable 3{3 isalsoputinthistable.Fromthistablewecanconcludethat 3.7 ,REMLestimatesworkbetterthanguaranteedconsistentempiricalBayesestimates.So,inpractice,werecom-mendusingthebestREMLestimatesselectedusingtheAICorBICcriterionwithasaturatingmodel.Whenwetriedtoanalyzethesimulateddatausingtraditionalfunctionalintervalmappingmethod,wefounditisverycomputationallyintensivebecauseofthenonlinearnatureofthefunctionalforms.Forexample,forthesamedata

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Figure3{14: TheHIVdynamicscurvesusedasthemeancurves. Table3{4: P-valuesandstandarddeviationofnonparametricfunctionalinter-valmapping(NPFIM)fromHIVdynamicsdataandgrowthdatafordierentcombinationofvariance-covarianceestimatorsandsamplesize.\EB"meanstheshrinkageestimatorwhichisguaranteedtobeaconsistentestimator.\REML"estimatorisobtainedfromSASProcMIXEDassumingeachsubjecthasadierentunderlyingmeancurve.\True"estimatoris,ofcourse,thematrixweactuallyusedtogeneratedata. VarianceHIVdynamicsdataGrowthdata EstimateN=61N=200N=61N=200

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Figure3{15: TheheritabilitycurveacrossallobservationtimepointsintheHIVsimulationstudy.

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settingfromsimulationforTable 3{2 ,the1000simulationteststodecidethecriticalvalueofLRTstatisticstakesover10timesmoretimewhenPFIMisused(WeusedthesimplexalgorithmtondtheMLEoffunctionalparameters)thanthetimeneededwhenNPFIMisused.Evenworse,thesimplexalgorithmsometimesgetsstuckinlocalmaximaandresultsinnegativelikelihoodratioteststatistics.Therefore,thesimilaritytosimplelinearregressionofourpurposednonparametricfunctionalmappingmethodhascomputationaladvantagescomparedtothetraditionalfunctionalmappingmethod.AsinChapter1Section1.4,itispointedoutthatB-splinesareveryattractivefornonparametricregression,buttheselectionoftheoptimalnumberandpositionofknotsdonothaveastandardrule.WhenweanalyzedthePoplardatasetandsimulateddatasets,weusedB-splineswithequidistantknotsandorder3.Ifoneneedsmorecontrolofexibility,onemayusedP-splinesasEilersandMarx(1996)proposed.Functionalmappingisparametricinnature,whichallowsonetoeasilytestmanydierentbiologicalhypothesesbytestingforequalityofparameters.Forexample,onecouldtestifaQTLstartsorceasestoexertaneectongrowthtrajectoriesbytestinghypothesisH0:a1=(1+b1er1t)=a2=(1+b2er2t)atagiventimet:Testingthishypothesisisequivalenttotestingthedierencebetweenthemodelwithnorestrictionandthemodelwiththerestriction:a1=(1+b1er1t)=a2=(1+b2er2t):Ourmethodcouldalsobeusedtotestsuchanhypothesis.Inthisexample,wecanrstndtheMLE'softhecoecientvectorsundertherestrictionm1(t)=m2(t)andcomparewhetherthereisasignicantdierencebetweenthetwocoecientvectors.Figure 3{16 showstheempiricaldistributionsoftheteststatisticsunderH0resultingfromthesimulationprocedureproposedinChapter3,fromwhichthecriticalpointshowingsignicantevidenceofexistingQTLisdetermined.

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Fromthisgurewecaneasilyseethat,withallotherinformationisthesame(samplesize,markerinformation,underlyingfunctionorbiologicalmechanism),theempiricaldistributionsareverysimilartoeachotherwhentheempiricalBayesprocedureisusedtoestimatethefourdierentcovariancematrixes.Thesamendingisobservedforthecasesusingtheexactunderlyingcovariancematrix.Thisinterestingphenomenamightbeduetotheorthogonalitybetweenthecoecientvectorssandthecovariancematrix:IfthereisnoQTL,thatis,=0or1,thereisnomixtureinthelikelihoodfunction:L/expf1 2n1+n2Xi=1(yiB1)01(yiB1)1 2NXi=n1+n2+1(yiB2)01(yiB2)g:Theo-diagonalelementsintheFisherinformationmatrixare(supposeistheunknownparameterincovariancematrix):E@2(2logL) 3{17 wherethetwoempiricaldistribu-tionsfordatasetssimulatedfromdierentbiologicaltrajectoriesbutwithequalsamplesizeareplotted.ThisisbecauseH0onlystatesthatwhenthereisnoQTL,theunderlyingfunctionsarethesamebutdoesnotspecifythefunctionalform.Butfordatasetswithdierentsamplesizes,evenifalltheotherinformationisthesame,thereisabigdierenceamongtheresultingempiricaldistributions.

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Figure3{16: TheempiricaldistributionoftheteststatisticsinthesimulationstudyofChapter3.Dataset1-4correspondtothefourmatrixesinTable 3{2 .\EB"meanstheEmpiricalBayesestimate.\True"representsthecovariancematrixusedtogeneratethedata.

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Figure3{17: TheleftplotindicatestheempiricaldistributionoftheteststatisticsinthesimulatedgrowthdataofChapter3withcovariancematrix1,61subjectswith11observationpoints.TherightoneistheonefromthesimulatedHIVdy-namicsdata,61subjectswith20observationpoints.

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3.10.1JointdistributionofGateachQunderH0NoticethatwearrangeYaccordingtowhichgroupyijisfrom,soatdierentpairsofmarkers,wewillhaveadierentvectorY:Tomakenotationuniform,wecanuseapermutationmatrixP.IfwedenoteY1astheYforrstpairofmarkers,thenforotherintervalsankedbysubsequentmarkers,k=2;:::;m;Yk=PkY1;wherePkisthepermutationmatrix.Ofcourse,foreachrelativetosamemarkerMk,YkisthesameandsoisPk.SupposethematrixAcorrespondingtokthpairofmarkersataxedislabelledasAk:Thus,ateachamongkthmarker'spair,thecorrespondingteststatisticisY0kAkYk=Y01P0kAkPkY1:Noticethat,var(PkY1)=Pk~P0k=~because~isblockdiagonalwithblocksizeTTandPkjustchangestherowsof~.SounderH0atposition;westillhaveY0kAkYk=Y01P0kAkPkY1k2l;withkdenotingthemultiplicitydeterminedbythekthmarkers'pairandhy-potheticQTLindexedby:RecallthatunderH0;m(~t)=m1(~t)=m2(~t);Y1MVN(1nm(~t);In)andwecanwriteY0kAkYk=Y0k~1 2~1 2Ak~1 2~1 2Yk=kY0k~1 2WkW0k~1 2Yk=kY01P0k~1 2WkW0k~1 2PkY1kZ0kZk;

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whereZk=W0k~1 2PkY1dMVN(W0k~1 2Pk~;W0kWk)andkWkW0karethespectraldecompositionsof~1 2Ak~1 2.ThematrixWkiscomposedoftheeigenvectorscorrespondingtothelnon-zeroeigenvalues,andithaslorthonormalcolumnsandisoforderNTl:Also,underH0thestructureofAkresultsin(W0k~1 2Pk~)0W0k~1 2Pk~=~0Ak~=k=0:andthus,underH0;Zkl1=W0k~1 2PkY1MVN(0;Il):Ateachintervalalongtheentiremap,weexaminenputativepositions.TheentirevectorZ=(Z101;:::;Zn01;Z102;:::;Zn0m)0hasdistributionZMVN(Z;)whereZji=Wj0i~1 2~;i=1;:::;m;j=1;:::;nand=0BBBBBBBBBBBBBB@W101~1 2...Wn01~1 2W102~1 2P2...Wn0m~1 2Pm1CCCCCCCCCCCCCCA~~1 2W11~1 2Wn1P02~1 2W12P0m~1 2Wnm;andunderH0,Z=0.IfweletBjk(x);j=1;:::;ndenotethel-dimensionballwithradiusequaltox=jk,thenusingthejointdistributionofGk'stoapproximatethejointdistribu-tionofeverylikelihoodratioteststatisticsatQTLpositionwithinkthpairof

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marker2logk's,wecangetP0(max1km;2logkx)P0(max1km;Y0kAkYkx)=P0(max1km;kZ0kZkx)=P0(Z101Z11x=11;;Zn0mZnmx=nm)=Z:::ZfZji2Bji(x)g1 2exp(1 2XiZj0i1Zji)dZ11dZnm:Thisprobability,whichisoneminusthep-valueforH0;canbedirectlycalculatedbysimulatingZMVN(0;):Again,alltheabovederivationsaremadeundertheassumptionthatweknow;whichistypicallyuntrueinpractice.WesuggestsubstitutingaREMLestimateofthevariance-covariancematrix^insteadof,and,inthatcase,theaboveformulaearecorrectasymptotically.

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Figure3{18: Graphofthepvalueandits95%condenceintervalfordatafromtwounderlyingatlineswithdierentdistancesusingnonparametricfunctionalintervalmapping. Basedonthemarkerinformation,100datasetsaregeneratedandanalyzed.Theunitofscanningthroughoutthelinkagemapis2cM.Sincetheunderlyingcurvearethetwoatlinesy=a0andy=a1;wecanuseja0a1jasameasureofdistance.Thepvaluetrendwithrespecttodierentdistancesbetweentwounderlyingatlines,andits95%condenceinterval,isgiveninFigure 3{18 .ItisclearthatthePvaluedecreasesasthecurvesmoveapart,whichisthebehaviorwewouldexpect.Theseconddatasetassumesonephenotypiccurveisat(m1(t)=1)andtheotherisquadratic(m2(t)=ax2+bx+c).Inthiscase,using25markerswith20cMapartfromnearestneighbors,50subjectsaregeneratedat12observationpoints

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Figure3{19: Graphofpvalueandits95%condenceintervalfordatafromtwounderlyingcurves:oneisatlineandtheotherisinquadraticformusingnonpara-metricfunctionalintervalmapping. usingthevariance-covariancematrixforeachyi=:1J12+Autoregressive(2=:05;rho=:5):Basedonthesamemarkerinformation,100datasetsaregeneratedandanalyzed.Tomeasuredistance,weusetheareabetweenthetwocurvesovertheobservedtimeinterval.Figure 3{19 showsthepvalueprolefordierentdistancesandits95%condenceinterval.Again,thebehavioriswhatwedesire,withthep-valuedecreasingasthecurvesgetfurtherapart.PoplarData

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Figure3{20: Graphofthelikelihoodratioteststatisticsalongthelinkagegroup10usingnonparametricfunctionalintervalmapping.

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AgainweapplythismethodtothesamePoplarDatasetasbefore.Becauseourmodelassumesthatalltobservationshaveequalvariance,wersttransformedtheoriginalobservationsbytakingthelog:Weuseorder3B-splineswithequidis-tantknotstotthegrowthcurveandscanthelinkagegroup102cMby2cM.Usingourmethodwiththevariance-covariancestructureautoregressive+simple(es-timatedbyPROCMIXEDas^=0:001236J11+Autoregressive(2=0:06155;=0:8945)),wefoundstrongevidencethatthereisaQTLbetweentherstandthesecondmarkeronlinkagegroup10whichcontrolsthegrowthtrajectoryofstemdiameterintheinterspecichybridsofpoplar(pvalue=5e4withstandarddeviation5e4).Thebiggestlikelihood-ratioteststatistics2logappearsintheintervalankedbyMarkerCA/CCC-640RandMarkerCG/CCC-825.Figure 3{20 showslikelihoodratioteststatisticsalongthelinkagegroup10.AgainthisndingagreeswiththeconclusioninMa,CasellaandWu(2002).ButtheresultingQTLlocus,i.e,wherethelargestlikelihoodratioteststatisticis,isslightlydierentfromourmethodandtheirmethod.Thismayduetoalltheapproximationswemade.

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allmarkersinformationhasbeenrecorded.Acompletedataanalysisispopularbe-causeitmapsaraggeddatamatrixintoarectangleone.Butaswestatedbefore,thereisnearlyalwaysasubstantiallossofinformationsothatalossofdetectingpowerfollows.Analternativewaytoobtainadatasetonwhichacompletedataanalysiscanbecarriedoutisllinginthemissingvalues,insteadofdeletingsubjectswithincompletesequences.Usingobservedvaluestoimputevaluesforthemissingvaluesistheprincipleofimputation.Formarkerdata,sinceweknowthegeneticdistancesbetweenthemarkers,wecanimputethemissingmarkerinformationbasedonitsnearestneighborandcorrespondinggeneticdistance.Thentheresultingdatasetisanalyzedasifitrepresentedthetruecompletedata.Athirdmethodisbasedontheprincipleofanalyzingtheincompletedataassuch.Asimplerepresentativeisavailabledataanalysis.Forthefunctionalintervalmapping,weconstructedalikelihoodratiotestintervalbyinterval.Soforeachinterval,wecanuseallsubjectswhohavebothvaluesoftwoendmarkerstoderivesuchLRtest.ThismaycausethenumberofsubjectsineachLRtesttobedierent.Obviously,thismethodismoreecientthanthecompletecasemethodformoreinformationisused. 3.1 .First,weconsiderthecasewhereatleastoneofthetwoneighborsofthemissingmarkerisrecordedinthedataset,thatis,foranymarkerinterval,say,bracketedbyMandN;allthesubjectsinthedatasetshaveinformationofeitherMorNorboth.ThosesubjectswithinformationofbothMandNmissingaredeletedwhenwedothelikelihoodratiotestinthisinterval.

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Withoutlossofgenerality,supposeonesubjecthasinformationonMbutinformationaboutNissomehowmissed.ThelikelihoodfunctionofonesubjectwithmeasurementvectorywhenMandNarebothobservedisf(yjM=M;N=N)=(Qq;)f(yjM=M;N=n)=(1)(Qq;)+(qq;)f(yjM=m;N=N)=(Qq;)+(1)(qq;)f(yjM=m;N=n)=(qq;);where(;)istheprobabilitydensityfunctionofMVN(;)andisdenedasinTable 1{3 .Asbefore,weassumeQq=m1(~t)=B1andqq=m2(~t)=B2:LetrdenotetherecombinationratebetweenMandN;thenwehaveP(N=NjM=M)=1rP(N=njM=M)=rP(N=NjM=m)=rP(N=njM=m)=1r:Thus,wecanwritethelikelihoodfunctionofyconditionalonmarkerMwhileitsneighbormarkerNismissingasthefollowing:f(yjM=M)=f(yjM=M;N=N)P(N=NjM=M)+f(yjM=M;N=n)P(N=njM=M)=(1r)(Qq;)+r[(1)(Qq;)+(qq;)]=(1r)(Qq;)+r(qq;);

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andf(yjM=m)=f(yjM=m;N=N)P(N=NjM=m)+f(yjM=m;N=n)P(N=njM=m)=(1r)(qq;)+r[(1)(qq;)+(Qq;)]=(1r)(qq;)+r(Qq;):Similarly,forsubjectswhohaveNinformationbutnoMinformation,thedensityfunctioncanbeobtainedwithfollowingexpressions:

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+n5Xi=n4+1log[(1r)(yi;B1;)+r(yi;B2;)]+n6Xi=n5+1log[(1r)(yi;B2;)+r(yi;B1;)]+n7Xi=n6+1logf[1r(1)](yi;B1;)+r(1)(yi;B2;)g+nXi=n7+1logf[1r(1)](yi;B2;)+r(1)(yi;B1;)g+constant: 4.2 wehavegottheloglikelihoodfunction,sowecangetthelikelihoodequationfor1as@logL @1=n1Xi=1B01(yiB1)+n2Xi=n1+1(1)(yi;B1;)B01(yiB1) (1)(yi;B1;)+(yi;B2;)+n3Xi=n2+1(yi;B1;)B01(yiB1) (1)(yi;B2;)+(yi;B1;)+n5Xi=n4+1(1r)(yi;B1;)B01(yiB1) (1r)(yi;B1;)+r(yi;B2;)+n6Xi=n5+1r(yi;B1;)B01(yiB1) [1r(1)](yi;B1;)+r(1)(yi;B2;)+nXi=n7+1r(1)(yi;B1;)B01(yiB1) (1t)(y;B1;)+t(y;B2;)

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Thenthelikelihoodequationfor1canberewrittenas@logL @1=n1Xi=1B01(yiB1)+n2Xi=n1+1P(yi;)B01(yiB1)+n3Xi=n2+1P(yi;1)B01(yiB1)+n5Xi=n4+1P(yi;r)B01(yiB1)+nXi=n5+1P(yi;1r)B01(yiB1)+n7Xi=n6+1P(yi;r(1))B01(yiB1)+nXi=n7+1P(yi;1r(1))B01(yiB1)Similarly,thelikelihoodequationfor2canberewrittenas@logL @2=nXi=n3+1B01(yiB2)+n2Xi=n1+1[1P(yi;)]B01(yiB2)+n3Xi=n2+1[1P(yi;1)]B01(yiB2)+n5Xi=n4+1[1P(yi;r)]B01(yiB2)+nXi=n5+1[1P(yi;1r)]B01(yiB2)+n7Xi=n6+1[1P(yi;r(1))]B01(yiB2)+nXi=n7+1[1P(yi;1r(1))]B01(yiB2)Thus,wehavethefollowingEMalgorithmtonumericallyndtheMLEsof1and2:EMAlgorithm:Forxedandknown;repeatthefollowingstepsuntiltheconvergencecriterionissatised.

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maxL(ky)=nXi=1kyiB^0k2n1Xi=1kyiB^1k2+2n2Xi=n1+1logf(1)exp(kyiB^1k2

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+r(1)exp(kyiB^2k2 4.2 ),wecanhave^1=(B01B)1B01W1Yand^2=(B01B)1B01W2Y;where(letnij=njni)W1=1n1(1)n12n230n34(1r)n45(r)n56[1r+r)]n67[rr)]n78IT

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Ofcourse,theMLEof0underH0canalsobeexpressedas(B01B)1B01W0YwithW0=1n1 1. Generate1000dierentdatasetsfromH0;thatis,havingsamevariance-covariancestructureandbasedonsameknownmarkerinformation.WecaneasilygenerateyiB^0;yiB^1;yiB^2,i=1;:::;n;foreachxed: Foreachintervalcalculatethelikelihoodratioteststatisticsatthesexedbasedonsimulateddata;Obtain1000dierentmaximumlikelihoodratioteststatisticsthroughthewholegenome(linkagegroup). 3. GetthecutopointvaluefromthesimulateddistributionofmaximumofLRTstatistics.Ifwetreatasanunknownparameter,wecanestimateitfromthesolutionof@logL @=n2Xi=n1+1(yi;B2;)(yi;B1;) (1)(yi;B1;)+(yi;B2;)+n3Xi=n2+1(yi;B1;)(yi;B2;) (1)(yi;B2;)+(yi;B1;)+n5Xi=n4+1r[(yi;B2;)(yi;B1;)] (1r)(yi;B1;)+r(yi;B2;)+n6Xi=n5+1r[(yi;B1;)(yi;B2;)] (1r)(yi;B2;)+r(yi;B1;)+n7Xi=n6+1r[(yi;B1;)(yi;B2;)] [1r(1)](yi;B1;)+r(1)(yi;B2;)+nXi=n7+1r[(yi;B2;)(yi;B1;)] [1r(1)](yi;B2;)+r(1)(yi;B1;)

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=n2Xi=n1+11P(yi;) 1+n3Xi=n2+1P(yi;1) 1+n5Xi=n4+11P(yi;r) 1r+n6Xi=n5+1P(yi;1r) 1r+n7Xi=n6+1rP(yi;r(1)) 1r(1)1P(yi;r(1)) 1+nXi=n7+1r[1P(yi;1r(1))] 1r(1)P(yi;1r(1)) 1=Pn2i=n1+1[1P(yi;)]+Pn3i=n2+1P(yi;1) 1Pn7i=n6+1[1P(yi;r(1))]+Pni=n7+1P(yi;1r(1)) 1Pn5i=n4+1P(yi;r)+Pn6i=n5+1[1P(yi;1r)] 1rPn7i=n6+1rP(yi;r(1))+Pni=n7+1r[1P(yi;1r(1))] 1r(1)A 4.3 :EMAlgorithm2:Supposeisknown.Repeatthefollowingstepsuntiltheconvergencecriterionissatised.

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Table4{1: P-valuesanditsstandarddeviationofnonparametricfunctionalin-tervalmapping(NPFIM)usingcase-wisedeleteddatasetandfulldataset,=0:3J11+Autoregressive(2=0:1;=0:8) VarianceEstimatecase-wisedeleteddatafulldata 3.7 wecreateddatasets(n=61)withvariance-covariancestructuresetas3=0:3J11+Autoregressive(2=0:1;=0:8):Hereweadd39subjectswhosemarkerinformationisincompleteintothosedatasets,thatisnowthedatasethasn=100subjects.WeuseaREMLestimateoftheunderlyingdependencematrixfromasaturatingmodelasrecommendedintheChapter3.WealsocalculatetheP-valueusingthetrueunderlyingvariance-covariancematrix3:Table 4{1 showstheresultfromonlysubjectswhosemarkerinformationiscompleteandthefulldatasetincludingsubjectsmissingmarkers.ThesmallerP-valuesresultedfromanalysisoffulldatasetobviouslysuggeststhepower-boostingadvantageofusingmoreinformation.Again,theperformanceoftheREMLestimateofisjustslightlyworsethanusingthetrueunderlying:Tostudytheinuenceofusingmoresubjects,especiallythosewithpartialgenotypicinformation,Figure 4{1 showsthehistogramsoftheLRTstatisticsunderH0fromdierentcombinationofmethodshandlingmissingdata:case-wisedeleteddataandfulldata,andmethodstoestimate:REMLestimateandtrue:Table 4{2 liststhecorrespondingcriticalvaluesfromthesefourdierentanalysisprocedures.FromthesewecanseethatthenulldistributionofLRTstatisticcalculatedfromcase-wisedeleteddatahaslongertailthantheonecalculatedusingthefullinformation,thatis,atthesamesignicancelevel,thethresholdvalueishigherifweusecase-wisedeleteddata.

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Figure4{1: HistogramsofLRTstatisticsfromdierentcombinationofmethodshandlingmissingdataandmethodstoestimateunderH0: Comparisonofcriticalvaluesfromdierentanalyzingmethods case-wisedeleteddatafulldatalevelREMLTrueREMLTrue .1012.6745414.2103111.2277212.63014.0514.3618716.0244413.2217314.81772.0117.7777119.7881917.9221219.67173

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Figure4{2: ThetrendofP-valuesandtheir95%condenceintervals(thedottedlines)fromusingcase-wisedeleteddata(thedashedline)andusingfulldataset(theblackline).Underlyingphenotypiclinesaretwoatlinesandtruecovariancematrixisusedinanalysis.

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Itisstraightforwardtohavepower-boostingadvantagewhensubjectswithpartialmarkerinformationareincluded,whichisFigure 4{2 demonstrates.Inthisgure,100datasetsusingsamemarkerleastheoneusedinTable 4{1 aresimulatedfromtwounderlyingatphenotypiccurves.Variance-covariancematrixis3=0:3J11+Autoregressive(2=0:1;=0:8)andisdirectlyusedinNPFIManalysis.Whenthedistancebetweenthesetwoatlinesincreases,wegetdecreasingP-valueseitherusingcase-wisedeleteddataonlyorusingfulldata.Butwhenweusefulldatatodotheanalysis,theresultedP-valuesareconsistentlysmallerthanthoseresultedfromusingcase-wisedeleteddataonly.

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where1;iand2;iarethedensityfunctionforyiMVN(Qq;)andyiMVN(qq;);I(x)isindicatorfunctionwithvalue1ifxistrue,and~Mdenoteallthemarkerinformationalongthegenome/linkagegroupforsubjecti.Ifforeverysubjectallthemarkersinformationisavailable,thelikelihoodfunctionforunknownQTLindexedbythekthinterval,k=1;:::;m;and;0<<1;canbeeasilyderivedasl(k;jy;~Mcomp)=mXl=1f(yj~Mcomp;k;)I(k=l)=mXl=1(nYi=14Xj=1fijI(k)j)I(k=l);where~Mcompdenoteseachsubject'sgenotypesofallmarkers.Foreachxedk;thelikelihoodfunctionforgivenallcompletedataisl(jy;~Mcomp;k)=f(yj~Mcomp;k;)=nYi=14Xj=1fijI(k)j:

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forthexedkthmarkerinterval,thelikelihoodfunctionforisl(jy;~Mobs;k)=f(yj;~Mobs;k)=n1Yi=14Xj=1fjI(k)jI(~Mobs=(Mk;Mk+1))n2Yi=n1+1f[(1r)1;i+r2;i]I(Mk=1)+[(1r)2;i+r1;i]I(Mk=0)gn3Yi=n2+1f[(1r+r)1;i+r(1)2;i]I(Mk+1=1)+[(1r+r)2;i+r(1)1;i]I(Mk+1=0)gnYi=n3+1(1;i=2+2;i=2)I(~Mobs=;):Thuswecanderivethedistributionofthemissingmarkers,~Mmis;inkthmarkerintervalconditionalonobserveddata,(y;~Mobs);by 4{2 )isjusttwodierentwaystofactorizethejointdistributionofyand~Mmisgiven~Mobsandotherparameters.Andf(~Mmisj~Mobs;k;)actuallydoesnotdependon;whichhasexpressionasfollowing:f(~Mmisj~Mobs;k) (4{3)=1I(~Mobs;k=(Mk;Mk+1))+(1r)I(~Mmis;k=Mk=1j~Mobs;k=Mk+1=1)+rI(~Mmis;k=Mk=0j~Mobs;k=Mk+1=1)+(1r)I(~Mmis;k=Mk=0j~Mobs;k=Mk+1=0)+rI(~Mmis;k=Mk=1j~Mobs;k=Mk+1=0)+(1r)I(~Mmis;k=Mk+1=1j~Mobs;k=Mk=1)+rI(~Mmis;k=Mk+1=0j~Mobs;k=Mk=1) (4{4)

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Table4{3: Distributionofmissingmarkerconditionalontheotherobservedmarkerforkthmarkerintervalforsubjecti. ObservedmarkerMissedmarker (1r)1;i+r2;iMk=0r[(1)2;i+1;i] (1r)2;i+r1;i(1r)2;i Conditionaldistributionofmissingmarkersofkthmarkerintervalforsubjecti. MissingmarkersConditionalprobability (1;i+2;i)MkMk+1=10r[(1)1;i+2;i] (1;i+2;i)MkMk+1=01r[(1)2;i+1;i] (1;i+2;i)MkMk+1=002;i(1r) (1;i+2;i) 4{2 ),itiseasytogettheconditionalprobabilitiesofmissingmarkertypesforeachsubjectiwhosemarkerinformationisincompleteinthekthmarkerintervalconditionalonallobservedphenotypicandgenotypicdata,whicharelistedintheTable 4{3 andTable 4{4

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Table 4{3 andTable 4{4 givetheconditionaldistributionf(~Mmisjk;;y;~Mobs):Oncewehavef(kj;~Mmis;y;~Mobs)andf(jk;~Mmis;y;~Mobs);wecanusethefol-lowingGibbssampleralgorithmtond(^k;^);whichspeciespossiblepositionsoftheQTL.GibbsSampler:Given(k(t);(t);~M(t)mis);generate 1. 2. 3.

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Ofcourseforobservedphenotypicvaluesywehavef(yij~Z;~Mcomp;k;)=1;iI(MkMk+1=11)+Zki1;i1Zki2;iI(MkMk+1=10)+Zki1;i1Zki2;iI(MkMk+1=01)+2;iI(MkMk+1=00);where1and2aretwomultivariatenormaldistributionscorrespondingtotwounderlyingphenotypiccurvesinbackcrossdesign.Therefore,wehavethejointdistributionof(yi;Zi)given(k;;~Mcomp)asthefollowingf(yi;Zijk;;~Mcomp)=1;iI(MkMk+1=11)+Zki1;i1Zki2;i(1)Zki1ZkiI(MkMk+1=10)+Zki1;i1Zki2;i(1)1ZkiZkiI(MkMk+1=01)+2;iI(MkMk+1=00);ThefollowingalgorithmisournewGibbsSamplerprocedure.GibbsSampler2:Given(k(t);(t);~M(t)mis;~Z(t)mis);generate 1. 2. 3. 4.

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Table4{5: Distributionofmissingmarkerconditionalontheotherobservedmarkerforkthmarkerintervalforsubjecti. ObservedmarkerMissedmarker 4{5 andTable 4{6 .2.f(kj;~Z;~Mcomp;y):

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Table4{6: Conditionaldistributionofmissingmarkersofkthmarkerintervalforsubjecti. MissingmarkersConditionalprobability (1r)1;i+r[1;i(1)]Zi(2;i)1Zi+r[2;i(1)]1Zi(1;i)Zi+(1r)2;iMkMk+1=10r[1;i(1)]Zi(2;i)1Zi

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[1;i(1)+2;i])andfori=nk2+1;:::;nk3;Zijk;;~Mcomp;y=Zkij;yBernoulli(1;i

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known.Butnoticethatifweputatprioron1;thenf(1;2j~Z;k;;~Mcomp;y) (4{5)/f(yj1;2;~Z;k;;~Mcomp)/f(yj1;2;~Z;k)/exp0@1 2X(Zki==1)kyB1k21 2X(Zki==0)kyB2k21A:SupposetherearenZksubjectswithZki=1andyisrearrangedsothatfori=1;:::;nZk;yihasvalue1forZkiwhilefori=nZk+1;:::;n;yihasvalue0forZki:Thusfrom( 4{5 )itisstraightforwardtoseethattheposteriordistributionsof1and2areindependenttoeachotherwiththefollowingmultivariatenormaldistributions:1j~Z;k;;~Mcomp;yMVN(B01B)1B01PnZki=1yi 1.

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theseventhmarker.100subjectswith15continuousseasons'growthdatasaregeneratedandtheheritabilityonyear8issettobe.4.Themathematicalformsforthetwounderlyinggrowthcurvesare20=[1+20exp(:6t)]and30=[1+27exp(:9t)]:Foreachmarker,werandomlycancelout10subjects'markertypes.AssuggestedinChapter3,whenweconsiderkthinterval,weuseinterval-wisedeletedREMLestimateofthecovariancestructure,thatis,theREMLestimatefromallsubjectswhohavebothinformationofmarkerkandmarkerk+1:TheGibbssamplersampled50,000times.Thealgorithmquicklyndsthe7thintervalandstaysthere.ThehistogramsofbasedonthelastnsamplesareshowninFigure 4{3 ,n=10;000;20;000;30;000;40;000:Aswecanseethatthealgorithmarrivesthestationarydistributionratherquickly.Themedianandmeanforis.915whiletheactualQTLisassumedat.917.WhenthereisnoQTL,thehistogramsoftheintervalindexkandcondi-tionalonkareillustratedinFigure 4{4 .TherearenoobviouspeaksinanyofthehistogramwhichindicatesthatthealgorithmdoesnotndanyQTL.InthisFig-ure,theunderlyingtruecovariancematrixisused.IfweusetheREMLestimatefrominterval-wisedeleteddata,theoutcomeisshowninFigure 4{5 andFigure 4{6 .Figure 4{5 suggests^k=1andinFigure 4{6 thehistogramofcorrespondingtok=1isverysimilartouniformdistribution.

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Figure4{3: Thisgureshowsthehistogramofgivenk=7fromthesimulateddatasetwhereQTLisassumedininterval7.TheREMLestimatefrominterval-wisedeleteddataisusedincalculation.\n"isthenumberoflastdrawnsamplesusedtoplotthehistogram.

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Figure4{4: ThehistogramsoftheintervalindexkandconditionalonkfromthesimulateddatasetunderH0:Thetruecovariancematrixisusedincalculation.

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Figure4{5: Thehistogramsoftheintervalindexkfromthelast25,000iterations(left)andthelast10,000iterations(right).TheREMLestimatefrominterval-wisedeleteddataisused.DatasetissimulatedunderH0:

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Figure4{6: Thehistogramsoffromthelast10,000iterations.TheREMLesti-matefrominterval-wisedeleteddataisused.DatasetissimulatedunderH0:

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Figure4{7: Thehistogramsof(k;)fromGibbssampler.TheleftgureshowsthehistogramofthepossibleintervalwhereQTLfallsin.Therightgureshowsthehistogramofgivenk=1: 4{7 .WecandirectlyseethattheQTLcandidateisintherstinterval.Themedianandmeanofgivenk=1is.606and.603,respectively,whichcorrespondstoabout7.8cMfromtherstmarker.The95%condenceintervalforfromthisempiricalposteriordistributionis[:4;:78];thatis,[5cM;10:3cM]fromtherstmarker.Thisresultisconsistentwithpreviousanalysis.

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naturallyhavemorepower.TherstmethodexplicitlyexpressestheconditionaldistributionofphenotypicobservationsygivenonallobservedmarkerssothatitiseasytousethelikelihoodprincipletondtheMLE'softheunknowncoecientvectorsandconstructthenalteststatisticforndingsignicantevidenceofexistingQTL.Thelikelihoodmapacrossthewholelinkagegroup/genomecanbeobtainedandhenceallQTLcandidatesimpliedbythepeaksoverthresholdvaluesaredetectedveryeasily.ThesecondmethodusesaGibbssampleralgorithmwhichcanprovidetheposteriordistributionofQTLspeciedbyintervalkandlocationinthatinterval:Theposteriorvarianceandtheempiricalcondenceintervalaboutisstraightforwardtocalculate.Thesignicancetestisalsoembeddedintheposteriordistributionoftheparametersandhenceitisnotnecessarytoobtainthecriticalvalueforateststatistictodeclaresignicance.ButsinceweonlyassumethatthereisoneQTLinthewholegenome,thisalgorithmmightnotndallQTLsiftheredoexistseveralQTLstogethercontrollingthedierentphenotypiccurves.Inthefollowingsimulationstudies,wetesttheperformanceofourproposedGibbssampleralgorithmwhentherearetwoQTLstogethercontrollingthephe-notypictrajectories.ThefourcurvescorrespondingtofourpossiblecombinationofthesetwoQTLsareshowninFigure 4{8 .Weassumethereare9intervalswithequaldistance20cM.15observationsfrom100subjectsarerecorded.Theheritabil-ityatthe8thmeasurementpointsissetas:4inFigure 4{9 and:1inFigure 4{10 ,Figure 4{11 .ThetwoQTLsaresettobe18cMand10cMfromtheirleftbracketedmarker,respectively.InFigure 4{10 andFigure 4{9 thetwoQTLsarelocatedinthesecondintervalandtheeighthintervalbutthealgorithmonlysuggeststhatinthesecondintervalthereexistsaQTL.Figure 4{11 wherethetwoQTLsareassumedtobeinthefourthintervalandthesixthintervalimpliesQTLlocatesinthesixthintervaleventhoughthealgorithmstartsfrominterval4.

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Figure4{8: TheunderlyingfourdierentphenotypiccurvescontrolledbytwoQTLstogether.

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Figure4{9: Theplotshowsthehistogramofgivenk=2:Medianofis:849whilethetrueis:917:TheREMLestimatefrominterval-wisedeleteddataisusedincalculation.Heritabilityvalueattheeightobservationpointissettobe.4.TwoQTLsareassumedtobeininterval2andinterval8,respectively.Startvalueforkis4.Thelast10,000samplesfrom50,000iterationsareusedtomaketheplots.

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Figure4{10: Thehistogramof(k;)fromthelast10,000samplesof50,000it-erations.TwoQTLsareassumedtobeininterval2andinterval8,respectively.Theleftplotshowsthehistogramofintervalindexwhichgives^k=2:Therightplotshowsthehistogramofgivenk=2:Medianofis:749whilethetrueis:917:TheREMLestimatefrominterval-wisedeleteddataisusedincalculation.Heritabilityvalueattheeightobservationpointissettobe.1.Startvalueforkis6.

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Figure4{11: Thehistogramof(k;)fromthelast10,000samplesof50,000it-erations.TwoQTLsareassumedtobeininterval4andinterval6,respectively.Theleftplotshowsthehistogramofintervalindexwhichgives^k=6:Therightplotshowsthehistogramofgivenk=6:Medianofis:77whilethetrueis:549:TheREMLestimatefrominterval-wisedeleteddataisusedincalculation.Heritabilityvalueattheeightobservationpointissettobe.1.Startvalueforkis4.

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122

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distributionofMdoesnotdependontheunobservedY;thatis,f(MjY)=f(MjYobs):Itisstraightforwardtogetthemoregeneralcategory:missingnotatrandom(MNAR).Kenwardetal.(2003)furthersplitMNARtonon-future-dependentandfuture-dependent.Innon-futuredependentmechanismsmissingnessisallowedtobedependentonpresentpossiblynon-responsebutnotonfuturemeasurements.ThisclasscontainstheimportantsubfamilyofMARmodelswhileputtingasensiblerestrictionforMNAR.Kenwardetal.(2003)arguedthatnon-futuremissingvaluesfamilyisnotequivalenttotheinteriorfamilynamedbyThijsetal.(2002),buttheybothhaveMARmodelsastheirintersection.Thepatternmixturemodelisnonidentiablebyconstructionbecausethemissingdataprovidesnoinformationaboutthedistributionofthemeasurementsconditionalontheincompletedatapattern.Tosolvethisproblem,Little(1993,1994)proposedidentifyingrestrictionsbysettingtheinestimableparametersofincompletedatapatternequaltothoseofcompletedatacases,whichwascalledthecomplete-casemissingvaluerestriction.Otherrestrictionsincludetheneighbouring-casemissingvaluesrestrictionwhichborrowsinformationfromclosestavailablepattern,andtheavailable-casemissingvaluesrestrictionwhichneedstoconsideraparticularlinearcombinationandisshowntobeequivalenttomissingatrandominaselectionmodel(Molenberghsetal.(1998)).Alltheserestrictionsareinthe'interior'familyofidentifying-restrictionssincetheyallusetheobservabledistributiontostudytheunobservabledistribution(Kenwardetal.(2003)).Thijsetal.(2002)studiedthesethreerestrictionsindetail.Themissingmechanismfornon-responsesisnotidentiedfromtheobserveddata.Sowehavetomakesomeunveriedassumptions.Manyresearchers(e.g.RosenbaumandRubin(1983,1985);Nordheim(1984);LittleandRubin(1987);Scharfsteinetal.(1999))havepointedouttheimportanceofconductingsensitivityanalysesforarangeofplausibleassumptions.Birminghametal.(2003)conducted

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asensitivityanalysisforthreepattern-mixture-typeunveriedrestrictionsundertheassumptionofsequentialignorabilitywhendatashowsmonotonemissingpatternsandshowedthatonlyonetypeofrestrictionsallowstheexaminationofthesensitivity.Itisworthwhiletointegratebothmissinggenotypicvaluessituationandmissingphenotypicvaluessituationintoournonparametricfunctionmappingframework.NaturallythemethodswhichmaybehelpfultomakeinferenceincludeEMalgorithm,Gibbssampler,multipleimputationoracombinationofthesemethods.Thepatternmixturemodelismoreattractiveforusbecausethosemissingpatternsmaybehighlycorrelatedto,ifnotdirectlycausedby,underlyingQTLespeciallyinmonotonedrop-outpatternsandpatternmixturemodelallowsusmodeltheconditionaldistributionofmissingmechanismsdirectly.Theremayexistalinkbetweenthispatternmixturemodelandtheselectionmodelwithsomepriordistribution.Toidentifypatternmixturemodel,unveriedidenticationrestrictionsareassumedsosensitivitystudieswillbeconducted.Inferenceaboutestimatesofparameterswillbestudiedtoo.

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PAGE 145

JieYangwasbornonJune5,1981,inLianyungang,JiangsuProvince,P.R.China,astheeldestdaughterofMr.MingfaYangandMrs.QiyunLi.Whenshewas15yearsold,sheenteredtheSpecialClassofGiftedYoung,UniversityofScienceandTechnologyofChina.FromthereshegainedalotofinterestinstatisticseldeventhoughshegraduatedwithaBachelorofEngineeringincomputerscience.Eagertofurtherintensifyherknowledgeinstatistics,JiedecidedtopursueaPh.D.degreeinstatisticsattheUniversityofFlorida.ShejoinedasaresearchassistantintheMaternalChildHealthandEducationResearchDataCenteratUniversityofFlorida.Inthesummerof2003,shewasawardedtheMerckBARDSGraduateFellowshipinBiostatisticsandstartedherdissertationworkwithDr.GeorgeCasellainnonparametricfunctionalmappingofquantitativetraitloci.Asidefromcompletingthestandardcurriculum,sheworkedasansummerinterninSummer2004atMerckResearchLaboratories,Rahway,NJ,andthepaperfromtheresearchshedidatMerckwonthe2005ENARstudentawardforpapercompetition.InSpring2005shevolunteeredtobealabinstructorforSTA2023,AnIntroductiontoStatistics.FromJan,2004,tillMay,2005,sheorganizedtheweeklystudentseminarintheDepartmentofStatistics,UniversityofFlorida. 131


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NONPARAMETRIC FUNCTIONAL MAPPING OF QUANTITATIVE TRAIT
LOCI
















By

JIE YANG


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2006

































Copyright 2006

by

Jie Yang


































I dedicate this to my parents, Mingfa Yang and Qiyun Li, my husband, Song Wu

and my son, David D. Wu.















ACKNOWLEDGMENTS

First, I would like to thank my parents, Mingfa Yang and Qiyun Li, my sisters,

Jing Yang, Hui Yang and Xiaomin Yang, for all their love and support shown at

all stages of my life. I owe the greatest debt of gratitude to my husband, Song Wu,

who has been at my side with his tireless love since my d4iv one in the USA. They

are great companions during the sunny and cloudy d-,i,- towards my Ph.D. And I

want to thank my son, David, for bring us so much joy and pride.

I express my warmest gratitude to my academic advisor and committee chair,

Dr. George Casella. Learning from his intuition and his clear understanding of

mathematical statistics has .i.- li, been a pleasure. As his student, I feel motivated

and energized after each research meeting with him, although at times I may

not know exactly what to do but I work more efficiently and am more creative

following a meeting with him.

My heartfelt thanks go to Dr. Rongling Wu for his generous counsel, help and

encouragement during all stages of my Ph.D. study here. I enjoy every talk with

him.

I would also like to extend my thanks to the other committee members: Dr.

Ramon Littell, Dr. Ronald H. Randles and Dr. Marta L. Wayne. I also thank

all other faculty and stuff members of the Department of Statistics, University of

Florida; whenever I needed advice or feedback, I felt welcome in their offices.

Equally important have been my friends at the University of Florida with their

humor, comradeship, and unwavering support. I will alr--,i- recall with fondness

the good times spent here during the graduate work.















TABLE OF CONTENTS
page

ACKNOWLEDGMENTS ................... ...... iv

LIST OF TABLES ...................... ......... vii

LIST OF FIGURES ..................... ......... ix

ABSTRACT ...................... ............ xiii

CHAPTER

1 INTRODUCTION ........................... 1

1.1 Genetic Basics . . . . . . . 2
1.2 Interval M apping ............................ 5
1.3 Functional Mapping .......... ................ 8
1.4 Splines ...................... .......... 12
1.5 Mixed Models ................... ....... 15
1.6 Motivation and Outline of the Dissertation ............. 17

2 DENSE-MAP CASE: WHEN THE QTL IS ON A MARKER ...... 19

2.1 Data Setting ....... .. ...... ... ...... 19
2.2 The Likelihood Ratio Test at One Marker . . ..... 21
2.3 The Likelihood Ratio Test over All Markers . . 22
2.4 Examples ............... ........... .. 25
2.4.1 Simulation .............. . ... 25
2.4.2 Poplar Data ..... ... ... ....... 27
2.5 Technical Details: Derivation of the Likelihood Ratio Test . 30

3 NONPARAMETRIC FUNCTIONAL INTERVAL MAPPING ...... 32

3.1 Data Setting ..................... .. ........... 32
3.2 Approximate Distribution of Test Statistics at Each Q under Ho 35
3.3 Precision of the Linear Approximation to LRT Statistics under Ho 41
3.4 Alternative Approach to Calculate P-value: Simulation ...... ..49
3.5 Estimation of Consistent Variance-Covariance Matrix with Small
Sample Size ....... .. ....... ...... 56
3.6 Application to Poplar Data Revisited . . ..... 61
3.6.1 A ('Ch iii..-..I"ii, -wise Analysis on Linkage Group 10 . 61
3.6.2 A Genome-wise Analysis Using REML Estimate ...... ..63









3.7 Simulation Study for Nonparametric Functional Interval Mapping 64
3.8 Further Evaluation of Empirical B i-;,- Estimate of Covariance Ma-
trix .... . . . .. . .. ...... . 69
3.9 Discussion ...... .. .. . .. .. ...... 71
3.10 Appendix: Performance of the Approximation G to Actual Likeli-
hood Ratio Test Statistics .................. .. 78
3.10.1 Joint distribution of G at each Q under H . ... 78
3.10.2 Examples. .................. ......... .. 80

4 MISSING MARKER INFORMATION ....... . ... 85

4.1 Simple Methods ........... . . ... 85
4.2 Modelling Incompleteness in Markers ..... . . 86
4.3 Likelihood Ratio Test when Missing Marker is Present ...... ..89
4.4 Simulation Procedure to Get Threshold Point . . 93
4.5 Application ...... ... ...... ............ 97
4.6 Another Way to Handle All Missing Marker Cases -Gibbs Sampler 100
4.6.1 Simulation ............... ........ .. 110
4.6.2 Application ... ............ ...... .. 111
4.7 Discussion ............... ........... .. 116

5 FUTURE STUDY ........... . . ...... 122

REFERENCES ............... ................. .. 125

BIOGRAPHICAL SKETCH ............. . . .. 131















LIST OF TABLES


Table page

1-1 Conditional probabilities of- i,' I.,rpe at a QTL bracketed by markers M.
and Af in a backcross population .. ............. . 5

1-2 Distribution of the phenotypic values for different genotype groups in a
backcross population; 01 7- and 02 ni( .. . . 6

1-3 Conditional probabilities of _. In.1rvpe at a QTL bracketed by markers M/
and AC in a backcross population, when rl is relatively small . 6

3-1 Analysis of Poplar data using nonparametric functional interval map-
ping (NPFIM) and parametric functional interval mapping (PFIM). "EB"
the empirical B i,--; estimator."REML", the REML estimator corre-
sponding to structure a2J +Autoregressive( 2, p). . . 64

3-2 P-values and its standard deviation of nonparametric functional interval
mapping (NPFIM) using different variance-covariance estimators under
different true Es. "EB" means the shrinkage estimator which is guaran-
teed to be a consistent estimator. "REML.-- estimator is the REML es-
timator corresponding to a wrong structure. "REML" estimator is the
one selected from SAS Proc MIXED using BIC and assuming each sub-
ject has different underlying growth curve. "True" estimator is of course
the matrix we actually used to generate data. .. . ..... 66

3-3 P-values and its standard deviation of nonparametric functional inter-
val mapping (NPFIM) using different variance-covariance estimators for
different sample size and E = E3. "EB" means the shrinkage estimator
which is guaranteed to be a consistent estimator. "REML,.- estimator
is the REML estimator corresponding to a wrong structure. "REML"
estimator is the one selected from SAS Proc MIXED using BIC and as-
suming each subject has different underlying growth curve. "True" esti-
mator is of course the matrix we actually used to generate data. . 68

3-4 P-values and standard deviation of nonparametric functional interval
mapping (NPFIM) from HIV dynamics data and growth data for dif-
ferent combination of variance-covariance estimators and sample size.
"EB" means the shrinkage estimator which is guaranteed to be a consis-
tent estimator. "REML" estimator is obtained from SAS Proc MIXED
assuming each subject has a different underlying mean curve. "True" es-
timator is, of course, the matrix we actually used to generate data. 72









4-1 P-values and its standard deviation of nonparametric functional inter-
val mapping (NPFIM) using case-wise deleted data set and full data set,
E 0.3 J1 + Autoregressive(r2 0.1,p 0.8) ............ ..97

4-2 Comparison of critical values from different analyzing methods . 98

4-3 Distribution of missing marker conditional on the other observed marker
for kth marker interval for subject i. ................ . 103

4-4 Conditional distribution of missing markers of kth marker interval for
subject i. . . . . . . . . 103

4-5 Distribution of missing marker conditional on the other observed marker
for kth marker interval for subject i. ................ . 107

4-6 Conditional distribution of missing markers of kth marker interval for
subject i. . . . . . . . . 108















LIST OF FIGURES
Figure page

1-1 Illustrations of one isolated B-spline and several overlapping ones with
degree 1 and 2, respectively. ............... ..... 13

2-1 Graph of the p-value and its 95'. confidence interval for data from two
underlying flat lines with different distances. .............. 26

2-2 Graph of p-value and its 95'. confidence interval for data from two un-
derlying curves:one is flat line the other is in quadratic form. . 27

2-3 Original poplar data profiles. ............... .... 28

2-4 The growth curve representing two groups of .:, ir. vpes at marker CA/CCC-
640R on linkage group 10 in the Populus deltoides parent map. . 29

3-1 The plot of 31+ Si and max,,+l jects n increases under simplest setting that all data from standard nor-
mal distribution. Recombination rate between two markers is .2 and 0
.5 .................. ..... ....... ....... 45

32 The plot of 3 S and max,+i increases under simplest setting that all data from standard normal dis-
tribution. Recombination rate between two markers is .2 and 0 = .5. 46

3-3 The plot of function f(x) = f (x) f2(x). ............... 47

3-4 The plot of '13 1 Si and max,+li<<3 |Sil when number of subjects n
increases. . . . . . .. . . 48

3-5 The plot of 3 ,1 Si and max1+li< increases. . . . . . .. . . 48

3-6 The plot of 3 ,+ Si and max1+li< increases using approximation f2(x)' 20(x + x2/2) 02(x + 2/2)2 49

3-7 Two simulated distributions of likelihood ratio test statistic based on
same 1000 datasets. ............... ..... .... 51

3-8 Two simulated distributions based on same 1000 datasets when treating
0 as unknown parameter. ............... ...... 54









3-9 Two simulated distributions based on different 1000 datasets while treat-
ing 0 as unknown parameter ............. .. .. 55

3-10 Fitted curves from using two different variance-covariance estimator. 63

3-11 The profile of the likelihood ratio test statistics between the full and re-
duced (no QTL) subject-specific model for the diameter growth trajec-
tories across the whole Populus deltoides parent map. The genomic po-
sitions corresponding to the peak of the curve are the MLEs of the QTL
localization ............... ............. .. 65

3-12 Heritability curves for the four simulated data sets using different Es. .. 67

3-13 The left plot shows the 10 heritability curves of simulated growth data
sets. The right plot shows the trend of P-values when heritability on year
4 increases. "EB" and "REML" means the Empirical BEi-- estimate
and REML estimate of covariance matrix, respectively. "True" repre-
sents the covariance matrix used to generate data. . . 70

3-14 The HIV dynamics curves used as the mean curves. . . 72

3-15 The heritability curve across all observation time points in the HIV sim-
ulation study ................ ............. .. 73

3-16 The empirical distribution of the test statistics in the simulation study
of C'! lpter 3. Dataset 1-4 correspond to the four E matrixes in Table
3-2. "EB" means the Empirical B-,-i estimate. "True" represents the
covariance matrix used to generate the data. .............. 76

3-17 The left plot indicates the empirical distribution of the test statistics in
the simulated growth data of Chapter 3 with covariance matrix EI, 61
subjects with 11 observation points. The right one is the one from the
simulated HIV dynamics data, 61 subjects with 20 observation points. 77

3-18 Graph of the p-value and its 95'. confidence interval for data from two
underlying flat lines with different distances using nonparametric func-
tional interval mapping. .................. .... 81

3-19 Graph of p-value and its 95'. confidence interval for data from two un-
derlying curves:one is flat line and the other is in quadratic form using
nonparametric functional interval mapping. .............. 82

3-20 Graph of the likelihood ratio test statistics along the linkage group 10
using nonparametric functional interval mapping. ........... ..83

4-1 Histograms of LRT statistics from different combination of methods han-
dling missing data and methods to estimate E under Ho. . ... 98









4-2 The trend of P-values and their 95' confidence intervals (the dotted lines)
from using case-wise deleted data (the dashed line) and using full data
set (the black line). Underlying phenotypic lines are two flat lines and
true covariance matrix is used in ,iln ii -i- ............ 99

4-3 This figure shows the histogram of 0 given k = 7 from the simulated
dataset where QTL is assumed in interval 7. The REML estimate from
interval-wise deleted data is used in calculation. "n" is the number of
last drawn samples used to plot the histogram. . . ..... 112

4-4 The histograms of the interval index k and 0 conditional on k from the
simulated dataset under Ho. The true covariance matrix is used in calcu-
lation . ............... ............... .. 113

4-5 The histograms of the interval index k from the last 25,000 iterations
(left) and the last 10,000 iterations (right). The REML estimate from
interval-wise deleted data is used. Data set is simulated under Ho. .... 114

4-6 The histograms of 0 from the last 10,000 iterations. The REML esti-
mate from interval-wise deleted data is used. Data set is simulated un-
derH. ................. ...... ............. 115

4-7 The histograms of (k, 0) from Gibbs sampler. The left figure shows the
histogram of the possible interval where QTL falls in. The right figure
shows the histogram of 0 given k = ................ . 116

4-8 The underlying four different phenotypic curves controlled by two QTLs
together .................... .............. 118

4-9 The plot shows the histogram of 0 given k = 2. Median of 0 is .849 while
the true 0 is .917. The REML estimate from interval-wise deleted data
is used in calculation. Heritability value at the eight observation point is
set to be .4. Two QTLs are assumed to be in interval 2 and interval 8,
respectively. Start value for k is 4. The last 10,000 samples from 50,000
iterations are used to make the plots. ............. . 119

4-10 The histogram of (k, 0) from the last 10,000 samples of 50,000 iterations.
Two QTLs are assumed to be in interval 2 and interval 8, respectively.
The left plot shows the histogram of interval index which gives k 2.
The right plot shows the histogram of 0 given k = 2. Median of 0 is .749
while the true 0 is .917. The REML estimate from interval-wise deleted
data is used in calculation. Heritability value at the eight observation
point is set to be .1. Start value for k is 6. ............... 120









4-11 The histogram of (k, 0) from the last 10,000 samples of 50,000 iterations.
Two QTLs are assumed to be in interval 4 and interval 6, respectively.
The left plot shows the histogram of interval index which gives k 6.
The right plot shows the histogram of 0 given k = 6. Median of 0 is .77
while the true 0 is .549. The REML estimate from interval-wise deleted
data is used in calculation. Heritability value at the eight observation
point is set to be .1. Start value for k is 4. ............ .. 121















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

NONPARAMETRIC FUNCTIONAL MAPPING OF QUANTITATIVE TRAIT
LOCI

By

Jie Yang

August 2006

C'!i ir: George Casella
Major Department: Statistics

Functional mapping is a tool for detecting i, i i '" genes responsible for different

Iph. ii,.I pic curves, developed by Ma, Casella and Wu. The methodology uses

a parametric functional form, usually derived from a biological law, to drive a

maximum-likelihood-based test for a significant QTL (quantitative trait loci).

However, in many situations there is no obvious functional form and, in such cases,

this strategy will not be optimal.

In this dissertation we propose to use nonparametric function estimation,

typically implemented with B-splines, to estimate the underlying functional form of

1!i, ph. ir pic trajectories, and then construct a nonparametric test to find evidence

of existing quantitative trait loci. Using the representation of a nonparametric

regression as a mixed model, we can easily derive a likelihood ratio test statistic.

Two situations are considered: one is based on the dense-map assumption that

QTL (quantitative trait loci) is on some marker and the other is the situation

where the actual genes responsible for different underlying ph. i, il. pic trajectories

might not just be on a marker locus, which is more realistic. For the dense-map

case, after obtaining the joint distribution of all test statistics at every putative









locus (each marker), we can then calculate the p-value directly. Simulation studies

show that our method is both powerful and quick. We also provide an application

to a real data set.

Nowd-l-1 with the cutting-edge biotechnology, more and more marker

information can be obtained. Along with this exciting human achievement, the

dataset has more possibility to have missing cells. In this dissertation we also take

into account this case.















CHAPTER 1
INTRODUCTION

The variations of many quantitative traits in human, plants and animals

can be attributed mainly to the segregation of multiple genetic factors. Sax

(1923) first investigated the gene affecting seed size in beans with pattern and

pigment markers. Since then, the theoretical principle of using markers to analyze

quantitative trait loci (QTL) is well established(Sax (1923); Jansen and Stem

(1994); Zeng (1993,1994); Kruglyak and Lander (1995); Kao et al. (1999); Ma,

Casella and Wu (2002)). In recent years, mapping of QTL is greatly facilitated

by fine-scale genetic marker maps constructed through modern molecular biology

techniques. After Lander and Botstein's (1989) pioneering work, many statistical

methodologies for mapping Quantitative Trait Loci (QTL) on a high-density

linkage map were proposed. These methods have been used to identify 1n i i"r genes

controlling different traits important to biomedicine, biology science, agriculture,

zoology and so on. In Section 1.2, a brief introduction to interval mapping, first

proposed by Lander and Botstein (1989) is given .

Typically these methods treat traits as discrete, even though the trait is

an infinite-dimensional characteristic (Kirkpatrick and Heckman 1989) or a

functional-valued trait (Pletcher and G-V-r 1999), which may be expressed as a

smooth function of one continuous variable. In real life, as a consequence of natural

selection, an infinite-dimensional characteristic may change its phenotype to be

more competitive. Therefore, embedding the underlying biological mechanism

and processes into a QTL mapping strategy is a reasonable approach. Functional

mapping, developed by Ma, Casella and Wu (2002), implements a parametric









function, usually derived from a universal biological law, to describe a phenotypic

curve. For example, a logistic growth curve can be described by a mathematical

form g(t) = +e -rt with parameters a, b, r. Because of its parametric nature, it is

straightforward to test numerous biologically important hypotheses by testing some

parameters or functions of parameters using the maximum likelihood principle.

This method is powerful in providing the estimated position of QTL governing the

character process. Section 1.3 provides an introduction of this functional mapping

framework. A short description of the comparison between traditional discrete

mapping methods and functional mapping methods is also included in Section 1.3.

In this dissertation, we extended parametric functional mapping to non-

parametric functional I'ij':,:''L because we propose to use splines to estimate the

underlying function and use the spirit of mixed models to characterize the complex-

ity of our beautiful nature. In Section 1.4 and Section 1.5 we briefly explain about

splines and mixed models.

1.1 Genetic Basics

The work of Gregor Mendel marked the beginning of modern genetics. (He

was an Austrian Monk who developed an interest in heredity, worked with peas

and published his work in the 1870's. His publication was discovered by three

independent investigators at the turn of the century, and all three quickly realized

the significance of the work. Unfortunately, Mendel died before his work was

rediscovered.) Genes are codes of life, which are some particular regions on a

chromosome, while the inter-genic materials on chromosomes carry no information.

He formed the basis of genetics:

1. The Law of Segregation: factors (called alleles) controlling the particular

trait separate cleanly one from the other and there is no mixing;

2. The Law of Independent Assortment: alleles of one gene segregate while

alleles of different genes assort independently.









Many traits of agronomic and horticultural interest are controlled by a single

gene and fall into a few distinct phenotypic classes. These classes can be used to

predict the genotypes of the individuals. And if we know the genotype we could

predict the phenotype. These type of phenotypes are called discrete traits. Other

traits may have a distribution that resembles the bell-shaped curve for a normal

distribution such as human IQ, crop yield, weight gain in animals, fat content of

meat etc. These types of traits are called continuous traits. Because continuous

traits are often given a quantitative value, they are often referred to as quantitative

traits and therefore the loci controlling these traits are called quantitative trait loci

or QTL.

The purpose of QTL mapping is to link quantitative phenotypic traits with

regions on a chromosome. Two questions are raised: (1) Are there significant

QTL effects? and (2) where are the QTLs if they do exist? For mapping a QTL,

a cross is arranged between two inbred lines (which means they are homozygotes

everywhere) which differ substantially in the quantitative trait of concern. Assume

there are two parents having alleles QQ and qq at a certain (unknown) place

(named locus) on the chromosome. The offspring of these parents are called the

FI generation which have the allele Qq. If the offspring is mated back to one of the

parents, -iv QQ, then the new generation produced has alleles QQ and Qq. Such

population structure is called a backcross and the new generation is called BC1.

Since our mapping method is to find the QTL Q or q, of course, that locus is

unknown. All we know are markers along the chromosome, -v M or m. Recom-

bination occurs when alleles cross over to another chromosome and recombination

rate r can be used as a measure of closeness between two genes or between gene

and marker. So we can use markers to locate QTL. If the recombination rate r

between marker and QTL is small, with zero being the limiting case of no recom-

bination, then marker M is closely linked to QTL Q, that is, when we see M we









hope Q is also there. If r is large with value 1/2 being the limiting case, then there

may be no linkage between AM and Q. The backcross is the simplest design in

which we get enough information to estimate r.

A simple statistical model directly follows. Assume Y is a random variable

that follows a normal distribution. The variance of Y is a2 and the mean of Y de-

pends on the alleles under consideration, pQQ, ,Qq or Pqq. Clearly in the backcross,

parents are N(pQQ, a2) (or N(pqq, a2)) and N(,Qq, a2) and generation BC1 is a

mixture of normals with means iQq and /QQ (or Pqq). If the QTL Q is known,

the mixing factor would be 1/2, but we can only observe the marker genotypes M

or m. For example, the parents of BC1 have alleles MQIMQ and alleles mqIMQ.

When there is no recombination between /M and Q, we will see alleles MQIMQ

and mq|MQ in generation BC1. But when recombination between M and Q oc-

curs, we will expect four possible genotypes MQIMQ, MqIMQ, mQIMQ and

mq|MQ. Now the distribution of genotypes in this backcross population is


Y|MM ( N(pQQ, a2), with probability 1 r
N(PQq, -2), with probability r

and

Y|Mm A N(pQQ, 72), with probability r
N(P/Qq, o2), with probability 1 r
The difference in means of the populations when categorized by the markers is


pMM Mm = (1 2r)(pQQ PQq).

Assuming /tQQ I-tq 0, a test of Ho : /MM M/m = 0 is equivalent to testing

Ho : r = 1/2, which means no linkage between /M and Q. This test can be carried

out with something as simple as a t-test or the more popular used likelihood

methods. Using this mixture model in likelihood analysis, we can not only test for

linkage but also can estimate r.






5


Table 1-1: Conditional probabilities of i_. In' pe at a QTL bracketed by markers
M and N in a backcross population

Marker QTL
Qq qq
MNlmn (1-r) (1-r2) rfr2
1-r 1-r
Munmn (1-r1)r2 rl(1-r2)
mNImn rl(1-r2) (1-r1l)r2
r r
mn mn2 (1-r) (1-r2)
ll _1-r 1-r


1.2 Interval Mapping

The principle of interval mapping was first established by Lander and Botstein

(1989). The success of interval mapping as a powerful method for linkage analysis

of a complex trait has roots in the rapid development of molecular technologies

to detect more and more markers of any organism and improved statistical and

computational techniques, such as the EM algorithm (Dempster et al. 1977), which

makes it possible to tackle complex genetic and genomic problems.

In an interval mapping procedure we typically use two markers to better locate

QTL Q and hence a somewhat more complex model. Assume the two markers

bracketing the unknown QTL are /M (with alleles M and m) and A" (with alleles

N and n). These two markers make four distinct genotype groups in a backcross

population

MN mn, Mn Imn, mNmn, mn Imn. (1-1)

Denote the recombination fraction between /M and AV by r, and that between

/M and QTL Q by ri, and that between Ai and QTL Q by r2. The numerical

relationship between r, ri and r2 is r = r1 + r2 r1r2.

In a genetic map, genetic distances D are used to imply the location of

detected markers. Genetic distances and recombination rates are related through

a map function (Ott 1991). We usually use the Haldane map function to link the

genetic distance D to corresponding recombination rate r : r = (1 e-2D)/2.









Table 1-2: Distribution of the phenotypic values for different genotype groups in a
backcross population; 01 = and 02 1-2)

Marker Distribution
MN Imn (1 01)N(PQq, a2) + 81N( qq, a2)
Mn mn (1 02)N(IQq, 72) + 02N(i q9 2)
mNImn 02N( Qq, a2) + (1- 02)N(qq, a2)
mnnmn 02N(IQ, 2) + (1- 2)(qq, 2)

Table 1-3: Conditional probabilities of .- i,..' vpe at a QTL bracketed by markers
/M and Af in a backcross population, when rl is relatively small

Marker QTL
Qq qq
MNImn 1 0
Mnlmn 1-0 0
mNlmn 0 1-0
mnlmn 0 1


The conditional probability of a QTL genotype, -iv Qq, given each of the four

two-marker genotypes is given in Table 1-1. Let 01 = and 02 = (1-r2), the

corresponding distributions of the phenotypic values for the four marker .-. in.1 pes

can be modeled, respectively, as stated in Table 1-2, where IQq, ,qq, .2 are the

unknown parameters contained in the mixture models and can be estimated using

likelihood principle.

When 0 = rl/r is small, we can use the following approximated conditional

probabilities listed in Table 1-3. In the dissertation, we use these approximate

conditional probabilities. Suppose that N1, N2, N3 and N4 are the sample sizes of

the four marker groups (1-1). If we define ni = N1, n2 = ni + N2, n3 = 2 + N3 and

n = n3 + N4, the log likelihood of the phenotype data conditional on the marker









information can be written as


(,Qq, fqq, 2) -log 2 (12)
2
i 1

k2 1 1
+ log[(1- )exp(- 2 (y- p)2) + 0exp(- 2 (y Pq)2)
nin+l
S1 1
+ log[0exp(-20 (i PQq)2) + (1- ) exp(- 22 ( I)2)
iin2+1


In
i=n3+1

In Lander and Botstein's (1989) model for interval mapping of QTL, the

profile of likelihood ratio test (LRT) statistics is constructed over the grid of

possible QTL locations, which is implied by 0, in a linkage group or an entire

genome. (The genome is the complete set of linkage groups or chromosomes.)

The maximum of the LRT statistics is used as a global test statistic. At a given

position of the QTL, that is, when 0 is fixed, the LRT statistic is .-ivmptotically

X2- distributed with degrees of freedom equal to number of associated QTL effects
under the null hypothesis that no QTL exists at this 0. However, under Ho : no

QTL, the QTL position is unidentified and therefore, the final global maximum

LRT statistic does not follow the standard X2-distribution .i-vmptotically. This is

exactly why there is no closed form to decide the critical threshold to declare the

statistical significance of a QTL. But several authors derived approximate formulas

to determine critical thresholds for a particular design based on the results of

Davies (1977, 1987) (Rebai et al. (1994); Doerge and Rebai (1996); Piepho (2001)).

To overcome the limitations due to the failure of the test statistic to follow a

standard statistical distribution, one can also use simulation studies (Lander and

Botstein (1989); Van Ooijen (1992); ('!C. i and ('C!. (2005)) and permutation tests

(see ('!,iti. !!! and Doerge (1994); Doerge and ('!ti.1 !1!! (1996)) to calculate the









threshold value throughout a genome. Lander and Botstein (1989) established

the finite-dimensional convergence of LOD score process, which arises from the

LRT, to the Orenstein-Uhlenbeck process. ('C!, i and ('C!, 1 (2005) established

a novel theorem for the .,-vmptotic distribution of the LRT statistic calculated

from one marker interval for backcross model. Then a distribution-free simulation

approach can be used to calculate critical values. Either the simulation-based or

permutation-based approach is a highly computational-demanding approach. For

permutation test, one need perform at least 10,000 permutations for the same

data set to obtain a reasonably accurate estimate of a critical threshold at a

genome-wide type I error rate of 0.01 (Doerge and Rehai (1996)).

Lander and Botstein's (1989) interval mapping method is based on a simplified

situation, that is, the segregation pattern of all markers strictly agrees with the

Mendalian laws and there is only one QTL on a chromosome controlling a trait

under study. T I i* authors extended and improved this work by including markers

from other intervals as covariates to control the overall genetic background (Jansen

and Stam (1994); Zeng (1994); Haley et al. (1994); Xu (1996)). Kao et al. (1999)

used multiple marker intervals simultaneously to map multiple QTL of epistatic

interaction throughout a linkage map.

1.3 Functional Mapping

As we know function-valued traits change as a function of some independent

and continuous variable, such as growth trajectories, allometric scalings, and norms

of reaction. A simple approach for mapping such infinite-dimensional character is

to treat it as a discontinuous trait, that is, link markers with phenotypes separately

for different ages, traits, or environments (C'!, v,. i ud et al. (1996); Nuzhdin et al.

(1997); Verhaegen et al. (1997); Emebiri et al. (1998); Wu et al. (1999)). However,

these separate analyses cannot provide effective estimates because they fail to









implement the information about the covariances of different traits or the same

trait measured at different ages or environments.

Multitrait mapping takes account of covariances among simultaneously

different traits or the same trait measured at different ages or environments. But it

is difficult to produce precise estimate when the number of traits increases (Jiang

and Zeng (1995); Korol et al. (1995); Ronin et al. (1995); Eaves et al. (1996);

Knott and Haley (2000)). To circumvent such difficulties, some authors attempted

to transform the initial trait space into a space of a lower dimension based on

principal component analysis (\! ,i_, _i et al. (1998); Korol et al. (2001)). However,

they still treat infinite-dimensional characters as discrete traits or eigenvalues and

do not consider the physiological mechanisms predisposing for the phenotypic

variation of functional-valued character.

The functional mapping method first introduced by Ma et al. (2002), incorpo-

rates the underlying physiological or developmental mechanisms of trait variation

into statistical ,n i1, i of QTL mapping framework, thus is more likely to produce

more accurate results in terms of biological reality. In their paper, an application

to map the growth of forest trees is used to illustrate their method. So we also use

that example to briefly introduce the functional mapping method.

The simplest backcross design is assumed so that there are only two groups

of genotypes at a locus. Of course, the genotypes of QTL also only have two

possibilities. In practice, instead of a continuum we only observe a finite set of

observations, y (1), ..., yj(m) for each tree i, i = ...n. Suppose this finite set of data

can be modelled as the logistic growth curve, which has a mathematical form


g(t) a
1+be-rt

For a particular .,. l irpe j (let j = 1 denote QTL genotype Qq and j = 2 for

QQ), the parameters describing corresponding growth curve are aj, bj and rj. To









determine whether and how the QTL affects growth trajectories is simply to test

the difference or some functions of these parameters.

Similar to interval r Il-,ppli. the phenotypes of the trait at all time points for

each QTL ,_. in, I vpe group follows a multivariate normal distribution:

1
(y =(2)/2 /2 exp[-(y gi)'E-(y g)/2],

where gj is the vector of the expected phenotypic value corresponding to QTL

genotype group j at t measurement times and E models the dependence structure

between observations from same subject. When we use logistic growth curve to

model y then

gj [gj(t)]x = -rjt lx

Ma et al. used an E = Autoregressive(a2, p) repeated measurement error structure

given by
1 p .. pm- 1

2 P 1 ... P"r-2
...2 p 1(13)

mn-l n-2 1

The likelihood function of the backcross progeny with m-dimensional measure-

ments can be expressed as a multivariate mixture model
n 2
L(aj, b, rj, bj,0, p, [) H= pifj(hf Yi ,
i= 1 .= 1

where pij, a function of parameter 0 as in Table 1-3, is the conditional proba-

bility of subject i having QTL genotype j given observed flanking markers. The

maximum-likelihood-based method implemented with the EM algorithm is used to

estimate all unknown parameters.

A lot of biologically meaningful hypotheses can be tested using such genetic

models. For example, the hypothesis to test the existence of a QTL affecting an









overall growth curve can be stated as


Ho : al = a2, b = b2, rl r2


against

HI : at least one of the qualities above does not hold.

To test if the detected QTL starts to exert or ceases an effect on growth curves at

particular time t* is essentially to test


Ho: gi(*) =g(t*)

against

H : gl(t*) g(t*).

The test statistic for testing such hypotheses is calculated as the log-likelihood

ratio of the full model over the reduced model. Ma et al. (2002) used 1000 per-

mutation tests to obtain the chromosome-wide empirical estimate of the critical

value.

This functional mapping method allows them to successfully detect one QTL

underlying poplar stem growth locating on linkage group 10 while traditional

interval mapping (Lander and Bostein (1989)) and composite interval mapping

(Zeng (1994)) failed to do so, which -ir--.- ; that by incorporating logistic growth

curves functional mapping procedure has greater power to detect a significant QTL

than the current methods. The increased detection power of functional mapping

results from the simultaneous use of repeated measurements that are correlated to

each other due to biological or/and environmental reasons.

Another advantage of functional mapping is that by treating phenotypic values

as a function of some variables a large number (theoretically unlimited number)

of measurements can be analyzed. An initially high-dimensional mapping model

becomes more tractable and the precision of the estimates of QTL parameters









gets improved. Also, a small sample size can have enough power of QTL detection

because the function modelled from repeated measurements extracts maximum

information about QTL effects and positions.

1.4 Splines

Nowad-, ,- exploratory analysis of data is more common and many datasets

are too "rich" to be modelled in parametric Sv- -i-. So smoothing gains a respectful

space in statistics. Its popularity can be demonstrated by the continuous ap-

pearance of many papers and a number of books (here to name a few: Silverman

(1986); Eubank (1988); Hastie and Tibshirani (1990); Hardle (1990); Wahba (1990);

Wand and Jones (1993); Green and Silverman (1994); R in- i,- and Silverman

(1997); Ruppert, Wand and Carroll (2003)). Smoothing methods can be applied

by running statistics like kernel smoothers (Silverman (1986); Hardle (1990)),

LOWESS (Cleveland (1979)), splines smoothers and so on. Spline smoothers

also have several varieties: smoothing splines, regression splines (Eubank (1988)),

B-splines (De Boor (1978); Dierckx (1993)), P-splines (Eilers and Marx (1996)).

B-splines are attractive for nonparametric modelling which is embedded in popular

statistical software such as R/Splus. Actually, the name "nonparametric" is not

well-chosen because splines are described by parameters, but these parameters have

no particular scientific interpretation. So in this dissertation we use B-splines to

model the underlying phenotypic function and still call our method "nonparametric

functional mapping .

Because B-splines are used in our proposed method, we will put emphasis

on introducing B-splines. A B-spline consists of polynomial pieces, connected at

certain values of x, the knots. A very simple example is shown at the upper part of

Figure 1-1: one B-spline of degree 1. It consists of two linear pieces: one is from

x = 1 to x = 2, the other is from x = 2 to x = 3. The knots are x = 1,2, 3. To the

left of x = 1 and to the right of x = 3 the value of this B-spline is zero. In the right














o
0






0
0


I I I I I I
0 2 4 6 8 10
x


I


Figure 1-1: Illustrations of one isolated B-spline and several overlapping ones with
degree 1 and 2, respectively.


/M


1









part of this upper figure in Figure 1-1, three more B-splines of degree 1 are shown

with each one based on three knots.

Also in the Figure 1-1, the lower figure illustrates B-splines with degree 2. We

can see that the leftmost B-spline consists of three quadratic pieces joined at two

knots x = 2 and x = 3. At the joint knots the values of these quadratic pieces

match and so do the first derivatives (but not their second derivatives). In the right

part, three overlapping B-splines with degree 2 are shown.

Note a first-degree B-spline can overlap with only at most two neighbors

and a second-degree B-spline overlaps with four neighbors. The leftmost and

rightmost splines have less overlap. These simple examples shed light on the

general properties of a B-spline of degree p :

It consists of p + 1 degree p polynomial pieces;

The polynomial pieces join at p inner knots;

Derivatives up to order p 1 are continuous at the inner knots;

B-spline is nonzero only on a domain spanned by p+2 knots while everywhere

else is zero;

A B-spline overlaps with 2p neighbor B-splines except at the boundaries;

At a given x, p + 1 B-splines are nonzero.

So n knots can determine n p 1 splines with degree p. In the R/Splus software,

order p + 1 is used in place of degree p. A common choice of B-splines is piecewise

cubic, that is, degree 3, which can provide a smooth fit.

De Boor (1978) gave an algorithm to compute B-splines of any degree recur-

sively from B-splines of lower degree, which is an simple algorithm for a zero-degree

B-spline in just a constant on one interval between two knots. Please see De Boor

(1978) for details.

It is straightforward to see that the number and position of knots determine

the smoothness and fit. But how to choose the optimal combinations of these









factors is a complex task. Some approaches start with a free choice of knot

locations and a rather dense set of knots and then eliminate unneeded knots by

an algorithmic procedure similar to variable selection techniques used in multiple

regression (see, for example, Friedman and Silverman (1989)). O'Sullivan (1986,

1988) proposed to use a relatively large number of knots but put a penalty on

the second derivative to restrict the flexibility of curve fitting. Eilers and Marx

(1996) followed this idea and used a simple difference penalty on the coefficients

themselves of .,Ii i.ent B-splines, which they call P-splines. P-splines can be

applied in any context where regression on B-splines is useful.

1.5 Mixed Models

Mixed models, or random-effects models have been extensively applied in

longitudinal data analysis, which is designed to investigate changes over time in a

characteristics measured repeatedly on the same subjects (Laird and Ware (1982);

Wolfinger (1993); Davidian and Giltinan (1995); Littell et al. (2000); Verbeke

and Molenberghs (2000); Diggle et al. (2002); Davidian and Giltinan (2003);

Kowalchuk et al. (2004)). Often, we cannot fully control the circumstances under

which the measurements are taken and there may be considerable variation among

individuals. A two-stage model is used to ain i1v. -. such data and a linear mixed

model is a result from such a two-stage model. In this formulation, the probability

distribution of serial measurements from the same subject has the same form for

each individual but the parameters of the distribution vary over individuals, so that

we can also view this as a subject-specific model.

Population parameters, individual effects, and within-subject variation are

modelled in the first stage while between-subject variation in the second stage.

Suppose yi are the ni serial observations from subject i, 3 is a p x 1 vector of

known population parameters, bi is a q x 1 vector of unknown individual effects.

Xi and Zi are known design matrices linking 3 and bi to yi, respectively.









Stage 1: For individual subject i,


Yi = Xi3 + Zibi + ci, (1-4)


where ci are independent from each other and usually assumed to have

a multivariate normal distribution with zero mean vector and variance-

covariance matrix Ri. Ri is a positive definite matrix with dimension ni x ni

and links to subject i only in this way. Each Ri is determined by the same

population parameters. Now, both 3 and bi are considered fixed.

Stage 2: Now bi are random variables independent from each other and from Ci,

which is therefore called random effects. Here we assume bi ~ MVN(O, D).

Such two-stage models have the following favorable feature:

1. No requirement for balance in data;

2. Explicitly modelling and analysis of within- and between-individual variation;

3. The individual parameters have a natural explanation and therefore facilitate

the exploratory analysis.

Unless in a B ,i, f -i i framework, the inference is based on least squares and

maximum likelihood principles from the marginal distribution of yi. From above

modelling marginally

yi MVN(X ,, ZDDZ + RJ).

However, if so, the hierarchical structure of the original model (1-4) is then not

taken into account. Indeed, the marginal model is not equivalent to the original

hierarchical model. Inferences based on marginal models do not explicitly assume

the presence of random effects representing the heterogeneity between subjects.

The purpose of our proposed method is to find the significant statistical

evidence of existing QTL which is reflected by the difference of average phenotypic

curves. Estimates for random effects show the subject-specific profiles which









deviate from the overall average profile and are needed for prediction of subject-

specific evolutions. So inference from the marginal distribution is enough for our

study. The selection of variance-covariance structure is a nontrivial step in the

model selection process, which will be further addressed in C'! Ilpter 3.

Several statistical packages are available for estimation and inference on all

parameters in the marginal model. Among these, the most flexible commercially

used one is the SAS procedure PROC MIXED. Littell et al. (2000) provided a

nice example to how to apply frequently used statements and options in PROC

MIXED. More detailed descriptions of all statements and options can be found in

SAS manuals.

1.6 Motivation and Outline of the Dissertation

In Section 1.3, we know the basic idea of functional mapping is using a

parametric form to model the phenotypic curve. However, in some cases there are

many different functions that describe same phenotypic trajectory, for example,

there are functions in 3 categories to describe a growth trajectory:exponential,

saturating and sigmoidal (Von Bertalanffy (1957), Niklas (1994)). Thus, it may

not be clear which one should be used, especially when there are not enough

observations for each subject to show obvious characteristics. Moreover, in many

situations, there are no obvious functional forms. Nonparametric functional

mapping avoids such problems of the original functional mapping method while

inheriting all its advantages. In this dissertation we propose a nonparametric

functional mapping procedure for different situations.

In C'! lpter 2 we consider the dense-map situation which assumes that the

QTL is on some marker. We use nonparametric function estimation, typically im-

plemented with B-splines, to estimate the underlying functional form of phenotypic

trajectories, and then construct a nonparametric test to find evidence of existing

quantitative trait loci. Using the representation of a nonparametric regression as a









mixed model, we can easily derive a likelihood ratio test (LRT) statistic. The joint

distribution of LRT statistics at each putative locus (each marker) is derived, so

that P-values can be exactly calculated using an importance sampling method. In

total, one real poplar data set and two simulated data sets are used for application.

In ('! Ilpter 3 we develop a nonparametric functional interval mapping proce-

dure for the situation where the actual genes responsible for different underlying

p1. i, I v.rpic trajectories might not just be on a marker locus, which is more real-

istic. LRT statistics are calculated from a nonparametric mixture-mixed model.

The joint distribution of all calculated LRT statistics along a genome or linkage

group can be approximately obtained, but the resulting P-value is a lower bound

of the exact P-value. So a simulation procedure is proposed to calculate the exact

P-value. Analysis from simulated data sets is performed to show the operating

characteristics of the nonparametric functional interval mapping procedure. Two

V--,V to estimate the covariance matrix are examined through simulation studies

also.

In ('!, lpter 4 we extend the nonparametric functional interval mapping

procedure to the missing .,. In..i pic data situation. The test statistic is obtained

from maximum likelihood principle and it critical value can also be determined

by a simulation approach. In this chapter, we also propose a B i.- i_ mapping

procedure to take into account of missing genotypic data.

In C'! Ilpter 5 we -i,---. -1 to extend the mapping procedure introduced in

C'! Ilpter 4 to handle both the missing genotypic and phenotypic data situation.















CHAPTER 2
DENSE-MAP CASE: WHEN THE QTL IS ON A MARKER

In this chapter we consider the simplest case-the dense-map case, which

assumes the .-; ir.1' pe is known at every point in the genome, that is, the QTL

locates on or very near to some marker locus. We use a subject specific model,

integrating a functional relationship estimated by B-splines to illustrate the

character process. The nonparametric test for evidence of existing QTL is derived

from the maximum likelihood approach. The joint distribution of likelihood ratio

test statistics at all putative loci is obtained exactly. Then we show how to directly

calculate p-values from the joint distribution of likelihood ratio test statistics at

every putative QTL locus under Ho. Applications to simulated and real data are

provided.

2.1 Data Setting

We assume that we have information about both phenotypic values and

genotypic values of N subjects. The phenotypic values are from some underlying

lih. ii,..i pic curves, which are functions of some variable t. For example, this

continuous variable t could be age (corresponding to a growth curve), or body size

(corresponding to an allometric law). In practice, we cannot measure continuous

ph,! i, ,I v.pic values but only those values recorded at some fixed t as a repeated

measure, t = (t, t2, tT)'. For simplicity, we consider a backcross design

from two contrasted l, ,i.... v.-ous inbred lines, that is, there are only two different

1ih. i. lvrpic trajectories, mi(t) and m2(t). Also, we make the assumption that the

QTL only appear at the marker loci. Using the marker information we can detect

the 1i, iri" genes responsible for different phenotypic curves.









We consider a subject-specific model, for each subject i, i = 1,..., N,


yi(t) = m(t) + ailT + Ei, (2-1)

where = (ti, t2, tT)', Ei is a parameter which accounts for the within

subject covariance structure of the observations on subject i, ailT models the

covariance structure of observations between subjects. The variables ai and Ei are

independently distributed as N(0, a2) and MVN(O, -2VT), respectively. So now

yi(t) ~ MVN(m(t), E), where = a2JT + T2V. Model (2-1) is equivalent to the
hierarchical model


yi(t) p= (t) + ai

/i(t) = mi(t) + rilT-

or the univariate model,

ij = pj + Eij, (2-2)

where i ...,N,j = ,...,T, = m(tj) and (Eil, ..,T)' ~ MVN(OT, 2T +

72V).

The univariate model (2-2) can also be viewed as a random coefficient model

that assumes the random deviation of a single curve with respect to population

curve at different time points may not be the same. This characteristic will be

determined by the covariance matrix. It is straightforward to see that the test for a

difference between two phenotypic curves is a test of the hypothesis H : mi = m2

at each marker vs H1 : m1 / m2 at some marker. We derive a maximum likelihood

test for this hypothesis. We assume that we can use 1 B-spline bases to estimate

the underlying functional, where 1 < T (This assures that the expression will be

unique). So we can write m = B(, where B is the basis matrix, and is the same for

different functions.









2.2 The Likelihood Ratio Test at One Marker

First we consider the likelihood-ratio test on one particular marker when

S= a2IT + r2V is known. Let 11 and 12 be the total number of subjects in each

group, with 1 + 12 = N. We assume


mi(t') B(1 and m2(t) BB2

with


Yi(t) ~ MVN(BI, E), i=1, ,11

yJ(t) ~ MVN(B2,E), i l+1,--- ,N.

The likelihood function for {(1, 2} is

1 1 N
L oc exp{- (yi Bi)'E- (yi B I) 1 (y- B2)'-1(Yi B2)}.
ii1 i=l1+1

To construct the likelihood-ratio test for HQ : m = m2 on one particular

marker, we first need to find the MLEs of the unknown parameters. Denote the

common coefficient vector under HQ by After taking the first derivative of the

log(L) we get the likelihood equations for ( :
N
E -'(yi B() 0.
i= 1
Now it is straightforward to see that the maximum likelihood estimate for under

H, is ( = (B'-'B)-1B'-'y, where y = i1Yi. Similarly, under H1,

~i (B'-1B)-1B'E-1yi, where Yi ELi y, and y2 z +i Y
The likelihood ratio test statistic is

maxH/ L
max L
exp{- Ei(yi B)'- (yi B})}
exp{- Ei i(yi Bi)'-i(yi Bi) Yi 1+1(yi B2)'Y-1(yi B2)}









The LRT rejects H' when A is small, which is equivalent to rejecting when
N 11
A' (yi B )' -(yi B) (yi- Bg1)'- (yi B1)
i= 1 i= 1
11+12
(yi- B2)'Z-I(yi- B2)
i=11+1

is large. A straightforward calculation (Section 2.5) shows that this is further

equivalent to rejecting Ho when


G (Y Yy2)'Ao(i Y2)


(2-3)


(2-4)


is large, where Ao = E-1B(B'-1B)-B'E-1.

It is straightforward to see that AoE = E-1B(B'E-1B)-1B' is idempotent, and


Yi ~ MVN(mi, E),
11
Y2 ~ MVN(m2, 1)

Y- Y2 ~ MVN(m m2(, ( + -
11 12

and hence,

(y y2)'Ao(y Y2) 2 (1 M M 2)
T~ x1 (ml- m2)'A0(ml- m2
2 2
where XT is a noncentral chi squared random variable with degrees of freedom

n = rank(Ao) = rank((B'E-1B)-'(B'-1B)) = rank(B) and noncentrality

parameter (1/2)(ml m2)'Ao(ml m2). Under Ho, C ~ X, a central chi squared

random variable.

2.3 The Likelihood Ratio Test over All Markers

If we write Y = (y1, Y1T, Y21,. YNT) where the first 11 T yijs' are from

group 1, then the test statistic could be written as a quadratic form G Y'AY,









where

A 1112 1 1112 2 Ao U D Ao (2-5)
N l 1 1 J2

(2 NxN
and var(Y) = IN 0 = E Also, one can easily verify that AE is idempotent. At

different markers, the test statistics have the same chi squared distribution but are

correlated with each other.

Notice that we arrange Y according to which group yi is from, so at different

markers, we will have a different vector Y. To make notation uniform, we can use

a permutation matrix P. If we denote Y1 as the Y for marker 1, then for other

markers, k= 2,..., m,

Yk PkY1,

where Pk is the permutation matrix and the matrix (2-5) corresponding to kth

marker is labelled as Ak. Thus, at each marker i, the corresponding test statistic is

Y'kAkYk = YI' AkPkY1.


Notice that, var(PkY1) PkEP = because E is block diagonal with block size

T T and Pk just interchanges the rows of E. So under Ho we still have


Y'AkYk Y'PAkPkY1 X-


Recall that under Ho, m(t = ml(t) m2, Y1 ~- MVN(1N0m(t,IN 0) )

MVN(p, E) and we can write


YAkYk= YZ' AkiYk

Y/ --Y-kWW 2lYk

IYP Z IkWZk:Z ,PkYl

ZkZk,









where Zk W/ Pkyi d MVN(W E- Pkip, WmWk) and WkWk' are the spectral

decompositions of SAk>I. The matrix Wk is composed of the eigenvectors

corresponding to the n non-zero eigenvalues, and it has n orthonormal columns and

is of order NT x n. Also, under Ho the structure of Ak results in

(W$k Pkp)'Wk PkP

= p'AkP= '(k iN) 0 (Ao m(t))

-= '0 0 (Ao m(tE) = 0.

and thus, under Ho, Zk, x W/-ZPkYl ~ MVN(O, I,).

The entire vector Z (Z/, Z,..., Z)' has distribution

Z MVN(pz, A)


where


Pz w ic: -


and


W' -2'
W' ,P


( _'iW P2ZtW2. PIn ) I


\\W'Z :Pm2
and under Ho, Iz = 0.

If we let B, denote the n-dimension ball with radius equal to x, then


Po( max YAkYk < x)
l SPo( max ZZk < x)
1 = Po(zlZl < X. ,Z',nZ _< x)


I 1ZEBI, ,21FA exp
J {ZieB^ 27}A|


SZZ A lZ J dZi...dZm.









This probability, which is one minus the p-value for Ho, can be directly calculated

by simulating Z ~ MVN(0, A) many times and counting how many Zi's fall in the

n-dimensional ball B,.

All the above derivations are made under the assumption that we know E,

which is typically untrue in practice. We -r-i-.- -1 substituting a REML estimate

of the variance-covariance matrix E instead of E, and, in that case, the above

formulae are correct .,-i-~.l!,il i ically (Wolfinger 1993, Littell et.al 2000). (Please

refer to C'i plter 3 for more details.)

2.4 Examples

In this section we apply our procedure to both simulated and real data. We

first use two simulated datasets to illustrate our method, and then analyze the

Poplar data of Ma et al. (2002).

2.4.1 Simulation

The purpose of this simulation is to examine how the test statistic behaves

as the underlying mean curves move apart from on another. We, of course, expect

that the p-value gets smaller when we move the underlying curves apart.

The first data set assumes the underlying phenotypic curves are two flat lines.

We choose these simple curves because it is easy to measure the distance between

them, and to quantify the notion of the curves getting further apart.

The data set has 10 markers and 100 subjects, with measurements generated

at 4 observation points using the variance-covariance matrix for each yi

.35 .33 .32 .31

.33 .35 .33 .32

.32 .33 .35 .33

.31 .32 .33 .35

Based on the marker information, 100 data sets are generated and analyzed. Since

the underlying curve are the two flat lines y = ao and y = al, we can use |ao all















o

S -



>









0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
distance



Figure 2 1: Graph of the p-value and its 95'. confidence interval for data from
two underlying flat lines with different distances.


as a measure of distance. The p-value trend with respect to different distances

between two underlying flat lines, and its 95'. confidence interval, is given in

Figure 2-1. It is clear that the p-value decreases as the curve moves apart, which

is the behavior we would expect.

The second data set assumes one phenotypic curve is flat (mi(t) = 1) and

the other is quadratic (m2(t) = ax2 + bx + c). In this case, using eight markers,

50 subjects are generated at 12 observation points using the variance-covariance

matrix for each yi


S= .1J12 + Autoregressive(T2 = .05, rho = .5),


where J is a matrix with all ones and the structure of matrix Autoregressive(r2, rho)

is shown as (1-3). Based on the same marker information, 100 data sets are gener-

ated and analyzed. To measure distance, we use the area between the two curves














c9



(D "' \


Co









0.0 0.5 1.0 1.5 2.0
distance



Figure 2-2: Graph of p-value and its 95'. confidence interval for data from two
underlying curves:one is flat line the other is in quadratic form.


over the observed time interval. Figure 2-2 shows the p-value profile for different

distances and its 95' confidence interval. Again, the behavior is what we desire,

with the p-value decreasing as the curves get further apart.

2.4.2 Poplar Data

This data set comes from an experiment of the triple hybridization of Populus

(poplar). In the spring of 1988, a total of 450 1-year-old rooted three-way hybrid

seedlings were planted at a forest farm near Xuchou City, Jiangsu Province,

C'li, The female parent is a Populus deltoides clone and the male parent is an

interspecific P.deltoides x P.nigra clone. At the end of each of the 11 growing

seasons the total diameters were measured(the original data are plotted in Figure

2-3: The left figure shows the original diameter growth trend for every tree. The

right one is the growth curve after taking logs.) A genetic linkage map was

constructed which comprises the 19 largest linkage groups for each parental map,















o

0




EI EI



C-






year year



Figure 2-3: Original poplar data profiles.


and represents roughly 19 pairs of chromosomes. We used our method to detect

QTL affecting diameter growth on linkage group 10 from the P.deltoides parent

map.

Because our model assumes that all t observations have equal variance, we first

transformed the original observations by taking the log. Using our method with the

variance-covariance structure autoregressive+simple (estimated by PROC MIXED

as t = 0.001236 J1 + Autoregressive(r2 = 0.06155, p = 0.8945)), we found strong

evidence that there is a QTL at a marker on linkage group 10 which controls the

growth trajectory of stem diameter in the interspecific hybrids of poplar ( p = .037,

se = 4.6e 4). The 'i-.-- -1 likelihood-ratio test statistics G appears at Marker

CA/CCC-640R. Figure 2-4 shows the estimated growth curves using B-splines for

each group indexed by this marker. This finding is consistent with the result in

Ma, Casella and Wu (2002).





























o

















0-
CO












C) -
E









I I I








2 4 6 8 10

Year




Figure 2-4: The growth curve representing two groups of genotypes at marker
CA/CCC-640R on linkage group 10 in the Populus deltoides parent map.









2.5 Technical Details: Derivation of the Likelihood Ratio Test

In what follows, we derive the likelihood-ratio test statistics from A' (2-3) to G

(2-6). Rewrite A' as


B)'YZ 1(yj


-B)'Z 1'(yi


B) (yji
i 1


B)


Bj1)'Z-l(yi I


i B2)/ -1(yi


N

i=11+1


6i)


B2)


Notice that


B)'Z 1(yj


11
(yi -
i= 1


i= 1


YI)/ -1(yi


B 1) -l(yl


with


i 1


11


i 1





B(1)- (B --


B)


Be).


B~1- )' 0.(B(I


1 B1)--1(B1




B) 0.


yi)Z 1(yi


B,)Z 1'(yi


11
B5j) + Y(B5j


B)'Y 1(BCj


11
A' (yi -
i 1
i=l1
N
+ (yii
i=11+1


A'/ + A.


11

Y(yi


B5)


B51)


Thus


- (Y
S1-

(yi
i= 1
11

i 1


B5).


B)'E- (yi B5)









Similarly, we can get


B1)


I11
Y(yi- B1l)'Y-(yi
i= 1


11
yi) + Z(yI
i-1


BI1)--1(yi


A/!- ( -
A'1 E (B
i= 1


11+12
xA'2 2 (
i=1+1


B)'Z 1(Bi1





-B)'YZ 1(B2


We than have,


A' A' + A/


B+) +12(B 2


B~)'Z 1(B~2


/1(yi y)'Ao(y y) + 12(Y2 y)'Ao(y2


Si(yi Y+ Y) Ao(y1
ilYl + 12Y2 )Ao(y
+ /2(Y2 + N
+ 12(Y2 Y 2 )'Ao (Y2

112 (Y1 Y2)'Ao(y, Y2),
N


IlYi + 12y2

N )
N

N


where Ao = -IB(B'E-IB)-1B'E.-.T

Therefore, the likelihood ratio test rejects Ho when


G = 1 (y y2)'Ao(Yi Y2)
x


is large.


11
E(yi
i= 1


y1)' /1(Yi


Therefore,


and


B-).


(2-6)


B) = 1 (B- B )'-'(BI -




- BO /2(B2 /)--l(B 2


Il(B61- B)'E--(B1














CHAPTER 3
NONPARAMETRIC FUNCTIONAL INTERVAL MAPPING

In C'! ipter 2, we introduced a nonparametric functional mapping procedure

with the assumption that marker data is everywhere. However nature has such a

complicated system that almost all markers in any type of linkage maps are not

even genes. So we extend that theoretical framework further to this case in this

chapter. The joint distribution of LRT statistics on each putative QTL locus can

be obtained approximately. But the resulting P-value from this approximate joint

distribution is a lower bound of the exact P-value. So a simulation procedure is

proposed to calculate the exact P-value.

3.1 Data Setting

As before, we assume for each subject i, i = 1, ..., n, that we have its informa-

tion about both phenotypic and genotypic profiles. On every design point tj (for

example, time point for growth data), j = 1, ..., T, we recorded yij, the observed

value of the underlying function at design point tj. To be simple, we only consider

the population structure from a backcross design, which means there would be two

groups with distinct phenotypes if a QTL does exist. Thus, those y's are actually

from two underlying phenotypic functionals mr(t) and m2(t). mk(t), k = 1, 2 could

be exactly expanded using 1 B-spline basis functions, I < T. Let B = {/3(t)}rTx,

then mk(t) = B(k. The null hypothesis is Ho : mi(t) = m2(t) for any t, i.e,

Ho: 1 = .2
Denote the putative QTL by Q with allele Q and q. Denote its outside

markers by M with allele M, m and N with allele N, n. The recombination rate

between M and AV is r and the recombination rate between M and Q is rl. Let









0 = r. When 0 is relatively small, the conditional probabilities of genotypes at a
QTL has been shown in Table 1-3. By the combination of two outside markers,
arrange the observations y's such that yi, y, is from marker class Mm/Nn,

Yni+l, .. Yn2 is from marker class Mm/nn, Y,2+1, .. YT3 is from marker class
mm/Nn, and Yn1+, -- is from marker class mm/nn.
Again we consider a subject-specific model, for each subject i, i = 1,..., n

yi(t) = m(t) + ailT + Ei, (3-1)

where t (t, t2, ,tTj)'. All notations are exactly as before. So now yi(t-)
MVN(m(t), E), where E a2T + T2V.

Denote I1, 1 = y'E-y and 4)(y; B, E) = exp{ k-B2 }. Then under the
above assumptions, the log-likelihood function for the unknown coefficient vectors

11 and 12 at each fixed 0 is

logL -+ log ((1 l-0)@(y; Bl, E) + 0K(y; B2, Z))
Y 2
i-1 i=nl+1

+ log((1 -0)4(y; B,;2)+ 0(yi; B~, ))+ i, -B2
i=n2+l i= ns+
+ constant.

The likelihood equation for 1 is

1ogL 2 (1 0)4(yi; Bg1, )B'g-l(yi B1)
B -I(yi-B1) (1 O)(:(yi; B,I)O + 0(yi; Bg2 )
n3 0((yi; B1, E)B' l(yi Bg1)
+ > (1 0)(y; B(2, ) + 0(yi; BB1, )*
i=n2+1
Denote
(1 t)@(y; B1, Y)
P(y; t) ( )
(1 t)K(y; B(1, Z) + t@(y; B(2, Z)









Then the likelihood equation for 1I can be rewritten as

0 log L n 2
^ B'E-'(yi- B,)+ + P(yi; )B'E-'(yi- BI)
i-1 i=nl+l
n3
+ P(yi; 1 O)B'- -(y -BBI).
i=n2+1

Similarly, the likelihood equation for 12 can be rewritten as

0 log L n n2
Sl = L B'E- (y- B2) + (1 P(yi; ))B'E- (y- B2)
i=n3+1 i=nl+1
n3
+ (1 P(y; 1 0))B'E- (yi- B2).
i=n2+l

Thus, it is straightforward to have the following EM algorithm to numerically

find the MLE of (1 and (2 :

EM Algorithm: For fixed 0 and known E, repeat the following steps until the

convergence criterion is satisfied.

Step k: Calculate P(yi; 1- 0)(k) and P(yu; 0)(k) using k and 2.

Step k + 1: Calculate

k+ (B'E-1B)-IB'_ 1 ( yi y+ E +2 P(yi; O)yi + YE +1 P(yi; 1 8)y
Y 1 + Z B1 P(yi; 0) + E:3+1 P(yi; 1- 0)


and (k+l (B'E-IB)-IBTE-1 Zn3+1 Yi n ~ (1-- n')Yi1 ))
and (BT B B^ n21_, *.+n3
2-3Ek n4-+n11 ) I1-P (yi;-O))
When the QTL doesn't exist, that is, under Ho, the MLE for the only un-

known coefficient vector is o (B'E-1B)-1B'E-1 E i. So the likelihood ratio









test statistics at each fixed 0 is

maxHo L((|y)
-2logA -2log maxH y)
maxL(|ly)

|ly Bgoll ly- Bgll|
ii i= 1
n2 B 12 -||y B21122
+2 Y log{(- 0)exp( yi 2 l)+0exp( It 2
i=ni+l
n3 2 -||y B2112
+ 2 log{0exp( + (1- ) exp( I )
i-n2+1
n+

E -|y2- 112
i=n3+l

According to classical interval mapping, a profile of likelihood ratio test statis-

tics for an entire linkage map is constructed by calculating it at each putative Q

along the map, where the position can be characterized by outside marker M, N

and 0. Usually the threshold value to reject Ho is obtained through computation-

ally intense v--,v- permutation or simulation. In the following sections, we will

derive the joint distribution of the likelihood ratio test statistics along the entire

linkage map under Ho to calculate the p-value directly.

3.2 Approximate Distribution of Test Statistics at Each Q under Ho



(I o)'B'g-'B(B( o) + 2(~i o)'B'-l (y, B 1). Thus we can rewrite
T2
-2 log A {(1 )'B'E-B( o) + 2(1 o)'B'-l(yi Bli)}
i=

+ E {(2 -0)'B'-lB(2 O) +2( 2 0)'B'-l(yi- B2)}
i=n1+l

i=n+2 2 B2
i#21+1
3 2
+2 logg{0exp(i 2 +2 )+(1-)}
i=n2+1









When -2 log A is big enough, Ho can be rejected. By Jensen's Inequality, we can

show that

log{1 + 0 exp(x)} > Ox.

(A simple proof can be found in Section ??). Let f(x) = log{1 0 + 0 exp(x)} Ox,

then a simple calculation shows that f(x) increases when x < 0 and decreases when

x > 0 with maximum value 0 at x = 0. So when x is in the neighborhood of 0, it is

reasonable to approximate log{1 0 + 0 exp(x)} by Ox.

Under Ho : 21 = 2, we expect 1 ~ so that |y, BlI|| |y, Bl|| is

around 0. Therefore, we can approximate the likelihood ratio test statistics -2 log A

with its lower bound
n2
-21og A > {( o)'B'-0 B( o) + 2( 1- o)'B'L-(yi- B )
i= 1

+ {(2 0o)'B'Z -B(2 0o) + 2( 2 o)'B'- (y B2)}
i=nz+l
+2 o0 JyY B 112 y -B211
2 2 2
i-n+l1
2 3 0 | y Bl- 2 11 yIl B 11||1

22 2
o rn+l \ /


i 1


fl2+l



n3
+ E O{(2 -1)'B-i B(2- ) 2(2 1)gB'-l(Y- 8B2)}


i=nl+l
+ 3 -( h2)'B' g (1 ) Y+2(+ 1 )'B'Z-I(y B-g1)}
ifz2+l

Denote this lower bound by G, which is a ;good" approximation to -2 log A under

Ho. Hence it's reasonable to use the distribution of G under Ho to approximate the

distribution of -2 log A under Ho.









Under Ho, G turns to be a quadratic form Y'AY, where Y is a nT x 1 vector

with yi, ..., y, stacking together and details about A can be found in Section ??.

Furthermore, the matrix A has rank(B) = 1 nonzero equal eigenvalues f. Under

Ho, Y is distributed as MVN(1, B, I, 0) =_ MVN(1, E), it is straightforward

to show that G ~ 32Xa (details in Section ??). Thus the distribution of -2 log A

under Ho is approximated by this multiple of a central Chi-square distribution.



Technical Details:

1. Proof of log{1- 0 +0 exp(x)} > Ox.

Proof: Let a discrete random variable Y with P(Y = x) = 1 P(Y =0) =0,

then E(eY) = 1 0 + Oexp(x). Because log(.) is concave, directly following

Jensen's Inequality we can get log{E(eY)} > E{log(e)} = E(Y) ,that is,

log{1 0 + exp(x)} > Ox.

2. Distribution of G under Ho Rewrite G as


G = n2( ( 0)'B'E-1B(~I )o)

+ (n n2) ( )'B'E- B(2 o)

+ 0(n2 n1)(2 1- 1)'B'E-1B(2 1)

+ 0(n3 n2)(1 2)'B'E-1B(B( 2


i= 1

n2
+ 2 ^ (^ 0)B'Z1(y, B^2)


+ 20 ( 2 1)'B'E- (y,- B2)
i=n1+1
i=n2+l


i=nz+l









Under Ho, the MLE for underlying coefficient


o = (B'E-1B)-1B'E

(B'Y-IB)-IB'S-

= (B'Z- B)- B'-

= (B'Z- B)- B'-


vector


n

n
1(l T)Y

l(wo0 IT)Y

IWoY,


where Y is a NT x 1 vector with yi, ..., y, stacking together.

Under Ho : 1 = 2 everywhere, after enough steps, P(.; 0) in the EM algorithm

should be equal to 1 0. That is,

(1 B'-B)-l~B/ yi+ (1 -) E l+ Yi y 0 3 2+1
n2 + (n3 n2 n2 + nl)O
+ 0 iYn +i + 3 1'n
S // Z +l i=nl+l yi (1 0) :I Z2+l Y
-(B B) Bn n2 (n3 n2 n2 +nl)0'

which now can be represented by a weighted sum of y's.

Denote 1 = (B'/-IB)-1B'/-lWI(Y)Y and 2 = (B'Z-1B)-1B'Z-1W2(Y)Y,

where


W1(Y)


(1ii 1-0 ..* 1-0 0 *.. 0 3On_ )

W w IT

and


W2(Y)

(0on 0 ..* 0 1-0 .. 1-0 1 _n- )


IT
n2 + (n3 n2 n2 + nl)O


IT
n -n2 -(n3 -n2 -n2 + nl)0


W2 0 IT-








Now we can rewrite G as

G n2Y'(W Wo)'-1B(B'E-1B)-1B'-1(W1 Wo)Y

+ (n n2)Y'(W Wo)'-1B(B'E- B)- B'E- (W1 Wo)Y

+ 0(n2 n)Y'(W2- W1)'Y-1B(B'E-1B)-B'-1(W2 W1)Y

+ 0(n3 n)Y'(WI W2)'/-1B(B'E-ZB)-1B'/E-(WC W2)Y

+ 2Y'(Wi Wo)'E/-B(B'E-ZB)- B'E- {V1 n2WiY

+ 2Y'(W2 Wo)'E-ZB(B'E-1B)- B'E-{V2 (n n2)W2}Y

+ 20Y'(W2 W1)'/-B(B'E -B)- B'E1{V3 ((2 nl)W2}Y

+ 20Y'(W1 W2)'-1B(B'E-1B)-1B'E1{V' (n3 n2)WjY,

Where

V (1, Ofl, ) IT V. IT

V2 = (O I,,, IT V2 IT

V1 = 0 fl2 _i 0,_ 2) IT v3 T

V4 (0n2 1,3-n2 on-n3) 1- V4 IT

Obviously, G can be expressed as quadratic form Y'AY, where under Ho, Y is
distributed as MVN(1T B, IT, 0 E) = MVN(7, E) and A = (A + A')/2 with

A1 = n2(W Wo)'-1B(B'-'1B)- B'Z- (1 i Wo)

+ (n- 2 (W2 Wo)'/-1B(B'-1B)-1B'-l(4W2 Wo)

+ 0(n2 n1)(W2- W1)'/-1B(B'-lB)- B' -l(W2 Wi)

+ 0(n3 n2)(W W2)'-1B(B'- B)- B'- (W1 W2)

+ 2(1W Wo)Y- B(B'-1B)- B'- 1{V1 n2W1}









+ 2(W2 Wo)'-1B(B'E- B)-1B'E-1{V2

+ 20(W2 W1)'-1B(B'E -B)-1B'E -{V

+ 20(W W2)'-1B(B'E -B)-1B'E -{V

S[n2(1 wo)'(W o) + (n n2)(W2 -

+ 0(n2 )(w w)'(w2 w) + 0(n -

+ 2(wl wo)'(v n21) + 2(w2- WO)'{V2

+ 20(w2- W)'{V3 (2 n1)w2} + 20(wi

0 E-1B(B'E-1B)- B'E-1

- A2 & -1B(B'E- B)-1B'E-1


-(n n2)2

3 (n2 nlW1 2

- (n3 12)WI

Wo)'(2 Wo)

n2)(wi 2)'(w1

- (n n2)W2-

- w2)'{v4 (n3


Since Y ~ MVN(Jt, E), the quadratic form Y'AY can be expressed as
Y'AY = i" k) iW,, where Wi X'2(7ii). ps, i 1, ..., k are those nonzero

distinct eigenvalues of AE(or AEz), where multiplicity for each eigenvalue Ai is
mi. Notice that AE = (A2 + A') -1B(B'E-1B)-1B'E-1E (A2 + A/)

E-1B(B'E-1B)-1B'. Because E-1B(B'E-IB)-1B' is idempotent with rank I and

rank('(A2 + A')) = rank(A2) = 1 due to the structure of wo and wl, A5 has

1 nonzero equal eigenvalues. Thus k = 1. Also, 7r is the noncentrality parameter
0 -= !'E-~ PP'E-' T, where P consists of I eigenvectors corresponding to 3





20
satisfying j3PP' Y1AYI Note that

e= 'E- I3PP')EZ
1=^/=


Thus G ~ fO3 follows. Furthermore, f > 0 because for all nonzero vectors z, the

quadratic form z'Tz ATz can finally be written as summation of nonnegative value

z'iEzz since E is positive definite.


- 2)



n2)wi}1









3.3 Precision of the Linear Approximation to LRT Statistics under Ho

When the marker map is not dense enough, it is more realistic to adopt the

idea of interval rn illpir_. which uses the two flanking markers to locate the QTL

position. At each fixed putative QTL position indexed by 0 within the kth marker

pair, 1 < k < m, the actual likelihood ratio test statistic -2 log A' has a lower

bound G', which can be written as a quadratic form Y'kAYk. Thus we use the

joint distribution of these G's at all putative QTL positions to estimate the joint

distribution of those actual likelihood ratio test statistics and hence again have a

direct way to get P-value estimate. Because G is .,1 ,- lower than -2log A, the

resulting P-value is lower than actual P-value. But we believe that under Ho, our

approximated P-value is i. ir" enough to the precise P-value. That is,

Conjecture: Under Ho, i.e, when no QTL exists,

Po( max -21ogA l
in probability, when the total number of subjects, n, goes to infinity.

Firstly we have the following lemmas which will be part of those steps to prove

this tentative theorem.

Lemma 1: Under Ho, the maximum likelihood estimate of the unknown coefficient

vector for smoother matrix B, is Vn-consistent for and hence ( -

o,(n-P), with 0 < p < .
Proof: The Vn-consistency of directly follows from the fact that On( ) 0)

is -i-mptotically normal with mean zero and covariance matrix [I(Q)]-1. (See

Lehmann and Casella (1998).)

For any c > 0, by the definition of On-consistency, there exists a constant M6,

so that we have

P(v nlk kl > M) <


for each k, 1 < k < length( ).









Obviously, for any p with 0 < p < ,

n2 | fk 1 k I Me
P c -1,Sf -TI > I <
n2 P n2 P

Obviously, when n -- oo, ,1 -- 0. So p oj(n-P), with 0 < p < A
n2 p
Lemma 2: Under H0, each term in the difference between the true LRT statistics

and its approximate contributed by subject i,

e yi B 1| J|yIlY BJ 11) Yi B I112| I|Yi B211
log{(1 8) + Oexp( 2 2 )} 0( 2 )
2 2 2 2
-0


almost surely in 0 E [0, 1] when the number of subjects, n -- oc. And the

convergence rate is op(n-7) with 1 > 7 > 0.

Proof: Under Ho with putative QTL position index 0 fixed, we have I1 ""s ao
Inyi-B II Ily-yiI t u-.8 0.
and (2 a- 0o. So if we let x -= 2| then under Ho, x 0.

Because f(x) = log[1 0 + 0exp(x)] Ox is a continuous function, we have

f () aa.s 0 as x a .s 0.
By Taylor series expansion, we have


ex = 1+x+x2

log(l + x) = x- x


where x|1 < |x| and |x2| < |x Thus,


f(x) log(1 0 + 0 + Ox + 0x}) 0x

log(1 + Ox + Ox ) Ox

SOx + Ox2 X Ox

SOx2 x2 < O2
1 2









where IX2 < Ox + Ox2 From Lemma 1, the convergence rates for 1 and 2 to are
both n-P with 0 < p < 1/2. That is,

n~( 1 -) 0

P( 2 ) p 0.

So we have

n2p2 2 [2( ) S-1yi + Bi Bl B2j2

S[2nP(2 1)'B' E-1y + n 1B' E-1B~ npB'E- 1B]2

--P [2( )'B'E-yi + n"'B'E- 1B n 'B' Z-'B]2 0.

Therefore, the convergence rate for x2 is n-7 = n-2*p. That is, 0 < 7 < 1. A
Notice that

2 log A G

2 E {log[(1- )+ exp(y B ~ y B2)


( 2 2
2 2

+2 {log[(1-0)+0exp( Y-22 i-2B )]
i=n2+1
yi B2 112 2i B |112
0( Iy)
2 2
n3 n3
S (< Ox OX'X)
i=nl+l i=nl+1

That is, the difference between the actual LRT statistics -2 log A and our linear
approximation G is a summation of many terms, Si. From these two lemmas we

can see that for each Si goes to 0 when the number of subjects, n gets 'i.-.1-r and
1i.v-.-r. But the number of those terms goes to infinity when the total number of

subjects goes to infinity. Is the Si goes to 0 faster than the summation of all Si? A









simple simulation shows the answer is heartbreaking "NO", which is implied in the

following two figures: Figure 3-1 and Figure 3-2. In the upper part of Figure 3-1

shows the wiggly pattern of the summation of all Si when the sample size increases,

that is, Zi Si does NOT go to zero even though it is not big. The wiggly pattern

again in the upper part of Figure 3-2 resulting from -i.- -.--r sample sizes than those

in Figure 3-1 confirms this. The lower parts in both Figure 3-1 and Figure 3-2

si--l. -1 Si does converge to zero when sample size goes to infinity.

So how about a better approximation in the sense of including higher order

terms when we use Taylor approximations to actual LRT statistics -2log A'?

That is, how about approximating fl(x) = 2log(1 + O(exp(x) 1)) by f2(x)

20(x + x2/2) 02( + 2/2)2 + 203(x + 2/2)3/3. We use the simplest setting

that all data are from standard normal distributions where the recombination rate

between two markers is .2 and 0 = .5.. Figure 3-3 plots the difference between

fl(x) and f2(x). We can see that when x is in the neighbor of 0, the difference
between two functions is also around 0. Figure 3-4 and Figure 3-5 show a simple

simulation result about the summation of the differences, which is the gap between

actual LRT and its following approximation, Go. Now the gap is small enough with

expression:


-2log A0 Go
.3
Z {21og[(1 0) + 0exp(x)] 20(x + xf/2)
i=ni+l

+ 02(x + /2)2 203(x + xf/2)3/
Ilyi-B 112 ly lYi-B IlYi-B1112
where xi B= 2 for i nl+1, n2 and x = -

for i = n + 1, ,ns.

Figure 3-6 shows the behavior of the other approximation, which also makes

the gap goes to 0 visually when n -i oo. But it is hard to find the joint distribution
































2I 5 8
2 e+05 4 e+05 6 e+O5 8 e+05


size


4 e+05 6 e+05


8 5
8 e+-05


size





Figure 3-1: The plot of Y 1+ ,1 Si and max~,+l1i<<3 Si 105 when number of
subjects n increases under simplest setting that all data from standard normal
distribution. Recombination rate between two markers is .2 and 0 = .5.


0 e+00


1 e
1 e+06


0 e+00


2 e+05


1 I
1 e+06


ry -- ----~

































0 I I
3000000 3500000 4000000


I I I I
1000000 1500000 2000000 2500000


size



Figure 3-2: The plot of Y :1+1 Si and max~1+li<, 3 S, when number of subjects
n increases under simplest setting that all data from standard normal distribution.
Recombination rate between two markers is .2 and 0 = .5.


1000000

01000000


I I I I I I
1500000 2000000 2500000 3000000 3500000 4000000








47

























o
0 -
0



o





O -


x

0
I I I




0









0 \
0
d


-1.0 -0.5 0.0 0.5 1.0


x





Figure 3-3: The plot of function f(x) = f(x) f2 (x).




















o

o


0


10000 20000 30000 40000 50000

size


U)
U)



.E


E o
S10000 20000 30000 40000 50000


size





Figure 34: The plot of ,1i+1 Si and maxi,,+i<<3 ISi when number of subjects

n increases.















S1 e+05 2 e+05 3 e+05 4 e+05 5 e+05 6 e+05


size
LD



CU)








0 0
E


















]0 1 e+05 2 e+05 3 e+05 4 e+05 5 e+05 6 e+05

size






Figure 3-5: The plot of C ,1+1 Si and max ,i+l<,, ISI when number of subjects

n increases.


T














()
Q>


C
0

0 0
Z 0
CU +
E w
E
= 0
U)


U o
( )


100000 150000 200000 250000 300000

size


E o
o 100000 150000 200000 250000 300000

size




Figure 36: The plot of 31 ,3 1 Si and max1, i S3 IS, when number of subjects
n increases using approximation f2(x)' 20(x + x2/2) 02( + 2/2)2



of the approximated test statistics or if we can, very time-consuming. So we have


to turn to other methods to find the exact P-value, which is stated in the following


section.


3.4 Alternative Approach to Calculate P-value: Simulation


Because we finally find that the linear approximation to the actual likelihood


ratio test statistic does not have good/acceptable properties, we have to turn to


other methods to decide the threshold showing significant evidence of existing


QTL. Notice that when Ho is actually true, after enough steps in EM algorithm,


the maximum likelihood estimate of two unknown coefficient vectors, E1 and 12


can be directly written as linear combinations of realized phenotypic vectors.


That is, using Y as a NT x 1 vector with yi, ..., y, stacking together we can have









i = (B'E-ZB)-1B'E-1W1(Y)Y and (2 = (B'E-1B)-1B'E-1W2(Y)Y, where

W1(Y)

01-0 -- 1-0 0 8 0 0IT
( ) IT
n2 + (n3 n2 n2 + n1)0

Wi 0 IT

and

W2(Y)

On, 0 ... 0 1-0 ... 1- 0 1_ --3 IT
Sn n (n3 n2 n2 + n1)
Sw2 0IT-

Of course, the MLE of o under Ho can also be expressed as a linear combination of

Y,

0o = (B'E-1B)-B'E-1 1


=(B'E-ZB)-1B'E- (1w0 IT)Y
(B'g-lB)-lB'E-l(w0 @ IT)Y

(B'Y-1B)-1BZ- 1WoY.

We can have following simulation procedure which can determine the cutoff
point for detecting QTL from a genome/linkage group:
1. Generate 1000 different data sets from Ho, that is, having same variance-
covariance structure and based on same known marker information. We can
easily generate y, Bio, y, B1, y, B 2 ,i = ..., n, for each fixed 0.
2. For each interval calculate the likelihood ratio test statistics at these fixed 0
based on simulated data; Obtain 1000 different maximum likelihood ratio test
statistics through the whole genome(linkage group).






51




Histogram of maximum of actual LRT statistics

0


I -




0 5 10 15




Histogram of maximum of simulated LRT statistics
0

o o
0 0
'I.0

I I I I
0 5 10 15





Figure 3-7: Two simulated distributions of likelihood ratio test statistic based on
same 1000 datasets.


3. Get the cutoff point value from the simulated distribution of maximum of

LRT statistics.

To check the validity of this simulation procedure, we get the distribution of

maximum of LRT statistics calculated using MLEs from EM algorithm. Figure 3-7

shows the two histograms resulted from each procedure, which confirms the validity

of this simulation procedure. In that figure, under Ho, the underlying phenotypic

curve is a flat line. 100 subjects are measured at 4 different points. Marker interval

is 25 C'l\ and scanning unit is 2 C'\ To be simple, we only assume this linkage

group has 3 markers, that is, two marker intervals.

By treating 0 as unknown parameter and finding its MLE, we need only

calculate one likelihood ratio test statistic for each marker interval, which is already









the maximum LRT statistic for that marker interval. The log-likelihood function

for the unknown coefficient vectors 1, 2 and 0 is
ni R/ II2 n2
logL -+ log {(1 0)4(y; B~1, E) + 0I(yi; B2, )}
2 + logL-2
i=1 i=nl+l

+ log {(1- 0)@(yi; B2, )+ 0(yi; B )}+ -B2
2
i=n2+1 i=n3+1
+ constant. (3


2)


The likelihood equations for (1 and (2 have been derived before. And using a

function defined as

P(y; 0 (1 t)K(Iy; BI, E)
P(y; t) =
(1 t);(Iy; B~I, E) + t ;(Iy; B2, Z)'

we can easily write out the EM algorithm for finding (1 and 12 when 0 is fixed.

Now when 0 is also an unknown parameter, its likelihood equation is

OlogL y K (y; B2, Y) 4(y; BI, E)
o0 (1 0)4(yi; BI, Y) + 0((yi; B2, Z)

+ 3 <:I(yi; BI, Y)-- (yi; B2, Y)
+ (1 0)4(yi; B2, Z) + 0(yi; B~I, E)
i=n2+1



+ (P(yi; -0) P(y; -0)
2+ 1-0
i=nz+l

Thus, it is straightforward to have this following EM algorithm to numerically

find the MLE of (1, 12 and 0 simultaneously:

EM Algorithm 2 : Suppose E is known. Repeat the following steps until the

convergence criterion is satisfied.

Step k: Calculate P(y); 1- 0)() and P(y; 0)(k) using 1, e and 0().









Step k + 1: Calculate


k+1) (B'Z -B)- B'/-1

(nl1 i n+ Z + P(yi; )(k)i + n2+ ; 1 0) (k)
n + E 1 P(y; O)(k) + EZ +1 P(; 1 )- ,(k)
(k+1) (B'-1B) 1B'-1

(Y .n3,+ Yi + Z2+[1 P(i; )() + Z1[1 P((i; 1 0(k)]i
fl3 + Z2 1 P(y,; 0)(k) Z-n+ P(y; 1- )(k) (

0(k+l) T 2 + E~ P(; i (k) En1+ P(y; O(k)
'3 1

Now the simulation procedure to determine the cutoff point of testing if there

is a QTL existing in this linkage group(genome) is stated below:

1. Generate 1000 different data sets from Ho.

2. For each dataset and each interval, find the MLEs of 0 and hence calculate

the maximum of LRT statistic among this interval by easily generating

Yi B0o, Yi BIy, yi B2 ,i = ,..., n.
3. For each dataset, move to each of the left intervals and repeat step 2. Obtain

1000 different maximum LRT statistics through the whole genome(linkage

group).

4. Get the cutoff point value from the simulated distribution of maximum of

LRT statistics.

Figure 3-8 shows two empirical distributions of the genome-wise maximum

LRT statistic. Under H0, the underlying phenotypic curve is a flat line. 100

subjects are measured at 4 different points. Marker interval is 25 C\ !. To be

simple, we only assume this linkage group has 3 markers, that is, two marker

intervals. The upper histogram is from the EM algorithm with fixed 0 and the

lower histogram is from the simulation procedure with 0 from the second EM

algorithm. The 95'. and 9' I. cutoff points from the upper procedure are 7.986






54

















Histogram of actual LRT statistics


o0
-a

0

0-
HI


5 10 15 20


Histogram of simulated LRT statistics


o

a

0-


5 10 15 20


Figure 3-8: Two simulated distributions based on same 1000 datasets when treat-
ing 0 as unknown parameter.












Histogram of actual LRT statistics

0
C


0 F- 1-_
I I I I I
0 5 10 15 20







0


0-









treating 0 as unknown parameter.
0 5 10 15 20




Figure 3 9: Two simulated distributions based on different 1000 datasets while
treating 0 as unknown parameter.


and 11.818, respectively. And the corresponding values from the lower procedure

are 7.944 and 11.802. These similar numbers with the similar distribution shapes

su-l--- -1 the correctness of above simulation procedure.

Remark:

1. When we generate 1000 data sets from Ho, the underlying phenotypic curve

under Ho can be randomly picked including /o = 0. Figure 3-9 shows two

empirical distributions from generating datasets from two different phenotypic

curves.

2. When Ho is true, the EM algorithm for finding MLE of 0 converges slow.

But when Ho is not true, the convergence is very fast. So I think for the

simulation procedure we can just use fixed 0 which is computationally less

time-consuming.









3.5 Estimation of Consistent Variance-Covariance Matrix with Small
Sample Size

In a general Gaussian linear model for n observations Y ,Y ~ N(XB/, E), the

precision of inference for an unknown vector 3, which is usually the main interest of

investigators, depends on the estimation of the unknown variance-covariance matrix

E. So how to estimate the unknown elements in E attracts a lot of attention. We

are principally concerned with the situation where the data consist of a collection

of independent sets from different subjects, that is, where E will be block-diagonal

(assume each block is the same, denoted by V). Obviously this type of model is

appropriate for repeated measurements experiments.

Restricted(residual) maximum likelihood (REML) has been well established af-

ter being introduced by Patterson and Thompson (1971) to estimate the unknown

parameter in E. The advantage of REML over a standard maximum likelihood

procedure is that REML takes account of the loss in degrees of freedom that results

from estimating fixed effects and it leads to unbiased estimators for a balanced

dataset. The REML estimators are defined as the maximum likelihood estimator

(\! .;) from the marginal likelihood of a linear transformed set of data Z = AY

such that the distribution of Z does not depend on Y. It turns out that the

marginal likelihood of Z does not depend on the choices of A and can be expressed

as (assume e is the vector of unknown parameters in E)


-2logL() = log{| Z} + log{lX'Z-1Xl}

+ Y'l{-1 l-1X(X'- 1X)-1X'- 1}Y.


The REML of e is the vector, denoted by O, which maximizes the above log-

likelihood function, whereas the standard MLE, O, maximizes this log-likelihood

function


-2logL(O) = log{| |} + Y'{E-1 1 -IX(X'-1X)-IX'Z-1}Y.









The only difference between this two log-likelihood function is the term log{|X'E-1X }.

Also, it is straightforward to see the form of the design matrix X must be correctly

specified, otherwise we may not even get consistent estimators for R. So Diggle,

Liang and Zeger (1994) recommended to use a saturating model for the mean

response profiles to get a guaranteed consistent estimate of variance-covariance

matrix.

For experiment where observed values from different subjects are measured at

same set of observation points, a robust estimate for V can be obtained using the

REML principle. Suppose for each of mi experiment units in ith of g experimental

treatment units, measurements are made at each of T observation-points tk. The

complete set of measurements are


vijk, i l1,...,g, j-1,...,m i, k-1,...,T.

The saturated model for mean response is E(yijk) = I-k. Thus, we could use the

ordinary least-squares fit to get a consistent estimate of V :


(9 ~1 9 mi
v= (- g) -- /Kyyy /4)'
i= 1 i= 1 j= 1

with i = (m-1 ji1 ylijl,, mT.. 1 -i1 ijT )'. This approach can be extended to

the situation when measurements times are not common to all units by estimating

each Vi in the block diagonal variance-covariance matrix E. This approach fails for

the extreme case where sets of measurements times are essentially unique for each

unit.

However, explicit modelling of the covariance structure has been popularly

considered for these following reasons. Firstly, when the true covariance matrix has

many fewer parameters than the unconstrained variance matrix, the estimate can

be made more accurately. Secondly, the objection to estimating 1T(T + 1) param-

eters in the covariance matrix gains force when T, the number of observations per









experimental unit is large. Thirdly, this robust approach uses replications across

experiment units to estimate the covariance matrix non-parametrically. When there

are a lot of missing values or sets of measurements times are essentially unique for

all units, a parametric modelling of covariance structure can avoid these problems.

M w!: statistical software provides general linear model fitting with a variety of

parameterized covariance structures. Littell et.al (2000) provided a tutorial of

using the MIXED procedure in SAS, which can model a rich selection of covariance

structures through the RANDOM and REPEATED statements.

One can choose the best fitting one using Akaike's Information Criterion (AIC)

or B ,-i, Information Criterion(BIC). The formulae for these two criteria are


AIC L(O)-q

BIC = L() (q/2)log(N*)


where L(O) is the maximized log-likelihood or restricted log-likelihood, q is the

number of parameters in the covariance matrix, p is the number of fixed parameters

and N* is the number of subjects: N for ML and N p for REML. Models with

larger AIC or BIC yield a better fit. The difference between these two criterion is

that BIC puts more penalty on the number of parameters used to fit covariance

matrix.

When the covariance stationarity assumption is likely to fail, one can use some

transformation to stabilize the dependence structure. Zimmerman and Ndnez-

Ant6n (2001) proposed a structured antedependence (SAD) model to estimate such

a nonstationary covariance matrix directly and showed such a model di- pl1 i- many

favorable properties. The variances on each observation point are not assumed to

be constant in this model, and the correlation between measurements equidistant in

time are not assumed equal.









Either as a guide to the formulation of a parametric model or not depending

on parametric assumption, non-parametric estimation of covariance structure has

also been proposed by a lot of authors. For example, Diggle and Verbyla(1998)

proposed to use kernel weighted local linear regression smoothing of sample

variogram ordinates and of squared residuals to get a nonparametric estimator for

the covariance structure.

For our nonparametric functional mapping procedure, we need find a consis-

tent estimator for the underlying covariance structure that behaves well when the

sample size is small. It is known that the sample covariance matrix is guaranteed

to a consistent estimator, however it can be very unstable with smallest estimated

eigenvalues being too small while largest estimated eigenvalues being too large,

especially for small sample sizes. Obviously standard estimators, like REML or

ML can gain some stability in estimating the matrix in small samples by assuming

some parametric covariance structure that involves estimation of fewer parameters.

But such estimators are only consistent when the hypothesized structures are

correct. No ..-i- !!I i ical properties of nonparametric estimators are studied by any

authors.

Daniels and Kass (1999) provided a better way to estimate the covariance

matrix by specifying an appropriate prior for the covariance matrix and choosing

an estimator based on a particular loss function. Their method first generically

stabilizes an unstructured estimate and then shrinks such estimates toward a

parsimonious, structured form of the matrix. How much shrinkage required is

decided by the data. Daniels and Kass (2001) extended this method to more

easily calculated estimators without using a fully MC'\ C approach. They called

such estimators empirical (or approximate) B.,l estimators. The final estimator

includes a combination of shrinking the eigenvalues and then shrinking toward

structure. The data decides the amount of shrinkage. These estimators induce









stability over the unstructured estimator of the covariance matrix while providing

robustness to misspecification of the structure. These estimators are consistent.

From their simulation study to evaluate the risk in estimating a covariance matrix,

Daniels and Kass recommended first shrinking the eigenvalues of the unstructured

estimator by replacing it with the Stein estimator and then shrinking the Stein

estimator toward a structure using a structured log eigenvalue or correlation

shrinkage estimator. This is because when the structure is far from correct,

the correlation shrinkage estimator is worse than the structured log eigenvalue

estimator. So we will adopt such an estimating procedure into our functional

mapping approach:

Step 1: Find an unstructured covariance matrix estimator, E, from a saturating

model.

Step 2: Shrink the eigenvalues of E, to obtain a more stable estimator, Stein's

estimator, Est = OA(A)OT, where O is the matrix of normalized eigenvectors,

A is the vector of sample eigenvalue, and A(A) diag(A (A), A;(A)) with


A^(A) nAj/aj

a = n p + 1+2A 2
i'j Aj Ai

Step 3: Fit the saturating model assuming some covariance structures, using AIC

or BIC to choose one. Get estimator Es. This step can be easily handled by

SAS, proc MIXED.

Step 4: Determine the amount of shrinkage r2 using Est and E8.
P
,2 (log(A,t) 0log(Ai,)/p)2/(p + 4) 2/n.
i= 1

This estimator corresponds to the posterior mode under a uniform shrinkage

prior on r2, 7r() = (2/n)/(2/n) + 72)2.









Step 5: Shrink Est toward chosen a structure using structured log eigenvalue

estimator sh = Odiag{A}OT, which is the final estimator. This step shrinks

the log of eigenvalues again toward the structure.


A = exp (2/^2 E log(,)/p 2/^2 l it
2/n + 2 2/n + 2o

This estimator corresponds to the posterior mean of Ai conditional on

estimates log(A) and 2. The prior distribution on the logarithm of eigen-

values log(Ai), i 1,,...,p, is normal: log(A)|IT2 _i.i.d N(log(A), r2).

The .i-i n,! .1 ic distribution of the logarithm of the sample eigenvalues is

log(Ai) N(log(Ai), 2/n). We use log(Ai,s)/p to estimate log(A).

3.6 Application to Poplar Data Revisited

3.6.1 A Chromosome-wise Analysis on Linkage Group 10

We apply this method to the same Poplar Data set as before, where 61 plants

all have information from 8 markers of linkage group 10 and 11 observations

from the same set of measurement time. To compare with previous ain 1, i--i;

using nonparametric functional interval mapping with linear approximation of

LRT statistics, we first log-transformed the original observations. We use order

3 B-splines because we need not to have smooth second derivative functions for

the underlying phenotypic curves. And the measurement time is at each end of

continuous 11 years. So we use equidistant inner knots at (1,3, 5, 7, 9, 11) to fit the

growth curve. As we know in Section 1.4, we need two more knots at each end of

the inner knots vector. Thus, we actually use 7 splines to fit the growth curve.

Procedure to find the consistent estimate of E: We will first get a

consistent variance-covariance matrix from a saturating model, then using this es-

timate to proceed following functional interval mapping approach. This estimating

procedure of E is adopted from Daniels and Kass(2001).









Step 1: Find the unstructured covariance matrix estimator E, from a saturat-

ing model, yi = k + Ei, i = 1, ,n and var(i) = E. k = 1, ,4 according to 4

possible combinations from the first two markers. E, has 11 distinct eigenvalues:

0.328, 0.086, 0.057, 0.011, 0.005, 0.001, 0.0009, 0.0006, 0.0002, 0.0001,6.6e 05.

Step 2: Shrink Z, to its Stein's estimator Es^, whose eigenvalues are

0.276, 0.072, 0.055, 0.010, 0.005, 0.0011,0.0010, 0.0007, 0.0002, 0.00017, 8.3e 05.

Step 3: Fitting the saturating model assuming some structure. We used the

previous estimator having structure autoregressive+simple chosen by SAS proce-

dure PROC MIXED, that is, Es, 0.001236*Jin+Autoregressive(r2 = 0.06155, p-

0.8945). This estimator has eigenvalues: 0.485, 0.107, 0.037, 0.018, 0.011, 0.008,

0.006,0.005, 0.004, 0.0037, 0.0035.

Step 4: Determine the amount of shrinkage r2 = 6.533 using Est and Es.

Step 5: Shrink Et using r2 to get final estimate Esh, whose eigenvalues now

are 0.272, 0.071, 0.054, 0.010, 0.0054, 0.0011, 0.0010, 0.0008, 0.00024, 0.00018, 8.6e -

05.

From the eigenvalues of each variance-covariance estimator we can see the

final estimator is closer to the unstructured variance-covariance estimator. Using

this empirical B-vi z variance-covariance estimator, we find the maximum of LRT

appears on the first interval with = .123 (about 3cM from the first marker

CA/CCC-640R). The maximum value of -2 log A is 15.487 with p value = .143.

The cutoff point for a-level at .1 is 16.84. (18.99 for a = .05 and 23.10 for a = .01.)

Therefore, we could not make a conclusion that a QTL governing stem growth

exists in linkage group 10 if we used EB estimator.

If we use the REML estimate of the underlying variance-covariance with

the same structure in Section 3.10.2, and use the simulation procedure to find

the cutoff points instead of using the approximate joint distribution of all LRT

statistics, the p-value is 0 with maximum LRT value 62.79 at 0 = .32 (about 8.5cM

















LO




0


0r
0)









2 4 6 8 10

year


Figure 3-10: Fitted curves from using two different variance-covariance estimator.


from the first marker CA/CCC-640R). The .01 a-level cutoff point from simulation

is 23.72. (17.26 for a = .1 and 19.26 for a = .05.) This finding are consistent with

previous analysis (p-value= 5e 4 with standard deviation 5e 4).

In Figure 3-10 the black lines are from Esh while the dashed lines are from

E,. Figure 3-10 shows that the fitted curve does not change much due to different

Z but the significance of the difference between two curves caused by the putative

QTL depends on how to estimate E.

3.6.2 A Genome-wise Analysis Using REML Estimate

For the Poplar data a genetic linkage map was constructed which comprises

the 19 largest linkage groups for each parental map, and represents roughly 19 pairs

of chromosomes. AR(1) repeated measurement errors are assumed to model the

within subject correlation and a log-transformation is applied to the raw data of









Table 3-1: Analysis of Poplar data using nonparametric functional interval map-
ping (NPFIM) and parametric functional interval mapping (PFIM). "EB" the
empirical Bw,--; estimator."REML", the REML estimator corresponding to struc-
ture a2J +Autoregressive(r2, p).

Variance Estimate NPFIM PFIM
REML <.001(62.69) <.001(37.06468)
EB .143(15.487) >.5(.858)


stem diameters to stabilize the age-dependent variance heteroscedasticity (Wu et.al

2004). Figure 3-11 shows the likelihood maps resulting from our nonparametric

functional mapping approach. The empirical estimate of the critical value is

obtained from 1000 simulation tests and we find the threshold value for declaring

the genome-wise existence of a QTL is 32.01 at the significance level P = 0.01. The

QTL candidate positions are the positions corresponding to the peaks of curves

higher than the critical value. From Figure 2, there is significant evidence showing

that several QTLs exist in linkage group 1, 2, 4, 7, 10, 14 & 18.

3.7 Simulation Study for Nonparametric Functional Interval Mapping

Table 3-1 shows the analysis result when using nonparametric functional

interval mapping (NPFIM) and parametric functional interval mapping (PFIM)

with two different variance-covariance estimators, respectively. The first number

in each cell is the P-value and the number in parenthesis is the maximum LRT

statistic along the linkage group 10.

Since the true underlying variance-covariance structure for the Poplar dataset

is unknown and the QTL location is also unknown, it is inconclusive regarding the

behavior of NPFIM and the behavior of the consistent estimator using procedures

introduced in Section 3.5. So we use the following simulated datasets to explore the

characteristics of our NPFIM procedure. We use the marker information of linkage

group 10 in the Poplar experiment and the true QTL exists in the first marker

interval, 8cM away from the first marker. Growth profiles of 61 subjects are from













0
4 -



0 0
b i


o



o 4-
















O m





















w4-
O o 0



oO
5 --



t c

m -E


o
2 0







a -&






o 5 5c ) oa












C^ _
^z k C1
.. a- "
s t-









Table 3-2: P-values and its standard deviation of nonparametric functional interval
mapping (NPFIM) using different variance-covariance estimators under different
true Es. "EB" means the shrinkage estimator which is guaranteed to be a consis-
tent estimator. "REML." estimator is the REML estimator corresponding to a
wrong structure. "REML" estimator is the one selected from SAS Proc MIXED us-
ing BIC and assuming each subject has different underlying growth curve. "True"
estimator is of course the matrix we actually used to generate data.

Variance Estimate NPFIM
1 E2 E3 E4
EB .0864(.0108) .288(.0223) .384(.0279) .458(.0293)
REMLw .00789(.0023) .0687(.0099) .143(.0165) .329(.0213)
REML .0011(.00044) .103(.0151) .209(.0238) .497(.0274)
True .00045(.00034) .0727(.0125) .197(.0240) .413(.0285)
The numbers in parentheses are the sampling errors of the P-values.

two underlying functions: 20/(1 + 20 exp(-.6 t)) and 30/(1 + 27 exp(-.9 t)).

Four randomly chosen different variance-covariance matrixes for log-transformed

data are


El1 = 0.1 Jl + Autoregressive(r2 = 0.06, p= 0.8)

E2 = 0.1 Jl + Autoregressive(r2 = 0.1, p = 0.8)

E3 = 0.3 Jl + Autoregressive(2 = 0.1, p = 0.8)

E4 1 J11 + Autoregressive(r2 = 0.6, p = 0.8),

where Jl is a square matrix with all ones with dimension 11 and Autoregressive(r2, p)

is defined as (1-3). Heritability curves corresponding to these four covariance struc-

tures are plotted in Figure 3-12.

Table 3-2 shows the analysis outcome of NPFIM using different variance-

covariance structures. From this table, we can see

The guaranteed consistent estimator is rather conservative. The final empir-

ical B,-.. ; covariance matrix estimator from the procedure stated in Section

3.5 merely shrinks to the true structure a little bit, that is, it is not very

stable because of more similarity to unstructured estimators.








































(0
0




0

't


C)


(N
0



0



0
0
C\


C)



'7
C)


2 4 6 8


Year


Figure 3-12: Heritability curves for the four simulated data sets using different Es.









Table 3-3: P-values and its standard deviation of nonparametric functional inter-
val mapping (NPFIM) using different variance-covariance estimators for different
sample size and E = E. "EB" means the shrinkage estimator which is guaranteed
to be a consistent estimator. "REML'.- estimator is the REML estimator corre-
sponding to a wrong structure. "REML" estimator is the one selected from SAS
Proc MIXED using BIC and assuming each subject has different underlying growth
curve. "True" estimator is of course the matrix we actually used to generate data.

Variance Estimate NPFIM
N 61 N 200
EB .384(.0279) .0239(.00787)
REMLw .143(.0165) .00165(.00056)
REML .209(.0238) .00105(.00064)
True .197(.0240) .00095(.00058)
The numbers in parentheses are the sampling errors of the P-values.

When the structure is correct, REML works fine even when the sample

size is not very big. The performance of the REML estimate from wrong

structure is worse than the performance of REML estimate corresponding

to true structure when the phenotypic curves show little variation from the

mean curve like the first column in Table 3-2. But when the variation is big,

"REML.-- estimator works better than "REML" estimate in terms of smaller

P-value. This may due to the small sample size and the variance-covariance

structure we chose to simulate the data set.

We also do the same NPFIM analysis for data from underlying variance-

covariance matrix E3 but the sample size has been increased to 200. Table 3-3

lists the change of P-values when the sample size increases. When the sample

size increases, because the underlying phenotypic curves are indeed different, the

smaller P-value is exactly what we expected. With larger sample sizes, the EB

estimate works fine. We can also see that when the sample size is big the REML

estimate of true structure outperforms the REML estimate of the wrong structure

in terms of smaller P-values.









3.8 Further Evaluation of Empirical Bayes Estimate of Covariance
Matrix

From the simulation studies in Section 3.7, we recommend to use the REML

estimate selected using AIC or BIC from a saturating model instead of the em-

pirical B-i,--i (EB) estimate even though it is a guaranteed consistent estimate.

Because empirical B-i,-- estimate is essentially obtained from shrinking an unstruc-

tured REML estimate toward some structure, it is rather conservative in the sense

that large sample sizes are required to make the EB estimate behaves like a true

covariance matrix. The following simulation studies are conducted to further study

the performance of the Empirical Biv-- estimate.

The first simulation study uses the 61 subjects' marker information of linkage

group 10 in the Poplar data set. The underlying functions are two logistic growth

curves: 12, -.6 and 3+-.9t where t = 1,..., 11. Autoregressive correlation is

assumed for any two observations. The covariance matrix is determined by letting

the heritability on year 4 (The genetic variance is the 'Li-:. -1 one in this year.)

equal to (.15, .2, .25, .3, .35, .4, .45, .5, .55, .6), respectively. The heritability curves

across all 11 years are shown in the left part of Figure 3-13. 100 datasets for each

heritability value are generated to perform the nonparametric functional interval

mapping (NPFIM) procedure. The average P-values are shown in the right part of

Figure 3-13 corresponding to each heritability value. This figure tells us that EB

performs better when heritability is ri. --r. But, in reality, we have no information

about how big the heritability is.

The second simulation study also uses the 61 subjects' marker information

of linkage group 10 in the Poplar data set as genotypic data. The two underlying

biological trajectories are from the HIV dynamics mechanism, which have double

exponential forms exp(12 .7 t) + exp(7.5 .05 t) and exp(ll .4 t) + exp(5 -

.03 t). The two mean curves are illustrated in Figure 3-14. Assume there are 20






70

















0 0


S- EB

/ \\\REML
STrue

0












o0


2 4 6 8 10 0.2 0.3 0.4 0.5 0.6

Time Heritability



Figure 3-13: The left plot shows the 10 heritability curves of simulated growth
data sets. The right plot shows the trend of P-values when heritability on year 4
increases. "EB" and "REML" means the Empirical B-.iv- estimate and REML es-
timate of covariance matrix, respectively. "True" represents the covariance matrix
used to generate data.
used to generate data.









observation point. The covariance matrix is randomly generated without putting

any known structure. Figure 3-15 plots the heritability value on each time points.

100 datasets with 200 subjects are generated to do the NPFIM analysis. NPFIM

analysis is also conducted for a sub-dataset containing 61 subjects randomly

selected from each datasets. The best structure picked by SAS Proc Mixed is

autoregressive moving average structure, ARMA(1,1). The result is in Table 3-4.

For easy comparison Table 3-3 is also put in this table. From this table we can

conclude that

When the sample size increases, the P-value gets smaller and so does the

standard deviation regardless of any covariance matrix estimate.

If the true covariance matrix has some structure such as autoregressive,

Toeplitz and so on, the REML estimate usually outperforms the empirical

B i.- estimate, as -i--.- -1 I by the result from the growth data set.

If the true covariance matrix is actually unstructured, the empirical Bayes

estimate is better than the REML estimate.

However, when analyzing a real data sets, there is typically some pattern in the

correlations among repeated measurements/longitudinal data. So the REML

estimate is still recommended even though the empirical B-.i estimate performs

well when the sample size is big.

3.9 Discussion

Given these simulation results in Section 3.7, REML estimates work better

than guaranteed consistent empirical Bayes estimates. So, in practice, we recom-

mend using the best REML estimates selected using the AIC or BIC criterion with

a saturating model.

When we tried to analyze the simulated data using traditional functional

interval mapping method, we found it is very computationally intensive because

of the nonlinear nature of the functional forms. For example, for the same data





















O
00

E





0
Q.


o
J -
(0 -


5 10 15 20


Time


Figure 3-14: The HIV dynamics curves used as the mean curves.



Table 3-4: P-values and standard deviation of nonparametric functional inter-
val mapping (NPFIM) from HIV dynamics data and growth data for different
combination of variance-covariance estimators and sample size. "EB" means the
shrinkage estimator which is guaranteed to be a consistent estimator. "REML"
estimator is obtained from SAS Proc MIXED assuming each subject has a different
underlying mean curve. "True" estimator is, of course, the matrix we actually used
to generate data.


Variance HIV dynamics data
Estimate N = 61 N = 200
EB 0.4018(0.0300) 0.00126(< .0001)
REML 0.1644(0.0169) 0.03346(0.0042)
True 0.0376(0.0167) < .0001(< .0001)
The numbers in parentheses are the sampling errors of


Growth data
N = 61 N = 200
.384(.0279) .0239(.00787)
.209(.0238) .00105(.00064)
.197(.0240) .00095(.00058)
the P-values.







73























0



o
0

d -









d0

5 10 15 20

Time



Figure 3-15: The heritability curve across all observation time points in the HIV
simulation study.









setting from simulation for Table 3-2, the 1000 simulation tests to decide the

critical value of LRT statistics takes over 10 times more time when PFIM is

used (We used the simplex algorithm to find the MLE of functional parameters)

than the time needed when NPFIM is used. Even worse, the simplex algorithm

sometimes gets stuck in local maxima and results in negative likelihood ratio test

statistics. Therefore, the similarity to simple linear regression of our purposed

nonparametric functional mapping method has computational advantages compared

to the traditional functional mapping method.

As in ('! Ilpter 1 Section 1.4, it is pointed out that B-splines are very attractive

for nonparametric regression, but the selection of the optimal number and position

of knots do not have a standard rule. When we analyzed the Poplar dataset and

simulated datasets, we used B-splines with equidistant knots and order 3. If one

needs more control of flexibility, one may used P-splines as Eilers and Marx (1996)

proposed.

Functional mapping is parametric in nature, which allows one to easily test

many different biological hypotheses by testing for equality of parameters. For

example, one could test if a QTL starts or ceases to exert an effect on growth

trajectories by testing hypothesis Ho : ai/(1 +ble-rit) = a2/(1+b2-r2t*) at a given

time t*. Testing this hypothesis is equivalent to testing the difference between the

model with no restriction and the model with the restriction: ai/(1 + ble-rlt*)

a2/(1 + b2e-r2t*). Our method could also be used to test such an hypothesis. In this

example, we can first find the MLE's of the coefficient vectors under the restriction

mi(t*) = m2(t*) and compare whether there is a significant difference between the

two coefficient vectors.

Figure 3-16 shows the empirical distributions of the test statistics under

Ho resulting from the simulation procedure proposed in ('! Ilpter 3, from which

the critical point showing significant evidence of existing QTL is determined.









From this figure we can easily see that, with all other information is the same

(sample size, marker information, underlying function or biological mechanism),

the empirical distributions are very similar to each other when the empirical B-iv.

procedure is used to estimate the four different covariance matrixes. The same

finding is observed for the cases using the exact underlying covariance matrix. This

interesting phenomena might be due to the orthogonality between the coefficient

vectors Is and the covariance matrix E.

If there is no QTL, that is, 0 0 or 1, there is no mixture in the likelihood

function:
i n1 +n2 2 N
L xo exp{- Y (yi B1)'- (yi B1) (y B2 -1yY B2)}.
i=1 i=nl+n2+1

The off-diagonal elements in the Fisher information matrix are (suppose a is the

unknown parameter in covariance matrix E):

02(-2 log L) E0E
aE a E U {y B) = 0.

So 1, 2 and E are orthogonal to each other. Cox and Reid (1987) stated that

the maximum likelihood estimates of 1 and 12 move slowly with respect to E.

(Convergence rate is O,(1).)

Furthermore, when the sample size is the same, even for testing QTL control-

ling different biological trajectories, the empirical distributions of the test statistic

are similar, which is illustrated in Figure 3-17 where the two empirical distribu-

tions for data sets simulated from different biological trajectories but with equal

sample size are plotted. This is because Ho only states that when there is no QTL,

the underlying functions are the same but does not specify the functional form.

But for data sets with different sample sizes, even if all the other information is the

same, there is a big difference among the resulting empirical distributions.




























Histogram of dataset 1,EB
0


S I I I I I I
0 10 20 30 40 50


Histogram of dataset 2,EB


Histogram of dataset 1,True


0 5 10 15 20 25 30 35


Histogram of dataset 2,True


0

I I

0 10 20


I I I
30 40 50


0 5 10 15 20 25 30 35
0 5 10 15 20 25 30 35


I I I II
0 10 20 30 40 50

LRT


Histogram of dataset 4,EB



AhzL___


I I I
0 10 20


30 40 50
30 40 50


Histogram of dataset 3,True



] I f If I t I I I I I

0 5 10 15 20 25 30 35

LRT


Histogram of dataset 4,True


( o
0 I I I I
0 5 10 15 20 25 30 35


Figure 3-16: The empirical distribution of the test statistics in the simulation

study of C'! lpter 3. Dataset 1-4 correspond to the four E matrixes in Table 3

2. "EB" means the Empirical B.--i estimate. "True" represents the covariance

matrix used to generate the data.


Histogram of dataset 3,EB
0

a J)


0


0

C 2
C 0-4
0







77




















o o
O O



P p -
O O
o o




O O
0 0

o0 0 -
0 0
















na 10 6t 30 40 20 observat points.
C C












I I I I I I I

0 10 20 30 40 50 0 10 20 30 40 50

LRT LRT



Figure 3-17: The left plot indicates the empirical distribution of the test statistics
in the simulated growth data of C'!i Iter 3 with covariance matrix El, 61 subjects
with 11 observation points. The right one is the one from the simulated HIV dy-
namics data, 61 subjects with 20 observation points.









3.10 Appendix: Performance of the Approximation G to Actual
Likelihood Ratio Test Statistics

3.10.1 Joint distribution of G at each Q under Ho

Notice that we arrange Y according to which group yij is from, so at different

pairs of markers, we will have a different vector Y. To make notation uniform, we

can use a permutation matrix P. If we denote Y1 as the Y for first pair of markers,

then for other intervals flanked by subsequent markers, k = 2,...,m,


Yk PkY1,

where Pk is the permutation matrix. Of course, for each 0 relative to same marker

1i,, Yk is the same and so is Pk. Suppose the matrix A corresponding to kth pair

of markers at a fixed 0 is labelled as A'. Thus, at each 0 among kth marker's pair,

the corresponding test statistic is


Y'kAYk YP= Y':PAPkY1.


Notice that, var(PkY1) = PkEPk = E because E is block diagonal with block size

T T and Pk just changes the rows of E. So under Ho at position 0, we still have


Y' AYk Y 1PA PkY, 1 ,


with of3 denoting the multiplicity determined by the kth markers' pair and hy-

pothetic QTL indexed by 0. Recall that under H0, m(t = mi(t m2(tm ,

Yi ~ MVN( 1,, m(t, I, & E) and we can write

Y'AY = Y'ZYIA2: I 2

k = Pk kL

t Y'l Pk$ Wk Wk 2PkY1









where ZT W f'Z-PkYl d MVN(WW '- PkI, Wf'W) and "WfWo' are
the spectral decompositions of EA YI. The matrix Wk is composed of the

eigenvectors corresponding to the I non-zero eigenvalues, and it has I orthonormal
columns and is of order NT x 1. Also, under Ho the structure of A results in

(Wk'Z Pkp)'Wf'Z 2Pktt

= 'A9p/j4 = 0.

and thus, under Ho, Z W/et-' PkY1 ~ MVN(O, I).
At each interval along the entire map, we examine no putative positions.
The entire vector Z (Z1',..., Z z,...', Z )' has distribution Z ~
MVN(pz, A) where


pIzJ0


wO%'z,_
Wi 2. ,t


i 1, ...,m j 1,..., no and





WIf (we let B (x), j 1, ..., denote the -dimension ball with radius equal to
A '1-- 1DW01 2 1 0n0 P,-_l^W1 t ^- W^9 ,





and under Ho, pz = 0.
If we let B, (x), j 1,..., n0 denote the 1-dimension ball with radius equal to

x/f3l, then using the joint distribution of Go's to approximate the joint distribu-
tion of every likelihood ratio test statistics at QTL position 0 within kth pair of









marker -2 log A's, we can get


Po max -2log A < x)
l SPo( max Y'Aek 1 Po( max Zo'Zo < x)
1 P (Zl1/z1 < X/ 10 / ,z o'ZO < X/Aeo )
0 0 p01 -1 Ono
= ... Iz exp 1 .ZA1Zo dZ' dZon


This probability, which is one minus the p-value for Ho, can be directly calculated

by simulating Z ~ MVN(O, A). Again, all the above derivations are made under

the assumption that we know E, which is typically untrue in practice. We --.i--- -I

substituting a REML estimate of the variance-covariance matrix E instead of E,

and, in that case, the above formulae are correct ..'-i,'i!l.1 ically.

3.10.2 Examples

Simulation

The purpose of this simulation is to examine how the test statistic behaves as

the underlying mean curves move apart from on another when applying this more

general method. We, of course, expect that the p-value gets smaller when we move

the underlying curves apart.

The first data set assumes the underlying phenotypic curves are two flat

lines as before. The linkage map has 20 markers totally with 25cM apart from its

neighbors. There are 100 subjects with measurements generated at 4 observation

points using the variance-covariance matrix for each yi

0.06 0.035 0.0225 0.01625

0.035 0.06 0.035 0.0225
>1
0.0225 0.035 0.06 0.035

0.01625 0.0225 0.035 0.06














o


co

o




a
0








0.0 0.2 0.4 0.6 0.8
distance



Figure 3 18: Graph of the p-value and its 95. confidence interval for data from
two underlying flat lines with different distances using nonparametric functional
interval mapping.


Based on the marker information, 100 data sets are generated and analyzed. The

unit of scanning throughout the linkage map is 2cM. Since the underlying curve are

the two flat lines y = ao and y = al, we can use lao all as a measure of distance.

The p-value trend with respect to different distances between two underlying

flat lines, and its 95'. confidence interval, is given in Figure 318. It is clear that

the P-value decreases as the curves move apart, which is the behavior we would

expect.

The second data set assumes one phenotypic curve is flat (ml(t) = 1) and the

other is quadratic (r2() = ax2 + bx + c). In this case, using 25 markers with 20cM

apart from nearest neighbors, 50 subjects are generated at 12 observation points


















Co










0 1 2 3 4


0




distance



Figure 3 19: Graph of p-value and its 95' confidence interval for data from two
underlying curves:one is flat line and the other is in quadratic form using nonpara-
metric functional interval mapping.


using the variance-covariance matrix for each y,


E .1J12 + Autoregressive(-2 .05, rho .5).


Based on the same marker information, 100 data sets are generated and analyzed.

To measure distance, we use the area between the two curves over the observed

time interval. Figure 3 19 shows the p- value profile for different distances and its

95' confidence interval. Again, the behavior is what we desire, with the p-value

decreasing as the curves get further apart.

Poplar Data



































0



O



Co
0 M



0

E


E \
C_

CO
0











0 20 40 60 80 100


QTL position




Figure 3-20: Graph of the likelihood ratio test statistics along the linkage group 10
using nonparametric functional interval mapping.









Again we apply this method to the same Poplar Data set as before. Because

our model assumes that all t observations have equal variance, we first transformed

the original observations by taking the log. We use order 3 B-splines with equidis-

tant knots to fit the growth curve and scan the linkage group 10 2cM by 2cM.

Using our method with the variance-covariance structure autoregressive+simple (es-

timated by PROC MIXED as E 0.001236*Jin+Autoregressive(r2 = 0.06155, p

0.8945)), we found strong evidence that there is a QTL between the first and the

second marker on linkage group 10 which controls the growth trajectory of stem

diameter in the interspecific hybrids of poplar(p-value= 5e 4 with standard

deviation 5e 4). The 'i.-.-- -1 likelihood-ratio test statistics -2 log A appears in the

interval flanked by Marker CA/CCC-64OR and Marker CG/CCC-, "'. Figure 3-20

shows likelihood ratio test statistics along the linkage group 10. Again this finding

agrees with the conclusion in Ma, Casella and Wu (2002). But the resulting QTL

locus, i.e, where the largest likelihood ratio test statistic is, is slightly different from

our method and their method. This may due to all the approximations we made.















CHAPTER 4
MISSING MARKER INFORMATION

The dataset used for QTL mapping contains not only genotypic values like

markers but also phenotypic values measured from some traits. In practice, missing

data situations are usually faced due to many reasons. To handle missing values in

the analysis of longitudinal or repeated measurement data, many authors proposed

statistical models to incorporate both complete cases and incomplete cases. Their

ideas can be similarly adopted to incorporate missing phenotypic cases into QTL

mapping. But little literature has appeared to incorporate missing marker cases

into QTL mapping. A popular way is to simply discard subjects with missing

markers, which directly results in loss of power in mapping QTL because of smaller

sample size. So in this chapter we will use full information from all subjects,

whether having complete marker information or not, to gain as much power as

possible. For simplicity we assume the phenotypic values from each subject are

complete.

4.1 Simple Methods

In order to incorporate incompleteness into the modelling process, we need

to reflect on the nature of the missing value mechanism. For marker information,

the loss of data is mainly from technical reasons or just simple recording mistakes.

So unlike the complicated missing mechanism in repeated measurement data, we

can view such missing mechanisms in marker information as missing completely at

random (\!CAR) (Little and Rubin 2002).

With this MCAR assumption there are a number of simple and valid tech-

niques for dealing with missing data. The computationally simplest techiques, a

complete data analysis, in which the analysis is restricted to the subjects for whom









all markers information has been recorded. A complete data analysis is popular be-

cause it maps a i .--. d data matrix into a rectangle one. But as we stated before,

there is nearly aliv-, a substantial loss of information so that a loss of detecting

power follows.

An alternative way to obtain a data set on which a complete data analysis

can be carried out is filling in the missing values, instead of deleting subjects with

incomplete sequences. Using observed values to impute values for the missing

values is the principle of imputation. For marker data, since we know the genetic

distances between the markers, we can impute the missing marker information

based on its nearest neighbor and corresponding genetic distance. Then the

resulting data set is analyzed as if it represented the true complete data.

A third method is based on the principle of analyzing the incomplete data

as such. A simple representative is available data analysis. For the functional

interval mapping, we constructed a likelihood ratio test interval by interval. So for

each interval, we can use all subjects who have both values of two end markers to

derive such LR test. This may cause the number of subjects in each LR test to be

different. Obviously, this method is more efficient than the complete case method

for more information is used.

4.2 Modelling Incompleteness in Markers

In this section, we directly put the missing markers into the model. The

data setting is exactly what has described in Section 3.1. First, we consider the

case where at least one of the two neighbors of the missing marker is recorded

in the data set, that is, for any marker interval, ;?-, bracketed by M and N, all

the subjects in the data sets have information of either M or AV or both. Those

subjects with information of both M and AC missing are deleted when we do the

likelihood ratio test in this interval.