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Bayesian Mapping of Multiple Qualitative Trait Loci

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Therearenowords,time,orspacetothankallthathavecontributedinsomewaytothisjourney.EveryonethatIcameacrossorspoketoalltheseyearsmaybeunawareoftheircontribution.Asimplegesture,aword,asmilearesometimestheturningpointinaparticularday.IrecognizetheCreatorastheSourceofallknowledgeandthankHimfortheopportunityofnishingthisdegreeandalltheblessingsthatIhavereceived.IthankmyadvisorGeorgeCasellaforsharing,sogenerously,hisknowledgeandtimewithme.Ithankhimforhissupport,encouragementandinspiration.HebelievedinmemorethatIbelievedinmyself,oneofthereasonsthatenabledmetoreachtheendofthisjourney.Ithankthemembersofmycommittee,JimHobert,SamWu,RonglingWuandJohnDavis,fortheirinterestinmyworkandfortheirsupport.Wehadexcellentcommitteemeetingsoutofwhichgreatideasemerged.IThankthemforsuchanopenlearningenvironment.IthankMatiasKirstforjoiningusandforhelpinginndingthedatasets.Ithankthefaculty,studentsandstaoftheDepartmentofStatisticsatUFforalltheirhelpandsupport.Ithankmyhusband,TimPorch,forallhissupportandencouragement.Ithankmyparentsandmysisterforalltheirlove,supportandprayers.Ithankmyhugefamilyandallmyfriendsforalltheirlove,supportandprayers.Icherisheverycall,meeting,ower,letter,meal,cupofcoeeandvisit.InallythankCarlosCastillo-ChavezforinvitingmetoparticipateathisInstituteatCornellUniversity,startingthechainofeventsthatbroughtmetonishingthisdegree. iv

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page ACKNOWLEDGMENTS ............................. iv TABLE ....................................... vii LISTOFFIGURES ................................ viii ABSTRACT .................................... x CHAPTER 1INTRODUCTION .............................. 1 1.1BasicConceptsofGenetics ....................... 2 1.2QTLAnalysis .............................. 4 2METHODSFORQTLMAPPING ..................... 6 2.1SingleQTLMethods .......................... 6 2.1.1SingleMarkerAnalysisandSimpleLinearRegression .... 6 2.1.2IntervalMappingandRegressionMapping .......... 8 2.2MultipleQTLMethods ......................... 10 2.2.1MultipleRegressiononMarkerGenotypesandCompositeIntervalMapping ........................ 10 2.2.2BayesianMethods ........................ 12 3SIMULTANEOUSQTLESTIMATION ................... 15 3.1Model1 ................................. 15 3.1.1GibbsSamplerforModel1 ................... 17 3.1.2Implementation ......................... 22 3.2Model2 ................................. 27 3.2.1ConditionsforaProperPosteriorDistribution ........ 29 3.2.2FullConditionalDistributions ................. 30 4PERFORMANCEEVALUATION ..................... 36 4.1SimulatedData ............................. 36 4.2ConvergenceandResultsPresentation ................ 37 4.3SimulationResults ........................... 37 4.4Model2withFullConditionalsConditionedon 40 4.5DataAnalysis .............................. 40 v

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.............. 53 APPENDIX APRIORSFORAND 57 BPARAMETERIZATIONOFQTLPOSITIONS .............. 61 B.1Rangeofthep1k'sandp2k'sandExplorationofTheirPriorDistri-bution .................................. 61 B.2ReparameterizationofthePosteriorDistributioninTermsoftheRecombinationFractions ........................ 62 CFULLCONDITIONALSFORMODEL2CONDITIONED 67 DRUNNINGMEANSFORPERFORMANCEANALYSYS ........ 68 REFERENCES ................................... 77 BIOGRAPHICALSKETCH ............................ 80 vi

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Table page 2{1BackcrossDesign,ModelAssumptions ................... 8 vii

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LISTOFFIGURES Figure page 3{1HistogramsofQTLpositions.TopPanel:Only r Q k ( k ) iscalculated.BottomPanel:Both r Q k ( k ) and r Q k ( k +1) arecalculated. ............ 33 3{2Simulateddata,equallyspacedmarkersatapproximately.26M.QTLat secondandforthintervalswithequaleectsof1. h 2 = : 94, 2 = : 04. Model1with 2 =1(topleft), 2 = 1 100 (topright)andModel2with 2 =1(bottom). ............................... 34 3{3Simulateddata,onlyafewindividualsexceed L =3000andnomore than.5%ofthetotalnumberofiterations. ................. 35 4{1Example 1:Simulateddata,equallyspacedmarkersatapproximately 26cM.QTLlocatedatthesecondandforthintervalwithequaleects of1.Toppanel: h 2 = :94, 2 = :04,Bottompanel: h 2 = :4, 2 =1,Left Panel:Model2,RightPanel:Compositeintervalmapping. ........43 4{2Example2:Model2onsimulateddataatequallyspacedmarkersatapproximately15cM.QTLlocatedatthesecondandforthintervalwith equaleectsof1. ...............................44 4{3Example2:Compositeintervalmappingonsimulateddataatequally spacedmarkersatapproximately15cM.QTLlocatedatthesecondand forthintervalwithequaleectsof1. ....................45 4{4Example3:Model2onsimulateddataatequallyspacedmarkersatapproximately15cM.QTLwitheects1,.1,1,.1locatedatmarkerintervals 2,5,7and8,respectively. ..........................46 4{5Example4:Model2onsimulateddataatequallyspacedmarkersatapproximately5cM.QTLwitheects.1,1,.1,1locatedatmarkerintervals 1,5,7and9,respectively. ...........................47 4{6Example4:Compositeintervalmappingonsimulateddataatequally spacedmarkersatapproximately5cM.QTLwitheects.1,1,.1,1located atmarkerintervals1,5,7and9,respectively. ................48 4{7Simulateddata,equallyspacedmarkersatapproximately26cMwith h 2 = :94.QTLatsecondandforthintervalswithequaleectsof1.Gibbs samplerwithfullconditionalsconditionedon ..............49 viii

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. 50 4{9RunningmeansforModel2onBarleydatausingallmarkers ....... 51 4{10RunningmeansforModel2onBarleydatausingselectedmarkers .... 52 B{1HistogramsofQTLpositions.Eachgraphshows5intervalsofequallength.TopPanel:OnlyrQk(k+1)iscalculated.BottomPanel:OnlyrQk(k)iscal-culated. .................................... 65 B{2HistogramsofQTLpositionsgeneratedusingthemixturedistributionforp1kjp2katdierentintervallengths.Eachgraphsshow5intervalsofequallength. ................................. 66 D{1Example2:h2=:4,n=250 ......................... 69 D{2Example2:h2=:2,n=250 ......................... 70 D{3Example2:h2=:2,n=500 ......................... 71 D{4Example2:h2=:1,n=500 ......................... 72 D{5Example3:h2=:4,n=250 ......................... 73 D{6Example3:h2=:3,n=600 ......................... 74 D{7Example3:h2=:3,n=800 ......................... 75 D{8Example4:h2=:3,n=600 ......................... 76 ix

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Wedescribeamethodforthesimultaneousestimationofthelocationsandtheeectsofquantitativetrailloci(QTL)inabackcrosspopulation.WeconsideramixedmodelthatincludesoneQTLperintervalandconsidersallmarkersascovariates.ByincludingonepossibleQTLperintervalitwaspossibletoexamineandaccountforalloftheQTLeects.ThemarkerinformationisincludedinthemodelusingatermthattakesintoaccounttheportionoftheeectsofthemarkersthatarenottakenintoaccountbytheQTL.Theeectsofthemarkerinformationareconsideredrandomeectsinthemodel.WeobtaintheposteriordistributionoftheQTLeectsalongthegenomeusingaGibbssampler.Todeterminethesignicanteects,95%posteriorcondenceintervalsareused.Oneadvantageofthisapproachisthatallmarkersareusedascovariates,whicheliminatestheconstraintofmarkerselection.Theperformanceofourmethodwasstudiedusingsimulateddataforequallyspacedmarkersatdierentintervallengths.Weconsideredexampleswithsixandtenmarkersandwithdierentheritabilitylevels.Wecomparedourresultstoresultsusingcompositeintervalmappingforsomeof x

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xi

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Manyagronomictraitsinplantsareclassiedasquantitativeinnature;i.e.,theobservedphenotypeisthejointresultoftheeectsofanumberofgeneticandenvironmentalfactors.Thegeneticsofquantitativetraitsarestudiedthroughestimatingtheeectsofthegenescontributingtothetraitsaswellasbydetermin-ingtheirlocationinthegenome.Onceamolecularlocationisdeterminedforthegenes,theyarecalledquantitativetraitloci(QTL).Knowledgeabouttheselociassistintheselectionofsuperiorgenotypesinapopulationfortraitimprovement(e.g.,yieldanddiseaseresistanceincrops).SeveralmethodsforQTLanalysishavebeendevelopedtodeterminethenumber,locationandeectsofQTL.Thesemethodsfallintwocategories,thosethatmodeltheeectsofsingleQTLandthosethatmodeltheeectsofmultipleQTL.ThebestapproachtosearchformultipleQTLremainsanopenproblem(SenandChurchill,2001).WedevelopedamethodforthesimultaneousestimationofQTLeectsandlocationsinthegenome.Wefocusedonexperimentalpopulations,particularlythebackcrossdesign. InChapter2wereviewsomeofthemethodsforQTLmappingpresentedintheliterature.OurmethodforsimultaneousQTLestimationinaBayesianframe-workispresentedinChapter3.Theperformanceofourmethodonsimulateddataaswellasonchromosome5ofaBarley(Hordeumvulgare)datasetispresentedinChapter4.AdiscussionoftheresultsandapresentationonfurtherresearcharepresentedinChapter5.First,weintroducesomebasicconceptsofgeneticsandabackgroundonQTLanalysis. 1

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Mostanimalandplanspeciesarediploid;thustherearetwocopiesofeachchromosome,andindividualshavetwocopiesofeachgene.GivenallelesA1andA2foraspecicgene,theindividualwillhaveoneofthreepossiblegenotypes:A1A1,A1A2orA2A2.AnindividualthathasgenotypeA1A1orA2A2iscalledhomozygous,indicatingthattheallelesareidentical.Otherwise,theindividualisheterozygous.Ifatraitisonlycontrolledbyasinglegene,theexpressionofthephenotypeisdeterminedbythedominancerelationshipbetweenthealleles.IfthealleleA1iscompletelydominant,theindividualswithgenotypeA1A1andA1A2willbeindistinguishableandwillexpresstheA1phenotype.IfA1isincompletelydominant,theheterozygoteswillhaveaphenotypeintermediatebetweenthe

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twohomozygotes.Inthiscase,theallelesaresaidtobecodominantbecauseheterozygotesandhomozygotesmaybedistinguished. Thetransmissionofgeneticinformationfromparentstoospringisthrougheggandspermcells.Thesecells,calledgametes,carryonecomplementofthechromosomesofeachparent.Duringtheformationofgametesintheprocessofmeiosiseachchromosomeinthepairduplicatesresultinginfourchromosomes.TheduplicatesinterchangeDNA,ineventscalledcrossovers,resultingintwonewchromosomesthatareamosaicoftheparentalchromosomes.Thefourchromosomesthenseparatetoformfournewcells(gametes),eachonewithonechromosome.Twogametesresultwithcopiesidenticaltooneofthechromosomesintheoriginalpair,theothertwowithchromosomesthatareacombinationofboth.Thelatterarecalledrecombinantgametes.Thediploidcopynumberinthecellisrestoredwhentheeggandthespermunite.Thus,recombinationisakeysourceofgeneticvariation. ConsidertwodiallelicgenesinanindividualwithgenotypeAaBb.ABareononechromosomeandabontheother.Thereare4possiblegametes:AB,ab,andtherecombinantsAbandaB.ByMendel'sruleofindependentassortment,eachofthesegameteswouldhavethesameprobability.Thisrulestatesthatallelesofdierentgenessegregateindependently.Itwasdiscoveredlaterthatthefrequencyofgametesdependsonthegeneticdistancebetweenthegenes.Genesthatareclosetoeachotheraremorelikelytoremaintogetherintheprocessofmeiosis.Genesthatarefurtherawayaremorelikelytoexperiencecrossoversandrecombination.Geneticdistanceisdeterminedthroughuseoflinkagebetweentwolocitocalculatetherecombinationfraction,theratioofthenumberofrecombinantgametestototalnumberofgametes.Recombinationbetweentwolocionthesamechromosomeismorelikelythefurtherthelociareapart.Forexample,therecombinationfraction,r=0,betweentwolocimeansthattheyarecompletelylinked,whiler=1 2means

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thattheysegregateindependentlyandareunlinked.Mendel'sruleofindependentassortmentappliestogenesthatareunlinked. MolecularmarkersaresmallregionsofDNAforwhichdetectableheritablevariationcanbeanalyzedforindividualsinapopulation.Ageneticmapconsistsoflinearlyorderedmolecularmarkersandthegeneticdistancebetweenthem.Thegeneticmapisconstructedbyanalyzingtherelationshipofthemarkergenotypesfortheindividualsinapopulationbyaprocesscalledlinkageanalysis(Liu1997).Geneticmarkersareplacedinlinkagegroupsbasedontheirlinkagerelationshipdenedbyrecombinationfractions.Recombinationfractionsarethentranslatedtogeneticdistancesusingamappingfunction(Haldane1919,Kosambi1944).Geneticmapsprovidearepresentationofgenomestructureandhavebeendevelopedformanyplantspecies. WefocusonthebackcrosspopulationstructureforthedevelopmentofamultipleQTLanalysismethod.Althoughthebackcrosspopulationmaybeconsideredsimpletoanalyze,itstillpresentsbigchallenges.Theextensionofthemethodsdevelopedforbackcrossexperimentstoothertypesofexperimental

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crossesisoftennotdicult.ForthepurposeofQTLanalysis,theparentallinesthatformthepopulationshouldbehomozygous,buttheymustdierforthetraitofinterest.ThetwoparentallinesarecrossedtogettheF1generation.EachF1individualreceivesacopyofachromosomefromeachofitstwoparents;thustheyareheterozygouswherevertheparentallinesdier.IndividualsintheF1generationaregeneticallyidentical.TheF1generationcanbebackcrossedtotheP1orP2parenttoobtaintheBC1orBC2population.Theindividualsinthebackcrosspopulationhaveoneoftwogenotypesateverylocus,homozygousorheterozygous.Afterthepopulationisgenerated,phenotypicinformationandthemarkergenotypesisobtainedforalltheindividualsinthepopulationaswellasthetwoparents. Thus,thegoaloftheQTLanalysisistodeterminetheassociationbetweentheindividualphenotypesandtheallelestheyreceivedfromtheirparentsatvariousmarkerlociusingthegeneticmap,themarkerinformation,andthephenotypicdata.

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SeveralmethodshavebeendescribedintheliteratureforQTLmapping.Thesemethodsfallessentiallyintwocategories,thosethatestimatetheeectsofsingleQTLandthosethatestimatetheeectsofmultipleQTL.Wewillbrieyreviewsomeofthemethodsinbothcategories.Someofthesemethodsperformtheanalysisatthemarkerlocationswhileothersusethemarkerinformationtoestimateeectsbetweenmarkers.MoreextensivereviewsarepresentedbyDoerge(2002),BromanandSpeed(1999)andDoergeetal.(1997). 2{1 ).ThemeansofthesepopulationsareattributedtotheQTLeect.Markersareassumedtohavenoeectonthetrait.ThebackcrosspopulationbetweenP1andF1willhavemarker-QTLgenotypesM1Q=M1Q,M1Q=M1q,M1Q=M2Q,M1Q=M2qwithprobability1rMQ 6

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betweenmarkerMandtheputativeQTL.Inthiscase,QrepresentstheQTLgenotypethatisnotobserved.Twoobservablemarkerclassesareobtainedwiththefollowingmixturedistributions:M1/M1:(1rMQ)N(QQ;2)+rMQN(Qq;2)M1/M2:rMQN(QQ;2)+(1rMQ)N(Qq;2): whichteststhepresenceofaQTLunlinkedtothemarkerunderconsideration.Fromtheexperimentaldesign,itisknownthatQQQq6=0andthatwestartedwithparentallinesthatdieratthetraitofinterest.However,theanalysisisconfoundedbytheeectoflocusQ,sinceitistheproduct(12rMQ)(QQQq)thatisbeingtestedfordeparturesfromzero.AQTLwithasmalleectthatisclosetothemarkerwillgivethesameresultasaQTLwithalargereectlocatedfurtherfromthemarker. Careshouldbetakenwiththedeterminationofcriticalvaluessincethedistributionofthepopulationsinconsideration,inthiscasetheobservablemarkerclasses,aremixturesofnormals.ChurchillandDoerge(1994)discusspermutationtheoryforthecalculationofempiricalthresholdvalues.

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Table2{1. BackcrossDesign,ModelAssumptions P1:M1Q/M1QxP2:M2q/M2q B1:M1Q/M1QM1Q/M1qM1Q/M2QM1Q/M2q Considerthelinearregressionmodel whereyiisthetraitvalue,xiiseither1or0(inabackcrossdesign)dependingonthemarkergenotype,homozygote(M1M1)orheterozygote(M1M2)respectivelyandiisaN(0;2)randomvariable.ThehypothesisH0:1=0isequivalenttotheabovehypothesissincetheregressioncoecient1isthedierencebetweenthemeanoftheobservablemarkerclasses,thatis,1=(12rMQ)(QQQq). 2{1 ).Forthismodel,thelikelihoodfunctionisL(0;1;2)=nYi=11 22(yi(0+1xi))2

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TotestthehypothesisH0:1=0theteststatistictypicallyusedisLOD=log10L(^0;^1;^2) where^^0and^^2aretheconstrainedMLEsobtainedunderthenullhypothesis.TheLODscoreindicateshowmuchmoreprobableitisthatthedataarisefromthesituationofhavingaQTLpresentversusabsent. Inintervalmapping,theinformationprovidedbythegeneticmapisusedtomarchalongthechromosomeandcalculatetheLODscoreatdierentpositionsinthegenome.Themodelisyi=0+1zi+i 3.1 ).TheprobabilitydistributionisbasedontherecombinationfractionbetweentheputativeQTLandtheankingmarkers.Onceataparticularposition,theserecombinationfractionsaredeterminedfromthegeneticmap.Thus,thelikelihoodis 22(yi0)2+P(zi=1)1 22(yi(0+1))2 MartinezandCurnow(1992)usedthesameapproachofmarchingalongthechromosome.Theirmodelisyi=0+1P(zi=1)+i

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NotethatP(zi=1)istheexpectedvalueofzi.Asbefore,onceataparticularposition,theseprobabilitiescanbecalculatedusingthegeneticmap;thusthisisasimpleregressionmodelandthestatisticalanalysisisstraightforward.TheminimumresidualsumofsquaresoverallstudiedpositionsinthegenomeisanindicationofasingleQTLifitissmallerthansomespeciedthresholdvalue. Thedrawbackofthesetwomethods,asdiscussedbyseveralauthorsincludingMartinezandCurnow(1992),isthattheydonotguardagainstghostQTL.GhostQTLoccurwhenaQTLislocatedinamarkerintervalandneighbouringregionsalsoexhibitsignicantteststatistics.TheproblemisthatthesemethodsdonottakeintoaccountthepresenceofotherpossibleQTLinthegenome.ThesemethodshavebeenshowntogiveaccurateestimatesoftheQTLpositionanditseectwhenthereisonlyoneQTLsegregatinginthepopulation.ThisproblembringsustothesecondcategoryofmethodswhichtakeintoaccountmultipleQTL. 2{1 ).Manymarkersareconsideredinsteadofonemarkeratatime.Lettbethenumberofmarkers.Themodelisdenedby

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whereyi,jandxijaredenedasinmodel( 2{1 ),exceptthatnowwehavetmarkers.Zeng(1993,1994)discussedpropertiesofthemultipleregressionanalysis.Amongthem,thepartialregressioncoecientparameterformarkerjdependsonlyonthoseQTLwhicharelocatedbetweenmarkersj1andj+1,undertheassumptionofnoepistasis.HearguesthatthisresultsinatestforthepresenceofQTLbetweenmarkersj1andj+1,regardlessofthepresenceofotherQTLinthegenome.Zeng(1993)discussedthattheuseofmultipleregressionaloneisnotappropriatesincetheestimatesofQTLeectsbythepartialregressioncoecientarebiased.Thus,heintroducedcompositeintervalmappingwhichisacombinationofintervalmappingandmultipleregression.Agenomescanusingintervalmappingisperformed,butmarkersoutsidetheintervalunderconsiderationareusedascovariatestocontrolfortheeectsofotherQTL.Ateachpositionunderconsideration,LODscoresarecalculated.Theselectionofthemarkerstouseascovariatesisaproblemwiththismethod.Conditioningonlinkedmarkerspotentiallyincreasestheprecisionofthetestandestimation,butwithapossibledecreaseinstatisticalpower(Zeng1993).IfthereareQTLintheintervalsimmediatelyadjacenttotheintervalunderconsideration,thismethodhasthepotentialtofalselyindicatethepresenceofaQTL(Zeng1994). BromanandSpeed(1999,2002)statedthatQTLmappingshouldbeviewedasaproblemofmodelselectionandnotofmultipletesting.Insteadofminimizingthepredictionerror,theyseektoidentifythesubsetofmarkersforwhichj6=0.TheyintroducedamodiedversionoftheBayesianinformationcriteria(BIC)formodelselection.Theyproceedintwostages.First,thespaceofmodelsissearchedinordertopickthebestones,thosethatwouldhavebeenchosenifallmodelsweretted.Second,themodelwithminimumBICischosenamongtheselectedmodels.Theydiscussedseveralmethodstoselectthebestmodels,including

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backwardandforwardselectionandanMCMCmethod,amongothers.DoergeandChurchill(1996)discussothermethodsformodelselection. Satagopanetal.(1996)andSillanpaaandArjas(1998)proposedmodelsthat,inabackcrossdesign,areequivalentto and respectively.Thexijsareasubsetofwmarkersselectedascovariatesfromthetotaloftmarkers,thezijsaretheunobservedQTLgenotypesandsisthenumberofQTL,alsounknown.Althoughthezijsareunknown,theirprobabilitydistributiongiventheankingmarkersisknown(seeSection 3.1 ).TheprobabilitydistributionisbasedontherecombinationfractionbetweentheputativeQTLandtheankingmarkers.BotharticlesdevelopedMCMCalgorithmstosamplefrom

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theposteriordistributionoftheparametersgiventhedata.Let=(1;:::;w),=(1;:::;s),y=(y1;:::;yn),X=fxijg,Z=fzikgand=(1;:::;s)(thevectorofQTLpositions).Satagopanetal.samplefrom;;2;Z;jy;XusingaGibbssamplerwithMetropolis-HastingsstepstoupdateQTLpositions.TheyuseBayesfactorstocomparemodelsofdierentsizes.SillanpaaandArjas(1998)samplefrom;;;2;Z;;sjy;XusingtheMetropolis-Hastingsalgorithmandreversible-jumpMCMCtomovebetweenmodelsofdierentsizes.SatagopanandYandell(1996)useasimilarapproach.SillanpaaandArjas(1998)usestepwiseregressiontochoosethemarkersthatwillbeusedatcovariates. Yi(2004)notedthatmethodsthatchangethedimensionalityoftheproblembychangingthenumberofQTLduringtheanalysishavethedisadvantagethattheinformationaboutaQTLislostassoonasitisremovedfromthemodel.TheauthoralsonotedthatreversiblejumpMCMCisusuallysubjecttopoormixingandslowconvergence.Yietal.(2003)presentedaGibbssamplerforthemultipleregressionmodel( 2{2 )(foranalysisatthemarkers)basedonavariableselectionmethodcalledstochasticsearchvariableselection,developedbyGeorgeandMcCulloch(1993).AsinBromanandSpeed(1999,2002),theobjectiveistoidentifythesubsetofmarkersforwhichj6=0(model( 2{2 )).Inthisapproach,thedimensionalityoftheproblemiskeptconstantbylimitingtheposteriordistributionofnonsignicantterms(markerswithnoeects)inaneighborhoodofzeroinsteadofremovingthemfromthemodel.Thatis,theydene=(1;:::;t),wherej=1or0representthepresenceorabsenceofthecovariatejinthemodel.Themarkereectsj,j=1;:::;taregivenapriordistributionjjj(1j)N(0;2j)+jN(0;c2j2j);

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(1;:::;t)ismultivariatenormal.Theauthorsdiscussedthatwithsparseandirregularlyspacedmarkers,themarkeranalysiswillbebiased,asnotedbyZeng(1993).Theyproposed,withoutimplementingorgivingdetails,twowaystodealwiththisproblem.OneofthemistoincorporateintheirprocedurethemultipleimputationmethodproposedbySenandChurchill(2001).Theirideaisthatifthecompletegenotypeinformationonadensesetofmarkersisknown,regressingthephenotypeoneachmarkerwillgiveinformationabouthowlikelyitisthataQTLisclosetothatmarker.Because,inpractice,themarkersmaybewidelyspacedacrossthegenome,theyproposedtocreateacompleteanddensesetofmarkersbyaddingwhattheycalledpseudomarkers.Thegenotypicinformationforthesepseudomarkerscanbeinferredusingtheirassignedpositionsinthegenome,thegeneticmapandtheavailablemarkerdata.SeveralversionsofthiscompletegenotypedataareconstructedandtheLODscoresobtainedfromeachofthemisthencombinedtomeasuretheevidenceinfavorofaQTLbeingnearanygivenpseudomarker.ToaccountformultipleQTL,theycomputedLODscoresforeachpairofpseudomarkersforatwo-dimensionalgenomescanandalsoimplementedathree-dimensionalgenomescan.

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OurobjectiveistodevelopagenomewidesearchthataccountsformultipleQTL.Asnotedpreviously,analysisatthemarkerswillbebiasedwhenthemarkersaresparseandirregularlyspaced.ItwasalsomentionedthatmethodsthatchangethedimensionalityoftheproblembychangingthenumberofQTLduringtheanalysishavethedisadvantagethattheinformationaboutaQTLislostassoonasitisremovedfromthemodel.Ourproposedmethodaddressesthesetwoproblemsbysamplingtheentiregenomewhilekeepingthedimensionoftheproblemxed.Atthispoint,weareworkingundertheassumptionsofnointerferenceandnoepistasis.Fornow,wearenotconsideringthecaseofmissingmarkerobservations.WewillbeworkinginaBayesianframework,thereforeourtargetistheposteriordistributionofthelocationsandtheeectsoftheQTL. Twomodelswillbeconsidered.Wewillpresentbothmodelsalongwithdetailedcalculationsandtheirimplementationinthenexttwosections,althoughModel2(section 3.2 )isournalmodel.ForadiscussionontheperformanceofModel2seeChapter 4 Thismodelisbasedonmodel( 2{4 )withallmarkersascovariatesandoneQTLpermarkerinterval,i.e.t1QTL.TheziksrepresenttheunobservedQTLgenotypes.Inabackcrossdesign,zikiseither1or0dependingontheQTL 15

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genotype,homozygote(QQ)orheterozygote(Qq).WewillconsiderthesasrandomeectsandwillkeepthedimensionofthemodelconstantbyconsideringoneQTLperinterval.WebelievethatincludingonepossibleQTLperintervalwillallowustoexamineandaccountforalltheireectsandthattheeectsthatareimportantwillbesignicant.Sincewearenotinterestedinprediction,wearenotthinkingabouttheproblemasmodelselection.Thismodelcanbewrittenasy=1+X+Z+whereXisantmatrix,isavectorofdimensiont,and1areofdimensionn,Zisan(t1)matrix,andisavectorofdimensiont1. Althoughtheziksareunobserved,theirdistributiongiventheankingmarkerinformation(xik;xi(k+1))canbederivedifthegeneticdistance,Qk(k),betweentheQTLkandoneoftheankingmarkers,saymarkerk,isknown.Undertheassumptionofnointerference,thegeneticdistanceQk(k)isrelatedtotherecombinationfractionrQk(k)(therecombinationfractionbetweenQTLkandmarkerk)bytheHaldanemappingfunctionrQk(k)=1 2(1e2Qk(k)).Thenfori=1;:::;nandk=1;:::;t1 where 1rk(k+1)ifxik=1;xi(k+1)=1p2k=(1rQk(k))rQk(k+1) andrk(k+1)istherecombinationfractionbetweenmarkerskandk+1,whichareassumedtobeknownfromthegeneticmap.Letbea2(t1)matrixofthep1ksandp2ks.

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Sinceourinterestistheposteriordistributionoflocationsandeects,wewilluseaGibbssampler.InSection 3.1.1 wewillshowtheposteriordistributionforModel1andwewillderivethefullconditionalposteriorforalltheparametersinthemodel.ThedetailsandchallengesonsamplingfromthesedistributionswillbediscussedinSection 3.1.2 3{1 ),notethattheposteriordistributionoftheparametersgiventhedatais =Zf(;;;Z;;2jy)d/Zf(yj;;;Z;;2)f(;;;Z;;2)d=Zf(yj;;;Z;2)f(Zj;;;;2)f(;;;;2)d=Zf(yj;;;Z;2)f()f()f(j2)f(2)f(Zj)f()d=Zf(yj;;;Z;2)f()f()f(j2)f(2)df(Zj)f() (3{4) Thisfactorizationshowsthat,conditionalontheQTLgenotypesZ,theprobleminvolvingy;;and2canbesolvedindependentlyfromtheprobleminvolving(SenandChurchill2001).Tondequation( 3{4 ),assumepriorsN(;v),N(;A),IG(a;b)andN(0;22It)for,,2and,respectively.2isconsideredtobexed.AssumeaBeta(c2;d2)priorforp2k.Forp1k,f(p1kjp2k)/pc111k(1p1k)d11withrange(1p2krk(k+1) 2andwithrange(p2k;1)withprobability1 2.FlatpriorsforandwerealsoconsideredforModel2,theconditionsneededtoobtainaproperposteriordistributionunderthesepriorsarepresentedinSection 3.2.1 andproofsinAppendix A .SeeAppendix B forgraphsthatexploretheshapeofthepriordistributionchosenforthep1k'sand

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B ,areparameterizationoftheposteriorintermsoftherecombinationfractionispresented.Theparameterizationintermsofthep1k'sandp2k'sturnedouttobesimplerthantheoneusingtherecombinationfractions.Withthepriorsmentionedabove,theterminsidetheintegralofequation( 3{4 )isf(yj;;;Z;2)f()f()f(j2)f(2)/1 22(y1XZ)0(y1XZ)e1 2()0A1()e1 2v()21 22201 2()0A1()e1 2v()2Ze1 22(y1XZ)0(y1XZ)e1 2220d=1 2()0A1()e1 2v()2e1 22(y1Z)0(y1Z)Ze1 22f0X0X20X0(y1Z)+0 2gd=1 2()0A1()e1 2v()2e1 22(y1Z)0(y1Z)Ze1 22f0(X0X+It 2)20X0(y1Z)gd=1 2()0A1()e1 2v()2e1 22f(y1Z)0(IXU1X0)(y1Z)gZe1 22f(U1X0(y1Z))0U(U1X0(y1Z))gd/1 2()0A1()e1 2v()2e1 22f(y1Z)0(IXU1X0)(y1Z)g(2)t 2

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whereU=X0X+It 3{4 )becomes, 22f(y1Z)0(IXU1X0)(y1Z)gjUj1 2e1 2()0A1()e1 2v()21 (3{5) Toobtainf(Zj)f(),fromequation( 3{2 ) Therefore(rangeofp1komittedforsimplicity), wheren1kisthenumberofindividualswithmarkergenotypesxik=1;xi(k+1)=1,i.en1k=Pni=1I(xik=1;xi(k+1)=1).Similarly,n2k=nXi=1I(xik=1;xi(k+1)=0)n3k=nXi=1I(xik=0;xi(k+1)=1)n4k=nXi=1I(xik=0;xi(k+1)=0)

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andz1k=nXi=1zikI(xik=1;xi(k+1)=1)z2k=nXi=1zikI(xik=1;xi(k+1)=0)z3k=nXi=1zikI(xik=0;xi(k+1)=1)z4k=nXi=1zikI(xik=0;xi(k+1)=0): 3{5 ),( 3{6 )and( 3{7 )wenowcalculatethefullconditionaldistributionsofalltheparametersinourmodel. 3{5 )that2j;;Z;;yIG(n 2+1 3{5 )f(j;Z;;2;y)/e1 22f(y1Z)0(IXU1X0)(y1Z)ge1 2v()2/e1 22f210(IXU1X0)1210(IXU1X0)(yZ)ge1 2v(22)=e1 2f2(10(IXU1X0)1 v)g=e1 2(10(IXU1X0)1 2(10(IXU1X0)1 3{5 )

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2f(y1Z)0(IXU1X0)(y1Z)ge1 2()0A1()/e1 20Z0(IXU1X0)Z 220Z0(IXU1X0)(y1) 2(0A120A1)=e1 20Z0(IXU1X0)Z 2+A120Z0(IXU1X0)(y1) 20Z0(IXU1X0)Z 2+A120Z0(IXU1X0)(y1) 2fT1Z0(IXU1X0)(y1) 2+A1.Thus,j;Z;;2;yN(T1Z0(IXU1X0)(y1) 3{5 )and( 3{6 ) 22f(y1Z)0(IXU1X0)(y1Z)gnYi=1t1Yk=1pzikk(1pk)1zik=e1 2f(Z(y1))0(IXU1X0) 3{7 ).Fork=1;:::;t1,p2kj;;;Z;2;yBeta(n3kz3k+z2k+c2;n2kz2k+z3k+d2) andp1kj;;;Z;2;y;p2k/pn4kz4k+z1k+c111k(1p1k)n1kz1k+z4k+d11

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withrange(1p2krk(k+1) 2andwithrange(p2k;1),withprobability1 2. 22f(y1Z)0(IXU1X0)(y1Z)gnYi=1t1Yk=1pzikk(1pk)1zik=e1 2f(Z(y1))0(IXU1X0) Thus, 22(y1XZ)0(y1XZ)e1 2220dnYi=1t1Yk=1pzikk(1pk)1zik=Ze1 22fPni=1(Pt1k=1zikkwi)2ge1 2220dnYi=1t1Yk=1pzikk(1pk)1zik=ZnYi=1"e1 22(Pt1k=1zikkwi)2t1Yk=1pzikk(1pk)1zik#e1 2220d

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wherewi=yiX(i)andX(i)representtheithrowofX.FromherewecanseethatgiventheZscanbegeneratedindependentlyforeachindividual.Forindividuali,thevector(zi1;:::;zi(t1))canbegeneratedasablock.Thiscanbedoneintwoways.Therstapproachistocalculatetheexactprobabilitydistributionof(zi1;:::;zi(t1))andsamplefromit.Thisisdonebycalculatingtheprobabilityofeachofthe2(t1)possiblevectors.SincethisisdoneforeachindividualateachiterationoftheGibbssampler,itcanbecomputationallyintensiveasthenumberofmarkersandindividualsincrease.AsecondwayistousetheAcceptRejectalgorithmwithtargetdistributione1 22(Pt1k=1zikkwi)2t1Yk=1pzikk(1pk)1zik: 2!2(azikkbwi)2pzikk(1pk)1zik 2!2(akbwi)2pkandP(zik=0)/e1 2!2(bwi)2(1pk).Thesupremumoftheratioofthetargetandcandidatedistributions,typicallycalledM,hastobeboundedfromabove,andshouldbecloseto1tohaveagoodacceptancerateinthealgorithm.a,band!arefreeparametersthatareusedtogetaclosedformofMtofacilitatetheimplementationofthealgorithmaswellastoassessitsperformance.Inthiscase,M=supzu1 22(Pt1k=1zikkwi)2 2!2Pt1k=1(azikkbwi)2;

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Notethatt1Xk=1(azikkbwi)2=t1Xk=1azikka Z+a Zbwi2=a2t1Xk=1zikk Zbwi2=a2t1Xk=1zikk b(t1)t1Xk=1zikkwi!2 Thus, supzu1! u2exp1 22Pt1k=1zikkwi2+(t1)b2 b(t1)Pt1k=1zikkwi2+a2 Weexaminedvariouschoicesfora,band!withthegoalofminimizingM.Forexample,onepossibilityisa=b(t1)and(t1)b2 u2e(t1) 22Sz 22,where0<<1.Then,M=supzu1! u2exp8<:1 22t1Xk=1zikkwi!2(1)+(t1) 22Sz9=; 22Sz<1

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thisis(t1)Sz whichimpliesthat 3.2 ),wewilluseamodiedversionofthisAcceptRejectalgorithm. Notethattocalculatetheposteriorandallthefullconditionals,weintegratedout.ButtosamplefromZwearegoingbacktotheexpressionintheintegral.Analternativeapproach,whichwetriedforModel2,istoconsideraspartoftheGibbssampler,i.e.,thefullconditionalposteriordistributionswillbeconditionedalsoon.ThefullconditionaldistributionsforthiscaseareshowninAppendix C .WeshowanexampleinSection 4.4 wheretheGibbssamplerdenedthiswaydidnotrecovertheeectsofthesimulatedQTL.

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andp1kj;;;Z;2;y;p2k/pn4kz4k+z1k+c111k(1p1k)n1kz1k+z4k+d11 2andwithrange(p2k;1),withprobability1 2. Thefollowingsamplingschemewasdevelopedwhileanalyzingsimulateddata.RecallthatourobjectiveistosampleaQTLpositionatintervalk.Oncewehaveavalueofp1kandavalueforp2k,wecalculatetherecombinationfractionrQk(k)orrQk(k+1).AscanbeseenfromthedenitionsofrQk(k)andrQk(k+1),eachresultindierentrestrictionsontherangeofthep's(seeAppendix B ).Westartedbysamplingp1k'susingtherange(1p2krk(k+1) 3{1 top).ThisisbecauseonlytheleftankingmarkerwasusedtodeterminethelocationoftheQTL.We,therefore,decidedtousetherightankingmarkeraswellbyalsocalculatingrQk(k+1)'s.Wenowdecideatrandomwhichankingmarkertouse.Usingthismethod,thedistributionoftheQTLpositionswilllooklikethebottompanelofFigure 3{1 Samplingfromp2kj;;;Z;2;yisstraightforward.Givenp2k,wesamplep1kusingtheAcceptRejectalgorithmwithauniformcandidateon(a;b),where(a;b)iseither(1p2krk(k+1)

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where sups>>><>>>>:1ifw1=0orw2=0(w1 3{8 ).ThismotivatedModel2whereweconsiderthemodel whereHz=Z(Z0Z)1Z0.TheideaistoaccountfortheportionoftheeectsofthemarkersthatisnotbeingtakenintoaccountbythematrixofQTLgenotypesZ.Althoughwearenotconsideringthiscasehere,ifthematrixZisnotoffullcolumnrank,theGibbssamplerusingHz=Z(Z0Z)Z0willstillwork.Thiscaseislikelytoariseinagenomewidesearchwherethenumberofmarkerintervalsislikelytobegreaterthanthenumberofindividuals. Figure 3{2 showsthegraphsofthe'sobtainedbyModel1with=1,=1 100andwithModel2with=1.TheseareresultsonsimulateddatawithtrueQTLatmarkerintervals2and4.Model1couldnotrecovertheQTLeectswhen=1.ThesimulationsetupwillbediscussedinChapter 4

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TosetupaGibbssamplerforthismodel,weassumedthesamepriorsasinModel1,exceptforforwhichaatpriorisassumed.Theposteriordistributionis wheref(Zj)f()isasbeforeandf(yj;;;Z;2)f()f()f(j2)f(2)/1 22(y1(IHz)XZ)0(y1(IHz)XZ)e1 2()0A1()1 22201 2()0A1()Ze1 22(y1(IHz)XZ)0(y1(IHz)XZ)e1 2220d=1 2()0A1()e1 22(y1Z)0(y1Z)Ze1 22f0X0(IHz)X20X0(IHz)(y1Z)+0 2gd=1 2()0A1()e1 22(y1Z)0(y1Z)Ze1 22f0(X0(IHz)X+It 2)20X0(IHz)(y1)gd=1 2()0A1()e1 22(y1Z)0(y1Z)e1 22(y1)0(IHz)XU1X0(IHz)(y1)Ze1 22f(U1X0(IHz)(y1))0U(U1X0(IHz)(y1))gd

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2()0A1()e1 22(y1Z)0(y1Z)e1 22(y1)0(IHz)XU1X0(IHz)(y1)(2)t 2 22(y1Z)0(y1Z)e1 22(y1)0(IHz)XU1X0(IHz)(y1)jUj1 2e1 2()0A1()1 (3{11) thatcanbewrittenasf(;;Z;;2jy)/e1 22(y1Z)0[I(IHz)XU1X0(IHz)](y1Z)jUj1 2e1 2()0A1()1 ThefullconditionaldistributionsarepresentedinSection 3.2.2 .Inthenextsection,wepresentresultsontheconditionsneededtoobtainaproperposteriordistributionunderatpriorsforand. A

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ForModel2,weareassumingaatpriorforandaN(;A)for(Theorem3.1).TheotherswerenotconsideredbecausethematrixZisnotnecessarilyoffullcolumnrank.ThiswillbethecaseiftherearelessindividualsthannumberofmarkerintervalsoriftheQTLgenotypesarethesameforallindividualsintwoormoremarkerintervals.However,wearenotconsideringanyofthesecasesinourmodelsexplicitly.Thesecondcaseoccurs,althoughrarely,whilesamplingfromthematrixZ.Sincewearenotusinggeneralizedinverse,wearedroppingtheseoccurencesandresampling. 3{11 ).ThecalculationsaresimilartotheonesperformedforModel1.Infact,p1kj;;;Z;2;y;p2kandp2kj;;;Z;2;ydonotchange.LetW=(IHz)XU1X0(IHz). 2+1 10(IW)1;2 2+A1. 2e1 22f(y1Z)0(IW)(y1Z)gnYi=1t1Yk=1pzikk(1pk)1zik:

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Samplingfromthislastdistributionisachallenge.AsinModel1,wehavegonebacktotheequation 22(y1(IHz)XZ)0(y1(IHz)XZ)e1 2220dnYi=1t1Yk=1pzikk(1pk)1zik=Z(e1 22(y1(IHz)XZ)0(y1(IHz)XZ)nYi=1t1Yk=1pzikk(1pk)1zik)e1 2220d: 22(y1Z)0(y1Z)nYi=1t1Yk=1pzikk(1pk)1zik 3.1.2 .Weshowedthatthevector(z1;:::;z(t1))canbesampledforeachindividualindependently.Inthiscase,wemustsampletheentirematrixZ,thegenotypesofallindividuals,tohaveacandidatedrawfortheMetropolis-Hastingsalgorithm.Aswementionedbefore,theAcceptRejectalgo-rithmpresentedinSection 3.1.2 giveslargeMforafewindividualswhilerunningtheGibbssampler.Sinceinthiscasethisdistributionisacandidatedistributionandnotthedistributionofinterest,wedecidedtosetamaximumnumberoftrialsfortheAcceptRejectalgorithm,sayL,basedonsimulations.WhilegeneratingthematrixZ,ifanindividualexceedL,thevector(z1;:::;z(t1))fromtheprevi-ousiterationiskeptinthematrixforthatindividual.Thismodicationleadstosamplingthevectorz=(z1;:::;z(t1))fromamixturedistributionoftheformqg(z)+(1q)zprev whereg(z)/e1 22(Pt1k=1zkkw)2t1Yk=1pzkk(1pk)1zk

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andzprevisaconstant.Thevalueofqisunknown,butisapproximately1whenLislargeenough.ForsimulateddatausingL=3000,Figure 3{3 showsthatin150000iterationsonlyafewindividualsexceedLandnomorethan.5%ofthetime.IndividualsIDareshowninthexaxisandthepercentofthetotaliterationsforwhichthatindividualexceedLintheyaxis.TheindividualwithID558exceedsL,keepingthethevectorzfromthepreviousiteration,inonly629outof150000iterations,foralittlebitmoreof.4%.FewotherindividualsexceedLmuchlessthan.4%ofthetime.MostofthemneverexceedLinanytrial. TheperformanceevaluationofModel2willbediscussedinChapter 4

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HistogramsofQTLpositions.TopPanel:OnlyrQk(k)iscalculated.BottomPanel:BothrQk(k)andrQk(k+1)arecalculated.

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Simulateddata,equallyspacedmarkersatapproximately.26M.QTLatsecondandforthintervalswithequaleectsof1.h2=:94,2=:04.Model1with2=1(topleft),2=1 100(topright)andModel2with2=1(bottom).

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Simulateddata,onlyafewindividualsexceedL=3000andnomorethan.5%ofthetotalnumberofiterations.

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Tosimulatethedatawerstdenedthelengthofthemarkerintervalsintermsofrecombinationfraction.Then,denedthelocationoftheQTL,bytherecombinationfractionfromtheirleftankingmarker.Then,themarkergenotypesfortheindividuals(matrixX)aregeneratedaswellasfortheQTLgenotypes(matrixZ).ForeachindividualwerstgeneratethegenotypeoftherstmarkerusingaBernoulli(1 2).Then,therestofthemarkersandQTLgenotypesforthatindividualaregeneratedsequentially.Ifthepreviousmarker(QTL)genotypewas1aBernoulli(1p)isusedtogeneratethecurrentgenotype.Ifthepreviousmarker(QTL)genotypewas0aBernoulli(p)isused.pistherecombinationfractionfromthepreviousleftankingmarkerorQTL,0p1 2. Phenotypes(y)aregeneratedbytheequationy=+Za+,whereisaxedvalue,aisthevectorofxedvaluesfortheQTLeectsandN(0;2E).ThemagnitudeoftheeectsoftheQTL(a)andtheenvironmentalvariance(2E)werechosensothatwewouldhavedatafrompopulationswithhighandlowheritability.Heritabilityisdenedastheratioofthegeneticvariancetothetotalphenotypicvariance,h2=2g

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whereforLQTLwitheectsai2g=1 4LXi=1a2i+1 4Xi6=j(12rij)aiaj;rijistherecombinationfractionbetweenQTLiandQTLj.ItisexpectedthattheeectsofQTLwillbeeasiertorecoverfromdatasetswithhighheritability. TheimplementationwasdoneinOxversion3.30(Doornik,2002)andthegraphswerecreatedusingthestatisticalsoftwareR. Fortheresultsweplottheposteriordistributionofthe's,i.e,theestimatedQTLeectsagainsttheirQTLpositions.Themeansofthe'sateachofthesmallwindowswillbeshownaswellascorresponding5and95percentcutos. D .Asexpected,itishardertoseparateQTLeectsforthecasesof

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lowheritability(.1,.2).Theeectivenessofourmethodwilldependonthenumberofindividualsinthesample.Increasingthenumberofiterationsmayimproveperformanceincertaincases.Wecomparedtheresultsofourmethodwiththoseofcompositeintervalmapping(CIM)inthreeoftheexamples. TheresultsobtainedfromModel2werecomparedwiththosefromcompositeintervalmappingusingallothermarkersascovariates.TheresultsareshowninFigure 4{1 .Thetoppanelshowsthecasewithheritability.94,thebottompanelthecasewithheritability.4.TheleftpanelshowstheresultsfromModel2.Thelast10,000of30,000iterationsareshownforthedatawithhighheritability,andthelast10,000of60,000iterationsforthedatawithlowheritability.Figure 4{1 showsthattheModel2wasabletoidentifytheeectsofthemultipleQTL,separatingtheireectssuccessfully.Themeansofthesatwindowsofsize2cMareshowninblackand5and95percentcutosinred.Compositeintervalmappingdidthesameforthemodelwithhighheritability,buttheLODscoresareabovethethresholdvaluealmostforallintervals.RecallthatCIMhasthepotentialtofalselyindicatethepresenceofaQTLifthereareQTLintheintervalsimmediatelyadjacenttotheintervalunderconsideration(Zeng1994).Forthedatawithlowheritabilitytheseparateeectswerenotrecovered.TheLODscoresareshowingapossibleQTLatmarkerinterval2,andatmarkerinterval3and4.

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heritability.2(2E=2:54).Forheritability.1(2E=5)only500individualswereconsidered. TheresultsfromModel2areshowninFigure 4{2 .QTLwereseparatedsuc-cessfullyforthecaseofheritability.4.Forheritability.2,Model2wassuccessfulwith500individuals.Forheritability.1ourmodelwasnotsuccessful.Thelast70thousandof150thousanditerationsareshown.CompositeintervalmappingdidnotseparatetheQTLeectsinanyofthecases(Figure 4{3 ).InthisexampleweareusingCIMwithbackwardandforwardregressiontoselectthemarkersthatareusedascovariates. TheresultsfromModel2areshowninFigure 4{4 .QTLoflargeeectwereseparatedsuccessfullyforthecaseofheritability.4.Forheritability.3,Model2wassuccessfulindetectingtheQTLoflargeeectswhenthesamplesizewas800.ItindicateapossibleQTLatinterval8thbutwithequaleectoftheoneininterval7,suggestingthatthemodelwasnotabletoseparatetheeects.Thelast70thousandof150thousanditerationsareshown.Increasingthenumberofiterationsmayimprovetheperformancewhenusing600individualsaswellaswith800.

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TheresultsfromModel2areshowninFigure 4{5 .QTLoflargeeectswereseparatedsuccessfully.CIMwithbackwardandforwardregressiontoselectthemarkersthatareusedascovariatesdidnotseparatetheQTLeectssuccessfully(Figure 4{6 ). 3.1.2 weintegratedout,i.e,allconditionalsweremarginalizedon.ButtosamplefromZwewentbacktotheexpressionintheintegral.AnalternativeapproachistoconsideraspartoftheGibbssampler,i.e.,thefullconditionalposteriordistributionswillbeconditionedalsoon.Weshowtheresultsusingthisapproachonthesimulateddatasetwithh2=:94andmarkersdistances26cMdescribedintheprevioussection.TwoQTLwitheectofmagnitude1weresimulatedinmarkerintervals2and4.Figure 4{7 showsthatwiththisapproachtheeectsarenotrecoveredafter30,000iterations.

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(EsparzaMartnezandFoster,1998).Thus,headingdateisarobusttraitthatissuitablefortestingQTLmethods. ThephenotypicvalueswereaveragedacrosstheenvironmentsandtocomparewiththeresultsobtainedbyYietal.(2003)werealsostandardized.Thereare14markersinthegeneticmapatlocations:0,10.9,18.5,78.2,91.2,111.2,114.7,121.7,125.2,138.8,143.7,150.7,154.2,159.9cM.Individualswithmissingmarkerinformationwereremovedfromtheanalysis.Fornow,ourmethodassumesnomissingmarkerinformation.Weanalyzedthedatausingallmarkers.Afterre-movingthemissingdata,thepopulationwasreducedfrom145to107individuals.TheresultsareshowninthetoppanelofFigure 4{8 .Thelasthundredthousandofsevenhundredthousanditerationsareshown.TherunningmeansareshowninFigure 4{9 ,thereisevidencethatmoreiterationsareneeded.NoQTLeectsweredetectedexceptformarkerinterval3thatisaverylonginterval. ThischromosomewasanalyzedbyYietal.(2003).Theydidaccountformissingmarkerinformationandusedallmarkers.RecallthattheirmethodassumesthattheQTLisatthemarker.Theyestimatedthatmarker10hasposteriorprobabilitygreaterthat.4withanestimatedeectbetween.2and.4.ThiscouldbeinterpretedassayingthattherearenoQTLeects. Notethatthemarkersareverycloseincertainregions,thuswedecidedtoanalyzeselectingmarkersthatwereatleast13cMapart.Themarkerdistanceswerenow:0,18.5,78.2,91.2,111.2,125.2,143.7,159.9cM.Afterremovingthemissingdata,thepopulationwasreducedfrom145to114individuals.TheresultsareshowninthebottompanelofFigure 4{8 .TherunningmeansareshowninFigure 4{10 .QTLaredetectedatallmarkerintervals,butifwelookatthechromosomeasawhole,andattheestimatedeects,itseemsthattheeectsoftheQTLarecancelingeachotheroutresultinginnooverallQTLeect.

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Thislatterobservationleadsustoreassesstheinterpretationofourresults.Wehavebeeninterpretingtheresultsintervalbyintervalwhenweshouldbeconcentratingontheoverallpicture.Quantitativetraitsarethejointeectsofanumberofgeneticandenvironmentalfactors.Ourmodelisseparatingthesefactorsandestimatingthegeneticones.Itseemsthattheappropriatewaytolookattheplotsoftheposteriordistributionoftheeectsisasawhole.Infurtherstudieswemustexplorethisinterpretationandthepossibilitiesitpresentsforpredictionandvalidationofourmodel. Also,fromthisanalysisitisevidentthatwemustbetterunderstandthepossibleeectsofcorrelationbetweenthemarkersintheestimationofQTLeects.

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43 Figure4{1. Example 1:Simulateddata,equallyspacedmarkersatapproximately 26cM.QTLlocatedatthesecondandforthintervalwithequaleects of1.Toppanel: h 2 = : 94, 2 = : 04,Bottompanel: h 2 = : 4, 2 =1, LeftPanel:Model2,RightPanel:Compositeintervalmapping.

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Example2:Model2onsimulateddataatequallyspacedmarkersatapproximately15cM.QTLlocatedatthesecondandforthintervalwithequaleectsof1.

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Example2:Compositeintervalmappingonsimulateddataatequallyspacedmarkersatapproximately15cM.QTLlocatedatthesecondandforthintervalwithequaleectsof1.

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Example3:Model2onsimulateddataatequallyspacedmarkersatapproximately15cM.QTLwitheects1,.1,1,.1locatedatmarkerintervals2,5,7and8,respectively.

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Example4:Model2onsimulateddataatequallyspacedmarkersatapproximately5cM.QTLwitheects.1,1,.1,1locatedatmarkerintervals1,5,7and9,respectively.

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Example4:Compositeintervalmappingonsimulateddataatequallyspacedmarkersatapproximately5cM.QTLwitheects.1,1,.1,1lo-catedatmarkerintervals1,5,7and9,respectively.

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Simulateddata,equallyspacedmarkersatapproximately26cMwithh2=:94.QTLatsecondandforthintervalswithequaleectsof1.Gibbssamplerwithfullconditionalsconditionedon.

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TopPanel:ResultsforModel2onBarleyDatausingallmarkers.Bot-tomPanel:ResultsforModel2onBarleyDatausingselectedmarkers.

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RunningmeansforModel2onBarleydatausingallmarkers

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RunningmeansforModel2onBarleydatausingselectedmarkers

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WeproposedamodelforthesimultaneousestimationofQTLeectsandlocationsinabackcrosspopulation.AGibbssamplerwasimplementedtoobtaintheposteriordistributionofQTLeectsalongthegenome.QTLwerelocatedateachmarkerintervalandtheirsignicancewasassessedbyexaminingtheposteriordistribution.ByconsideringoneQTLperinterval,thedimensionoftheproblemwaskeptxed.Wealsoincludedallthemarkersascovariates.Asidefromaccountingfortheinformationthatthemarkersprovide,usingallmarkersascovariateseliminatesthemarkerselectionproblem,whichisonedisadvantageofmanyoftheexistingmethodsofQTLanalysis. Theperformanceofourmethodwasstudiedusingsimulateddatawithdierentnumbersofmarkersatdierentspacing.Wealsoconsidereddierentheritabilitylevels.OurmethodwasgenerallyeectiveindetermininglocationandindierentiatingQTLwithlargeeects.Inthecaseswherethemethodwasnotsuccessful,theposteriordistributionoftheQTLeectswasinformativeandshowedpatternsthatsuggestsignicanceintheintervalsforwhichQTLweredeclarednonsignicant.WeobservedthatincreasingthenumberofiterationsmaybecrucialforthesuccessfuldetectionofQTLthatwereundetected.QTLofsmalleectsweretypicallynotdetected,whichisalsothecaseformanyotherexistingmethodsofQTLanalysis.Asexpected,incasesoflowheritabilitythepowertodetectQTLislow.Forsomeofthesimulatedexamples,weempiricallydeterminedaneectivesamplesizethatwasrequiredforourmodeltosuccessfullydetecttheQTL.Theminimumsamplesizewillvaryfromproblemtoproblemandcouldbeatopicoffurtherstudy. 53

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SamplingfromthefullconditionaldistributionofthematrixZofQTLgenotypeswasachallengebecauseoftheircovariancestructure.WeusedaMetropolis-Hastingsalgorithmwithacandidateforwhichthiscovariancestructurewasreducedtotheindividuallevel.Thegenotypesoftheindividualsweresampledasblocks,theindividualsbeingindependentfromeachother.Evenwiththissimplication,samplingtheindividualsgenotypeswasnotstraightforward.WeusedanAcceptRejectalgorithmwithadierentcandidateforeachindividual.Inpractice,thenumberoftrialsbeforeacceptanceforfewindividualsturnedouttobelargethuswelimitedthenumberofrejectionstoaxednumber.Ifthisnumberwasreachedthegenotypesfromthepreviousiterationofthesamplerforthatindividualwerekept.ThismethodwasemployedbecausetheAcceptRejectalgorithmwasnotbeingusedtosamplefromthedistributionofinterest,buttosamplefromthecandidateoftheMetropolis-Hastingalgorithm.Satagopanetal.(1996)proposedtosampleeachindividualQTLgenotypeindependently.RecallthatinourmodelweareconsideringoneQTLpermarkerintervalthus,thisapproachcouldbecomputationallyintensiveasthenumbersofQTLconsideredandthenumberofindividualsincreases.Also,thespeedofconvergenceoftheGibbssamplermaybereduced.WeplantocomparetheperformanceoftheMetropolis-HastingsalgorithmandtheGibbssamplerapproachintermsofthespeedofconvergenceandexecutiontime. WederivedaninnovativewaytosamplefromtheposteriordistributionoftheQTLpositions.WeusedanAcceptRejectalgorithmtosamplefromtheconditionalprobabilitiesofthegenotypesgiventheankingmarkers().Fromthere,wecalculatedtheQTLpositionsforeachintervalateachiterationoftheGibbssampler.ThecalculationoftheQTLpositionsusingtherecombinationfractionsimposedrestrictionsontherangeof.Theserestrictionsinconjuctionwiththefactthatwestartedbyusingonlyoneankingmarkertocalculatedthe

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QTLpositions,createdmixingproblems.Partoftheintervalswererarelyvisited.Wesolvedthisproblembyusingtheinformationfrombothankingmarkersatrandom. Inpractice,somemarkergenotypeswillbemissing.Thus,allowingformissingmarkerinformationinourmodelwillbeimportant.Thiscouldbedonebyincorporatingthemissingmarkerinformationasparametersinthemodel.Thefullconditionaldistributionofthemissingmarkerinformationgivenalltheotherparametersinthemodelcouldbecalculatedandthemissingmarkergenotypecouldbesampledateachiterationofthesampler.Wesuspectthataccountingformissingmarkerinformationwouldnotbeadicultadditiontomaketothemethod. Asinanyregressionmodel,thecorrelationbetweenindependentvariablesdeservessomeattention.Collinearitycreatesseriousproblemsintheestimationofregressioncoecients(Rawlingsetal.,2001).Importantvariablesmaybeincorrectlyidentied.Moreover,ifallpotentialvariablesareincludedinthemodel,allofthemmaybeidentiedasnotsignicant.Inourcontext,asmoreandmoremarkersareincludedinthegeneticmapandthedistancesbetweenmarkersarereduced,linkagebetweenmarkersincreases.Linkedmarkers,inourunderstanding,translatetocorrelatedindependentvariablesinthemodel.SincethedistributionoftheQTLgenotypesdependsonthemarkergenotype,thecorrelationbetweenmarkerswillinducecorrelationinthesampledQTLgenotypes(matrixZ)andwillcreatedicultiesintheestimationoftheQTLeects(). Withincreasingnumbersofgeneticmarkersonthegeneticmapsitisimpor-tanttoknowhowtoeectivelyuseallofthisinformationwithoutreducingtheeectivenessofthestatisticalmethods.SomeauthorscommentabouthavingadensemapasanadvantagefortheaccuratelocalizationoftheQTLinthecontextoftheirmodels(SenandChurchill,2001),butthereisnotmuchdiscussionabout

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theimpactontheestimationofQTLeects.Thisisaninterestingareaforfurtherresearch.Fromourmodelperspective,apromisingandinterestingapproachwillbetoconsideraxednumberofmarkersselectedatrandom,spacedataxeddistanceateveryiterationoftheGibbssampler.Thedistancebetweenthemarkerswouldpossiblyhelptoreducecorrelation.ConsideringdierentsetsofmarkersmayhelpseparateQTLeectinadjacentintervals. Thecorrelationproblemcouldleadusbacktothesinglemarkeranalysisapproach.OneideaistousetheinformationofasinglemarkertoinfertheQTLgenotypeatalocationinanintervalaroundthatparticularmarker.Thewidthofthisintervalwilldependonthedistancefromthemarkerunderconsiderationanditsadjacentmarkers.Athoughtistoconsidertheintervalthatishalfwaytotheadjacentmarkers.HowtodeterminethelocationoftheQTLintheintervalandhowtoaccountforallQTLeectssimultaneouslymuststillbeexplored.WesuspectthatoncewehavethepositionsandtheQTLgenotypesthesimultaneousestimationoftheQTLeectsmustnotbethatdierenttowhatwehavealreadydone. WhatevermodelisusedforthemappingofmultipleQTL,itwillbeofinteresttoexploretherobustnessofthemethodsagainstnoninformativemarkers.Otherareasforcontinuedresearchincludetheextensiontoothertypesofexperimentalcrosses,inclusionofepistaticeectsandthepossibleapplicationoffunctionalmappingtotheparameter.

PAGE 68

3 weconsideredproperpriorsforand.HereweconsidercombinationsofatandnormalpriorsforandinModel2andndtheconditionstoensurethattheposteriordistributionisproper.ThepriorsforallotherparametersremainasinChapter 3 .Itissucienttoshowthattheintegrationoftheposteriorovertheparameter(s)withaatpriorisbounded.Fromequation( 3{4 ),theintegrandistheposteriorZf(yj;;;Z;2)f()f()f(j2)f(2)df(Zj)f() But,becausetheparameterhasaproperpriorsucestoshowthattheintegra-tionof (A{1) overtheparametersofinterestisbounded.

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22f1(y(IHz)XZ))0(1(y(IHz)XZ)gd=e1 22(y(IHz)XZ)0(y(IHz)XZ)Ze1 22fn2210(y(IHz)XZ)gd=e1 22(y(IHz)XZ)0(y(IHz)XZ)Zen 22(y(IHz)XZ)0(y(IHz)XZ)en 22(y(IHz)XZ)0(y(IHz)XZ)e1 22(10(y(IHz)XZ))2 22(y(IHz)XZ)0hI110 BecausethematrixI110 A{2 )isboundedby1.Thus,theintegrationofequation( A{1 )over,and2isnite.

PAGE 70

22f1(y(IHz)XZ))0(1(y(IHz)XZ)gd=e1 22(y1(IHz)X)0(y1(IHz)X)Ze1 22f0Z0Z20Z0(y1(IHz)X)gd=e1 22(y1(IHz)X)0(y1(IHz)X)Ze1 22f0Z0Z20Z0(y1(IHz)X)(y1(IHz)X)0Hz(y1(IHz)X)gd=e1 22(y1(IHz)X)0[IHz](y1(IHz)X)Ze1 22f((Z0Z)1Z0(y1(IHz)X))0[Z0Z]((Z0Z)1Z0(y1(IHz)X))gd/e1 22(y1(IHz)X)0[IHz](y1(IHz)X)j(Z0Z)12j1 2: A{1 )over,and2isnite. Proof:

PAGE 71

TheintegrationoverisgivenonTheorem3.2.SinceZisoffullcolumnrankZZf(yj;;;Z;2)f()f()f(j2)f(2)f(Zj)f()dd/Ze1 22(y1(IHz)X)0[IHz](y1(IHz)X)d=e1 22(y(IHz)X)0(IHz)(y(IHz)X)Ze1 22f210(IHz)1210(IHz)(y(IHz)X)gd=e1 22(y(IHz)X)0(IHz)(y(IHz)X)Ze10(IHz)1 222210(IHz)(y(IHz)X) 10IHz)110(IHz)(y(IHz)X) 10IHz)12d=e1 22(y(IHz)X)0h(IHz)(IHz)110(IHz) 10(IHz)1i(y(IHz)X)Ze10(IHz)1 22n10(IHz)(y(IHz)X) 10IHz)1o2d/e1 22(y(IHz)X)0h(IHz)(IHz)110(IHz) 10(IHz)1i(y(IHz)X)2 2: 10(IHz)1issymmetricandidempotenthencepositivesemidenite(Searle1982)theproductisboundedif10(IHz)16=0.Thentheintegrationofequation( A{1 )overand2isnite.

PAGE 72

1rk(k+1) Substituting( B{2 )in( B{1 )weobtainthat (B{3) whichimpliesthat Substituting( B{4 )in( B{2 )weobtainthat SincerQk(k)1p2krk(k+1) Also,sincerQk(k+1)
PAGE 73

ThistellusthatifwewanttocalculaterQk(k)oncewehaveavalueforp2k,p1kmustbegreaterthan1p2krk(k+1) B{1 .WhileifonlyrQk(k+1)'sarecalculated,thepriorwilllookliketheoneonthetop.Eachgraphshows5intervalsofequallength.TohaveamoreuniformdistributionoftheQTLpositionsattheintervals,amixturedistributionisconsideredforp1kgivenp2k.WeassumeaBeta(c2;d2)priorforp2kandforp1k,f(p1kjp2k)/pc111k(1p1k)d11withrange(1p2krk(k+1) 2andf(p1kjp2k)/pc111k(1p1k)d11withrange(p2k;1)withprobability1 2.Figure B{2 showhistogramsoftheQTLpositionsgeneratedatdierentintervallengthsusingthisprior.Forallthegraphsinthegure,c1=1andd1=1,i.e,p1kjp2kisgeneratedfromauniformdistributiononthecorrespondingrange.Forthefourgraphsonthetop,c2=2andd2=2.Forthe98cMintervalsc2=1andd2=1seemstobeabetterchoice(graphatthebottom). 3{6 )f(Zj)/nYi=1t1Yk=1f(zikjpk;xik;xi(k+1))/nYi=1t1Yk=1pzikk(1pk)1zik=t1Yk=1pn4kz4k+z1k1k(1p1k)n1kz1k+z4kpn3kz3k+z2k2k(1p2k)n2kz2k+z3k 3{2 )

PAGE 74

1rk(k+1)n4kz4k+z1krQk(k)rQk(k+1) Undertheassumptionofnointerferencerk(k+1)=rQk(k)+rQk(k+1)2rQk(k)rQk(k+1) B{5 )canbewrittenas(1rQk(k))(1rk(k+1)rQk(k))n4kz4k+z1krQk(k)(rk(k+1)rQk(k))n1kz1k+z4k(1rQk(k))(rk(k+1)rQk(k))n3kz3k+z2k1 1rk(k+1)n1k+n4krQk(k)(1rk(k+1)rQk(k))n2kz2k+z3k1 12rQk(k)n 2(1e2).Thereforeapriorforrisf(r)=f1 2log(12r)1 12r=1 (ba)(12r)/1 12r

PAGE 75

Thus,f(rQk(k)j;;;Z;2;y)f(rQk(k))/(1rQk(k))(1rk(k+1)rQk(k))n4kz4k+z1krQk(k)(rk(k+1)rQk(k))n1kz1k+z4k(1rQk(k))(rk(k+1)rQk(k))n3kz3k+z2k1 1rk(k+1)n1k+n4krQk(k)(1rk(k+1)rQk(k))n2kz2k+z3k1 12rQk(k)n+1

PAGE 76

HistogramsofQTLpositions.Eachgraphshows5intervalsofequallength.TopPanel:OnlyrQk(k+1)iscalculated.BottomPanel:OnlyrQk(k)iscalculated.

PAGE 77

HistogramsofQTLpositionsgeneratedusingthemixturedistributionforp1kjp2katdierentintervallengths.Eachgraphsshow5intervalsofequallength.

PAGE 78

2j;;Z;;yIG(n+t 2+0 2+A1. 22f(y1(IHz)XZ)0(y1(IHz)XZ)gnYi=1t1Yk=1pzikk(1pk)1zik:

PAGE 79

TherunningmeansfortheexamplesinChapter4areshownhere.Notethatthescaleinthey-axisaredierent. 68

PAGE 80

Example2:h2=:4,n=250

PAGE 81

Example2:h2=:2,n=250

PAGE 82

Example2:h2=:2,n=500

PAGE 83

Example2:h2=:1,n=500

PAGE 84

Example3:h2=:4,n=250

PAGE 85

Example3:h2=:3,n=600

PAGE 86

Example3:h2=:3,n=800

PAGE 87

Example4:h2=:3,n=600

PAGE 88

Broman,K.W.andSpeed,T.P.(1999).AreviewofmethodsforidentifyingQTLsinexperimentalcrosses,StatisticsinMolecularBiologyIMSLectureNotes-MonographSeries33:114{142. Broman,K.W.andSpeed,T.P.(2002).Amodelselectionapproachfortheidenticationofquantitativetraitlociinexperimentalcrosses,JournaloftheRoyalStatisticalSociety,SeriesB64:641{656. Churchill,G.A.andDoerge,R.W.(1994).Empiricalthresholdvaluesforquantitativetraitmapping,Genetics138:963{971. Doerge,R.W.(2002).Mappingandanalysisofquantitativetraitlociinexperi-mentalpopulations,NatureReviewsGenetics2:43{52. Doerge,R.W.andChurchill,G.A.(1996).Permutationtestsformultiplelociaectingaquantitativecharacter,Genetics142:285{294. Doerge,R.W.,Zeng,Z.B.andWeir,B.S.(1997).Statisticalissuesinthesearchforgenesaectingquantitativetraitsinexperimentalpopulations,StatisticalScience12:195{219. Doornik,J.A.(2002).Object-OrientedMatrixProgrammingUsingOx,3edn,London:TimberlakeConsultantsPressandOxford.www.nu.ox.ac.uk/Users/Doornik. EsparzaMartnez,J.H.andFoster,A.E.(1998).Geneticanalysisofheadingdateandotheragronomiccharactersinbarley(HordeumvulgareL.),Euphytica99:145{153. George,E.I.andMcCulloch,R.E.(1993).VariableselectionviaGibbssampling,JournaloftheAmericanStatisticalAssociation88:881{889. Haldane,J.B.S.(1919).Thecombinationoflinkagevaluesandthecalculationofdistancebetweenthelocioflinkedfactors,JournalofGenetics8:299{309. Haley,C.S.andKnott,S.(1992).Asimplemethodformappingquantitativetraitlociinlinecrossesusingankingmarkers,Heredity69:315{324. Jansen,R.C.andStam,P.(1994).Highresolutionofquantitativetraitsintomultiplelociviaintervalmapping,Genetics136:1447{1455. 77

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Kasha,K.J.andKleinhofs,A.(1994).MappingofthebarleycrossHarringtonTR306,BarleyGeneticsNewsletter23:65{69. Kleinhofs,A.,Kilian,A.,SaghaiMaroof,M.A.,Biyashev,R.M.,Hayes,P.,Chen,F.Q.,Lapitan,N.,Fenwich,A.,Blake,T.K.,Kana-zin,V.,Ananiev,E.,Dahleen,L.,Kudrna,D.,Bollinger,J.,Knapp,S.J.,Liu,B.,Sorrells,M.,Heun,M.,Franckowiak,J.D.,Homan,D.,Skadsen,R.andSteenson,B.J.(1993).Amolecular,isozymeandmorphologicalmapofthebarley(Hordeumvulgare)genome,TheoreticalandAppliedGenetics86:705{712. Kosambi,D.D.(1944).Theestimationofmapdistancesfromrecombinationvalues,AnnEugen12:172{175. Lander,E.S.andBotstein,D.(1989).MappingMendelianfactorsunderlyingquantitativetraitsusingRFLPlinkagemaps,Genetics121:185{199. Liu,B.(1997).StatisticalGenomicsLinkage,MappingandQTLAnalysis,CRCPressLLC. Ma,Z.,Steenson,B.J.,Prom,L.K.andLapitan,N.L.V.(2000).MappingofquantitativetraitlociforFusariumheadblightresistanceinbarley,Phy-topathology90:1079{1088. Martinez,O.andCurnow,R.N.(1992).Estimatingthelocationandsizeoftheeectsofquantitativetraitlociusingankingmarkers,TheoreticalandAppliedGenetics85:480{488. Rawlings,J.O.,Pantula,S.G.andDickey,D.A.(2001).AppliedRegressionAnalysis,AResearchTool,2edn,NewYork:Springer-VerlagInc. Satagopan,J.M.,Yandell,B.S.,Newton,M.A.andOsborn,T.C.(1996).ABayesianapproachtodetectquantitativetraitlociusingmarkovchainMonteCarlo,Genetics144:805{816. Searle,S.R.(1982).MatrixAlgebraUsefulforStatistics,NewYork:JohnWiley&Sons,Inc.,p.321. Sen,S.andChurchill,G.A.(2001).Astatisticalframeworkforquantitativetraitmapping,Genetics159:371{387. Sillanpaa,M.J.andArjas,E.(1998).Bayesianmappingofmultiplequantitativetraitlocifromincompleteinbredlinecrossdata,Genetics148:1373{1388. Soller,M.andBrody,T.(1976).Onthepowerofexperimentaldesignsforthedetectionoflinkagebetweenmarkerlociandquantitativelociincrossesbetweeninbredlines,TheoreticalandAppliedGenetics47:35{39. Yi,N.(2004).AuniedmarkovchainMonteCarloframeworkformappingmultiplequantitativetraitloci,Genetics167:967{975.

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DamarisSantanaMorantwasborninRoPiedras,PuertoRico.SheisthesecondoftwodaughtersofJuanManuelSantanaandFranciscaMorant.SheearnedaB.S.incomputationalmathematicsfromtheUniversityofPuertoRicoinHumacaoin1989andaM.S.inappliedmathematicsfromtheUniversityofPuertoRicoinRoPiedrasin1995.Inthesummerof1997,sheparticipatedintheMathematicalandTheoreticalBiologyInstituteatCornellUniversitywhichinspiredhertopursueagraduatedegreeinstatistics.In2001,sheobtainedaM.S.degreeinbiometryfromCornellUniversityandin2005aPh.D.instatisticsfromtheUniversityofFlorida,bothwithDr.GeorgeCasellaasheradvisor.Herinterestsincludestatisticalgeneticsandenvironmentalstatistics. 80


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BAYESIAN MAPPING OF MULTIPLE QUANTITATIVE TRAIT LOCI


By

DAMARIS SANTANA MORANT

















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


2005


































Copyright 2005

by

Ddmaris Santana Morant
















To Tim, my parents and sister















ACKNOWLEDGMENTS

There are no words, time, or space to thank all that have contributed in

some way to this journey. Everyone that I came across or spoke to all these years

may be unaware of their contribution. A simple gesture, a word, a smile are

sometimes the turning point in a particular d w. I recognize the Creator as the

Source of all knowledge and thank Him for the opportunity of finishing this degree

and all the blessings that I have received. I thank my advisor George Casella

for sharing, so generously, his knowledge and time with me. I thank him for his

support, encouragement and inspiration. He believed in me more that I believed

in myself, one of the reasons that enabled me to reach the end of this journey. I

thank the members of my committee, Jim Hobert, Sam Wu, Rongling Wu and

John Davis, for their interest in my work and for their support. We had excellent

committee meetings out of which great ideas emerged. I Thank them for such an

open learning environment. I thank Matias Kirst for joining us and for helping in

finding the data sets. I thank the faculty, students and staff of the Department of

Statistics at UF for all their help and support. I thank my husband, Tim Porch,

for all his support and encouragement. I thank my parents and my sister for all

their love, support and prayers. I thank my huge family and all my friends for all

their love, support and prayers. I cherish every call, meeting, flower, letter, meal,

cup of coffee and visit. I finally thank Carlos Castillo-C'l i, .; for inviting me to

participate at his Institute at Cornell University, starting the chain of events that

brought me to finishing this degree.















TABLE OF CONTENTS
page

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

TA BLE . . . . . . . ..... . vii

LIST OF FIGURES ................... ......... viii

ABSTRACT ... .. .. .. ... .. .. .. .. ... .. .. .. ... .. .. x

CHAPTER

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

1.1 Basic Concepts of Genetics ........... ........... 2
1.2 QTL Analysis .. ... .. .. .. .. ... .. .. .. ... .. .. 4

2 METHODS FOR QTL MAPPING ................. .... 6

2.1 Single QTL Methods .......................... 6
2.1.1 Single Marker Analysis and Simple Linear Regression .... 6
2.1.2 Interval Mapping and Regression Mapping .......... 8
2.2 Multiple QTL Methods ................... ..... 10
2.2.1 Multiple Regression on Marker Genotypes and Composite
Interval Mapping ............... .. .. .. 10
2.2.2 B --i Methods. .............. .. .. .. 12

3 SIMULTANEOUS QTL ESTIMATION .................. .. 15

3.1 Model 1 .......... ......... ........ .... .. 15
3.1.1 Gibbs Sampler for Model 1 .... . . 17
3.1.2 Implementation .... ......... ... ... .. .. 22
3.2 Model 2 ....... ..... ..... ...... 27
3.2.1 Conditions for a Proper Posterior Distribution . ... 29
3.2.2 Full Conditional Distributions ................ .. 30

4 PERFORMANCE EVALUATION .................. ... 36

4.1 Simulated Data .................. ........ .. .. 36
4.2 Convergence and Results Presentation ............... .. 37
4.3 Simulation Results .................. ........ .. 37
4.4 Model 2 with Full Conditionals Conditioned on . ... 40
4.5 Data Analysis ............... ........... .. 40









5 CONCLUSIONS AND FURTHER RESEARCH ............ ... 53

APPENDIX

A PRIORS FORp AND 7 .......... .................. 57

B PARAMETERIZATION OF QTL POSITIONS ............. 61

B.1 Range of the plk's and p2k's and Exploration of Their Prior Distri-
bution ............... ............. ...... 61
B.2 Reparameterization of the Posterior Distribution in Terms of the
Recombination Fractions ............ . .. 62

C FULL CONDITIONALS FOR MODEL 2 CONDITIONED . 67

D RUNNING MEANS FOR PERFORMANCE ANALYSIS ........ 68

REFERENCES .............. ............... ..... 77

BIOGRAPHICAL SKETCH ............................ 80















TABLE
Table page

2-1 Backcross Design, Model Assumptions ................. 8















LIST OF FIGURES


Figure page

3-1 Histograms of QTL positions. Top Panel: Only rQm(k) is calculated. Bot-
tom Panel: Both rQm(k) and rQm(k+1) are calculated. . . 33

3-2 Simulated data, equally spaced markers at approximately .26M. QTL at
second and forth intervals with equal effects of 1. h2 .94, a2 .04.
Model 1 with 72 1 (top left), 72 (top right) and Model 2 with
r2 1 (bottom). ............... ........... 34

3-3 Simulated data, only a few individuals exceed L = 3000 and no more
than .5'. of the total number of iterations. ............... 35

4-1 Example 1: Simulated data, equally spaced markers at approximately
26cM. QTL located at the second and forth interval with equal effects
of 1. Top panel: h2 = .94, a2 = .04, Bottom panel: h2 .4, a2 = 1, Left
Panel: Model 2, Right Panel: Composite interval mapping. . ... 43

4-2 Example 2: Model 2 on simulated data at equally spaced markers at ap-
proximately 15cM. QTL located at the second and forth interval with
equal effects of 1. ............... ........... .. 44

4-3 Example 2: Composite interval mapping on simulated data at equally
spaced markers at approximately 15cM. QTL located at the second and
forth interval with equal effects of 1. ................ .. .. 45

4-4 Example 3: Model 2 on simulated data at equally spaced markers at ap-
proximately 15cM. QTL with effects 1,.1,1,.1 located at marker intervals
2, 5, 7 and 8, respectively. .................. .... 46

4-5 Example 4: Model 2 on simulated data at equally spaced markers at ap-
proximately 5cM. QTL with effects .1,1,.1,1 located at marker intervals
1,5,7 and 9, respectively. .................. .... 47

4-6 Example 4: Composite interval mapping on simulated data at equally
spaced markers at approximately 5cM. QTL with effects .1,1,.1,1 located
at marker intervals 1,5,7 and 9, respectively. .............. 48

4-7 Simulated data, equally spaced markers at approximately 26cM with h2
.94. QTL at second and forth intervals with equal effects of 1. Gibbs
sampler with full conditionals conditioned on . . ...... 49









4-8 Top Panel: Results for Model 2 on Barley Data using all markers. Bot-
tom Panel: Results for Model 2 on Barley Data using selected markers.. 50

4-9 Running means for Model 2 on Barley data using all markers ...... .51

4-10 Running means for Model 2 on Barley data using selected markers . 52

B-1 Histograms of QTL positions. Each graph shows 5 intervals of equal length.
Top Panel: Only rQm(k+l) is calculated. Bottom Panel: Only rQm(k) is cal-
culated .................. ................. .. 65

B-2 Histograms of QTL positions generated using the mixture distribution
for plk P2k at different interval lengths. Each graphs show 5 intervals of
equal length. . . . . . .. .. . . 66

D-1 Example 2: h2 = .4, n 250 .................. ... .. 69

D-2 Example 2: h2 = .2, n 250 .................. ... .. 70

D-3 Example 2: h2 = .2, n 500 .................. ... .. 71

D-4 Example 2: h2 = .1, n 500 .................. ... .. 72

D-5 Example 3: h2 = .4, n 250 .................. ... .. 73

D-6 Example 3: h2 = .3, n 600 .................. ... .. 74

D-7 Example 3: h2 = .3, n 800 .................. ... .. 75

D-8 Example 4: h2 = .3, n 600 .................. ... .. 76















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

BAYESIAN MAPPING OF MULTIPLE QUANTITATIVE TRAIT LOCI

By

Ddmaris Santana Morant

December 2005

C'!I ,i: George Casella
Major Department: Statistics

We describe a method for the simultaneous estimation of the locations and

the effects of quantitative trail loci (QTL) in a backcross population. We consider

a mixed model that includes one QTL per interval and considers all markers as

covariates. By including one possible QTL per interval it was possible to examine

and account for all of the QTL effects. The marker information is included in the

model using a term that takes into account the portion of the effects of the markers

that are not taken into account by the QTL. The effects of the marker information

are considered random effects in the model. We obtain the posterior distribution

of the QTL effects along the genome using a Gibbs sampler. To determine the

significant effects, 95'. posterior confidence intervals are used. One advantage

of this approach is that all markers are used as covariates, which eliminates the

constraint of marker selection. The performance of our method was studied

using simulated data for equally spaced markers at different interval lengths. We

considered examples with six and ten markers and with different heritability levels.

We compared our results to results using composite interval mapping for some of










the examples. The analysis of Barley (Hordeum ;;i ,ljre) chromosome five is also

presented.















CHAPTER 1
INTRODUCTION

Many agronomic traits in plants are classified as quantitative in nature; i.e.,

the observed phenotype is the joint result of the effects of a number of genetic

and environmental factors. The genetics of quantitative traits are studied through

estimating the effects of the genes contributing to the traits as well as by determin-

ing their location in the genome. Once a molecular location is determined for the

genes, they are called quantitative trait loci (QTL). Knowledge about these loci

assist in the selection of superior genotypes in a population for trait improvement

(e.g., yield and disease resistance in crops). Several methods for QTL analysis

have been developed to determine the number, location and effects of QTL. These

methods fall in two categories, those that model the effects of single QTL and those

that model the effects of multiple QTL. The best approach to search for multiple

QTL remains an open problem (Sen and ('!,i, !,111 2001). We developed a method

for the simultaneous estimation of QTL effects and locations in the genome. We

focused on experimental populations, particularly the backcross design.

In C'!i lpter 2 we review some of the methods for QTL mapping presented in

the literature. Our method for simultaneous QTL estimation in a B ,-.i I, frame-

work is presented in ('!i lpter 3. The performance of our method on simulated data

as well as on chromosome 5 of a Barley (Hordeum ;;,ll ire) dataset is presented in

('! Ilpter 4. A discussion of the results and a presentation on further research are

presented in ('!i lpter 5. First, we introduce some basic concepts of genetics and a

background on QTL analysis.









1.1 Basic Concepts of Genetics

We may ask ourselves, why do individuals look different? Aside from envi-

ronmental factors, the differences between individuals are defined by their genetic

makeup. This genetic information is coded in deoxyribonucleic acid (DNA). DNA

is composed of four repeating units called nucleotides and forms double helix

molecules, termed chromosomes. The information in the DNA is expressed through

the translation of DNA into amino acids, which in turn form proteins that are

responsible for the function and structure of cells. It is believed that only a small

part of the DNA encodes for proteins. DNA that encodes for information is called

a gene and the location of the gene on the chromosome is called a locus. Genes are

therefore the inherited factors that control a trait's phenotype, or the observable

form of the trait. A trait can be controlled by one gene or by multiple genes (quan-

titative traits). The variation in the phenotype of a particular trait corresponds to

different forms of the gene, or alleles. C'!i ,,' in the sequence of nucleotides that

compose a gene results in the formation of a new allele. Therefore, a gene may have

many alleles.

Most animal and plan species are diploid; thus there are two copies of each

chromosome, and individuals have two copies of each gene. Given alleles A, and

A2 for a specific gene, the individual will have one of three possible genotypes:

A1A1, A1A2 or A2A2. An individual that has genotype A1A1 or A2A2 is called

homozygous, indicating that the alleles are identical. Otherwise, the individual is

heterozygous. If a trait is only controlled by a single gene, the expression of the

p1 i,. .r pe is determined by the dominance relationship between the alleles. If the

allele A, is completely dominant, the individuals with genotype A1A1 and A1A2

will be indistinguishable and will express the Al phenotype. If A, is incompletely

dominant, the heterozygotes will have a phenotype intermediate between the









two homozygotes. In this case, the alleles are said to be codominant because

heterozygotes and homozygotes may be distinguished.

The transmission of genetic information from parents to offspring is through

egg and sperm cells. These cells, called gametes, carry one complement of the

chromosomes of each parent. During the formation of gametes in the process of

meiosis each chromosome in the pair duplicates resulting in four chromosomes.

The duplicates interchange DNA, in events called crossovers, resulting in two

new chromosomes that are a mosaic of the parental chromosomes. The four

chromosomes then separate to form four new cells (gametes), each one with one

chromosome. Two gametes result with copies identical to one of the chromosomes

in the original pair, the other two with chromosomes that are a combination of

both. The latter are called recombinant gametes. The diploid copy number in the

cell is restored when the egg and the sperm unite. Thus, recombination is a key

source of genetic variation.

Consider two diallelic genes in an individual with genotype AaBb. AB are on

one chromosome and ab on the other. There are 4 possible gametes: AB, ab, and

the recombinants Ab and aB. By Mendel's rule of independent assortment, each

of these gametes would have the same probability. This rule states that alleles of

different genes segregate independently. It was discovered later that the frequency

of gametes depends on the genetic distance between the genes. Genes that are close

to each other are more likely to remain together in the process of meiosis. Genes

that are further away are more likely to experience crossovers and recombination.

Genetic distance is determined through use of linkage between two loci to calculate

the recombination fraction, the ratio of the number of recombinant gametes to total

number of gametes. Recombination between two loci on the same chromosome is

more likely the further the loci are apart. For example, the recombination fraction,

r 0, between two loci means that they are completely linked, while r = means









that they segregate independently and are unlinked. Mendel's rule of independent

assortment applies to genes that are unlinked.

1.2 QTL Analysis

Linkage is the basis for QTL analysis. Genetic maps are constructed based

on the recombination fraction between genetic markers. The relationship be-

tween phenotypes and the genotypes of these genetic markers is explored. If an

association exists, it .-. -. -; that a gene controlling the trait is located near the

marker. Experimental populations, like backcross populations and F2 populations,

derived from the cross of pure inbred lines offer the ideal setting for the study

of associations between phenotypes and markers because the prc..--iv of the first

filial generation (Fi) are genetically identical. Because the population structure is

controlled and the parental genotypes are known, genetic questions can be precisely

determined.

Molecular markers are small regions of DNA for which detectable heritable

variation can be analyzed for individuals in a population. A genetic map consists

of linearly ordered molecular markers and the genetic distance between them. The

genetic map is constructed by analyzing the relationship of the marker genotypes

for the individuals in a population by a process called linkage analysis (Liu 1997).

Genetic markers are placed in linkage groups based on their linkage relationship

defined by recombination fractions. Recombination fractions are then translated to

genetic distances using a mapping function (Haldane 1919, Kosambi 1944). Genetic

maps provide a representation of genome structure and have been developed for

many plant species.

We focus on the backcross population structure for the development of

a multiple QTL analysis method. Although the backcross population may be

considered simple to analyze, it still presents big challenges. The extension of

the methods developed for backcross experiments to other types of experimental






5


crosses is often not difficult. For the purpose of QTL analysis, the parental lines

that form the population should be homozygous, but they must differ for the trait

of interest. The two parental lines are crossed to get the F1 generation. Each F1

individual receives a copy of a chromosome from each of its two parents; thus

they are heterozygous wherever the parental lines differ. Individuals in the F1

generation are genetically identical. The F1 generation can be backcrossed to the

P1 or P2 parent to obtain the BC1 or BC2 population. The individuals in the

backcross population have one of two i, -: 1 ir pes at every locus, homozygous or

heterozygous. After the population is generated, phenotypic information and the

marker genotypes is obtained for all the individuals in the population as well as the

two parents.

Thus, the goal of the QTL analysis is to determine the association between the

individual phenotypes and the alleles they received from their parents at various

marker loci using the genetic map, the marker information, and the phenotypic

data.















CHAPTER 2
METHODS FOR QTL MAPPING

Several methods have been described in the literature for QTL mapping.

These methods fall essentially in two categories, those that estimate the effects of

single QTL and those that estimate the effects of multiple QTL. We will briefly

review some of the methods in both categories. Some of these methods perform

the analysis at the marker locations while others use the marker information to

estimate effects between markers. More extensive reviews are presented by Doerge

(2002), Broman and Speed (1999) and Doerge et al. (1997).

2.1 Single QTL Methods

We will consider several methods: single marker analysis (Soller and Broody

1976), simple linear regression, interval mapping (Lander and Botstein 1989) and

regression mapping (Haley and Knott 1992, Martinez and Curnow 1992).

2.1.1 Single Marker Analysis and Simple Linear Regression

Single marker analysis involves studying single genetic markers one at a time.

Based on the putative QTL genotype of each individual, the population can be

separated into two groups in the backcross design and their respective trait means

can be compared. Unfortunately, the QTL genotypes are unknown. Thus the

analysis is performed at the markers where the genotypes are known. We will

assume that the phenotypes of parent P1, parent P2 and the first filial (Fl) are

distributed N(pQQ, J2), N(qq, 7a2) and N(PQq, a2), respectively (Table 2-1). The

means of these populations are attributed to the QTL effect. Markers are assumed

to have no effect on the trait. The backcross population between P1 and Fl will

have marker-QTL ._. nI, ,'pes M1Q/M1Q, MiMq, MQ/iq Q/, l_Q M1Q/ 1_[ with

probability 1-r, rM 1-,r respectively. rMQ is the recombination fraction









between marker M and the putative QTL. In this case, Q represents the QTL

genotype that is not observed. Two observable marker classes are obtained with the

following mixture distributions:


Mi/M : (1 rrMQ)N(QQ, a2) + rMQN(Qq, a2)

M1/3. : rMQN(IQQ,) + -(1- Q)N(I q, U2).

The difference in the means of these two populations is


IM M1 IM M2 = (1 2rMQ)(IQQ PQq)-

The usual t-test will then test the hypothesis


Ho : I -pMml = pMi H1 : pmMiM / PMiM


which is equivalent to

Ho: rMQ = .5 HI : rMQ < .5

which tests the presence of a QTL unlinked to the marker under consideration.

From the experimental design, it is known that /IQQ -Qq / 0 and that we started

with parental lines that differ at the trait of interest. However, the analysis is

confounded by the effect of locus Q, since it is the product (1 2rMQ)(JQQ PQq)

that is being tested for departures from zero. A QTL with a small effect that is

close to the marker will give the same result as a QTL with a larger effect located

further from the marker.

Care should be taken with the determination of critical values since the

distribution of the populations in consideration, in this case the observable marker

classes, are mixtures of normals. C('hi, !!! and Doerge (1994) discuss permutation

theory for the calculation of empirical threshold values.









Table 2-1. Backcross Design, Model Assumptions


PI: MIQ/MiQ x P2: 1iq/i-_q
N(pQQ, a2) N( ,a2)
4 \Fl: MIQ/13[q
N N(Pq, a2)

Bl: MIQ/MIQ MiQ/Miq MiQ/- _Q MiQ/- _q
1-rMQ rMQ rMQ 1-rMQ
2 2 2 2
T T
observable marker class 1 observable marker class 2


Consider the linear regression model


yi = 0 + 3lxi + ci (2-1)

where y is the trait value, xi is either 1 or 0 (in a backcross design) depending on

the marker genotype, homozygote (MIM1) or heterozygote (M 11.) respectively

and ci is a N(0, -2) random variable. The hypothesis Ho : 31 = 0 is equivalent to

the above hypothesis since the regression coefficient 31 is the difference between the

mean of the observable marker classes, that is, 31 = (1 2rMQ)(/QQ IfQq).

2.1.2 Interval Mapping and Regression Mapping

Lander and Botstein (1989) introduced interval mapping to remedy the

problems presented by single marker analysis, including the confounding of

genetic locations of QTL and phenotypic effects. An additive model is assumed

for the phenotype, with no epistasis (interactions among QTL). The phenotype

results from summing the effects of individual QTL and normally distributed

environmental noise. Interval mapping extends the idea of maximum likelihood in

model (2-1). For this model, the likelihood function is

12) 2
L(o,/ 1,2) -$- (i- I 2
i= 12









To test the hypothesis Ho : 31 = 0 the test statistic typically used is



L(3o,0, 72)

where /o and 02 are the constrained MLEs obtained under the null hypothesis. The

LOD score indicates how much more probable it is that the data arise from the

situation of having a QTL present versus absent.

In interval ir: pplli-r' the information provided by the genetic map is used to

march along the chromosome and calculate the LOD score at different positions in

the genome. The model is

Yi = 0o + A-i + ei

where zi is the genotype of the putative QTL. When calculating the LOD score

at positions other than the markers, zi is unknown. By using the genetic map,

the probability distribution of zi given the flanking markers is known (see Section

3.1). The probability distribution is based on the recombination fraction between

the putative QTL and the flanking markers. Once at a particular position, these

recombination fractions are determined from the genetic map. Thus, the likelihood

is

L(/o, 01, a2)

1 _2 1 -1
r P(z 0) 27(2 27-(Y-)2 + F>z = j
i= 1
Parameter estimation is done using the EM algorithm. The maximum LOD score

over all studied positions in the genome is an indication of a single QTL if it is

larger than some specified threshold value.

Martinez and Curnow (1992) used the same approach of marching along the

chromosome. Their model is


Yi = so + 3P(zi = I) + ci









Note that P(zi = 1) is the expected value of zi. As before, once at a particular

position, these probabilities can be calculated using the genetic map; thus this

is a simple regression model and the statistical analysis is straightforward. The

minimum residual sum of squares over all studied positions in the genome is an

indication of a single QTL if it is smaller than some specified threshold value.

The drawback of these two methods, as discussed by several authors including

Martinez and Curnow (1992), is that they do not guard against ghost QTL. Ghost

QTL occur when a QTL is located in a marker interval and neighboring regions

also exhibit significant test statistics. The problem is that these methods do

not take into account the presence of other possible QTL in the genome. These

methods have been shown to give accurate estimates of the QTL position and its

effect when there is only one QTL segregating in the population. This problem

brings us to the second category of methods which take into account multiple QTL.

2.2 Multiple QTL Methods

Methods that account for multiple QTL allow for the separation of the effects

of linked QTL and for the study of interactions among QTL. We will focus on

B i., i i models which is the framework for our research. We will first describe

briefly multiple regression and composite interval mapping (Zeng 1993, 1994,

Jansen and Stam 1994).

2.2.1 Multiple Regression on Marker Genotypes and Composite
Interval Mapping

Multiple regression on marker genotypes is an extension of the model (2-1).

iM ,i: markers are considered instead of one marker at a time. Let t be the number

of markers. The model is defined by
t
yi = /3o+ /3i + (2-2)
j=1









where yi, cj and xij are defined as in model (2-1), except that now we have t

markers. Zeng (1993,1994) discussed properties of the multiple regression analysis.

Among them, the partial regression coefficient parameter for marker j depends

only on those QTL which are located between markers j 1 and j + 1, under the

assumption of no epistasis. He argues that this results in a test for the presence of

QTL between markers j 1 and j + 1, regardless of the presence of other QTL

in the genome. Zeng (1993) discussed that the use of multiple regression alone

is not appropriate since the estimates of QTL effects by the partial regression

coefficient are biased. Thus, he introduced composite interval mapping which

is a combination of interval mapping and multiple regression. A genome scan

using interval mapping is performed, but markers outside the interval under

consideration are used as covariates to control for the effects of other QTL. At

each position under consideration, LOD scores are calculated. The selection of

the markers to use as covariates is a problem with this method. Conditioning on

linked markers potentially increases the precision of the test and estimation, but

with a possible decrease in statistical power (Zeng 1993). If there are QTL in the

intervals immediately .,1i i,:ent to the interval under consideration, this method has

the potential to falsely indicate the presence of a QTL (Zeng 1994).

Broman and Speed (1999,2002) stated that QTL mapping should be viewed

as a problem of model selection and not of multiple testing. Instead of minimizing

the prediction error, they seek to identify the subset of markers for which Oj / 0.

They introduced a modified version of the B i, -i ,i information criteria (BIC6) for

model selection. They proceed in two stages. First, the space of models is searched

in order to pick the best ones, those that would have been chosen if all models

were fitted. Second, the model with minimum BIC6 is chosen among the selected

models. They discussed several methods to select the best models, including










backward and forward selection and an MC'\ C method, among others. Doerge and

C('li !111!! (1996) discuss other methods for model selection.

2.2.2 Bayesian Methods

In general, the methods presented in the literature that use a B li-,-i in

approach consider the problem of mapping multiple QTL as a model selection

problem. In some methods, the number of QTL is considered a parameter of the

model and the dimension of the problem varies through the analysis. In other

methods, the dimensionality is kept constant and models with different numbers

of QTL are compared. The goal of B li-, -i i, methods is to obtain the posterior

distribution of the parameters in the model (QTL position, QTL effects and in

some approaches, QTL number) given the phenotype and marker information.

B i-, -i io methods allow for the case of missing markers _-.' Ir,,Ipe, treating them as

parameters in the model. The methods described below take into account missing

marker information.

Satagopan et al.(1996) and SillanpLa and Arjas (1998) proposed models that,

in a backcross design, are equivalent to
S
yi = + + 7kZi + i (2-3)
k=1

and
w 8
li + jXij + + 7X+ kz? + (2-4)
j=1 k=1

respectively. The xijs are a subset of w markers selected as covariates from

the total of t markers, the zijs are the unobserved QTL genotypes and s is the

number of QTL, also unknown. Although the zijs are unknown, their probability

distribution given the flanking markers is known (see Section 3.1). The probability

distribution is based on the recombination fraction between the putative QTL and

the flanking markers. Both articles developed MC'\ C algorithms to sample from









the posterior distribution of the parameters given the data. Let 3 = (01,..., 0,),

7 = (Yi,..,7s), y (Y1, .. ,yn), X = {xij}, Z = {zik} and A = (A1,... ,A) (the
vector of QTL positions). Satagopan et al. sample from p, y, a2, Z, Aly, X using

a Gibbs sampler with Metropolis-Hastings steps to update QTL positions. They

use B -,-- factors to compare models of different sizes. Sillanpaa and Arjas (1998)

sample from p, /, 7, a2, Z, A, s y, X using the Metropolis-Hastings algorithm and

reversible-jump MC'\ C to move between models of different sizes. Satagopan and

Yandell (1996) use a similar approach. Sillanpiaa and Arjas (1998) use stepwise

regression to choose the markers that will be used at covariates.

Yi (2004) noted that methods that change the dimensionality of the problem

by changing the number of QTL during the a nil i have the disadvantage that

the information about a QTL is lost as soon as it is removed from the model.

The author also noted that reversible jump MC'\ C is usually subject to poor

mixing and slow convergence. Yi et al. (2003) presented a Gibbs sampler for the

multiple regression model (2-2) (for analysis at the markers) based on a variable

selection method called stochastic search variable selection, developed by George

and McCulloch (1993). As in Broman and Speed (1999,2002), the objective is to

identify the subset of markers for which /j / 0 (model (2-2)). In this approach, the

dimensionality of the problem is kept constant by limiting the posterior distribution

of nonsignificant terms (markers with no effects) in a neighborhood of zero instead

of removing them from the model. That is, they define 6 = (1,..., 6), where

6j = 1 or 0 represent the presence or absence of the covariate j in the model. The

marker effects 3j, j t are given a prior distribution

P3l |- ~ (1 6,)N(0, r2) + 6,N(0, c2),

where c2 and rf are chosen so that rf is small and c-,2 is large. If j = 0, the

effect 3j is pushed toward zero. Based on this prior, the posterior distribution of









(/i,... /t) is multivariate normal. The authors discussed that with sparse and
irregularly spaced markers, the marker analysis will be biased, as noted by Zeng

(1993). They proposed, without implementing or giving details, two v--i-s to deal

with this problem. One of them is to incorporate in their procedure the multiple

imputation method proposed by Sen and 0('i1 !.11!! (2001). Their idea is that if

the complete genotype information on a dense set of markers is known, regressing

the ph1 i', r vrpe on each marker will give information about how likely it is that

a QTL is close to that marker. Because, in practice, the markers may be widely

spaced across the genome, they proposed to create a complete and dense set of

markers by adding what they called pseudomarkers. The genotypic information for

these pseudomarkers can be inferred using their assigned positions in the genome,

the genetic map and the available marker data. Several versions of this complete

genotype data are constructed and the LOD scores obtained from each of them is

then combined to measure the evidence in favor of a QTL being near any given

pseudomarker. To account for multiple QTL, they computed LOD scores for each

pair of pseudomarkers for a two-dimensional genome scan and also implemented a

three-dimensional genome scan.















CHAPTER 3
SIMULTANEOUS QTL ESTIMATION

Our objective is to develop a genomewide search that accounts for multiple

QTL. As noted previously, analysis at the markers will be biased when the markers

are sparse and irregularly spaced. It was also mentioned that methods that change

the dimensionality of the problem by changing the number of QTL during the

analysis have the disadvantage that the information about a QTL is lost as soon as

it is removed from the model. Our proposed method addresses these two problems

by sampling the entire genome while keeping the dimension of the problem fixed.

At this point, we are working under the assumptions of no interference and no

epistasis. For now, we are not considering the case of missing marker observations.

We will be working in a B ,,-i i, framework, therefore our target is the posterior

distribution of the locations and the effects of the QTL.

Two models will be considered. We will present both models along with

detailed calculations and their implementation in the next two sections, although

Model 2 (section 3.2) is our final model. For a discussion on the performance of

Model 2 see ('! Ilpter 4.

3.1 Model 1

We consider the model
t t-1
Yi /I + Y OIXij + Y 7kctk + C (3-1)
j=1 k=1

This model is based on model (2-4) with all markers as covariates and one QTL

per marker interval, i.e. t 1 QTL. The Ziks represent the unobserved QTL

genotypes. In a backcross design, zik is either 1 or 0 depending on the QTL









genotype, 1..',..i. v-ote (QQ) or heterozygote (Qq). We will consider the ps as

random effects and will keep the dimension of the model constant by considering

one QTL per interval. We believe that including one possible QTL per interval

will allow us to examine and account for all their effects and that the effects that

are important will be significant. Since we are not interested in prediction, we are

not thinking about the problem as model selection. This model can be written as

y = ipl + Xp + Z7 + e where X is a n x t matrix, f is a vector of dimension t, e

and p1 are of dimension n, Z is a n x (t 1) matrix, and 7 is a vector of dimension

t- 1.

Although the Ziks are unobserved, their distribution given the flanking marker

information (xik, Xi(k+l)) can be derived if the genetic distance, AQ (k), between

the QTL k and one of the flanking markers, -v marker k, is known. Under

the assumption of no interference, the genetic distance AQo(k) is related to the

recombination fraction rQm(k) (the recombination fraction between QTL k and

marker k) by the Haldane mapping function rQm(k) = (1 e-2A (). Then for

i = ,...,n and k= 1,...,t -1


f (zi Pk, Xi(k+)) k Pk) -zik (3-2)

where


(1-rQk(k))(1-rQk(k+l)) if k 1, i(k )
P1k 1-rk(k+ )
2k = --r(k))(k) if Xik 1, Xi(k+l) 0 1

Pk= P2k -c(kc1) (3 3)
SP2k (k)(l-rQ(k+)) if Xk 0, Xi(k+l) 1
rk(k+l)
Pk (k)rQ(k+) if Xk -0, Xi(k+l) 0
Slrk(k +l)
and rk(k+l) is the recombination fraction between markers k and k + 1, which are

assumed to be known from the genetic map. Let p be a 2 x (t 1) matrix of the

plks and p2ks.








Since our interest is the posterior distribution of locations and effects, we will
use a Gibbs sampler. In Section 3.1.1 we will show the posterior distribution for
Model 1 and we will derive the full conditional posterior for all the parameters in
the model. The details and challenges on sampling from these distributions will be
discussed in Section 3.1.2.
3.1.1 Gibbs Sampler for Model 1
To set up a Gibbs sampler for the model in equation (3-1), note that the
posterior distribution of the parameters given the data is
f/(l, 7Z, p, a2 y)

I f(P1, i3, ,Zp,72 ly)dj3

o [f(ylp, 3, 7 ,p, 2) f(p,3,%7,Z,p, )] d

J [f(y ,13, -7, Z,72)f(Z Ip, 0,7, ,p,72)f(p, 3, y, p,72)] d

= [f(Y p /,3, 7, Z,7 2) /(l,) f() f 2) f (2) f(Z p)f(p)] d3

{ [f(y p, 7, Z, 2)/, P)f(7)f(03 2) (,2)] d} f(Zp)f(p)
(3-4)

This factorization shows that, conditional on the QTL genotypes Z, the
problem involving y, /, 7 and a2 can be solved independently from the problem
involving p (Sen and C('!i, 1!111 2001). To find equation (3-4), assume priors
N(q, v,), N(, A), IG(a,b) and N(0, r2a2It) for pI, 7, a2 and 0, respectively. r2 is
considered to be fixed. Assume a Beta(c2, d2) prior for f 2k. For plk, f(plk P2k) oc
Pk 1-(1 Plk)dl-1 with range (1 p2k 1r() 1k ) 1) with probability and with range
(p2k, 1) with probability -. Flat priors for p and 7 were also considered for Model
2, the conditions needed to obtain a proper posterior distribution under these
priors are presented in Section 3.2.1 and proofs in Appendix A. See Appendix B
for graphs that explore the shape of the prior distribution chosen for the plk's and










p2k's. The mixture provides a more uniform distribution of QTL locations in the

intervals. Also, in Appendix B, a reparameterization of the posterior in terms of

the recombination fraction is presented. The parameterization in terms of the plk's

and p2k's turned out to be simpler than the one using the recombination fractions.

With the priors mentioned above, the term inside the integral of equation (3-4) is


f(,ll/, 3,7, Z, a2)f(pf()f (/l 2)f(a2)

OC2 (y-p -X -Z)(y-pL-X -)'A -1 )72 () q)2


2 a+1
022 22;)


Thus,


f f(y I/3, 7, f fZ, 2)f( 7)f(f)f 2 )d

( n 1- \ +1
oc e e ()'(A-L (- ) e ( )




( 2) nt+a+ 1
72

x e 'Xp- 3-Z'X'(y-l-- e'Zy)+ d





(-) n-t+ a +2
-




( 1) g+ +1U







( -) g Up+- 1-- -
X g2 2
\ 2






x ~ f ( It-)'( (y-2 p-Z')2
2 711 \ ,q2







2 7,1 -,q)( 2-, l- 0^( ^

X 2a22 2


Zj)-'(y-Pl-Z-)


Z-)'(y-P -Zy)






19


where U = X'X + 4. The last line follows since the term in the integral is the


kernel of a NV(U-1X'(y


p Zy), 2U -1). Therefore equation (3-4) becomes,


f (pI, Z, p, 1Y) C e {(y-p-Z)'(I-XU 1X')(y- 1 Zy )}U 2 eg (-)'A l (1y- )
-1 21 \ (a+1
x e -7) () a ea f(Z|p)f(p) (3


To obtain f(Zlp)f(p), from equation (3-2)

n t-1 n t-1
f(Z|p) oC fJ f(zickk, Xk, Xi(k+)) o kIIp (1 Pkl -zik (3
i k= 1 i- 1k 1


Therefore (range of plk omitted for simplicity),

f(Zlp)f(p)


t-1
OC np (1- 1 pl '2nk-z k


x f(1
k=l


P2kc ) pk


x f pc(1
k=1
t-1


lk ~c-z 4+zk +c-1 -
k 1


plk d-1 c2-1(
Pilk) !J2k i


P2k)d2-


P1k )nlk-zl +z4 dl -1


x p2k
k-1


3k +z +c2 1-(1


where n1k is the number of individuals with marker .-,- i' pes xik = Xi(k+l)


i.e nl1k =- 1 I(ikc


, xi(k+l) 1). Similarly,


i= 1


i= 1

iik
i= 1


-5)


6)


p2k) s2c


SP1k) 4 k


P2k )12k -z2 +z +d2


(3-7)


, Xi(k+) = 0)


0, Xi(k+l) 1)


0, Xi(k+l) = 0)










and

n
zk Zik I(ik Xi(k+l) 1)
i-1
n
Zk Z ikaik 1, i(k+l) 0)
i=1
n
z4k = Zik (xik = 0, Xi(k+l) = 1)



i=
i-i

Using equations (3-5), (3-6) and (3-7) we now calculate the full conditional

distributions of all the parameters in our model.

Distribution of 21 p, Z, p, y

It is clear from equation (3-5) that

2 n (y p1 Zy)'(I- XU-1X')(y 1 Z) 1
U2pY, Z, p,y IG( + a, + ).
2 2 b

Distribution of p7, Z, p, 12, y

To obtain the distribution of p7y, Z, p, a2, y note that from equation (3-5)


f(p Z,p, ,y) oC e {(y-l-Zy)(I-XU- X')(y-Pl-ZY)} (P-0)2

O {21'(I-XU- X')1-2pl'(I-XU- X')(y-Zy)}2 (p-2pq)
1 (2 1'(I xU-1x)1 1 )-2p((-x -x)(y -




oc e2

2 v '(I-XU 1X')(y-Zy)+U)2 Z 2
Therefore, 17, Z, p, a2, y N( V1(I -U -1X')+2 /(I -xu -ix,)l++2)-

Distribution of I, Z, p, e2, y

To obtain the distribution of 7 l, Z, p, y 2, y note that from equation (3-5)









f(7, Z, p, 2, y)

O e 1{-(y-p -Zy)'(I-XU 1X ')(y-p -Z-y)} c ( 2 -)'A-1(Q-)
2 1 Z'(I-x- 1x')Z) -2'Z'(I-xU lx')(Y- ) (~yA- 7-2A 1A-l)

= ^ -xU-ix')z A-1 7-2'Z'(I-xU- x1')(yt-) -27/A-l
(, (z'(I-XU-x')Z ,A-1 7-27, (I X U X ')(y- 1l) A,1


7-T 1 zl(I-xU-lx/)(Y-/l)+A-1)T( Ty-T-1 z'(I-xU x)(-l)+A-l(}
2 (ZI(.(I XU IX')ZyA 1)'>- (ZI(I XU 1X/a)(y /+A 1




where T '= (I-XU X)z + A-1. Thus,

Zp, J2, y N(T-1 (Z(I XU-1X')(Y -/ + ) T1).

Distribution of ZIp, 7, p, 2,y
To obtain the distribution of Z 7, p, a 2, y note that from equations (3-5) and
(3-6)

f(Z|,,p, p2, y)
n t-1


S {(Zy-(Y-pl)),'(IxU2 1X')(Zy-(Y-)} H _ik Pk) -zik
^ 2{ -(Z y (-l)) I(IX X)( ( pi)i- t1
i= 1 k= 1
Distribution of plk |p, 3, Z, a7, Y,p2k and p2k P, Z, 17, y
The distribution of plk/j, P, 7, Z, 92, y and p2kl ,, 3, 7, Z, 2, y is obtained using
equation (3-7). For k = ,...,t- 1,


P2k II, 7, Z, 2, y Beta(n3k c k + ZX + C2, 2 Zk + Z3k + d2)

and


PlkU, /, 7,Z, 2 Y,P2k --1 +d -









with range (1 p2k 1- ) r 1), with probability and with range (p2k, 1), with

probability .

3.1.2 Implementation

Sampling from the distributions of a2 p, 7, Z, p, y, p <, Z, p, a2, y and

7|p, Z, p, e2, y is straight forward. Sampling from the distribution of ZI, 7, p, P2, y,

plk f,/, 7,Z, 2, Y, P2k and p2k l/1, 7, 2, Z /, y is more challenging.
Sampling from the distribution of ZIp, 7, p, a2,

The distribution of ZI/, 7, p, a2, y is

f(Z, 7, p,2, y)
n t-1
O C {(y-- YI-XU-1X')(y-p-)} z k Pk)1-zi
i 1k 1
n t-l
S {(Z-(y- 'Zy- )) ii} (1 )lPk) -z
i-1 k-1
To sample from this distribution we went back to the expression

f (p, 7, Z, p, 2y)

oc [f(y 3, 7, Z, 2)f ()f(3,2)(7)f (2)] d} f(Z|p)f(p)

Thus,

f(Z|p, 7,p,72,y)

oc { [f(Y I /3,' Z, 72)/(32 2)] di f (Z p)
n t-1
o f e(y-pl-xp- zy)'(-p-1-3 I) ( p 1-zi

1 { r { 1 ikk- e i }n ik(1 [D)-zl=

Si=i=1 k= 1
Sn t-l1
iJ 1l l"i -- )2i Jpk(l k)l-zi r e2 33d3 (3 8)
i 2 1 k 1
k-1










where = yi p X(i)3 and X(i) represent the ith row of X. From here we

can see that given 3 the Zs can be generated independently for each individual.

For individual i, the vector (zil,..., zi(t-1)) can be generated as a block. This

can be done in two v--v. The first approach is to calculate the exact probability

distribution of (il,..., zi(t-i)) and sample from it. This is done by calculating

the probability of each of the 2(t-1) possible vectors. Since this is done for each

individual at each iteration of the Gibbs sampler, it can be computationally

intensive as the number of markers and individuals increase. A second way is to use

the Accept Reject algorithm with target distribution
t-1
2,L2 !^'/-- ) k-(1 -- c)kl-zi(
22 Pk Pk)
k= 1

A candidate distribution will be
t-1

k= 1

which implies that the Zs are independent with P(zik = 1) oc e6T (ah -bwipk
1 w 2
and P(zik = 0) oc e (bwi (1 pk). The supremum of the ratio of the target

and candidate distributions, typically called M, has to be bounded from above,

and should be close to 1 to have a good acceptance rate in the algorithm. a, b

and u are free parameters that are used to get a closed form of M to facilitate the

implementation of the algorithm as well as to assess its performance. In this case,

1 I {Ft1 zk k _Wi)2
M sup -i t-(ai-Y--bw)2
1 w t 1 azi -bwi)
ejj


where ul and u2 are constants.









Note that


t-1
k 1
(aZik7k
k-1
t-1
Y (azik7k
k 1
t-1
= a2 (Zik7k
k=1
t-1
=a2 Y (Zik7k
t-1

where Z7 = k t-1k7k

Thus,


'll' )2


aZ7 + aZ7


',,,.) 2


Z )2 + (t )( aZ ,)2


-_ ) + (t


M=
ul1 t--1 2 (t-1)b2 (t-1 )2 + a2
sup, exp (k Zik 7k )2 + 2W2 (( E- Zik t 2 + }S

where Sz = I (Zikyf Zy) .

We examined various choices for a,b and w with the goal of minimizing M. For

example, one possibility is a = b(t 1) and (t 2 giving

U lfj (t -) S
M sup --eC 2


which can give large values for M as seen in simulation studies. After examination

of choices with the goal of making the value in the exponent as small as possible,

we decided to set a b(t 1) and (t-' = K where 0< <. Then,
ulwe -1 t-1- (-
Si t t K( t 1 )
sup exp zik~ k
z U2 2a 2 2a
k-1

Finally, to ensure a negative exponent, we obtained K such that



K2( < 2 min zik7k -
2m 2 2(- \k 1


t 2t-1 2

t) k-1
1)b2 b(t 1) zkk -1t
Zk=1- l









this is
(t 1) S< 1
<--1
mm {(E 40-k 12} K
min, Zik7k it 2
which implies that

m { Z 't- i )21
min, ( Zik7k 'I '
Ilil( < -- -- --- 21- -
mm!^ { (Zl1 f )2} + (t 1)S

for all z. Thus, we choose


min, (Ec Zik7k W
min { (Z k7f -t )2 + (t -1) max{S}

Now, choosing c = a2, we have that

M<^.
U2

As seen in simulations, this method could give large M for few individuals

while running the Gibbs sampler. For Model 2 (see Section 3.2), we will use a

modified version of this Accept Reject algorithm.

Note that to calculate the posterior and all the full conditionals, we integrated

3 out. But to sample from Z we are going back to the expression in the integral.

An alternative approach, which we tried for Model 2, is to consider 3 as part of the

Gibbs sampler, i.e., the full conditional posterior distributions will be conditioned

also on 3. The full conditional distributions for this case are shown in Appendix C.

We show an example in Section 4.4 where the Gibbs sampler defined this way did

not recover the effects of the simulated QTL.

Sampling from the distribution of plk ,, 7, Z, 2,y,p2k and

p2k~ I 7, Z, a2, y
For k = ,...,t- 1,


p2k p, 7, Z, 72, y ~ Beta(n3k k k + C2, n2k +d2)









and


2,p^.Z^ y oc p"4-z 4k+z lk+ cl n-^-+z4+dl I
P1k |/, 3, 7, Z, IP2k k> k -1 P1k ik+ z+d-1

with range (1 p2k (f1 ) 1), with probability and with range (p2k, 1), with

probability .

The following sampling scheme was developed while analyzing simulated data.

Recall that our objective is to sample a QTL position at interval k. Once we have

a value of Plk and a value for p2k, we calculate the recombination fraction rQm(k)

or rQm(k+1). As can be seen from the definitions of rQm(k) and rQm(k+1), each result

in different restrictions on the range of the p's (see Appendix B). We started by

sampling plk'S using the range (1 p2k- l) 1) and calculating rQs(k)s only.

We noted that as the interval length increases, this method was not mixing well.

Mostly the right side of the interval was visited by the algorithm and rarely the left

side (see Figure 3 1 top). This is because only the left flanking marker was used to

determine the location of the QTL. We, therefore, decided to use the right flanking

marker as well by also calculating rQm(k+l)'S. We now decide at random which

flanking marker to use. Using this method, the distribution of the QTL positions

will look like the bottom panel of Figure 3-1.

Sampling from p2k Ip, 3, 7, Z, O2, y is straightforward. Given p2k, we sample plk

using the Accept Reject algorithm with a uniform candidate on (a, b), where (a, b)

is either (1 p2k 1k+- ) 1) or (p2k, 1). In this case, M is easier to calculate. Let
I k(k +l)
s = b a, wi = n4k Zk + Z 1 1 and w2 ilk k + k + dl 1, then

p~ (1 plk)
M sup l-(1 p)d
s









where

1 if wl 0 or w2 = 0



s (1 s)2 if < s
W W1+W2
vl+t2


3.2 Model 2

While examining the performance of Model 1 in simulations, we noted that

the fixed parameter 72 had to be very small to be able to recover the simulated

QTL effects in problems where the marker intervals were small. It seems that the

information in the matrix of QTL genotypes Z is very similar to the information

in the matrix of marker genotypes X and the QTL effects are unidentifiable.

('! -...- ig a very small 72 forces the ps to be zero while generating from the

mixture in equation (3-8). This motivated Model 2 where we consider the model


y = p+(I- H)XPf+Zy+c (3-10)


where H, = Z(Z'Z)-1Z'. The idea is to account for the portion of the effects of

the markers that is not being taken into account by the matrix of QTL -,. ._i vpes

Z. Although we are not considering this case here, if the matrix Z is not of full

column rank, the Gibbs sampler using Hz = Z(Z'Z)-Z' will still work. This case

is likely to arise in a genomewide search where the number of marker intervals is

likely to be greater than the number of individuals.

Figure 3-2 shows the graphs of the 7's obtained by Model 1 with 1,r =
100
and with Model 2 with r 1. These are results on simulated data with true

QTL at marker intervals 2 and 4. Model 1 could not recover the QTL effects when

'- 1. The simulation set up will be discussed in C'!i Ilter 4.









To set up a Gibbs sampler for this model, we assumed the same priors as in

Model 1, except for p for which a flat prior is assumed. The posterior distribution

is

f(p, Z, p, 21y)


{ J [f(y\ 7, Z, f) f (7)f(f 12)f(2)] d f(Zp)f (p)

where f(Zpp)f(p) is as before and

f(y \, Z, f 2) f (7) f( (/31,2) f (2)




X \ 2
1 H 2 (y-2l-(8-X)'(-H)/(y-L-(-)X-) (-0Al(-)

( 2)1 \z g ( ) 1 \ 11

Thus,

f f(y\1 /, 7, Z, 72)f (/) f (7) f(/3 2) f (72)d/


\ l+a+l1

2/
X e1(y-pl-( I-H~)X -y) (l-(I-H)X/~-Zy)e2-

/ b +1
X 1{#'X'(I-Hz)Xp-20'X'(I-Hz)(y-pl-Z 7)+ /}--}





X /, e 1 /y'(X'(I-Hz)X+'t )/3-2/3 'X'(I-Hz)(y-pl)} /
Se +a+ ( -_)'A-(^) l 2 (y-pl-Zy)'(y-pl-Zy)



x #-(Y pl)'(I-Hz)XU 1X'(I-H,)(y-pI)

X g2 -{(- ( 1X'(I-H, )(-Ipl))'U(-U 1X'(I-H)- ))}y









n 12 \t+a+l
S ( 2-E )'A-(A ) (y-p1-ZlZy)'(y-pl-Zy)
O2
X e 2~(y- l)'(I-Hz)XU 1I-H)(y- 2) 2U


where U = X'(I H )X + 4. The last line follows since the term in the integral

is the kernel of a Nt(U-1X'(I H,)(y pl),2U -1). Therefore, the posterior

distribution of the parameters is,

f (p,,Z, -, o2 y) O (y- pl-Zy)'(y-pl-Z)e2(y-pl)'(I-Hz)XU 1X'(I-Hz)(y-pl)
f(p, 7, Z, p o21y) oc e72a -P1-Z7)eyP 1))C-a
1 1 1 ) ( a+1
X |UI2 2 e 2 e erb f(ZIp)f(p)

(3-11)


that can be written as


f(p7, Z, p, 2 y) O 22 (Y-Pl-ZY)'[I-(I-Hz)XU 1X'(I-Hz)](y-pl-Zy)

xU e- (- )'A(-) 2 e f 2bf(Z p)f(p)


The full conditional distributions are presented in Section 3.2.2. In the next

section, we present results on the conditions needed to obtain a proper posterior

distribution under flat priors for p and 7.

3.2.1 Conditions for a Proper Posterior Distribution

The conditions needed to obtain a proper posterior distribution under flat

priors for p and 7 are summarized in the following theorems. Proofs can be found

in Appendix A.

Theorem 3.1 The posterior distribution is proper if a flat prior on p and a

N(, A) on 7 are assumed.
Theorem 3.2 The posterior distribution is proper if a N(rl, v,) on p and a flat

prior on 7 are assumed and Z is of full column rank.

Theorem 3.3 The posterior distribution is proper if a flat prior on p and 7 is

assumed and Z is of full column rank and 1'(I H,)1 / 0.









For Model 2, we are assuming a flat prior for p and a N(A, A) for 7 (Theorem

3.1). The others were not considered because the matrix Z is not necessarily of

full column rank. This will be the case if there are less individuals than number of

marker intervals or if the QTL -,. .n ripes are the same for all individuals in two or

more marker intervals. However, we are not considering any of these cases in our

models explicitly. The second case occurs, although rarely, while sampling from

the matrix Z. Since we are not using generalized inverse, we are dropping these

occurences and resampling.

3.2.2 Full Conditional Distributions

We now present the full conditional distributions for Model 2 which are

obtained from equation (3-11). The calculations are similar to the ones performed

for Model 1. In fact, plk I P, 7, ,Z, a2, p2k and p2k ,p, f, 7, Z, a2, y do not change.

Let W = (I H,)XU-1X'(I H,).

Distribution of a2 p, 7, Z, p, y

2n, (y P1 Zy)'(y Z) (y )'W(y 1
a2|\l,7, Z, p,y ~ IG( +a, ).
2 2 b

Distribution of p 7, Z, p, a2, y

I'(lI W)(y Zy) a2
1t(1 W~t) 1t/(1 W)1

Distribution of- I|, Z,p, a2, y


7pZ, p, a2, y N(T-1 Z'(y + AP T-1).

where T = Z- + A-1
Distribution of Zp, 7, p, a2,


f(Z|, ,p,a2,y)
n t-1
SX'(I H,)X + e{(y-p-Z'(I-w)(y-p1- } pZ(1- lk)1-zi
i=1 k=1









Sampling from this last distribution is a challenge. As in Model 1, we have

gone back to the equation

f(Zp, 7, p,2, y)

{j S > (~y-p1 H )X- zj'(y-p1 i-H )X3Z'di3} -pl

f 2(y-pl-(I-Hz)X/3-Z)'(y-pl-(I-Hz)X/3-Z) d2(1 p k )1- 2 z.

v i lk 1 J

Thus, given a 3 we must sample from the distribution inside the integral. To this

end we are using the Metropolis-Hastings algorithm with candidate distribution
n t-1
1( y-Pl-z Y'(Yy- I-Z) (1 -zi
i k= 1
Recall that we described how to sample from this candidate using the Accept

Reject algorithm in Section 3.1.2. We showed that the vector (zi,..., Z(t-1)) can

be sampled for each individual independently. In this case, we must sample the

entire matrix Z, the genotypes of all individuals, to have a candidate draw for the

Metropolis-Hastings algorithm. As we mentioned before, the Accept Reject algo-

rithm presented in Section 3.1.2 gives large M for a few individuals while running

the Gibbs sampler. Since in this case this distribution is a candidate distribution

and not the distribution of interest, we decided to set a maximum number of trials

for the Accept Reject algorithm, w L, based on simulations. While generating

the matrix Z, if an individual exceed L, the vector (zi,..., z(t-1)) from the previ-

ous iteration is kept in the matrix for that individual. This modification leads to

sampling the vector z = (zi,..., z(t-)) from a mixture distribution of the form


qg(z) + (1 q)6prev

where
t-1
g(z) 2oc eg l -w)1 pz(1 pk) 1)l-z
k=1









and 6 prev is a constant. The value of q is unknown, but is approximately 1 when

L is large enough. For simulated data using L = 3000, Figure 3-3 shows that in

150000 iterations only a few individuals exceed L and no more than .5' of the

time. Individuals ID are shown in the x axis and the percent of the total iterations

for which that individual exceed L in the y axis. The individual with ID 558

exceeds L, keeping the the vector z from the previous iteration, in only 629 out of

150000 iterations, for a little bit more of !' Few other individuals exceed L much

less than !' of the time. Most of them never exceed L in any trial.

The performance evaluation of Model 2 will be discussed in C'!i pter 4.





































I I I I
0.05 0.10 0.15 0.20 0.25


QTL Position (M)


0.10
0.10


I
0.20


0.25
0.25


QTL Position (M)


Figure 3-1. Histograms of QTL positions. Top Panel: Only rq,(k) is calculated.
Bottom Panel: Both rQ,(k) and rQm(k+1) are calculated.


0.00
0.00


0.00


0.05
0.05












Model 1


QTL Position (M)


Model 2


QTL Position (M)


Model 1


QTL Position (M)


Figure 3-2.


Simulated data, equally spaced markers at approximately .26M. QTL
at second and forth intervals with equal effects of 1. h2 .94, a2 .04.
Model 1 with 72 1 (top left), 72 10 (top right) and Model 2 with
72 1 (bottom).



























































Indivudual ID


Figure 3-3. Simulated data, only a few individuals exceed L
than .5'. of the total number of iterations.


3000 and no more


P I^ I















CHAPTER 4
PERFORMANCE EVALUATION

4.1 Simulated Data

The performance evaluation was done on data simulated from a backcross

experiment. Several examples were generated for equally spaced marker at different

interval lengths. We considered examples with six and ten markers with different

heritability levels.

To simulate the data we first defined the length of the marker intervals in

terms of recombination fraction. Then, defined the location of the QTL, by the

recombination fraction from their left flanking marker. Then, the marker genotypes

for the individuals (matrix X) are generated as well as for the QTL genotypes

(matrix Z). For each individual we first generate the genotype of the first marker

using a Bernoulli( ). Then, the rest of the markers and QTL genotypes for that

individual are generated sequentially. If the previous marker (QTL) genotype was 1

a Bernoulli(1 p) is used to generate the current genotype. If the previous marker

(QTL) .,. I. .Irpe was 0 a Bernoulli(p) is used. p is the recombination fraction from

the previous left flanking marker or QTL, 0 < p < .

Phenotypes (y) are generated by the equation y = p+Za+c, where p is a fixed

value, a is the vector of fixed values for the QTL effects and c ~ N(0, a). The

magnitude of the effects of the QTL (a) and the environmental variance (ak) were

chosen so that we would have data from populations with high and low heritability.

Heritability is defined as the ratio of the genetic variance to the total phenotypic

variance,

2
2 + aE
-g E









where for L QTL with effects ai

1
a 4 4 -2ri)aia,
i 1 ij

rij is the recombination fraction between QTL i and QTL j. It is expected that the

effects of QTL will be easier to recover from data sets with high heritability.

The implementation was done in Ox version 3.30 (Doornik, 2002) and the

graphs were created using the statistical software R.

4.2 Convergence and Results Presentation

We are interested in the effects of QTL at different locations in the genome for

a specific trait of interest. We are exploring convergence by looking at the running

means of the 7s in small windows of size 1cM or 2cM. Recall that at each iteration

of the Gibbs sampler a position for a QTL is generated in each interval as well

as a corresponding 7. Thus, the total number of 7s varies in each window since

at every iteration of the Gibbs sampler a 7 will not be necessarily generated on

that window. The running mean is weighted by the number of 7s in the particular

window and not by the number of iterations in the Gibbs sampler. The running

means of all windows in an interval will be shown in the same panel, thus the x

axis that corresponds to the iteration number is also scaled.

For the results we plot the posterior distribution of the 7's, i.e, the estimated

QTL effects against their QTL positions. The means of the 7's at each of the small

windows will be shown as well as corresponding 5 and 95 percent cutoffs.

4.3 Simulation Results

We present the results obtained from Model 2 in simulated data. We consid-

ered examples with 6 and 10 markers spaced at 26cM, 15cM and 5cM as well as

different levels of heritability. We will present each example and its results sepa-

rately. Graphs for the running mean of the 7s for each one of examples are shown

in Appendix D. As expected, it is harder to separate QTL effects for the cases of









low heritability (.1,.2). The effectiveness of our method will depend on the number

of individuals in the sample. Increasing the number of iterations may improve

performance in certain cases. We compared the results of our method with those of

composite interval mapping (CIM) in three of the examples.

Example 1: We considered simulated data for one chromosome with 6 equally

spaced markers at 26cM distance. QTL were located at marker intervals 2 and 4,

both with effect equal to one. Phenotypic data was generated for 250 individuals.

Heritability .94 ( = .04) and .4 (E 1) were considered.

The results obtained from Model 2 were compared with those from composite

interval mapping using all other markers as covariates. The results are shown

in Figure 4-1. The top panel shows the case with heritability .94, the bottom

panel the case with heritability .4. The left panel shows the results from Model 2.

The last 10,000 of 30,000 iterations are shown for the data with high heritability,

and the last 10,000 of 60,000 iterations for the data with low heritability. Figure

4-1 shows that the Model 2 was able to identify the effects of the multiple QTL,

separating their effects successfully. The means of the 7s at windows of size

2cM are shown in black and 5 and 95 percent cutoffs in red. Composite interval

mapping did the same for the model with high heritability, but the LOD scores

are above the threshold value almost for all intervals. Recall that CIM has the

potential to falsely indicate the presence of a QTL if there are QTL in the intervals

immediately .,i1] i:ent to the interval under consideration (Zeng 1994). For the data

with low heritability the separate effects were not recovered. The LOD scores are

showing a possible QTL at marker interval 2, and at marker interval 3 and 4.

Example 2: We considered simulated data for one chromosome with 6 equally

spaced markers at 15cM distance. QTL were located at marker intervals 2 and

4, both with effect equal to one. 250 individuals were considered for the case of

heritability .4 (a = 1). 250 and 500 individuals were considered for the case of









heritability .2 (a = 2.54). For heritability .1 (E = 5) only 500 individuals were

considered.

The results from Model 2 are shown in Figure 4-2. QTL were separated suc-

cessfully for the case of heritability .4. For heritability .2, Model 2 was successful

with 500 individuals. For heritability .1 our model was not successful. The last 70

thousand of 150 thousand iterations are shown. Composite interval mapping did

not separate the QTL effects in any of the cases (Figure 4-3). In this example we

are using CIM with backward and forward regression to select the markers that are

used as covariates.

Example 3: We considered simulated data for one chromosome with 10

equally spaced markers at 15cM distance. QTL were located at marker intervals 2,

5, 7 and 8, with effects 1, .1, 1, and .1, respectively. Cases with heritability .4 and

.3 were considered. 250 individuals were considered for the case of heritability .4

(E = .82). 600 and 800 individuals were considered for the case of heritability .3

(J = 1.27).
The results from Model 2 are shown in Figure 4-4. QTL of large effect were

separated successfully for the case of heritability .4. For heritability .3, Model

2 was successful in detecting the QTL of large effects when the sample size was

800. It indicate a possible QTL at interval 8th but with equal effect of the one

in interval 7, --Ii -Iii-; that the model was not able to separate the effects. The

last 70 thousand of 150 thousand iterations are shown. Increasing the number of

iterations may improve the performance when using 600 individuals as well as with

800.

Example 4: We considered simulated data for one chromosome with 10

equally spaced markers at 5cM distance. QTL were located at marker intervals 1,

5, 7 and 9, with effects 1, .1, 1, and .1, respectively. The case with heritability .3

(a = 1.31) was considered with 600 individuals.









The results from Model 2 are shown in Figure 4-5. QTL of large effects were

separated successfully. CIM with backward and forward regression to select the

markers that are used as covariates did not separate the QTL effects successfully

(Figure 4-6).

4.4 Model 2 with Full Conditionals Conditioned on 3

Recall that to calculate the posterior and the full conditionals in Section

3.1.2 we integrated 3 out, i.e, all conditionals were marginalized on 3. But to

sample from Z we went back to the expression in the integral. An alternative

approach is to consider 3 as part of the Gibbs sampler, i.e., the full conditional

posterior distributions will be conditioned also on 3. We show the results using

this approach on the simulated data set with h2 = .94 and markers distances

26cM described in the previous section. Two QTL with effect of magnitude 1 were

simulated in marker intervals 2 and 4. Figure 4-7 shows that with this approach

the effects are not recovered after 30,000 iterations.

4.5 Data Analysis

We analyzed chromosome five of the "Harrington" x "TR306" population

of Barley. This population is composed of 150 doubled haploid (DH) lines. The

parents in this population are closely related and thus the level of polymorphism is

relatively low. Harrington, a 2-row barley variety, has high malting quality. TR306

is a high yielding line, but is a non-malting type. From the population, a random

sample of 150 doubled haploid lines were selected for genotypic analysis in order

to generate the molecular map (Kleinhofs et al. 1993; Kasha et al. 1994) and for

lph.i i, r vpic analysis of traits such as heading date in replicated field trials. D-i- to

heading is a measurement of the number of d4i-< between planting and flowering.

The phenotypic data for this trait was collected in 29 different environments for

145 DH lines in the population. Previous work has found that the heritability for

heading date in barley is high, for example 0.923 ( \! et al., 2000) and 0.42 to 0.86









(Esparza Martnez and Foster, 1998). Thus, heading date is a robust trait that is

suitable for testing QTL methods.

The phenotypic values were averaged across the environments and to compare

with the results obtained by Yi et al. (2003) were also standardized. There are

14 markers in the genetic map at locations: 0, 10.9, 18.5, 78.2, 91.2, 111.2, 114.7,

121.7, 125.2, 138.8, 143.7, 150.7, 154.2, 159.9 cM. Individuals with missing marker

information were removed from the analysis. For now, our method assumes no

missing marker information. We analyzed the data using all markers. After re-

moving the missing data, the population was reduced from 145 to 107 individuals.

The results are shown in the top panel of Figure 4-8. The last hundred thousand

of seven hundred thousand iterations are shown. The running means are shown in

Figure 4-9, there is evidence that more iterations are needed. No QTL effects were

detected except for marker interval 3 that is a very long interval.

This chromosome was ,in i.-. .1 by Yi et al. (2003). They did account for

missing marker information and used all markers. Recall that their method

assumes that the QTL is at the marker. They estimated that marker 10 has

posterior probability greater that .4 with an estimated effect between .2 and .4.

This could be interpreted as v,-iing that there are no QTL effects.

Note that the markers are very close in certain regions, thus we decided to

analyze selecting markers that were at least 13cM apart. The marker distances

were now: 0, 18.5, 78.2, 91.2, 111.2, 125.2, 143.7, 159.9 cM. After removing the

missing data, the population was reduced from 145 to 114 individuals. The results

are shown in the bottom panel of Figure 4-8. The running means are shown

in Figure 4-10. QTL are detected at all marker intervals, but if we look at the

chromosome as a whole, and at the estimated effects, it seems that the effects of

the QTL are canceling each other out resulting in no overall QTL effect.









This latter observation leads us to reassess the interpretation of our results.

We have been interpreting the results interval by interval when we should be

concentrating on the overall picture. Quantitative traits are the joint effects of a

number of genetic and environmental factors. Our model is separating these factors

and estimating the genetic ones. It seems that the appropriate way to look at the

plots of the posterior distribution of the effects is as a whole. In further studies we

must explore this interpretation and the possibilities it presents for prediction and

validation of our model.

Also, from this analysis it is evident that we must better understand the

possible effects of correlation between the markers in the estimation of QTL effects.



















Model 2


C QTL on (M)
QTL position (M)


eC CD
C CN


0 o I CN CO
fN 16 r
Co C3 N 0 EN
o a o a T


Model 2


CN C
0 C


o I- CA C0
SLO -
o a a
M


Figure 4-1.


Example 1: Simulated data, equally spaced markers at approximately

26cM. QTL located at the second and forth interval with equal effects

of 1. Top panel: h2 = .94, O2 = .04, Bottom panel: h2 .4,2 2

Left Panel: Model 2, Right Panel: Composite interval mapping.


0 I D IN I
L position (M
C) pD C C3
QTL position (M)


















h -0.4 n=250

















QTL Position (M)


h2 0.2 n=250


m
E










E










E
-


I9
E
E -
10
*J


QTL Position (M)


E
E
10


h2 0.2 n=500

















QTL Position (M)


h2 01 n=500

















QTL Position (M)


Figure 4-2. Example 2: Model 2 on simulated data at equally spaced markers at

approximately 15cM. QTL located at the second and forth interval

with equal effects of 1.


(1
E
E
'a


0 f












h2= .4, n=250


h2 =.2n=500


0 75


h2=.2, n=250


h2 =.1, n=500


Figure 4-3. Example 2: Composite interval mapping on simulated data at equally
spaced markers at approximately 15cM. QTL located at the second
and forth interval with equal effects of 1.














2.4 n250
h =0.4 n=250


o Co Q LO O LO O no C (M
d d C d d C d 0 -
QTL Position (M)


h2 = 0.3 n=600


h2 = 0.3 n=800


O -


o o LO 0 LO 0 o Co
S QTL Position (M)
QTL Position (M)


o o L o C oion 0 o LO
S QT Position (M)
QTL Position (M)


Figure 4-4. Example 3: Model 2 on simulated data at equally spaced markers
at approximately 15cM. QTL with effects 1,.1,1,.1 located at marker
intervals 2, 5, 7 and 8, respectively.


C( -


C -


O -

















h2= 0.3 n=600


*i*"* U... ***..sU


m a

6 urn...


0 0 0 C a 0 C 0
QTL Position (M)


Co d


Figure 4-5. Example 4: Model 2 on simulated data at equally spaced markers
at approximately 5cM. QTL with effects .1,1,.1,1 located at marker
intervals 1,5,7 and 9, respectively.


I I I I I I I I







48






















21D











IIM






42




OiJ
0 4S


Figure 4-6. Example 4: Composite interval mapping on simulated data at equally
spaced markers at approximately 5cM. QTL with effects .1,1,.1,1 lo-
cated at marker intervals 1,5,7 and 9, respectively.













Model 2 (Gibbs with Beta)


(N -


ro
E
-
E 0


LfI I I I I
O CO CO
O CM LO O NCM

QTL Position (M)


Figure 4-7. Simulated data, equally spaced markers at approximately 26cM with
h2 = .94. QTL at second and forth intervals with equal effects of 1.
Gibbs sampler with full conditionals conditioned on 3.












Barley: All Markers


0 6 CO C6i 0 06 0
QTL Position (M)
QTL Position (M)


Barley: Selected Markers


QTL Position (M)
QTL Position (M)


Figure 4-8. Top Panel: Results for Model 2 on Barley
tom Panel: Results for Model 2 on Barley


Ot' 0
-- -0 -



Data using all markers. Bot-
Data using selected markers.


C=:=

















Interval 1


02 0.4 06 0.8 10
Iterations (scaled)


Interval 4


0.2 0.4 0.6 0.8 1.0
Iterations (scaled)


Interval 7







02 0.4 06 0.8 10

Iterations (scaled)


Interval 10







02 0.4 06 0.8 10
Iterations (scaled)


Interval 2


u)

O, C

C o
nJ


u,)
r t
(U
0)

0) a
c
1= 6
d


02 04 06 08 10
Iterations (scaled)


Interval 5


0.2 0.4 0.6 0.8 1.0
Iterations (scaled)


Interval 8


02 04 06 08 10

Iterations (scaled)


Interval 11


02 04 06 08 10
Iterations (scaled)


Interval 3


02 04 06 08 10
Iterations (scaled)


Interval 6


0.2 0.4 0.6 0.8 1.0
Iterations (scaled)


Interval 9


02 04 06 08 10

Iterations (scaled)


Interval 12


02 04 06 08 10
Iterations (scaled)


Interval 13


0.2 0.4 0.6 0.8 1.0
Iterations (scaled)


Figure 4-9. Running means for Model 2 on Barley data using all markers



















Interval 1


0.2 0.4 0.6 0.8 1.0

Iterations (scaled)



Interval 4


0.2 0.4 0.6 0.8 1.0

Iterations (scaled)



Interval 7


Interval 2


U 0

0) to
c~ "
C
C 0
n


0.2 0.4 0.6 0.8 1.0

Iterations (scaled)



Interval 5


Interval 3


(U


n0)

r


0.2 0.4 0.6 0.8 1.0

Iterations (scaled)


i






0.2 0.4 0.6 0.8 1.0

Iterations (scaled)



Interval 6


0.2 0.4 0.6 0.8 1.0

Iterations (scaled)


I I I I1
0.2 0.4 0.6 0.8 1.0

Iterations (scaled)


Figure 4-10. Running means for Model 2 on Barley data using selected markers


02

0J P
0),
.9
C~


i0
(D
0
0















CHAPTER 5
CONCLUSIONS AND FURTHER RESEARCH

We proposed a model for the simultaneous estimation of QTL effects and

locations in a backcross population. A Gibbs sampler was implemented to obtain

the posterior distribution of QTL effects along the genome. QTL were located

at each marker interval and their significance was assessed by examining the

posterior distribution. By considering one QTL per interval, the dimension of the

problem was kept fixed. We also included all the markers as covariates. Aside

from accounting for the information that the markers provide, using all markers as

covariates eliminates the marker selection problem, which is one disadvantage of

many of the existing methods of QTL analysis.

The performance of our method was studied using simulated data with

different numbers of markers at different spacing. We also considered different

heritability levels. Our method was generally effective in determining location and

in differentiating QTL with large effects. In the cases where the method was not

successful, the posterior distribution of the QTL effects was informative and showed

patterns that -i-i- -1 significance in the intervals for which QTL were declared

nonsignificant. We observed that increasing the number of iterations may be crucial

for the successful detection of QTL that were undetected. QTL of small effects

were typically not detected, which is also the case for many other existing methods

of QTL analysis. As expected, in cases of low heritability the power to detect QTL

is low. For some of the simulated examples, we empirically determined an effective

sample size that was required for our model to successfully detect the QTL. The

minimum sample size will vary from problem to problem and could be a topic of

further study.









Sampling from the full conditional distribution of the matrix Z of QTL

genotypes was a challenge because of their covariance structure. We used a

Metropolis-Hastings algorithm with a candidate for which this covariance structure

was reduced to the individual level. The genotypes of the individuals were sampled

as blocks, the individuals being independent from each other. Even with this

simplification, sampling the individuals genotypes was not straight forward. We

used an Accept Reject algorithm with a different candidate for each individual.

In practice, the number of trials before acceptance for few individuals turned out

to be large thus we limited the number of rejections to a fixed number. If this

number was reached the .-_, Ir.1 pes from the previous iteration of the sampler for

that individual were kept. This method was employ, l1 because the Accept Reject

algorithm was not being used to sample from the distribution of interest, but

to sample from the candidate of the Metropolis-Hasting algorithm. Satagopan

et al.(1996) proposed to sample each individual QTL _., in i.1pe independently.

Recall that in our model we are considering one QTL per marker interval thus, this

approach could be computationally intensive as the numbers of QTL considered

and the number of individuals increases. Also, the speed of convergence of the

Gibbs sampler may be reduced. We plan to compare the performance of the

Metropolis-Hastings algorithm and the Gibbs sampler approach in terms of the

speed of convergence and execution time.

We derived an innovative way to sample from the posterior distribution of

the QTL positions. We used an Accept Reject algorithm to sample from the

conditional probabilities of the _-, ir .1 Jpes given the flanking markers (p). From

there, we calculated the QTL positions for each interval at each iteration of the

Gibbs sampler. The calculation of the QTL positions using the recombination

fractions imposed restrictions on the range of p. These restrictions in conjuction

with the fact that we started by using only one flanking marker to calculated the









QTL positions, created mixing problems. Part of the intervals were rarely visited.

We solved this problem by using the information from both flanking markers at

random.

In practice, some marker ., Irvpes will be missing. Thus, allowing for

missing marker information in our model will be important. This could be done

by incorporating the missing marker information as parameters in the model. The

full conditional distribution of the missing marker information given all the other

parameters in the model could be calculated and the missing marker _.. ,I'vpe

could be sampled at each iteration of the sampler. We suspect that accounting

for missing marker information would not be a difficult addition to make to the

method.

As in any regression model, the correlation between independent variables

deserves some attention. Collinearity creates serious problems in the estimation

of regression coefficients (Rawlings et al.,2001). Important variables may be

incorrectly identified. Moreover, if all potential variables are included in the model,

all of them may be identified as not significant. In our context, as more and more

markers are included in the genetic map and the distances between markers are

reduced, linkage between markers increases. Linked markers, in our understanding,

translate to correlated independent variables in the model. Since the distribution

of the QTL genotypes depends on the marker genotype, the correlation between

markers will induce correlation in the sampled QTL .1 .irvpes (matrix Z) and will

create difficulties in the estimation of the QTL effects (7).

With increasing numbers of genetic markers on the genetic maps it is impor-

tant to know how to effectively use all of this information without reducing the

effectiveness of the statistical methods. Some authors comment about having a

dense map as an advantage for the accurate localization of the QTL in the context

of their models (Sen and C('!ni !,n!! 2001), but there is not much discussion about









the impact on the estimation of QTL effects. This is an interesting area for further

research. From our model perspective, a promising and interesting approach will

be to consider a fixed number of markers selected at random, spaced at a fixed

distance at every iteration of the Gibbs sampler. The distance between the markers

would possibly help to reduce correlation. Considering different sets of markers

may help separate QTL effect in .,l1i ient intervals.

The correlation problem could lead us back to the single marker analysis

approach. One idea is to use the information of a single marker to infer the QTL

genotype at a location in an interval around that particular marker. The width

of this interval will depend on the distance from the marker under consideration

and its .,li went markers. A thought is to consider the interval that is halfway to

the .,11] went markers. How to determine the location of the QTL in the interval

and how to account for all QTL effects simultaneously must still be explored. We

suspect that once we have the positions and the QTL genotypes the simultaneous

estimation of the QTL effects must not be that different to what we have already

done.

Whatever model is used for the mapping of multiple QTL, it will be of interest

to explore the robustness of the methods against noninformative markers. Other

areas for continued research include the extension to other types of experimental

crosses, inclusion of epistatic effects and the possible application of functional

mapping to the parameter p.













APPENDIX A
PRIORS FOR p AND 7
In C'!I ipter 3 we considered proper priors for p and 7. Here we consider
combinations of flat and normal priors for p and 7 in Model 2 and find the
conditions to ensure that the posterior distribution is proper. The priors for
all other parameters remain as in ('! Ilpter 3. It is sufficient to show that the
integration of the posterior over the parameters) with a flat prior is bounded.
From equation (3-4), the integrand is the posterior


{ J [f(y p, 7y, Z 2)f(/)f(7)f(02) f2)] d3} f(Zp)f(p)

But, because the parameter 3 has a proper prior suffices to show that the integra-
tion of

f(yI p/, 3, 7, Z, O2)f(/)f ()f(/3 a2)f(a2)f(Z| p)f(p) (A-1)

over the parameters of interest is bounded.
Theorem 3.1 The posterior distribution is proper if a flat prior on p and a

N(, A) on 7 are assumed.






58


Proof.


f (y /, 7, Z, (2)() f (7) f (/372) f (72) f ( p) f (p)} dp

O {pl- (-(I-Hz)X--Zy))'(pl-(y-(I-Hz)X-Zy)}d,

= 6 (y-(I-H,)X 3-Zy)'(y-(I-H,)X 3-Z) {np2-2pl'(y-(I-Hz)X--ZY)}d

S(I H 2)X8o Z'(y (I Hz)X8 ZY) I
(y-(I-H )X-Z y)'(y-(I-H,)X -Z


x -(y(IH)xz(y-(I-Hz)Xp-Zy) n (y-(-zy)-HZ )2
(y-(I-H )X -ZY)'(y-(I-H )X -Z) ) 7 1'(n-(I-)x-zn)

x e2{' dp

C (y-(I-Hz)X -Zy)'(y-(I-Hz)XP-Zy)e (1'(-(I -H)X -Z-))2

'( (y-(I-H)X -ZY)'l I- 1 (y-(I-H)XZy) (A2)


Because the matrix I- is symmetric and idempotent hence positive semidef-

inite (Searle 1982), (A-2) is bounded by 1. Thus, the integration of equation (A-1)

over 3, 7 and a2 is finite.

Theorem 3.2 The posterior distribution is proper if a N(rl, v,) on p and a flat

prior on 7 are assumed and Z is of full column rank.









Proof:


I f f(y 1 3, Z J2) f () f (7) f ( 2) f (2) f (Z p) f (p)} d

OC J {pl-(y-(yI-HX-Zy)) '(p- (y-(I-H-)XP-Zy)}

S ,(y-pl-(I-H )X8,)'(y-pl-(I-Hz)X) J {7'Z'Z7-27'Z'(y-pl-(I-Hz)X )}d.

S(y-l-(I-Hz)X)'(y-pl-(I-Hz)X-)

X f e{Y'Z'Z -27'Z'(y-l-(I-H )XI)-(y-4l-(I-Hz)X-S)'H (y-~l-(I-Hz)X1)}d
2 (y-pl-(I-Hz)XP)'[I-Hz](y-pl-(I-Hz)XP)

X [e--{(7-(Z'Z) Z'(y- -(I-Hz)Xp))'[Z'Z](q-(Z'Z) Z'(y-l-(I-Hz)Xp))}d

OC e (y-pl-(I-Hz)X3)'[I-Hz](y-pl-(I-Hz)X/3)I(Z'Z)-la2[.

The last line follows if Z'Z is positive definite, i.e., when Z is of full column
rank, the product is then bounded because the matrix I H, is idempotent hence
positive semidefinite (Searle 1982). Thus, the integration of equation (A-1) over 3,
p and a2 is finite.
Theorem 3.3 The posterior distribution is proper if a flat prior on p and 7 is
assumed and Z is of full column rank and 1'(I H)1 / 0.
Proof:









The integration over 7 is given on Theorem 3.2. Since Z is of full column rank

f f Z, ,2)f()f()f(ff 23)f(f2)f(Zp)f(p)} dd

SJ c (y-pl-( I-H)X)'[I-Hz](y-pl-(I-Hz)X )dp
-i
2 (y- (I-Hz)XP)'(I-HZ)(y-(I-Hz)X3)

X / {p21'(I-Hz)1-2pl'(I-Hz)(y-(I-Hz)X3)}d
S(y- (I-Hz)X3)'(I-H z)(y-(I-HZ)X)3)
-1'(I-Hz)1 2 ,2 l/(I Hz)(y-(I z)XH ) l'(I-Hz)(y-(I-Hz)X3) 21



/ -1'(I-Hz)l i'. T ..T ,


S -2 (y-(I-Hz)X3)' (I-H)-r 1," (y-(I-Hz)X8)
,1'(I H)1H

Because (I H) '(-Hz) is symmetric and idempotent hence positive
semidefinite (Searle 1982) the product is bounded if 1'(I H,)1 / 0. Then the
integration of equation (A-1) over 3 and a2 is finite.















APPENDIX B
PARAMETERIZATION OF QTL POSITIONS

B.1 Range of the plk's and p2k's and Exploration of Their Prior
Distribution

By definition,


(1 rQ,(k))(1 rQ(k+l))
1 rk(k+1)
(1 rQ,(k))rQ,(k+l) = 1
rk(k+l)


P2krk(k+l)
rQk(k) rQ
r<2 (k+1)


Substituting (B-2) in (B 1) we obtain that


rQk(k+l) (1 -rk(k+l))


which implies that


P2krk(k+l)
(1 rk(k+l))Plk + P2krk(k+l)


Substituting (B-4) in (B-2) we obtain that


rQk(k) 1 [(1 rk(k+1))Plk +P2krk(k+1)].

Since rQk(k) < rk(k+l), from (B-5)


1 rk(k+1) < (1 k(k+l))Plk + P2krk(k+l) 9 P1k > 1




Also, since rQk(k+l) < rk(k+l), from (B-4)


P2krk(k+l)
(1 rk(k+l))Plk + P2krk(k+l)


(B-5)


rk(k+l)
P2k rk(k+- )
1 rk(k+l)


< rk(k+l)


which implies that plk > P2k.


(B-1)

(B-2)


P2krk(k+1) (1


rQk(k+l))


(B-3)


rQk(k+l)


(B-4)









This tell us that if we want to calculate rQm(k) once we have a value for p2k, Plk

must be greater than 1 p2k r(+l) On the other hand, if we want to calculate

"Qm(k+l) once we have a value for p2k, Plk must be greater than p2k. If P2k's and
plk's are generated and only rQm(k)'s are calculated, the prior will look like the one
in the bottom panel of Figure B 1. While if only rQm(k+l)'s are calculated, the prior

will look like the one on the top. Each graph shows 5 intervals of equal length. To

have a more uniform distribution of the QTL positions at the intervals, a mixture

distribution is considered for plk given p2k. We assume a Beta(c2, d2) prior for p2k

and for plk, f(Plk l2k) oc pl (1 plk)d-1 with range (1 p2k r~ + 1) with

probability and f (plk P2k) oc p (1 Plk)l-1 with range (p2k, 1) with probability

. Figure B 2 show histograms of the QTL positions generated at different interval

lengths using this prior. For all the graphs in the figure, cl = 1 and d 1, i.e,

Plk P2k is generated from a uniform distribution on the corresponding range. For
the four graphs on the top, 2 = 2 and d2 = 2. For the 98 cM intervals 2 = 1 and

d2 = 1 seems to be a better choice (graph at the bottom).

B.2 Reparameterization of the Posterior Distribution in Terms of the
Recombination Fractions

From equation (3-6)
n t-1 n t-1
f(ZIp) X IJf (zik Xik i(k+l)) N jp (1 Pk) -zk
i=1k 1 i k= 1
t-1
4-++l- i ,-z ,1- z k-z*+ 2( -2k) -+
J 4k ( Pl1k 1k 4k 2k( P2k22k 3k
k= 1
Thus, we can consider each interval independently. Consider interval k from the

previous equation and equation (3-2)









rQ k (k) W, P0 Z, 7 Z O

PXQ n14c-
1 rk(k+l)
r ( Q k(k))( Qk(k+l1)
rk(k+l)


t rk(k+1)-l
L2k r TQ(k)(l- TQk(k+l)) 2
rk(k+1) j


Under the assumption of no interference


rk(k+1) = rQ(k) + rTQ(k+l) 2rQ((k)rQ,(k+1)


which implies that


1 rk(k+l) rQ(k)
rQ(k+l) -- 2rQ(k)
1 2rQy(kc)


and


1 rQ(k+l)


1 rk(k+l) Q- rQ(k)
1 2rQ,(k)


Therefore equation (B-5) can be written as


[(1 rQk(k))( fk(k+l) Qk(k))] 4k- k rQk(k)(rk(k+l) rQk(k))] -z
1 Q())(k(k Q() I +n4k

x [Q( )(Q k(k+1) rQ(k))] nk +k n2

S t 2rQk (k) IT
1 "


since nlk + n2k + nsk + n4k = n, the number of individuals. If the prior for the

location (A) of the QTL at the interval is assumed to be U(a, b), where a and b

are the location of the flanking markers, by the Haldane mapping function, its

corresponding recombination fraction, v- r, is r (1 e-2). Therefore a prior
for r is


f(r) = (2 log(1- 2r) 12


1 1
(b -a)(- 2r) 1 2r


(B-6)









Thus, f(rQ (k) I, P, Z, 2, y)f(rQk(k)) o


[(t1 rQk(k))1 r- fk(k+) TQ(k))] 4k- 4k k [rk(k)(rk(k+) rQk(k) )]l-zk z





x 1 2rQk(k)
1 n+1



Sampling rQk(k) from this distribution turned out to be complicated, therefore

we explored the parameterization with the plk's and p2k's which turned out to be

easier to sample from.














8cM marker intervals


QTL Position (M)



8cM marker intervals


QTL Position (M)



Figure B 1. Histograms of QTL positions. Each graph shows 5 intervals of equal
length. Top Panel: Only rTQ(k+1) is calculated. Bottom Panel: Only

rQm(k) is calculated.


S I .II I I .I I


I I I I I


















26cM marker intervals


00 0.1 02 03 04

QTL Position (M)



34cM marker intervals


o
o


5 g

&
eq









o


0.0 02 04 06 08 10 1.2

QTL Position (M)



98cM marker intervals


0 -I


00 0.5 10 15

QTL Position (M)


0 1 2 3

QTL Position (M)


98cM marker intervals


0 1 2 3

QTL Position (M)


4 5


Figure B-2.


Histograms of QTL positions generated using the mixture distribution

for plk Pi2k at different interval lengths. Each graphs show 5 intervals of

equal length.


4 5


8cM marker intervals


8
o
N


G
5 g
~ S
e
LL


O
8

o















APPENDIX C
FULL CONDITIONALS FOR MODEL 2 CONDITIONED P

The distributions of Plk P, 3,7, Z, 02, Y,P2k and p2k l/, 3, 7, Z, 02, y do not


change.

Distribution of j2 |p, 3, 7, Z, p, y

2 Di 7,t Z, p, y ~ IG( +a, (
Distribution of /I\, 7, Z, p, a7, y


p 7, Z, p,2, N(v1'(Y


-pl-(I-Hz)XP-Zy)'(y-pl-(I-Hz)X3-Zy/) /' 1
2 27-2 + b



-(I H)XP Zy) + y 2 2
nv + -2 nv + 2
nvp + ao + 2


Distribution of p- I Z, p, 2, y


IPL, Z, p,, 2y N(T-


1 Z'(y- ) + A- ,T-).
a-2


where T = + A-1

Distribution of l|7, p, Z, p, -2 y


Z, P, 2, y N(W-1X'(I H)(y 1p), W- 2).

where W = X'(I H,)X + .

Distribution of Zip, 3, 7, p, ,-2 y


f(ZP,7,p,72, Y)
n t-1
C 21 {(y-l-(I-H )Xf-Z)'(y- l-(I-H )X-Z)} (tJp pk)1-zi
i1 k- 1















APPENDIX D
RUNNING MEANS FOR PERFORMANCE ANALYSIS

The running means for the examples in C'! plter 4 are shown here. Note that

the scale in the y-axis are different.




















Interval 2


0.2 0.4 0.6 0.8 1.0

iterations (scaled)



Interval 3


0.2 0.4 0.6 0.8 1.0

iterations (scaled)



Interval 5


0.2 0.4 0.6 0.8 1.0

iterations (scaled)



Interval 4


0.2 0.4 0.6 0.8 1.0

iterations (scaled)


0.2 0.4 0.6 0.8 1.0

iterations (scaled)


Figure D 1. Example 2: h2 = .4, n 250


E





1-
o


Interval 1





















Interval 2


0.2 0.4 0.6 0.8 1.0

iterations (scaled)



Interval 3






















0.2 0.4 0.6 0.8 1.0

iterations (scaled)



Interval 5


0.2 0.4 0.6 0.8 1.0

iterations (scaled)



Interval 4
V^VA^--- ---





































0.2 0.4 0.6 0.8 1.0

iterations (scaled)
Iterations (scaled)


0.2 0.4 0.6 0.8 1.0

iterations (scaled)


Figure D-2. Example 2: h2


K
in

c
r
o

o
,5


c *I

c e'
er

o


.2, n=250


Interval 1




















Interval 2


E o-
en
c


0 -

0







*^-


0.2 0.4 0.6 0.8 1.0

iterations (scaled)



Interval 5


0.2 0.4 0.6 0.8 1.0

iterations (scaled)



Interval 4


0.2 0.4 0.6 0.8 1.0

iterations (scaled)


0.2 0.4 0.6 0.8 1.0

iterations (scaled)




Figure D-3. Example 2: h2 = .2, n 500


I I I I I
0.2 0.4 0.6 0.8 1.0

iterations (scaled)



Interval 3





1I


~-2sr"z--
~~-----,
*~r/Yr^`T~cl"~Z--r*
y~h,

~sc--------------------------

----I-=----


e!
o


o

o
K o





?

e3
?


Interval 1























Interval 1


-i i iBOT^ -a- ^- -









0.2 0.4 0.6 0.8 1.0

iterations (scaled)



Interval 3


22
o C
(n o

c I

2 N
o




















(D
0



rn
I


g>o
c
1- (
0

CM

0


Interval 2


0.2 0.4 0.6 0.8 1.0

iterations (scaled)



Interval 4

























0.2 0.4 0.6 0.8 1.0

iterations (scaled)


0.2 0.4 0.6 0.8 1.0

iterations (scaled)





Figure D-4. Example 2: h2 = .1, n 500


.I I.

0.2 0.4 0.6 0.8 1.0

iterations (scaled)



Interval 5








-^ ------=rz= ^


n o



I
E
-0 ?

?



n
0




















Interval 2









141A.,~c


2
o

o
o



S 8


= i


0.2 0.4 0.6 0.8 1.0

iterations (scaled)



Interval 4


0.2 0.4 0.6 0.8 1.0

iterations (scaled)


Interval 7


E


0o
<0










0.2 0.4 0.6 0.8 1.0

iterations (scaled)


0.2 0.4 0.6 0.8 1.0

iterations (scaled)



Interval 5


0.2 0.4 0.6 0.8 1.0

iterations (scaled)



Interval 8


0.2 0.4 0.6 0.8

iterations (scaled)


0.2 0.4 0.6 0.8 1.0

iterations (scaled)



Interval 6


0.2 0.4 0.6 0.8 1.0

iterations (scaled)



Interval 9


I I I I I
0.2 0.4 0.6 0.8 1.0

iterations (scaled)


Figure D-5. Example 3: h2 = .4, n 250


Interval 1


Interval 3






















Interval 1


0.2 0.4 0.6 0.8 1.0

iterations (scaled)


Interval 4


0.2 0.4 0.6 0.8 1.0

iterations (scaled)


Interval 2


0.2 0.4 0.6 0.8 1.0

iterations (scaled)



Interval 5






D


O

01


o







0.2 0.4 0.6 0.8 1.0

iterations (scaled)


Interval 8


0





E o
en
c
c
2


Interval 9


I I I I
0.2 0.4 0.6 0.8 1.0

iterations (scaled)


I I I I I
0.2 0.4 0.6 0.8 1.0

iterations (scaled)


0.2 0.4 0.6 0.8 1.0

iterations (scaled)


Figure D-6. Example 3: h2 = .3, n 600


Interval 3























0.2 0.4 0.6 0.8 1.0

iterations (scaled)



Interval 6























0.2 0.4 0.6 0.8 1.0

iterations (scaled)


o
0

2 65


2 o
0) ']


Interval 7


r~
0







in


c
2
o^


0


~8iF~ii~~





















Interval 1


0.2 0.4 0.6 0.8 1.0

iterations (scaled)


Interval 4


0.2 0.4 0.6 0.8 1.0

iterations (scaled)



Interval 7


Interval 2






















0.2 0.4 0.6 0.8 1.0

iterations (scaled)



Interval 5


0.2 0.4 0.6 0.8 1.0

iterations (scaled)



Interval 8


Interval 3


0.2 0.4 0.6 0.8 1.0

iterations (scaled)



Interval 6


0.2 0.4 0.6 0.8 1.0

iterations (scaled)



Interval 9


I I I I I
0.2 0.4 0.6 0.8 1.0

iterations (scaled)


0.2 0.4 0.6 0.8 1.0

iterations (scaled)


Figure D-7. Example 3: h2


8
o

n
o
o
in
1
I 0
sz i


ss
E o
=)
o 0
o


.3, n 800


0.2 0.4 0.6 0.8 1.0

iterations (scaled)


















Interval 1



















0.2 0.4 0.6 0.8 1.0

iterations (scaled)


Interval 4


0.2 0.4 0.6 0.8 1.0

iterations (scaled)


Interval 7


0 o
0


Interval 2



















0.2 0.4 0.6 0.8 1.0

iterations (scaled)


Interval 5


0.2 0.4 0.6 0.8 1.0

iterations (scaled)


Interval 8


Interval 3



















0.2 0.4 0.6 0.8 1.0

iterations (scaled)


Interval 6


0.2 0.4 0.6 0.8 1.0

iterations (scaled)


Interval 9


I I I I I
0.2 0.4 0.6 0.8 1.0

iterations (scaled)


0.2 0.4 0.6 0.8 1.0

iterations (scaled)


Figure D-8. Example 4: h2


7~1~.~

'~R-----t----,


0.2 0.4 0.6 0.8 1.0

iterations (scaled)


.3, n-600















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BIOGRAPHICAL SKETCH

Damaris Santana Morant was born in Rio Piedras, Puerto Rico. She is the

second of two daughters of Juan Manuel Santana and Francisca Morant. She

earned a B.S. in computational mathematics from the University of Puerto Rico

in Humacao in 1989 and a M.S. in applied mathematics from the University of

Puerto Rico in Rio Piedras in 1995. In the summer of 1997, she participated in

the Mathematical and Theoretical Biology Institute at Cornell University which

inspired her to pursue a graduate degree in statistics. In 2001, she obtained a

M.S. degree in biometry from Cornell University and in 2005 a Ph.D. in statistics

from the University of Florida, both with Dr. George Casella as her advisor. Her

interests include statistical genetics and environmental statistics.