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Bayesian Functional Mapping of Complex Dynamic Traits

Permanent Link: http://ufdc.ufl.edu/UFE0021375/00001

Material Information

Title: Bayesian Functional Mapping of Complex Dynamic Traits
Physical Description: 1 online resource (98 p.)
Language: english
Creator: Liu, Tian
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: bayesian, dynamic, functional, mapping, mcmc, qtl
Statistics -- Dissertations, Academic -- UF
Genre: Statistics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Many quantitative traits of fundamental importance to agricultural, evolutionary, and biomedical genetic research can better be described as dynamic processes. Understanding the genetic control of such dynamic or longitudinal traits (such as growth curves, HIV dynamics and drug response) has been a long-standing challenge because of their intrinsic developmental complexity. More recently, a general statistical framework, called functional mapping, has been proposed to map quantitative trait loci (QTLs) that regulate the developmental pattern and process of dynamic traits. Functional mapping has proven to be biologically relevant because it is incorporated by fundamental biological principles to test the genetic and developmental mechanisms for trait changes based on tractable mathematical functions. Original functional mapping was derived within the maximum likelihood (ML) context and implemented with the EM algorithms. Although ML-based functional mapping has many favorable statistical properties for parameter estimation, it has quickly become limited in capacity when a high-dimensional longitudinal problem, as commonly seen in systems biology, is encountered. In my research, I derive a general functional mapping framework for QTL mapping of dynamic traits within the Bayesian paradigm. The Markov Chain Monte Carlo (MCMC) techniques were implemented for functional mapping to estimate biologically and statistically sensible parameters that model the structures of time-dependent genetic effects and covariances. The Bayesian approach is useful to handle difficulties in constructing confidence intervals as well as the identify ability problem, enhancing the statistical inference of functional mapping. By comparing the Bayes factors from separate models, the actual number of QTLs that are involved in the dynamic variation of a trait can be estimated. The model framework was extended to estimate the effects of epistatic interactions between different QTLs on dynamic traits in various developmental stages. I have undertaken extensive simulation studies to investigate the statistical behavior of the new statistical model and used a real example for the F2 mice to validate model utilization. Bayesian-based functional mapping via MCMC algorithms estimates parameters that determine the shape and function of a particular biological process, thus providing a flexible platform to test biologically meaningful hypotheses regarding the complex relationships between gene actions or interactions and developmental processes.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Tian Liu.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Wu, Rongling.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021375:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021375/00001

Material Information

Title: Bayesian Functional Mapping of Complex Dynamic Traits
Physical Description: 1 online resource (98 p.)
Language: english
Creator: Liu, Tian
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: bayesian, dynamic, functional, mapping, mcmc, qtl
Statistics -- Dissertations, Academic -- UF
Genre: Statistics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Many quantitative traits of fundamental importance to agricultural, evolutionary, and biomedical genetic research can better be described as dynamic processes. Understanding the genetic control of such dynamic or longitudinal traits (such as growth curves, HIV dynamics and drug response) has been a long-standing challenge because of their intrinsic developmental complexity. More recently, a general statistical framework, called functional mapping, has been proposed to map quantitative trait loci (QTLs) that regulate the developmental pattern and process of dynamic traits. Functional mapping has proven to be biologically relevant because it is incorporated by fundamental biological principles to test the genetic and developmental mechanisms for trait changes based on tractable mathematical functions. Original functional mapping was derived within the maximum likelihood (ML) context and implemented with the EM algorithms. Although ML-based functional mapping has many favorable statistical properties for parameter estimation, it has quickly become limited in capacity when a high-dimensional longitudinal problem, as commonly seen in systems biology, is encountered. In my research, I derive a general functional mapping framework for QTL mapping of dynamic traits within the Bayesian paradigm. The Markov Chain Monte Carlo (MCMC) techniques were implemented for functional mapping to estimate biologically and statistically sensible parameters that model the structures of time-dependent genetic effects and covariances. The Bayesian approach is useful to handle difficulties in constructing confidence intervals as well as the identify ability problem, enhancing the statistical inference of functional mapping. By comparing the Bayes factors from separate models, the actual number of QTLs that are involved in the dynamic variation of a trait can be estimated. The model framework was extended to estimate the effects of epistatic interactions between different QTLs on dynamic traits in various developmental stages. I have undertaken extensive simulation studies to investigate the statistical behavior of the new statistical model and used a real example for the F2 mice to validate model utilization. Bayesian-based functional mapping via MCMC algorithms estimates parameters that determine the shape and function of a particular biological process, thus providing a flexible platform to test biologically meaningful hypotheses regarding the complex relationships between gene actions or interactions and developmental processes.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Tian Liu.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Wu, Rongling.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021375:00001


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TherearelotsofpeopleIwouldliketothankforahugevarietyofreasons.Firstly,Iamgratefultomythesisadvisor,Dr.RonglingWu,forhisguidanceandsupport.IcouldnothaveimaginedhavingabetteradvisorandmentorformyPhD,andthisdissertationwouldnothavebeenpossiblewithouthisknowledge,perceptivenessandexpertguidance.Notonlywashereadilyavailableforme,ashesogenerouslyisforallofhisstudents,buthealwaysreadandrespondedtothedraftsofeachchapterofmyworkmorequicklythanIcouldhavehoped.IwasintheclassesofDr.MalayGhoshandDr.RamonLittellandIreallyappreciatetheirkindnessandconsiderablementoring.ManythanksalsogotocommitteemembersDr.XueliLiuandDr.HartmutDerendorf.Ineverysense,noneofthisworkwouldhavebeenpossiblewithouttheirhelps.IalsothankalltherestoftheacademicandsupportstaoftheDepartmentofStatisticsattheUniversityofFlorida.Muchgratitudeisgiventomyocemates,andfriends,HongyingandSongforputtingupwithmeforalmostthreeyears.Apenultimatethank-yougoestomymotherandmymother-in-law.ForalwaysbeingtherewhenIneededthemmost,andhelpingmetotakegoodcareofmynew-bornbaby.theydeservefarmorecreditthanIcanevergivethem.Mynal,andmostheartfelt,acknowledgmentmustgotomyhusbandShenghua.Hissupport,encouragement,andcompanionshiphasturnedmyjourneythroughgraduateschoolintoapleasure.Forallthat,andforbeingeverythingIamnot,hehasmyeverlastinglove. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 11 CHAPTER 1INTRODUCTION .................................. 13 1.1GeneticsofQuantitativeTraits ........................ 13 1.2NatureofQuantitativeVariation ....................... 15 1.3GeneticMapping:fromStatictoDynamic .................. 15 1.4MappingApproaches:fromFrequentisttoBayesian ............. 17 1.5DissertationGoals ............................... 20 2AGENERALBAYESIAN-FUNCTIONALMAPPINGFRAMEWORK ..... 21 2.1Introduction ................................... 21 2.2FunctionalMapping .............................. 24 2.2.1LinearModel .............................. 24 2.2.2ModelingtheMean-CovarianceStructures .............. 24 2.3Likelihood .................................... 26 2.4ParameterEstimation ............................. 28 2.5Implementation ................................. 29 2.6EstimationIssues ................................ 32 2.7ModelingtheCovarianceMatrixbyaReferencePrior ............ 33 2.8ModelingtheStructureoftheCovarianceMatrix .............. 35 2.8.1ModelingtheStructureoftheWithin-SubjectCovarianceMatrixbytheFirst-OrderAutoregressiveModel ............... 35 2.8.2ModelingtheStructureoftheWithin-SubjectCovarianceMatrixbytheStructuredAntedependenceModel ............... 38 2.9AWorkedExample ............................... 40 2.9.1AnimalMaterial ............................. 40 2.9.2Results .................................. 40 2.10MonteCarloSimulation ............................ 46 2.11Discussion .................................... 53 3ABAYESIANMODELFORMAPPINGEPISTATICQTLSTHATREGULATEDYNAMICPROCESSES .............................. 59 3.1Introduction ................................... 59 3.2Model ...................................... 60 5

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............................... 63 3.3.1Material ................................. 63 3.3.2Results .................................. 63 3.4MonteCarloSimulation ............................ 69 3.5Discussion .................................... 82 4PROSPECTS ..................................... 83 4.1Introduction ................................... 83 4.2Parametric,NonparametricandSemiparametricModeling ......... 83 4.3TowardaComprehensiveBiology ....................... 85 4.3.1JointModelsofLongitudinalTrajectoriesandTime-to-Events ... 85 4.3.2MultivariateLongitudinalTrajectories ................. 86 4.4StatisticalConsiderations ........................... 86 4.4.1ModelSelection ............................. 86 4.4.2SensitivityStudiesandOtherEstimationIssues ........... 87 4.4.3TestingQTLEects ........................... 87 4.5Conclusions ................................... 88 REFERENCES ....................................... 90 BIOGRAPHICALSKETCH ................................ 98 6

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Table page 2-1Thesamplevariance-covariancematrixforthemice-body-massdata. ...... 56 2-2Thesamplevariance-covariancematrixforthetransformedmice-body-massdata. ............................................. 56 2-3BayesianestimatesofgrowthcurvesfortheQTLgenotypesdetectedonmousechromosome6.Numbersintheparenthesesarethe95%equal-tailcondenceintervals. ....................................... 56 2-4BayesianestimatesofgrowthcurvesfortheQTLgenotypesdetectedonmousechromosome6.Numbersintheparenthesesarethe95%equal-tailcondenceintervals. ....................................... 57 2-5BayesianestimatesofgrowthcurvesfortheQTLgenotypesdetectedonmousechromosome7.Numbersintheparenthesesarethe95%equal-tailcondenceintervals. ....................................... 57 2-6BayesianestimatesofgrowthcurvesfortheQTLgenotypesdetectedonmousechromosome10.Numbersintheparenthesesarethe95%equal-tailcondenceintervals. ....................................... 57 2-7Resultsfromasimulationstudybyassumingnonecovariancestructure.Numbersintheparenthesesarethe95%equal-tailcondenceintervals. .......... 57 2-8ResultsfromasimulationstudybyassumingthecovariancestructuretobeSAD(1).Numbersintheparenthesesarethe95%equal-tailcondenceintervals. .... 58 2-9ResultsfromasimulationstudybyassumingthecovariancestructuretobeAR(1).Numbersintheparenthesesarethe95%equal-tailcondenceintervals. .... 58 2-10Resultsfromasimulationstudybyperformingtraditionalmaximum-likelihoodtypedmethod.Numbersintheboxbractsarethegivenvaluesoftheparameters. ............................................. 58 3-1BayesianestimatesofgrowthcurvesfortheQTLgenotypesdetectedonmousechromosome6and7.Numbersintheparenthesesarethe95%equal-tailcondenceintervals. ....................................... 66 3-2BayesianestimatesofgrowthcurvesfortheQTLgenotypesdetectedonmousechromosome6and10.Numbersintheparenthesesarethe95%equal-tailcondenceintervals. ....................................... 70 3-3BayesianestimatesofgrowthcurvesfortheQTLgenotypesdetectedonmousechromosome7and10.Numbersintheparenthesesarethe95%equal-tailcondenceintervals. ....................................... 71 7

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.... 72 3-5Resultsfromasimulationstudy,wheres=2andH2=0:1.Numbersintheparenthesesarethe95%equal-tailcondenceintervals.Numbersintheboxbractsarethegivenvaluesoftheparameters. ................... 75 3-6Resultsfromasimulationstudy,wheres=2andH2=0:4.Numbersintheparenthesesarethe95%equal-tailcondenceintervals.Numbersintheboxbractsarethegivenvaluesoftheparameters. ................... 81 8

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Figure page 2-1Autocorrelationfunctionofthelocus()atdierentlags.Intherstplot,theMCMCsamplesweresubsampledatevery30thcycles.Inthesecondplot,theMCMCsamplesweresubsampledatevery60thcycles. .............. 42 2-2Traceplotsfortheestimatedparameters. ...................... 43 2-3AproleofEstimatedmarginalposteriordistributionoftheQTLlocationbyassumingthatexactlyoneQTLislocatedononeofthechromosomerespectively. 44 2-4FittedgrowthcurvesforthethreeQTLgenotypesassumingasingleQTLislocatedonmousechromosome6. .......................... 45 2-5FittedgrowthcurvesforthethreeQTLgenotypesassumingtheQTLislocatedonmousechromosome7. ............................... 46 2-6FittedgrowthcurvesforthethreeQTLgenotypesassumingtheQTLislocatedonmousechromosome10. .............................. 47 2-7DynamicchangesoftheadditiveanddominanteectduetotheQTLlocatedonmousechromosome6,7,and10respectively. ................... 48 2-8EstimatedmarginalposteriordistributionoftheQTLlocationonmousechromosome6,7,and10respectively. ................................ 49 2-9EstimatedmarginalposteriordistributionoftheQTLlocationonamousechromosomeinasimulationstudy,assumingnocovariancestructures. ............. 50 2-10EstimatedmarginalposteriordistributionoftheQTLlocationonamousechromosomeinasimulationstudy,assumingthecovariancestructureisSAD(1). ....... 51 2-11EstimatedmarginalposteriordistributionoftheQTLlocationonamousechromosomeinasimulationstudy,assumingthecovariancestructureisAR(1). ........ 52 2-12TheproleoftheLRbetweenthefullandreduced(noQTL)modelestimatedfromtheSAD(1)modelforbodymassgrowthtrajectoriesinasimulationstudy.Astothesettingsofthissimulation,weassumethatasingleQTLcontrolbodymassgrowthtrajectoriesofmice. .......................... 53 3-1EstimatedmarginalposteriordistributionoftheQTLlocationonmousechromosome6,7,and10respectively. ................................ 65 3-2Estimatedmarginalposteriorsofthetwolocibasedonatwo-QTL-model. ... 68 3-3Fittedgrowthcurvesforthe9QTLgenotypesassumingtwoQTLarelocatedonmousechromosome6and7. ........................... 69 9

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....................... 70 3-5Fittedgrowthcurvesforthe9QTLgenotypesassumingtwoQTLarelocatedonmousechromosome6and10. ........................... 71 3-6Fittedgrowthcurvesforthe9QTLgenotypesassumingtwoQTLarelocatedonmousechromosome7and10. ........................... 72 3-7Estimatedmarginalposteriordistributionsofthetwolocionasamechromosomeinasimulationstudy.Astothesettingsofthissimulation,weassumethattwoQTLcontrolbodymassgrowthtrajectoriesofmice,andtheheritabilityis0.1. 77 3-8TheproleoftheLRbetweenthefullandreduced(noQTL)modelestimatedfromtheSAD(1)modelforbodymassgrowthtrajectoriesinasimulationstudy.Astothesettingsofthissimulation,weassumethattwoQTLcontrolbodymassgrowthtrajectoriesofmice,andtheheritabilityis0.1. ........... 78 3-9Estimatedmarginalposteriordistributionsofthetwolocionasamechromosomeinasimulationstudy.Astothesettingsofthissimulation,weassumethattwoQTLcontrolbodymassgrowthtrajectoriesofmice,andtheheritabilityis0.4. 79 3-10Fittedandgivengrowthcurvesforthe9QTLgenotypesinasimulationstudy,assumingtheheritabilityis0.4. ........................... 80 10

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Manyquantitativetraitsoffundamentalimportancetoagricultural,evolutionary,andbiomedicalgeneticresearchcanbetterbedescribedasdynamicprocesses.Understandingthegeneticcontrolofsuchdynamicorlongitudinaltraits(suchasgrowthcurves,HIVdynamicsanddrugresponse)hasbeenalong-standingchallengebecauseoftheirintrinsicdevelopmentalcomplexity.Morerecently,ageneralstatisticalframework,calledfunctionalmapping,hasbeenproposedtomapquantitativetraitloci(QTLs)thatregulatethedevelopmentalpatternandprocessofdynamictraits.Functionalmappinghasproventobebiologicallyrelevantbecauseitisincorporatedbyfundamentalbiologicalprinciplestotestthegeneticanddevelopmentalmechanismsfortraitchangesbasedontractablemathematicalfunctions.Originalfunctionalmappingwasderivedwithinthemaximumlikelihood(ML)contextandimplementedwiththeEMalgorithms.AlthoughML-basedfunctionalmappinghasmanyfavorablestatisticalpropertiesforparameterestimation,ithasquicklybecomelimitedincapacitywhenahigh-dimensionallongitudinalproblem,ascommonlyseeninsystemsbiology,isencountered. Inmyresearch,IderiveageneralfunctionalmappingframeworkforQTLmappingofdynamictraitswithintheBayesianparadigm.TheMarkovChainMonteCarlo(MCMC)techniqueswereimplementedforfunctionalmappingtoestimatebiologicallyandstatisticallysensibleparametersthatmodelthestructuresoftime-dependentgeneticeectsandcovariances.TheBayesianapproachisusefultohandlediculties 11

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65 ][ 66 ][ 93 ][ 103 ]andhumangeneticsaimedtodetectgenesforcomplexhumandiseases[ 67 ].Quantitativetraitsarethoughttobecontrolledbymultiplegenes,eachwithasmalleectandsegregatingaccordingtoMendel'slaws,andcanalsobeaectedbytheenvironmenttovaryingdegrees[ 53 ].AccordingtothisargumentestablishedbyR.A.Fisher[ 24 ],theobservedphenotypeofaquantitativetrait(y)canbeexpressedasalinearcombinationofgenetic(g),environmental(e)andgenotypeenvironmentinteractioneects,i.e., whereisthepopulationmeanandistheresidualerror.Dependingondierentpurposesofplantbreeding,theenvironmentaleectcanbeduetodierentclimatesorlocations[ 69 ],whichareusuallycalled\macroenvironments"inlightoftheirevidentvaryingpatterns[ 102 ].Themacroenvironmenteectcanbediscrete,likelocation,orcontinuous,suchastemperature,moistureandnutrient,innature[ 91 ].Theresidualerrorisduetostochasticuctuations,i.e.,\microenvironments"[ 102 ].Undertheprerequisitethatthefamiliesorgenotypesstudiedhavemultiplereplicatesinspace,statisticalapproachesbasedonregressionmodelsandanalysisofvariancehavebeenavailabletoestimatethevariancesduetothegenetic,environmentandresidualeects,as 13

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Withthedevelopmentofmodernmolecularmarkers,classicalquantitativegeneticshasbeendevelopedtoapointatwhichindividualgeneticlociunderlyingaquantitativetrait,calledquantitativetraitloci(QTLs),canbemappedandidentied[ 53 ].BydissectingaquantitativetraitintoatotalofmpossibleQTLseachsegregatinginaMendelianratio,thephenotypicvalueofthetraitcanbeexpressedas wherePmr=1andPmr6=sdenotethesummationsassociatedwiththemainandepistaticgeneticeects,respectively,amongtheQTLsestimatedfromalinkagemapconstructedbymolecularmarkers.Equation 1{2 presentsageneralstatisticalmodelforQTLmapping[ 35 ][ 42 ][ 45 ][ 46 ][ 49 ].ThedetectionoftheunderlyingQTLforaquantitativetraitisbasedonasegregatingpopulationofprogenyderivedfromcrossinggenotypescontainingdierentallelesatphenotypicallyimportantloci.Thecrossedparentsshouldbeadequatelydivergenttoidentifydiscretemolecularmarkersthatsamplethegenomeatsucientlydenseintervals.StatisticalprinciplesandmethodsformappingQTLswiththelinkagemapconstructedfromgenotypedmolecularmarkershavebeenwellestablished[ 46 ][ 50 ][ 54 ][ 69 ][ 70 ][ 107 ].Inpractice,QTLmappingapproacheshavebeeninstrumentalforthecharacterizationanddiscoveryofthousandsofQTLsresponsibleforavarietyoftraitsinplants,animalsandhumans.Inarecentstudy,Lietal.[ 52 ]wereabletocharacterizethemolecularbasisofthereductionofgrainshattering{afundamentalselectionprocessforricedomestication{atthedetectedQTL.ManyotherexamplesforthesuccessofQTLmappingincludethepositionalcloningofQTLresponsibleforfruitsizeandshapeintomato[ 26 ]andforbranch,orescenceandgrainarchitectureinmaize[ 19 ][ 28 ][ 98 ]. 14

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22 ].Thetraditionalpolygenictheoryofquantitativetraits[ 57 ]envisagedafairlylargenumberofloci,eachwithrelativelysmallandequaleects,actinginalargelyadditiveway.Overtheyearsithasindeedbeenobservedthataquantitativetraitmaydisplaycomplicatedgeneticarchitecture[ 2 ][ 56 ],describedbelow: (1) Itmaybecontrolledbyafairlylargenumberofloci;forexample,oftheorderof50,accordingtotheworkofShrimptonandRobertson[ 79 ][ 80 ]; (2) Genesactinwayswhichmaybeadditive,dominant,epistaticandinteractivewithenvironmentalfactors; (3) Themagnitudeoftheeectproducedbyeachlocuscanvaryconsiderably; (4) Thesamegenesmayaectdierentphenotypictraitsthroughpleiotropiceects; (5) Thegenesaectingthetraitmaybedistributedoverthegenomeatrandomorinacertainpattern. Withtheuseofgeneticmappingtoanalyzequantitativetraits,increasingevidencehasbeenobservedforthethirdpoint,whichsuggeststhattypicallyasmallnumberoflociaccountforaverylargefractionofthevariationinthetrait.Forthisreason,thetraditionalpolygenicmodelmaybereplacedbyanewoligogenicmodelinwhichasmallnumberofmajorgeneseachwithalargeeect,combinedwithmanyminorgeneseachwithasmalleect,determinethegeneticvariationofaquantitativetrait(see[ 55 ]foranexcellentreview).Accordingtotheoligogenicmodel,thedistributionofgeneticeectsmaybeapproximatedbyageometricseries[ 50 ].WhenincorporatedintoaQTLmappingmodel,suchanapproximationcansignicantlyincreasethepowerofQTLmappingandtheprecisionofparameterestimation. 15

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Currently,mostgeneticapproachesdescribeabiologicaltraitbythevaluemeasuredatasingletimepointduringitsontogenyorasinglestateinanenvironmentalspace.Thisislargelyinsucientand,rather,aprecisedescriptionofthetraitshouldincludeaseriesofmeasurementstakenatmultiplediscretetimepointsorstates.Empiricalanalysesoftime-dependentgrowthdatasuggestthatgrowthtrajectories(changesofthetraitvalueovertime)followparticularexponentiallawsthatcanbemathematicallyelucidatedbytheso-calledgrowthfunctions[ 4 ].Basedonfundamentalprinciplesbehindbiologicalorbiochemicalnetworks,[ 99 ]havemathematicallyproventheuniversalityofthesegrowthequations.Apartfromthegrowthlaw,manymathematicalequationshavebeenwellconstructedtodescribevariouslifeprocessesofparamountimportanceinagricultural,biomedicalandhealthsciences[ 38 ][ 68 ]. Thegeneticmappingofaquantitativetraitexpressedasatrajectoryorcurvepresentsoneofthemostchallengingissuesingeneticresearchbecauseoftheinnite-dimensional[ 48 ]orfunction-valuednatureofthecurve[ 71 ].However,someofkeydicultiesinmappinghavebeenovercomebyR.Wuandcolleagues([ 54 ][ 104 ][ 105 ][ 106 ],andreviewedin[ 107 ]).Theyhaveproposedageneralstatisticalframework,i.e.,functionalmapping,togenome-widemapspecicQTLthatdeterminethedevelopmentalpatternofacomplextrait. Thebasicrationaleoffunctionalmappingliesintheconnectionbetweengeneactionorenvironmentaleectsanddevelopmentbyparametricornonparametricmodels.FunctionalmappingmapsdynamicQTLthatareresponsibleforabiologicalprocessthatismeasuredatanitenumberoftimepointsthroughestablishedmathematicalmodelsfordeningthedevelopmentalprocessofabiologicalphenotype.WithmathematicalfunctionsincorporatedintotheQTLmappingframework,functionalmappingestimates 16

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Fromastatisticalperspective,functionalmappingisaproblemofjointlymodellingmean-covariancestructuresinlongitudinalstudies,anareathathasrecentlyreceivedaconsiderableinterestinthestatisticalliterature[ 64 ][ 72 ][ 73 ][ 101 ].However,dierentfromgenerallongitudinalmodelling,functionalmappingintegratestheparameterestimationandtestprocesswithinabiologicallymeaningfulmixture-basedlikelihoodframework.FunctionalmappingisthusadvantageousintermsofbiologicalrelevancebecausebiologicalprinciplesareembeddedintotheestimationprocessofQTLparameters.Theresultsderivedfromfunctionalmappingwillbeclosertobiologicalreality. where$=($1;;$J)arethemixtureproportions(i.e.,QTLgenotypefrequencies)whichareconstrainedtobenon-negativeandsumtounity,=(u1;;uJ)isavectorthatcontainstheparametersspecictocomponent(orQTLgenotype)jandvincludestheparameterscommontoallcomponents.Eachdensityismodelledbyamultivariatenormaldistribution. 17

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1{3 .Themaximumlikelihoodestimates(MLEs)oftheunknownparametersunderthemixtureQTLmodelcanbecomputedbyimplementinganEMalgorithm[ 16 ][ 58 ].However,therecansometimesbeproblemswhenMLisusedforQTLmixturemodels.First,forsomechoicesofparametricfamilies,f,thelikelihoodisunbounded.Second,incomplexsituationsthelikelihoodfunctioncanhavemanylocalmaxima,eachofwhichmaygivedierent(andpossiblyreasonable)plug-inestimatesforquantitiesofinterest.Inthesecasesitcouldbedicultinchoosingoneofthesepointestimatesoftheparametersabovetheothers.Third,toperformsignicancetestsandobtaincondenceintervalestimatesoftheestimators,substantialcomputationonrepeatedsamplingthroughpermutationtests[ 12 ]orbootstrapping[ 92 ]isrequired.Furthermore,theseapproachesdonotproperlyaccountforuncertaintiesintheotherparameters,makingitunreliabletoclaimcoverageprobabilitiesofthecondenceintervals. Fourthandmostimportantly,whenaQTLmappingstrategyisincorporatedbyautocorrelatedlongitudinaldata,wewillencounteramuchhigherdimensionalspacefortheunknownparametersthantraditionalQTLmodelers.Althoughthemereexistenceofahigh-dimensionalparameterspaceisnotnecessarilydetrimental,extracaremustbetakeninsearchingfortheMLestimator.Anextracomplication(notonlyofML)isthattheuncertaintyabouttheactualnumberofQTLforalongitudinalquantitativetraitresultsinextradicultyinmodelttingandselection. TheBayesianmethodcanavoidmanyoftheproblemsdescribedaboveforML.InML,theunknownparametersaretreatedasunknownvariables(unobservables)andthelikelihoodfunctionismaximizedinthesevariables.IntheBayesianparadigm,eachunobservableparameterisgivenapriordistribution,andwetheninfertheposteriordistributionofeachunobservableconditionalonthedata(theobservables).Thesummarystatisticsoftheposteriordistribution,e.g.,themean,themodeorthemedian,canbe 18

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8 ].Theintervalestimatecanbeobtainedsimplybyexaminingtheposteriordistribution.Letusdenotetheobservablesbyavectory(thedatavector)andunobservablesbyaparametervector=(;;v).Theposteriordistributionisp(jy)=p(y;)=p(y)=p(yj)p()=p(y)/p(yj)p() wherep()isagenericexpressionforaprobabilitydensity,p(yj)isthelikelihoodandp()isthepriorprobabilitydistributionoftheunobservables.Becausethedominatorisjusttheprobabilitydensityofy,notafunctionoftheparameters,itcanbeignored.Wepartitionthevectorinto=[`fg`]where`isasingleelementoftheunobservablesand`istherestoftheunobservablesthatexclude`.Themarginalposteriordistributionof`isexpressedbyp(`jy)=Zp(`;`jy)d`/Zp(yj`;`)p(`;`)d` 77 ].ThepotentialoftheBayesianapproachimplementedwiththeGibbssamplerorMetropolis-Hastingsalgorithmforgenomemappinghasbeenexploredforseveralrelativelysimplegeneticdesigns[ 78 ][ 81 ].Inparticular,becausewearenowabletoexaminetheentireposteriordistributionofeachparameter,wewillbebetterabletodealwithproblemssuchasmulti-modalityofthelikelihoodfunction.Baysianinferenceforgenemappingwasrstintroducedby[ 39 ],[ 40 ],and[ 86 ][ 88 ].ModelsforimplementingmultipleQTLweredevelopedbySatagopanet.al.[ 78 ],Heath[ 37 ],UimariandHoeschele[ 89 ],StephensandFisch[ 85 ],SillanpaaandArjas[ 81 ][ 82 ].Thedeterminationofthe 19

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(1) Incorporatebiologicallymeaningfulparametersthatdenethegrowthcurveofadynamictraitintofunctionalmapping,andthenderiveaBayesianprocedurefortheirestimation; (2) Modelthestructureofcovariancematrixamongtime-dependentobservationswithcommonlyusedstatisticalapproaches,suchasthestationaryrst-orderautoregressive,AR(1),ornonstationaryrst-orderstructuredantedependence,SAD(1),andimplementaBayesianapproachtoestimatethecovariance-structuringparameters; (3) DeriveageneralapproachforthegenomewideenumerationofQTLthatcontroladynamictraitwithintheBayesianparadigm; (4) ExtendtheBayesianapproachtoestimateepistaticinteractionsofQTLforadynamictrait,providingaquantitativeframeworkfortestingtheroleofepistasisindevelopment. IwillperformextensivesimulationstudiestoinvestigatethestatisticalpropertiesofmyBayesainfunctionalmappingmodelintermsofitsconvergencerate,estimationprecisionandpowerforQTLdetection.ArealexamplefortheF2mouseprogenywillbeusedtodemonstratetheutilizationofthemodelandvalidateitsusefulnessinapracticalgenomicprojectofdynamicQTL. 20

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49 ][ 65 ][ 66 ][ 67 ][ 103 ].However,becausemostoftheseapproachesignorethedevelopmentalordynamicfeatureofaquantitativetrait,theirapplicationtorevealthegeneticanddevelopmentalbasisfortraitvariationwillbeverylimited. TheQTLmappingofcomplexdynamictraitscannowbemadepossiblewithanewstatisticalapproach,calledfunctionalmapping[ 54 ][ 107 ].Functionalmappingincorporatesfundamentalbiologicalprinciplesbehindtraitgrowthanddevelopmentintoamappingframework.Itcapitalizesonthemathematicalaspectsofaubiquitousgrowthlawthateverybiologicaltraitexperiencesagrowthanddevelopmentalchangewithtimethroughalteringtheratiooftheanabolicormetabolicrateandtherateofcatabolism[ 4 ][ 99 ].Theadvantagesofthisapproachlieinitsincreasedbiologicalrelevancebyembeddingbiologicalprinciplesintotheestimationprocessanditsexibilitytogenerateanumberoftestablehypothesesaboutthedevelopmentalandgeneticregulationofgrowthinaquantitativeway.Fromastatisticalperspective,functionalmappingestimatesparametersthatdeterminetheshapeofagenotype-specicgrowthcurveand/orthecovariancestructure,insteadofdirectlyestimatingindividualtime-dependentelementsinthemeanvectorsandcovariancematrix,whichthusstrikinglyreducesthenumberofparameterstobeestimatedandincreasethestatisticalpoweroffunctionalmapping. 21

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15 ],[ 58 ].AlthoughML-basedapproacheshavemanyfavorablestatisticalpropertiesforparameterestimation,theyhavealsosomesignicantproblemswhenappliedtoapracticaldataset.First,thelikelihoodofmixturemodelsisunboundedforsomechoicesofparametricfamilies.Second,theremaybemanylocalmaximaforthelikelihoodfunctioninsomecomplexsituations,whichleadstomanydierentdierent(andpossiblyreasonable)plug-inestimatesforquantitiesofinterest.Itisdicultinchoosinganoptimalsetofpointestimatesoftheparametersfromallpossibleestimates.Third,substantialcomputationonrepeatedsamplingthroughpermutationtests[ 14 ]orbootstrapping[ 92 ]isrequiredtodeterminesignicancetests.Itisalsocomputationallyexpensivetoobtaincondenceintervalestimatesoftheestimators.Furthermore,theseapproachesdonotproperlyaccountforuncertaintiesintheotherparameters,makingitunreliabletoclaimcoverageprobabilitiesofthecondenceintervals. AlltheproblemsrelatedtoML-basedapproachesabovewillbecomemoreseriouswhenwedealwithfunctionalmappingofautocorrelatedlongitudinaldata,inwhichamuchhigherdimensionalspacefortheunknownparametersthantraditionalQTLmodelersneedtobefaced.Becausetheparametersthatmodelthemeanandcovariancestructuresarerelatedinanonlinearmathematicalform,itisextremelydicultorimpossibletoderivethelong-likelihoodequationsfortheseparameters.Thelast,butnotonlyML-specic,issueisabouttheactualnumberofQTLinvolvedinalongitudinalquantitativetrait.InML-basedQTLmapping,Kaoetal[ 45 ]adoptedvariableselectionviastepwiseregression,butthishasbeenshowntobehighlycomputationallyexpensiveformodelttingandselection. ManyoftheproblemsdescribedaboveforMLcanbeavoidedbyusingtheBayesianmethod.Insection1.4,thedierencesinmodelformulationandinterpretationbetweentheMLandBayesianapproachesweredescribed.Somekeyissuesarehighlightedherefor 22

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8 ].Theintervalestimatecanbeobtainedsimplybyexaminingtheposteriordistribution.ThemeanofthismarginalposteriordistributionisacandidateBayesianestimatorofanunknownparameter.Thismarginaldistributionrarelyhasanexplicitform,andnumericalintegrationisoftenprohibitedbecausethedimensionalityofparametersmaybehigh.However,aMarkovchainMonteCarlo(MCMC)algorithmcanbeusedtosimulatethesamplefromthejointposteriordistribution[ 77 ].ThepotentialoftheBayesianapproachimplementedwiththeGibbssamplerorMetropolis-Hastingsalgorithmforgenomemappinghasbeenexploredforseveralrelativelysimplegeneticdesigns[ 78 ][ 82 ].ModelsforimplementingmultipleQTLweredevelopedbySatagopanetal.[ 78 ],Heath[ 37 ],UimariandHoeschele[ 89 ],StephensandFisch[ 85 ],SillanpaaandArjas[ 81 ][ 82 ].ThedeterminationoftheactualnumberofQTLismadebycomparingtheBayesfactorsfromseparatemodelsorbyapplyingareversiblejumpalgorithm. Inthischapter,IwilldevelopageneralBayesianframeworkforfunctionalmappingofcomplexdynamictraitsbasedonparametricmodelingofthemean-covariancestructures.ThisframeworkisconstructedbyamixturemodelinwhichmultiplemixturecomponentscorrespondingtothegenotypesoftheunderlyingQTLareinvolved.IwillimplementtheMCMCalgorithmtoestimatetheposteriordistributionofeachparametercontainedwithinthemixturemodel.WewillperformextensivesimulationstudiestoinvestigatethestatisticalpropertiesofmyBayesainfunctionalmappingmodelintermsofitsconvergencerate,estimationprecisionandpowerforQTLdetection.ArealexamplefortheF2mouseprogenywillbeusedtodemonstratetheutilizationofthemodelandvalidate 23

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2.2.1LinearModel AssumethatthereisabiallelicputativeQTLwithgenotypesqq(0),Qq(1)andQQ(2)aectingtheshapeofgrowthcurves.Ataspecictimepointt,thephenotypicvalueofthetraitforeachindividualiduetotheQTLmaybegivenbythelinearmodelasfollows: whereijisanindicatorvariableforindividualitocarryaQTLgenotypeanddenedas1ifaparticularQTLgenotypejisindicatedand0otherwise,uj(t)istheexpectedphenotypicvalueforQTLgenotypejattimet,andi(t)isindependentlyandidenticallydistributedasN(0;2(t)).Notethatmeasurementswithinanindividualaremorelikelytobecorrelatedacrosstimes,withcovariance(t1;t2)betweentimest1andt2(t1;t2=1;:::;).Thesevariancesandcovariancesforma()matrix. 24

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4 ]hadextensivediscussionsonthefundamentalbiologicalaspectsofmathematicalmodelingofgrowthcurves.Amongthevariousmodels,thesigmoidal(orlogistic)growthfunctionisregardedasbeingnearlyuniversalinlivingsystemstocaptureage-specicchangeingrowth[ 99 ].Specically,thisS-shapedcurveforaQTLgenotypejcanbemathematicallyexpressedas: wherej=(j;j;j)isasetofparametersthatdeterminetheshapeofthecurves.Parameterdeterminesthelimitingvalueofgrowthastimetgoestoinnity;=(1+)givestheinitialvalueofgrowth,andisdenedastherelativegrowthrate.Sincetheshapeofthecurvecanbecharacterizedbyasetofparametersj,asignicantdierenceamongj(j=0;1;2)fordierentgenotypesofaputativeQTLimpliesthatthisQTLhasaneectongrowthcurves. Anumberofstatisticalmethodshavebeenderivedtomodelthestructureofcovariancesbetweenlongitudinalmeasurements[ 17 ].TheAR(1)modelhasbeensuccessfullyappliedtomodelthestructureofthewithin-subjectcovariancematrixforfunctionalmapping.Thismodelusestwosimpliedassumptions,i.e.,variancestationarity{theresidualvariance(2)isconstantovertime,andcovariancestationarity{thecorrelationbetweendierentmeasurementsdecreasesproportionally(in)withincreasedtimeinterval. Inpractice,thetwosimpliedassumptionsoftheAR(1)modelmaynotholdsothattheelegantexpressionsofthematrixcannotbeusedforfunctionalmapping.TomakelongitudinaldatawellsuitedtotheAR(1)model,sometreatmentsareneeded.Forexample,inordertoremovetheheteroscedasticproblemoftheresidualvariance,CarrollandRupert's[ 9 ]transform-both-sides(TBS)modelisembeddedintothegrowth-incorporatednitemixturemodel[ 106 ],whichdoesnotneedanymoreparameters. 25

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TheTBS-basedmodeldisplaysthepotentialtorelaxtheassumptionofvariancestationarity,butthecovariancestationarityissueremainsunsolved.ZimmermanandNu~nez-Anton[ 117 ]proposedaso-calledstructuredantedependence(SAD)modeltomodeltheage-specicchangeofcorrelationintheanalysisoflongitudinaltraits.TheSADmodelhasbeenemployedinseveralstudiesanddisplaysmanyfavorablepropertiesforgeneticmappingofdynamictraits[ 114 ]. Although,inpractice,onlythephenotypicvaluesyiandthemarkergenotypesMiareobservable,theprobabilitydistributionoftheQTLgenotypesforF2populationscanbeexpressedintermsofthelocationoftheputativeQTL,themarkergenotypesandthedistancebetweenthemarkers.SupposethisQTLislocatedbetweenmarkerskandk+1.Then,theQTLgenotypedistributionsoftheithindividualthatreectthemixtureproportionsinthemixturemodel 1{3 canbeexpressedas 26

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Sincethedatay=fyigni=1canberegardedasindependentobservations,thejointlikelihoodforallnindividualsisaproductoftheindividuallikelihoods,i.e., Inthemixturemodel 2{5 ,=fjg2j=0isanunknownvectorthatdeterminestheparametricfamilyfj,whichpresentsamultivariatenormaldistributionwiththegenotype-specicmeanvectorexpressedas andthecovariancematrix. Oneaimistomakeinferenceabout,,andviathislikelihood, 2{5 .Butwhenwedealwithlongitudinaldata,itisusuallynottrivialtoevaluatethislikelihood.Inaddition,itispossiblethatnotalltheunknownparametersinthemodel 2{5 canbeidentied.Insteadofattemptingtooptimizethelikelihoodsurface,IinvestigatedaBayesianapproachtotacklethesediculties.Iwillintegratethelikelihoodwithpriorknowledgetoproduceinferencesummariesforallthecomponentsinthemodel.Furthermore,BeyesianmodelselectioncanbeperformedtoestimatethenumberofQTLeitherbyusingBayesfactorsorareversiblejumpMCMCalgorithm.TheproceduresfordeterminingthenumberofQTLaregiveninthesectionMultipleQTLModel. 27

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where(yjQ;;)=Q(yijQ=qi;qi;)denotestheprobabilitymassoftheobservationygiventheQTLgenotypes;(Qj)=Q(Qij)istheprobabilitymassoftheQTLgenotypesofallnindividualsgiventheirmarkergenotypesandtheQTLlocus(Mand);and(;;)isthepriorimposedonthegeneticparameters.Itisreasonabletoassumethepriorsareindependentfortheparameters.Thus, Usually,thereisnoinformationavailablefortheQTLlocation,auniformdistributionon[0;Dm]isanaturalchoiceforit.Theinformationaboutjcanbeobtainedrelativelyreliably,forwhichmultivariatenormalpriorswithmoderatevariancesareused.Thestandardpriordistributionfortheinverseofthecovariancematrix1istheWishart(R;)[ 10 ][ 20 ],wherethesocalledscalematricC=R1representspriorstructuralinformationaboutandisthedegreeoffreedom,whichmustbegreaterthanT1tohaveaproperprior.Asmallvalueofgivesarelativeatdistribution.TheWishartpriorwithlowdegreesoffreedomandaspeciedRisregardedasareference(or 28

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IntheBayesianapproach,statisticalinferencefortheunknownparametersarebasedontheirmarginalposteriordistributions.Theoretically,themarginalposteriorscanbeobtainedfromthejointposterior 2{7 byintegratingovertheotherunknowns.Unfortunately,inpractice,theevaluationofsuchhigh-dimensionalintegralsinclosedformisnotpossible.However,itisoftenstraightforwardtoderiveeitherthefullconditionalposteriordistributionsforsomeparameters,ortheexplicitexpressionsthatareproportionaltothecorrespondingfullconditionalposteriordistributionsforotherparameters,i.e., 2Pnjqi=jh(y(j)ig(j))01(y(j)ig(j))i1 2(j)01(j)o and 2expn1 2trhR11+P2j=0Pnjqi=j(y(j)ig(tjj))(y(j)ig(tjj))0iovWi(D1;n+) (2{10) wherej=fj0:j0=0;1;2;j06=-jg;y(j)i'srepresentfortheobservationsfromthoseindividualswhichhavegenotypej;andD=R1+P2j=0Pnjqi=j(y(j)ig(tjj))(y(j)ig(tjj))0. 2{7 ofunknownparameters.Inthiswork,ahybridschemeofGibbssamplerandMetroplolis-Hastings(M-H)algorithm[ 29 ][ 36 ][ 87 ]willbeapplied. 29

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78 ],isupdatedbyusingthemetropolisalgorithm.AnewvalueofisgeneratedfromUniform(max(0;);min(+;Dm)),andthisproposeddistributionisdenotedbyq(;).Thisproposedisacceptedwithprobabilitymin(;1),andthestateremainscurrentvalueiftheproposalisreject.Whereisgivenasbelow: Notethat, (2{12) Similarly, 30

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2{11 canbesimplifyto Oneoftheimportantimplementationissueshereisthatthetuningparameter>0needstobechosencarefully,sinceitdeterminesthevarianceoftheproposal.Ifthevarianceoftheproposeddensityistoolarge,alargeproportionofproposedmoveswillberejected,whichmayresultinineciency.Ontheotherhand,atoosmallvariancegivesahighacceptratewhereasaslowmovementontheparameterspace,whichalsoleadstoineciency.Satagopanetal[ 78 ]suggestedthatmorethanoneupdatesofbetweenotherupdatesmaybeabletoimprovethemixingoftheMarkovchain. Hence,ateachcycle,wecansampletheQTLgenotypeQidirectlyfromthisfullconditionaldensity. Notethat,thechoiceoftheMetropoliskernelqisessentiallyarbitrary,andasymmetricqinthesensethatq(j;j)=q(j;j)isusuallypreferred.Andinthatcase,theratioq(j;j)=q(j;j)iscanceledintheaboveexpression 2{16 .Here,fortheproposeddensity,weuseamultivariatenormaldistributioncenteredatthecurrent, 31

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94 ]whoseinversehas(u;v)thelement 2@ @j,u@ @j,v(j)01(j):(2{17) Thisexpressioncombinestheexpectedinformation(therstterm)withthepriorinformation(thesecondterm)oersanadvantageofavoidingsingularinformationmatrices.Unfortunately,tediousinitialanalysishastobedonetoobtainestimatesjandfromwhichtoevaluate 2{17 .So,initiallytheMetropolisalgorithmdescribedbeforecanbecarriedoutbyusinganarbitraryvariance-covariancematrix.Theposteriormeanofjandisthenpluggedinto 2{17 fromthesubsequentanalysis. 2{10 foritsfullconditionalposteriordistribution. 87 ]thatforanyfunctionofunknownf,whichisasquareintegrablewithrespecttothestationarydistribution,if((k);(k);Q(k);(k))arethesamplesthatwecollectfromtheMarkovChain, fN=1 Inotherwords,theempiricalaveragesoftheircorrespondingMCMCsamplesmayberegardedastheconsistentestimatorsfortheunknownparameters. Forthemarginalposteriordensitiesoftheseparameters,kerneldensityestimator[ 23 ]canbeused,sincetheclosedformoftheirfullconditionalposteriorsarenotavailable.Forexample,ifthesimplestkerneldensityestimatorisutilized,thenthehistogramkernel 32

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^(jy)=1 Anotherimportantestimationissueistoobtainthecondenceintervalsfortheunknowns.BoxandTao[ 5 ]suggestedthathighestposteriordensity(HPD)regionscanbeconstructedtogivethecondenceintervalsfortheparametersofinterestandthedetailedmethodfordevelopingapproximatedHPDviaMCMCsamplescanbeseenin[ 76 ].Alternatively,wecanobtaintheapproximateHPDfortheparametersdirectlybyfromtheircorrespondingsmoothdensityestimators. Finally,inordertoestimatetheMonteCarloerrorviatheCLT,Geyer[ 31 ]suggestedthreetypesofconsistentestimators,thewindowestimators,themethodofstandardizedtimeseriesandthespecializedMarkovChainestimators.Amongthose,thewindowsestimatorsprobablyprovidesthebestestimates,althoughitrequiresmoreworkandstrongerregularityconditionstobeconsistent. 15 ]and[ 84 ]that,inpractise,ifthetruecovariancematrixisclosedtoI,theeigenstructureofcanbesystematicallydistortedbytheestimator,sothisconventionalpriorcanbehavepoorly,especiallywhenthesamplesizeissmallorthedataaresparse.Toovercomethesedrawbacks,severalothermoreexiblepriorshavebeenintroduced,includingalog-matrixprior[ 51 ],areferencenoninformativeprior[ 109 ],andaconstrainedWishartprior[ 21 ].Meanwhile,severalrecentpapers[ 13 ][ 14 ][ 72 ]proposedstrategiesofmodelingthecovariancematrixwithadierentparameterization.Allthesepapersarebasedonthekeyideathatacovariancematrixforlongitudinaldatacanbediagonalized,i.e., 33

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Withoutnicepropertiesduetotheconjugatepriors,theresultingfullconditionalposteriorofnolongerhasanexplicitform.Often,oneevenhavetogeneratecomponentwisebyusingGibbssampling.Consequently,Iwillfocusoncomputationallyeasymethodsthatcangeneratetheentireatatime,giventheproblem'scomplexity.Amongvariouschoices,thereference(noninformative)priorwasrstintroducedbyBergerandBernado[ 3 ]andthoroughlydiscussedbyYangandBerger[ 109 ].Asanoptiontofurtherimprovetheestimationforthecovariancematrix,IfurtherinvestigatedtheimplementationofthisreferencepriorinthecontextofQTLfunctionalmapping. ThemostcommonlyusedreferencepriormightbetheJerey'sprior(Jereys1961), However,caremustbetakenwhenusingthisprior,asitcanleadtoanimproperposteriordistribution,anditmightfailtoshrinktheeigenvaluesappropriatelysometimes.However,theapproachproposedbyYangandBerger[ 109 ]hasproventobeabletoovercometheseinadequaciesoftheJerey'spriorremarkably.Notethatcanbedecomposedas=ODO',whereOisorthogonalwithpositiveentriesintherstrow,andD=diag(d1;d2;:::;dp),withd1>d2>;:::>dp.Hence,providedthesemonotonicallyorderedfdig,thereferencepriorfor(D;O)isgivenby andtheresultingposteriordistributionis 2nOD1O0(P2j=0Pnjqi=j(y(j)ig(tjj))(y(j)ig(tjj))0))] ComparingthereferencepriorwiththeJerey'sprior,itisnotedthattheposteriorgiveninequation 2{22 isalwaysproper.Alsonotethat,sincethisreferencepriorput 34

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Unsurprisingly,itisverydiculttoanalyticallyevaluatetheposterior 2{22 .YangandBergersuggestedusingaMetroplisizedhit-and-runsampleralgorithmtoobtaintheintegration.Thedetailsamplingprocedureatthekthiterationisgivenasfollows: 2-3 displaysthematrixofsamplevarianceandcorrelationscorrespondingtherealdataforthebodymassofmicemeasuredat10consecutiveweeks.Clearly,thevariancesarenothomogeneous,andtendtoincreaseovertime.Inordertostabilizedthe 35

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9 ]intoourmodeltofurtherpreservethebiologicalmeansforthesecurvatureparameters.Table 2-2 displaysthesamplevariance-covariancematrixforthetransformeddata.Noticethat,rst,afterthetransformation,thevariancestendtobeaconstantovertime.Second,thecorrelationsexistapparently,andtheyaremostlypositive.Foranygivencolumn,correlationstendtodecreasetozero.Finally,theserialcorrelationseemstobepresent,sincethecorrelationsdecreaseastheelapsedtimebetweenmeasurementsincreases. Stationaryautoregressive(AR)models[ 43 ][ 44 ]areperhapsthemostpopularparametricmodelsforserialcorrelations.Thesemodelsarebasedontwoassumptions,variancesareconstantovertimeandcorrelationsbetweenmeasurementswithequaltimelagareequal.Particularly,inourproblem,theautoregressivemodelwithorder1,i.e.theAR(1)modelisconsidered.Asaresult,thecovariancematrixforobservationsyi=(yi1;:::;yiT)canbeconstructedasfollows and Asaspecialcaseofparametricmodel,Bayesiananalysiscanbeperformedtoobtaintheestimatesofthesematrixstructureparameters.Again,itisreasonabletoassumetheindependencybetweenthepriorsof2and.Asaconventionalchoice,theinversegammapriorisgivento2,andaninformativepriorrestrictedon[1;1]isimposedon.Insuchacase,theposteriordensityof,,2,,andQisgivenby: where(2)=IG(;)and()=Uniform(1;1). 36

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21 2P2j=0Pnjqi=j(yif(j))01(2;)(yif(j))g andfor, 2j(2;)jn 2P2j=0Pnjqi=j(yif(j))01(2;)(yif(j))g Basedonaboveexpressions,thecorrespondingMetroplolis-Hastingsstepscanbedevelopedtoupdate2andwithintheMCMCestimationschemedescribedinsection2.5. IneachMCMCcycle,acandidatevalueof2denotedby2isgeneratedfromitsproposaldistribution,whichcanbespeciedas:q(2j2)=IG(1 Theproposaldistributionofcanbespeciedasauniformwithinmoderaterangearoundthecurrentvalueof,inotherwords,q(j)=Uniform(max(1;);min(+;1)).Anewvalueof,,isgeneratedfromthisproposaldistributionandisacceptedwithprobabilitymin(;1),with 37

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27 ]ifytgivenyt1;:::;y2;y1onlydependsonyt1;:::;ytr,foralltr.Basedontheassumptionthatobservationswithinindividualsareantedependent,Nu~nez-AntonandZimmerman[ 63 ]proposedausefulclassofmodelscalledstructuredantedependence(SAD)models.Theideaofthisapproachistotthecovariancestructurebyparametricmodelingtheantedependencecoecientsandinnovationvariances.Henceasaresult,theSADmodelhasagreatadvantageofreducingthenumberofparametersconsiderablycomparedwiththetraditionalunstructuredantedependence(UAD)models. Tobespecic,ifanr-orderSADmodelisassumed,theautoregressivecoecientsfollowaBox-Coxpowerlawandtheinnovationvariancesarespeciedasparsimoniousfunctionsoftime,i.e., where andhereg()isarelativelysimplefunctionwithfewerparameter,sayapolynomialwithlow-order. 38

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41 ], Therefore,dierentfromanAR(1)model,evenforthesimplestSADmodelwhichhasconstantinnovationvariancesovertime,neitherthevarianceoftheobservedprocessnorthecorrelationfunctionisstationary,i.e.thevariances2(j)canchangewithtime,and2(j;k)doesnotdependonlyonthelagtimejjkj. WithintheframeworkofBayesiananalysis,athick-tailedinverse-gammapriorcanbegiventotheinnovationvariance2,andanormalpriorcanbegiventotheantedependenceparameter.Intheactualanalysis,thepriorsareselectedas,(2)=IG(;)=IG(1;1)and()=N(;)=N(0;10).Asbefore,forbothparameters,theexplicitexpressionsthatareproportionaltotheircorrespondingfullconditionalposteriorscanbederivedasfollows, 21 2P2j=0Pnjqi=j(yif(j))01(2;)(yif(j))g andfor, 2j(2;)jn 2()21 2P2j=0Pnjqi=j(yif(j))01(2;)(yif(j))g: Aslongaswespeciedtheproposaldistributions,thedetailedMetroplis-Hastingsstepsupdating2andcanbederivedasbelow, 39

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IneachMCMCcycle,acandidatevalueof2denotedby2isgeneratedfromitsproposaldistribution,whichcanbespeciedas:q(2j2)=IG(1 Theproposaldistributionofcanbespeciedasauniformwithinmoderaterangearoundthecurrentvalueof,inotherwords,q(j)=N(;V).Anewvalueof,,isthengeneratedfromthisproposaldistributionandisacceptedwithprobabilitymin(;1),with 2.9.1AnimalMaterial 90 ]constructedalinkagemapwith96microsatellitemarkersfor1043F2mice(503malesand540females)derivedfromtwostrains,theLarge(LG/J)andSmall(SM/J).Thetotallengthofthismapis1780cM(inHaldane'sunits)andanaveragemarkerintervallengthis23cM.TheF2progenywasmeasuredfortheirbodymassat10weeklyintervalsstartingatage7days.Therawweightswerecorrectedfortheeectsofeachcovariateduetodam,littersizeatbirth,parityandsex[ 90 ].Overall,about10%ofthemarkergenotypeswererandomlymissing.Themicewithmissingdatawereexcludedfromtheanalyses. 113 ]andZhaoetal[ 114 ]rstanalyzedthisdatasetbyusingfunctionalmappingwithmaximum-likelihoodbasedmethods.TheyshowedthatbodymassesintheF2micefollowalogisticcurve,butdisplaysubstantialvariationintheshapesof 40

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AssuggestedbyGemanandGeman[ 29 ],onelongrunmaybemoreecientwithconsiderationsofusingthefollowingtwostrategies:(1)discardanumberofinitial"burn-in"simulations,andconsideronlytheremainingsamples,sinceitwouldbeunlikelyforthoseinitialsimulationscamefromthestationarydistributiontargetedbytheMarkovchain;(2)subsamplingthechainatequalintervalstoreducetheserialcorrelationamongsamples.Also,basedonthedetaileddiscussionsinGelfandandSmith[ 30 ],severalgraphicaltechniquescanbeappliedheretochecktheconvergence.Initially,aspilotruns,severalshortMarkovChainsarestartedatdierentQTLlocationsalongalinkagegroup.Itappearsthatalloftheseshortchainsconvergetoarelativelysamestationarydistributionafterroughly10,000cycles.Also,forthesetestruns,theautocorrelationfunctionofthesinglelocusaresampledatdierentspans;asdisplayedingure 2-1 .TheestimatedautocorrelationfunctionofMCMCsamplesatlagkcanbeexpressedas ^R(k)=1 (nk)2nkXt=1[xt][xt+k]:(2{38) Weobservedfromtheseplotsthatautocorrelationbetweentheevery60thsub-sampledecreasedquickly,andisclosedto0atlag10. Accordingtotheresultsfromthesepilotruns,thestartingvalueforthesingleputativelocuswassettobe0=40cMforalllinkagegroups.Foreachanalysis,the 41

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Autocorrelationfunctionofthelocus()atdierentlags.Intherstplot,theMCMCsamplesweresubsampledatevery30thcycles.Inthesecondplot,theMCMCsamplesweresubsampledatevery60thcycles. MarkovChainran60,000cyclesandsampledevery60cycles,whichyieldedaworkingsetwith1000states.Thesestateswereregardedassamplesfromthetargetedposteriordistributionsoftheunknowns. Aninformalcheckofparameteridentiabilityisconductedviaongraphicaltechniquesbasedontimeseriestheories[ 17 ].AsshowninFigure 2-2 ,theconsecutivesamplesmoverandomlytowardsdierentdirections,whichindicatesthattheMCMCsamplerisnot"sticky".Thismeansthattheparameterscanberegardedasbeingidentiable. EstimatedmarginalposteriorsofQTLlocationforall19chromosomesaredisplayedingure 2-3 .Atarstglance,wenoticedthatposteriorsonchromosome6,7,8,10,11,and15haveobviousspikes,whereasthoseonotherchromosomesdisplayrelativelyatter 42

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Traceplotsfortheestimatedparameters. patternsalongthewholechromosome.BycomparingtheBayesfactorsofM0andM1,singleQTL'saredetectedonchromosome6,7and10respectively.WherethelogarithmicscaledBayesfactorscomputedby 3{5 is12.91forchromosome6,13.47forchromosome7,and7.99forchromosome10respectively.AsingleQTLwasestimatedon82:7cMbetweenmarker3and4ofchromosome6,oron46:8cMbetweenmarker2and3ofchromosome7,oron77:7cMbetweenmarker3and4ofchromosome10.TheseresultsaregenerallyconsistentwiththoseinZhaoetal[ 114 ].Itisstraightforwardtomakeinferenceontheunknownparametersbasedonthesemarginalposteriors.Table 2-4 2-5 ,and 2-6 ,showthesummarizedresultsfromtheanalysisinwhichtheestimatedparametersaregivenbytheir 43

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Figure2-3. AproleofEstimatedmarginalposteriordistributionoftheQTLlocationbyassumingthatexactlyoneQTLislocatedononeofthechromosomerespectively. EstimatedcurveparametersarelistedinTable 2-4 2-5 ,and 2-6 ,assumingtheactualQTLislocatedonchromosome6,7,or10.Andgivenasetofestimatedcurveparameters,wecanobtainthegrowthcurvesbyusingthelogisticgrowthfunction 2{2 .Figure 2-4 2-5 ,and 2-6 illustratethegrowthcurvesofthethreegenotypesateachofthedetectedQTL.Basedonquantitativegenetictheory,wecanpartitiontime-dependentgenotypicvalue,j(t)(j=0;1;2),intotheadditiveanddominanteectsduetotheQTL.Tobespecic,thedynamicchangesofadditivegeneticeectsoftheQTLcanbeexpressedas 2[2(t)1(t)];(2{39) andthedynamicchangesofdominantgeneticeectsoftheQTLcanbeexpressedas 2[21(t)2(t)0(t)]:(2{40) 44

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2-7 .TheestimatedadditiveeectispositiveandincreaseswithageforallthreedetectedQTLs.Ontheotherhand,theestimateddominanteectdisplaysdierentpatternsdependingontheQTLdetected.Andalsonotethat,themagnitudeofdominanteectismuchsmallercomparingtotheadditiveeects. Figure2-4. FittedgrowthcurvesforthethreeQTLgenotypesassumingasingleQTLislocatedonmousechromosome6. Byttingsingle-QTLmodel,Idetect3QTLonchromosome6,7,and10separately,andthisfactsuggeststhattheremightactuallyexistmorethanoneQTLcontrollingthegrowthcurvesofmice.Tofurtherexplorethispossibility,Ialsotthedatawithamulti-QTLmodel,whichwillbedescribedinthenextchapter. 45

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FittedgrowthcurvesforthethreeQTLgenotypesassumingtheQTLislocatedonmousechromosome7. SimulationexperimentswereperformedtoinvestigatethestatisticalpropertiesofthisBayesianmodelproposedforfunctionalmappingforcomplexdynamictraits.Theexperimentwasdesignedasfollows:AnF2populationof450individualswassimulated, 46

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FittedgrowthcurvesforthethreeQTLgenotypesassumingtheQTLislocatedonmousechromosome10. andIassumethatthereisexactlyoneQTLlocatedonalinkagegroupwith11equallyscatteredmarkers.Thetotallengthofthislinkagegroupis100cM,andIassumethatthereisexactlyoneQTL,anditislocatedat34cM.Thetruevaluesofthecurvatureparameterswerechosetobe: Thesamplecovariancefromthedog-body-massdatawassettobethetruecovariancematrix.Giventhissettingofcovariancematirxandthecurvatureparametersofthe3genotypes,theheritabilityoftheQTLisabout0:1.Finally,vectorsofphenotypicvaluesaresimulatedat10evenlyspacedtimepointsforthese450F2individuals. 47

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DynamicchangesoftheadditiveanddominanteectduetotheQTLlocatedonmousechromosome6,7,and10respectively. InordertoimplementtheMCMCtechniquesdescribedinsection2:5,suitablepriorshavetobechosen.Thecurveparametersjisgivenamultivariatenormaldistributioncenteredat(30;10;0:7)withcovariancematrix=diagf10;5;4g.Sincewedon'thaveanypriorinformationabouttheQTLlocation,issimplyassumedtohaveuniformprioralongtheentirelinkagegroup.Threemethodswereusedtoestimatethecovariancematrixofthephenotypicalvalues.Astheconventionalchoice,method1imposedaninverseWhishartpriorwithdegreeoffreedomT=10on.Basedonsection2:7:2,method2utilizedtheSAD(1)tomodelthecovariancestructure.IimposedIG(1;1)asapriorontheparameterofinnovationvariance2,andNormal(0;10)asapriorontheantedependenceparameter,respectively.Finally,method3modeledthecovariancestructurebytheAR(1)model.Basedonwhatwedescribedinsection2:7:1,WeimposedIG(10;1)asaprioronthevariance2,andUniform(1;1)asapriorontheparameterofcorrelation,respectively. 48

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EstimatedmarginalposteriordistributionoftheQTLlocationonmousechromosome6,7,and10respectively. ForeachMCMC-implementedBayesiananalysis,10,000initial"burn-in"iterationswerediscard,andfortherest60,000cycles,samplesarecollectedforevery60cycles,whichresultinaworkingsetof1000states.Again,inordertoensurethechainhadconvergedtothesamestationarydistribution,IrepeattheMCMCexperimentseveraltimeswiththesamesimulatedphenotypicaldataset.Thosereplicatesproducedresultsthatweregenerallyconsistentforeachmethod.Hence,forthepurposeofdemonstration,Ipresentoneofthereplicatestoshowthegeneralbehaviorofthese3dierentmethods. Estimatesyieldedbyusingmethod1method3aredisplayedintable 2-7 2-8 ,and 2-9 respectively.ThetruelocationofQTLandthetruevaluesoftheparametersthatdeningthegrowthcurvearegiveninsquarebrackets,alsothe95%empiricalHPD(highestposteriordensity)condenceregionsaregivenintheparentheses. 49

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2-9 2-10 ,and 2-11 ).Astothecomputationalload,method1hadagreatadvantagetorequireonly3 4ofthecomputationaltimeofmethod3. Figure2-9. EstimatedmarginalposteriordistributionoftheQTLlocationonamousechromosomeinasimulationstudy,assumingnocovariancestructures. Forthepurposeofcomparison,Ialsoanalyzedthesamesimulateddatabyusingtheconventionalmaximum-likelihoodbasedmethod.StandardtestprocedureswereperformedtodeterminethelocationforaQTL.TheLikelihoodratioteststatisticwascomputedfor 50

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EstimatedmarginalposteriordistributionoftheQTLlocationonamousechromosomeinasimulationstudy,assumingthecovariancestructureisSAD(1). every2cMalongthewholelinkagegroup.Figure 2-12 showstheproleofthesevaluesofLikelihoodratioteststatistic.Thisresultingprolehadtwopeeks.AlargerpeekwithLRscore635:71waslocatedat34cM,andarelativelysmallerpeekwithLRscore628:91waslocatedat44cM.Therefore,thetraditionalmethodwasalsoabletodetectthelocationofthesimulatedQTLaccurately,butwithahighriskofdetectingfalseQTLs.Althoughtheestimatesofcurvatureparameters(seeTable 2-10 )weregenerallyresealable,theseestimateshadsignicantlybiggerbiasesthanthosegivenbythenewmethodweproposed.Mostimportantly,byasinglescan,thetraditionalmethodisincapableofproducingcondenceintervalsoftheestimates,ordeterminethecriticalvalueoftheLRtest.Apermutationtestcanbeperformedtodeterminethecriticalvalue,butrequiresextremely 51

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EstimatedmarginalposteriordistributionoftheQTLlocationonamousechromosomeinasimulationstudy,assumingthecovariancestructureisAR(1). extensivecomputationaltimewhichcancost20timesofthecomputationaltimebyusingournewapproach. Tofurtherinvestigatethebehaviorsoftheseestimators,morereplicatesofthissimulationexperimentneedtobeconducted.Andforeveryanalysis,althoughIdidn'tencounteranycomplications,weneedtoalwayskeepitinmindthatasensitivityanalysisisnecessarytoexaminethedependenceoftheparameterestimatesonthepriorsandinitialvalues.UnderthesameMCMCsamplingscheme,wecaninitiatethechainwithseveraldierentstartingpointand/orwithdierentchoicesofpriors[ 74 ].Iftheestimatedparametersarenotsensitivetothepriorsandtheinitialvalues,thoserunsshouldexhibitsimilarandstablebehavior. 52

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TheproleoftheLRbetweenthefullandreduced(noQTL)modelestimatedfromtheSAD(1)modelforbodymassgrowthtrajectoriesinasimulationstudy.Astothesettingsofthissimulation,weassumethatasingleQTLcontrolbodymassgrowthtrajectoriesofmice. 107 ].Originalmodelsforfunctionalmappingwerederivedwithinthemaximumlikelihood(ML)contextandimplementedwiththeEMalgorithm.AlthoughML-basedapproachespossessmanyfavorablestatisticalpropertiesinparameterestimation,theymaynotbepowerfulenoughtohandlethecomplexityofhigh-dimensionalQTLmappingmodels,asoftenseeninfunctionalmapping.Asanincreasinglypopularapproach,Bayesianmethodsdisplay 53

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39 ][ 40 ],and[ 86 ][ 88 ][ 78 ][ 81 ][ 111 ]. Inthischapter,IderivedageneralBayesianframeworkforfunctionalQTLmappingofdynamictraitsandimplementedMarkovchainMonteCarlo(MCMC)algorithmstolocategenomicpositionsofQTLs,andestimatethemathematicalparametersthatdeneabiologicalprocessandthestatisticalparametersthatmodelthecovariancestructure.TheBayesian-basedmodelallowstheestimationoftheseparametersandtheircondenceintervalsbasedonposteriordistributions,andhasgreatpowertohandlecomplexestimationissuesrelatedtofunctionalmappinginaneectiveway.Likeoriginalparametricfunctionalmapping[ 54 ],thenewmodelallowstoapproximatetheontogeneticchangesofthegeneticeectstriggeredbyaQTL.Becausemanybiologicalprocesses,suchasgrowth,followaparticularpatternofdevelopment[ 99 ],theontogeneticcontrolofaQTLcanbemathematicallydescribedand,thereby,testedbyestimatingtheparametersthatdeneabiologicalprocess.Thenewmodelalsotakeanadvantageoffunctionalmappingtomodelthestructureofcovariancematrixbyastationaryornonstationaryapproach.Becauseofitscomputationalsimplicity,theAR(1)isadvantageousforstructuringthecovariancealthoughitneedsthevarianceandcovariancestationarityassumptions.TheTBS-basedmodelcanrelaxtheassumptionofvariancestationarity[ 106 ],butithasnotresolvedthecovariancestationarityissuewhenembeddedintotheAR(1)model.Aso-calledstructuredantedependence(SAD)model,advocatedbyZimmermanandNu~nez-Anton[ 117 ],canbeusedtosimultaneouslymodelthetime-dependentchangesofandvarianceandcorrelationintheanalysisoflongitudinaltraits.TheSADmodelisfoundtodisplaymanyfavorableproperties[ 117 ]. TheuseofBayesianapproachestofunctionalmappingwasalsoconsideredbyotherauthors.YangandXu[ 110 ]integratedBayesianshrinkageapproachestomapdynamicQTL,buttheirmodelwasbasedonanonparametricLegendrepolynomialtting.Althoughthistreatmentmaybestatisticallyexible,itsbiologicalrelevancemay 54

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113 ][ 114 ].Yet,thenewmodelprovidesestimatesofcondenceintervalsofcurveparameters,thusallowingbetterstatisticalinferencesaboutthegeneticcontrolofdynamicQTLs.SimulationstudiesshowthatthenewmodelisrobustinthatitprovidesreasonableestimatesofQTLeectsandpositionsinawiderangeofparameterspace. Themodelproposedinthischaptercanbemodiedbyconsideringanetworkofgeneticcontrol.Asabasicframework,theone-QTLisnotadequatetoexploretheeectsofinteractionbetweendierentQTL[ 46 ]andQTLandenvironments[ 113 ]onvariationinadynamictrait.OneofthesignicantadvantagesofBayesianapproachesliesintheestimationoftheoptimalnumberofQTLsinvolved.VariableselectionviastepwiseregressionisusedinMLmapping[ 45 ],butitishighlycomputationallyexpensive.Correspondingtothisvariableselectionprocedure,reversiblejumpMCMCisproposedinBayesiananalysis[ 33 ][ 34 ],althoughitissubjecttopoormixingandaslowconvergencetothestationarydistribution([ 6 ][ 32 ][ 34 ].MoreecientmethodsbasedonBayesianshrinkageanalysis[ 108 ][ 96 ]andstochasticsearchvariableselection[ 112 ]havenowbeenproposed.Thesemethodsdonotrelyuponanyexplicitformofvariableselection;rathertheyproceedimplicitlybyshrinkingtheeectsofexcessiveQTLstozero.Themodiedmodelwillcertainlyproveitsvalueinelucidatingthegeneticarchitectureofdynamictraitsandwillprobablybethebeginningofdetectingthedrivingforcesbehinddynamicgeneticsanditsrelationshiptotheorganismasawhole. 55

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Thesamplevariance-covariancematrixforthemice-body-massdata. t12345678910 10.4200.5070.6830.8890.8010.7530.7910.7650.7560.76720.5071.0601.2881.5461.4771.3701.4691.4701.4751.52430.6831.2882.2922.5912.5002.3272.4812.4612.5552.57040.8891.5462.5914.5874.4144.1844.4254.5644.6064.69150.8011.4772.5004.4145.4675.4175.8846.2006.3716.58760.7531.3702.3274.1845.4176.4917.0907.5937.8668.23170.7911.4692.4814.4255.8847.0908.7329.3569.84010.28880.7651.4702.4614.5646.2007.5939.35611.24211.54212.13490.7561.4752.5554.6066.3717.8669.84011.54213.13513.423100.7671.5242.5704.6916.5878.23110.28812.13413.42315.190 Table2-2. Thesamplevariance-covariancematrixforthetransformedmice-body-massdata. t12345678910 10.0190.0140.0120.0110.0070.0060.0060.0060.0050.00520.0140.0170.0140.0110.0080.0070.0070.0060.0060.00630.0120.0140.0160.0120.0090.0070.0070.0070.0070.00640.0110.0110.0120.0140.0100.0090.0080.0080.0080.00750.0070.0080.0090.0100.0090.0080.0080.0080.0080.00860.0060.0070.0070.0090.0080.0090.0090.0090.0090.00970.0060.0070.0070.0080.0080.0090.0100.0110.0100.01080.0060.0060.0070.0080.0080.0090.0110.0120.0120.01190.0050.0060.0070.0080.0080.0090.0100.0120.0120.012100.0050.0060.0060.0070.0080.0090.0100.0110.0120.013 Table2-3. BayesianestimatesofgrowthcurvesfortheQTLgenotypesdetectedonmousechromosome6.Numbersintheparenthesesarethe95%equal-tailcondenceintervals. ParameterQTLgenotype 36.09(35.20,37.04)34.94(34.36,35.52)33.12(32.36,33.93)11.93(11.44,12.45)11.58(11.16,12.03)11.07(10.65,11.51)0.65(0.64,0.66)0.65(0.64,0.66)0.65(0.64,0.67) 56

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BayesianestimatesofgrowthcurvesfortheQTLgenotypesdetectedonmousechromosome6.Numbersintheparenthesesarethe95%equal-tailcondenceintervals. ParameterQTLgenotype 36.09(35.20,37.04)34.94(34.36,35.52)33.12(32.36,33.93)11.93(11.44,12.45)11.58(11.16,12.03)11.07(10.65,11.51)0.65(0.64,0.66)0.65(0.64,0.66)0.65(0.64,0.67) Table2-5. BayesianestimatesofgrowthcurvesfortheQTLgenotypesdetectedonmousechromosome7.Numbersintheparenthesesarethe95%equal-tailcondenceintervals. ParameterQTLgenotype 36.55(35.50,37.73)35.61(34.56,36.50)33.38(32.54,34.33)11.83(11.43,12.34)11.27(10.90,11.73)11.25(10.76,11.70)0.65(0.63,0.66)0.64(0.63,0.65)0.65(0.63,0.66) Table2-6. BayesianestimatesofgrowthcurvesfortheQTLgenotypesdetectedonmousechromosome10.Numbersintheparenthesesarethe95%equal-tailcondenceintervals. ParameterQTLgenotype 35.41(34.33,36.52)34.71(33.67,35.70)33.59(32.61,34.42)11.67(11.23,12.19)11.47(11.23,11.81)11.01(10.61,11.44)0.65(0.64,0.66)0.64(0.63,0.66)0.65(0.63,0.66) Table2-7. Resultsfromasimulationstudybyassumingnonecovariancestructure.Numbersintheparenthesesarethe95%equal-tailcondenceintervals. ParameterQTLgenotype 36.67(35.81,37.46)36.03(35.47,36.63)33.67(32.67,34.31)True36.7035.6033.4011.83(11.95,12.64)11.22(10.97,11.50)11.30(10.94,11.60)True11.9011.2011.200.66(0.64,0.67)0.64(0.63,0.65)0.64(0.63,0.66)True0.650.640.65 57

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ResultsfromasimulationstudybyassumingthecovariancestructuretobeSAD(1).Numbersintheparenthesesarethe95%equal-tailcondenceintervals. ParameterQTLgenotype 36.58(35.85,37.36)35.88(35.37,36.39)33.81(33.09,34.55)True36.7035.6033.4012.04(11.65,12.45)11.27(11.01,11.54)11.46(11.09,11.83)True11.9011.2011.200.66(0.65,0.67)0.64(0.63,0.65)0.65(0.64,0.65)True0.650.640.65 Table2-9. ResultsfromasimulationstudybyassumingthecovariancestructuretobeAR(1).Numbersintheparenthesesarethe95%equal-tailcondenceintervals. ParameterQTLgenotype 36.60(35.59,37.66)35.57(34.88,36.35)33.61(32.65,34.54)True36.7035.6033.4012.04(11.61,12.48)11.23(10.99,11.56)11.42(11.05,11.83)True11.911.2011.200.65(0.64,0.66)v0.63(0.62,0.65)0.66(0.64,0.67)True0.650.640.65 Table2-10. Resultsfromasimulationstudybyperformingtraditionalmaximum-likelihoodtypedmethod.Numbersintheboxbractsarethegivenvaluesoftheparameters. ParameterQTLgenotype 37.22[36.70]37.76[35.60]33.38[33.40]12.56[11.90]9.87[11.20]11.23[11.20]0.66[0.65]0.63[0.64]0.64[0.65] 58

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25 ][ 59 ].Thus,tobetterunderstandthebiologyoftraitdevelopment,adetailedgeneticarchitectureofhowgenesactandinteracttocontrolvariousstagesofdevelopmentmustbequantied.Unfortunately,manycurrentstatisticaltechniquesusedingeneticresearchassumetheadditivecontrolofgenes,aimedtofacilitatedataanalysisandmodeling,whichcertainlyprovidemisleadingresultswhengeneticinteractionsorepistasisactuallyoccur. Thegeneticeectorvarianceofquantitativetraitloci(QTL)includestwocomponents,additive,duetothecumulationofbreedingvalues,andnonadditive,duetoallelic(dominant)ornonallelic(epistatic)interactions.Epistaticinteractionsbetweendierentlocicanbefurtherpartitionedintodierenttypes,additiveadditive,additivedominant(ordominantadditive)anddominantdominant.Thepresenceofepistasisimpliesthattheinuenceofageneonthephenotypedependscriticallyuponthecontextprovidedbyothergenes.Inthepast,theestimationoftheadditiveandnonadditivegeneticarchitectureofaquantitativetraitwasbasedonthephenotypesofrelatedindividuals[ 53 ],althoughthishasminimalpowertodetectthenonadditivegeneticvariances,especiallyepistaticvariancebecauseepistasiscontributeslittletotheresemblanceamongrelatives[ 11 ]. TheadventofDNA-basedlinkagemapsopensanovelavenueforpreciselyestimatingthegeneticarchitectureofdevelopmentaltraits[ 90 ].CurrentstatisticalmethodsproposedtodetectthemainandinteractioneectsofQTLarebasedonthephenotypesofaquantitativetraitmeasuredatalimitedsetoflandmarkages.Morerecently,Wuetal.[ 104 ][ 105 ][ 106 ][ 107 ]andMaetal.[ 54 ]havederivedapowerfulfunctionalmapping 59

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99 ]andthestructuredresidual(co)variancematrixamongdierenttimepoints(see[ 48 ][ 71 ]).Thismethodhasproventobestatisticallymorepowerfulandmoreprecisebecauseofareducednumberofparametersbeingestimated,andtobebiologicallymoremeaningfulduetotheconsiderationofbiologicalprinciplesunderlyingtraitdevelopment[ 106 ].However,thismodelhasnotincorporatedtheestimationprocessofepistaticinteractionsand,thus,cannotexaminetheroleoftheentiregeneticarchitectureindevelopmentaltrajectories. Inthischapter,IextendthefunctionalmappingmethodtomapanyQTL(includingadditive,dominantandepistatic)thattransformsallelicand/ornonalleliceectsintonalphenotypesduringacontinuousprocessofdevelopmentrepresentedasontogenetictrajectoriesorapaththroughphenotype-timespace[ 1 ][ 100 ].Iderivespecialprocedurestoestimateandtesttheimpactofepistasisontraitgrowthbecauseagrowingbodyofevidencenowshowsthatepistasisplaysamoreimportantroleindeterminingdevelopmentalchangesthanoriginallythought[ 75 ][ 100 ].IuseaBayesian-basedmethod,implementedwiththeMarkovchainMonteCarlo(MCMC)algorithm,toestimateQTLlocationsandgeneticeectsongrowthdierentiation.Comparedwithcurrentmappingmethods,ourmethodofincorporatinggrowthtrajectoriestendstobemorepowerfulandmorepreciseinQTLdetectionandeectestimation,asdemonstratedinanexampleusingmouseF2data. 60

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Thegenotypeofkthlocusofindividualicanbeupdatedconditioningontheupdated,theobservedphenotypicvalueyi,andthegenotypesforotherQTLlocusofindividuali.Andspecically,thefullconditionalthatgenotypeofkthlocusofindividualibeingj;(j=0;1;:::;3s)is: GiventheupdatedQTLgenotypes,inasamewayasdescribedinsection2:5,wecanupdatethe3sblocksofoneatatime.Finally,dependingondierentassumptionsonthecovariancestructure,wecaneitherupdatethecovariancematrixasawholebyGibbssampling,orupdatethecovariancematrixthroughupdatingthematrix-structuralparameters.Inthisanalysis,Imadenoassumptiontothestructureofthecovariancematrix.Therefore,aGibbssamplingstepwasappliedtoupdatethecovariancematrixaccordingitsfullconditionalmarginalposteriordistribution. Oneoptiontodeterminethevaluesisbyrunningdierentmodelsundertheassumptionthatthereare0;1;2;:::QTLsrespectively,andthencomparingthemwithBayesfactors.TheBayesfactorfortwomodelsisdenedastheratioofmarginalprobabilitiesofygiventhetwomodels: 61

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Inpractice,aBayesfactorlargerthan100canoftenberegardedasanevidencesupportingmodel1. Whencomparingtwoormoremodels,weperformsignicanceteststorejectoracceptacertainhypothesis,byafrequentistapproach.Alternatively,Bayesfacotersnotonlyoerusevidenceinfavorofacertainhypothesis,butalsoenableustoincorporatethepriorinformationwithregardtothehypothesisofinterest.Forexample,wecanimposehigherdenseonthemodelassumingtwoQTLsaectthetrait,ifpreviousstudiessuggestedso.Unlikefrequntistapproaches,thisexternalinformationwillaecttheresultofmodelselection.Meanwhile,KassandRaftery[ 47 ]providedanapproximationbetweenLODscoreandtheBayesfacor: 2(dimM2dimM1)log10(n):(3{5) WheredimM1anddimM2arethenumbersofparametersinmodel1andmodel2respectively;nisthesamplesize.Giventhisapproximation,wecancomparethesetwocriterionsstraightforwardlyinbothdirections. Inourproblem,itisimpossibletoevaluatethelikelihoodsofparametersanalytically,andwehavetoresorttovariousnumericalapproximations.OnepossiblerecipeisproposedbyNewtonandRaftery[ 62 ],theysuggestedanempiricalestimatorofBFgiventheMCMCsamples: ^(yjmodels)=N ThisisanconsistentestimatoroftheBayesfactor,andconvergesalmostsurelytothecorrectvalue.However,itisgenerallytoounstabletosatisfyacentrallimitTheorem, 62

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3{6 ,itisveryeasytocalculate.Moreover,bytransformingtotheLODscoreby 3{5 ,itusuallyyieldsresultsthatareaccurateenoughforthepurposeofinterpretation.Hence,inthiswork,Iuse 3{6 toestimatetheBayesfactorsandinterpretthemonthelogscale 3{5 3.3.1Material 90 ]constructedalinkagemapwith96microsatellitemarkersfor1043F2mice(503malesand540females)derivedfromtwostrains,theLarge(LG/J)andSmall(SM/J).Thismaphasatotalmapdistanceof1780cM(inHaldane'sunits)andanaverageintervallengthof23cM.Foreachofthe19linkagegroups,around10%ofthemarkergenotypeswererandomlymissing.TheF2progenywasmeasuredfortheirbodymassat10weeklyintervalsstartingatage7days.Therawweightswerecorrectedfortheeectsofeachcovariateduetodam,littersizeatbirth,parityandsex[ 90 ]. 113 ]andZhaoetal[ 114 ]rstanalyzedthisdatasetbyusingfunctionalmappingwithmaximum-likelihoodbasedmethods.Resultsfromhisanalysisshowedthat,insteadofasingleQTL,multipleQTLsmayexistaectingthegrowthpatternofbodymassofmice.AndthepossiblelocationfortheseQTLsareonchromosome6,7and10respectively.Inouranalysis,wettedmodelsunderdierentassumptionswithregardtototalnumberofQTLs(M0;M1;:::;Ms)separately.Then,theBayesfactorsonalogarithmicscale 3{5 areusedtocomparethesemodels,andhence,todeterminethenumberofQTL. TheBayesianformulationoftheproblemrequiresspecifyingpriordistributionsonthesetofmodelparameter,QTLlocusandthecovariancematrix.Accordingtoseveralrelatedstudies,someinformationareavailablefor,andthepriorsforj;(j=1;2;:::;3s)aregivenbyamultivariatenormal,centeredat(30;10;0:6)Tandwithalarge 63

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Therearetwopossibleassumptionforthetwo-QTLmodel,1)thesetwoQTLarelocatedonasamechromosome;or2)thesetwoQTLarelocatedondierentchromosomes.Weexaminedthedataunderbothassumptionsforchromosome6,7,and10.AstotheMCMCimplementation,thelengthofaMarkovchainconsistedof80,000cycles.Aftertherst20,000cycles(burn-inperiod),thechainwastrimmedbykeepingsub-samplesinevery60cycles.Thustheposteriorsamplescontained1000observationsforpost-MCMCanalysis. Byttingthesingle-QTLmodel,weobservedingure 3-1 thatthemarginalposteriordensityofQTLlocationhasonemodebyassumingtheQTLislocatedonchromosome6orchromosome10.Ontheotherhand,byassumingtheQTLislocatedonchromosome7,themarginalposteriordensityofQTLlocationisbimodal,butthetwomodesareveryclosedtoeachother.Thissuggeststhat,iftherearetwoQTL,itismorelikelythattheyarelocatedondierentchromosomes. 3and1 2ofthetotallengthofalinkagegroup.Unsurprisingly,theestimatedlociareveryclosedtoeachother.Besidesthat,bothadditiveanddominanteectsforthetwoQTLareveryclosed,whichfurthersupportoursuspicionthatatmostoneQTLlocatedoneachchromosome.Statistically,accordingtotheBayesianmodelselectioncriterion,BayesfactoronthelogarithmicscaleofM1vs:M2wasestimatedtobe15:76,whichissubstantiallysupportM1. 64

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EstimatedmarginalposteriordistributionoftheQTLlocationonmousechromosome6,7,and10respectively. correspondingto3possibilities.TheseareM201,the2QTLarelocatedonchromosome6andchromosome7;orM202,the2QTLarelocatedonchromosome6andchromosome10;orM203,the2QTLarelocatedonchromosome7andchromosome10.Ineachcase,thestartingvaluesforthelociwerechosetobes=40cM(s=1;2)forbothchromosomes. First,let'sconsidermodelM201.Therstlocus1wasestimatedat53:4cMbetweenmarker2andmarker3onchromosome6,andthesecondlocus1wasestimatedat43:4cMbetweenmarker2andmarker3onchromosome7.Theestimatedcurveparametersfor9genotypesandtheir95%condenceintervalsarepresentedintable 3-1 .Basedonthetheseestimatedparameters,growthcurvesfor9genotypescanbettedasshowningure 3-3 .Examiningtheplot,wecanseethat9growthcurvesarewellapart 65

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BayesianestimatesofgrowthcurvesfortheQTLgenotypesdetectedonmousechromosome6and7.Numbersintheparenthesesarethe95%equal-tailcondenceintervals. ParameterQTLgenotype 37.03(36.13,38.07)35.56(34.88,36.13)33.51(32.67,34.32)11.61(11.10,12.05)11.07(10.61,11.53)11.22(10.60,11.81)0.64(0.63,0.66)0.66(0.65,0.67)0.66(0.64,0.67) 35.54(34.95,36.30)35.42(34.94,36.10)32.92(32.15,33.42)11.47(11.07,11.84)11.18(10.79,11.42)11.36(11.03,11.76)0.66(0.65,0.67)0.65(0.64,0.66)0.65(0.64,0.66) 35.82(34.81,36.80)33.43(32.77,34.15)31.70(30.90,32.41)11.53(11.05,12.07)10.67(10.25,11.06)10.18(9.75,10.69)0.64(0.63,0.65)0.65(0.63,0.66)0.65(0.64,0.67) fromeachother,whichsuggestedatwo-QTLmodelmightbeappropriate.Also,theplotofmarginalposteriorsofthetwolociisillustratedingure 3-2 .Somewhatconsistentwiththeirrelativeeects,theQTLlocatedonchromosome7hasahigherpeek,andhenceisheavierconcentrated.Ontheotherhand,themarginalposteriordistributionfortheQTLlocatedonchromosome6hasamuchheaviertail. Moreover,when2QTLarepresented,inasimilarmannerasdescribedbefore,wecanpartitiontime-dependentgenotypicvalue,jw(t)(j=0;1;2;w=0;1;2),intotheoverallmean(),theadditiveanddominanteectsduetotwoQTL(a1;a2;d1;d2),aswellastheinteractionbetweenthetwoQTL(iaa;iad;ida;idd), ^(t)=1 9(22+21+20+12+11+10+02+01+00);(3{7) ^a1(t)=1 2(2202);(3{8) ^a2(t)=1 2(0200);(3{9) ^d1(t)=1 2(12+102^(t));(3{10) 66

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2(21+012^(t));(3{11) ^iaa(t)=1 2(22+002^(t));(3{12) ^iad(t)=1 2(21012^a1(t));(3{13) ^ida(t)=1 2(12102^a2(t));(3{14) ^idd(t)=11^d1(t)^d2(t)^(t);(3{15) Dynamicchangesinadditive,dominant,andinteractioneectsduetothetwoQTLaredisplayedingure 3-4 .WeobservedthattheQTLlocatedonchromosome7hasamuchlargeradditiveduringthewholegrowthprogress.ThedominanteectoftheQTLonchromosome6isverysmall,whichisalmostnegligible,whilethedominanteectofQTLonchromosome7isnegativeinasmallmagnitude.OnegreatadvantagebyusingourapproachisthatwecanenumeratemultipleQTLsimultaneously,andevaluatealltypesofinteractioneectsbetweentheseQTL.Asillustratedingure,,and,theinteractioneectsdoexists.Amongthem,thedominant-dominantinteractioneectisparticularlystronganditevenoverpowerstheadditiveeectoftheQTLlocatedonchromosome6.Thedominant-additiveinteractionisalsostrongandthedirectionchangesovertime. Comparedtotheresultsproducedbyttingasingle-QTLmodel,themarginalposteriordistributionsofthetwolociareconsiderablymoreconcentrated,henceresultinmuchnarrowercondenceintervalsfortheQTLlocations.Besidesthat,theestimatedposteriorsalsohavebettershapes,theyaremuchsmoother,andmoreimportantly,withonlyasinglemode.Wealsonoticedthatcondenceintervalsforcurveparametersareslightlynarrowerthanthecondenceintervalsproducedbyttingasingle-QTLmodel.TheBayesfactorofM1-1(asingleQTLislocatedonchromsome6)vs.M2,-1,(twoQTLarelocatedonchromosome6andchromosome7respectively)wasestimatedtobe3.48,andtheBayesfactorofM1-2(asingleQTLislocatedonchromsome7)vs.M2'-1,wasestimatedbe2.14.Thereby,wewouldsubstantiallyfavorM2'-1overasingle-QTLmodel. 67

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Estimatedmarginalposteriorsofthetwolocibasedonatwo-QTL-model. Also,weexploredotherpossibilitiesforatwo-QTLmodelbyttingM2'-2andM2'-3.Figure 3-2 illustratedtheestimatedmarginalposteriorsofthetwoloci,whichwereassumedtobelocatedonchromosome6and10oronchromosome7and10.Generally,theestimatedQTLlocationsareconsistentwiththoseyieldedfromttingthesingle-QTLmodel.Also,thelengthofthecondenceintervalsremainedaboutthesame,whichsuggestedthattheprecisionoftheestimateswasnotsignicantlyimproved.Resultsofestimatedcurveparametersforallgenotypesaredisplayedintable 3-2 and 3-3 .Forthese 68

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Fittedgrowthcurvesforthe9QTLgenotypesassumingtwoQTLarelocatedonmousechromosome6and7. Fittedgrowthcurvesfor9genotypesareshowedingure 3-5 and 3-6 inbothcases.WeobservedthattheestimationprecisionevendroppedalittlebitifM2'-2orM2'-3wasused.Furthermore,logarithmicscaledBayesfactorswereestimatedanddisplayedintable 3-4 ,andthesestatisticalevidencedidn'tsupportthese2modelseither.Inbothcases,wewereinfavorofasingle-QTLmodelovermodelM2'-2orM2'-3. 69

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Dynamicchangesoftheadditive,dominantandinteractioneectsduetothetwoQTLbasedonatow-QTL-model. Table3-2. BayesianestimatesofgrowthcurvesfortheQTLgenotypesdetectedonmousechromosome6and10.Numbersintheparenthesesarethe95%equal-tailcondenceintervals. ParameterQTLgenotype 37.03(34.39,37.72)36.03(34.97,37.25)34.70(33.19,35.94)11.84(11.03,12.74)11.59(10.80,12.13)11.39(10.66,12.16)0.67(0.64,0.69)0.65(0.63,0.66)0.66(0.64,0.69) 35.11(33.98,36.19)35.11(33.98,36.19)33.64(32.50,34.81)11.54(11.03,12.74)11.59(10.80,12.13)11.39(10.66,12.16)0.66(0.64,0.68)0.65(0.63,0.66)0.64(0.62,0.66) 34.47(32.87,36.08)31.93(30.96,33.03)32.47(31.01,33.87)10.93(10.18,11.75)10.70(10.14,11.22)10.73(9.97,11.52)0.64(0.61,0.66)0.66(0.64,0.68)0.65(0.63,0.68) 70

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Fittedgrowthcurvesforthe9QTLgenotypesassumingtwoQTLarelocatedonmousechromosome6and10. Table3-3. BayesianestimatesofgrowthcurvesfortheQTLgenotypesdetectedonmousechromosome7and10.Numbersintheparenthesesarethe95%equal-tailcondenceintervals. ParameterQTLgenotype 35.84(34.28,37.77)36.59(35.53,37.87)34.15(32.63,35.73)11.52(10.58,12.56)11.92(11.26,12.60)11.42(10.57,12.346)0.66(0.64,0.69)0.64(0.63,0.66)0.66(0.64,0.69) 36.13(34.71,37.43)34.77(33.69,35.45)33.62(32.56,35.57)11.52(10.79,12.28)11.11(10.40,11.91)10.90(10.16,11.62)0.65(0.63,0.68)0.65(0.63,0.68)0.66(0.64,0.67) 34.01(32.56,35.57)31.62(30.66,33.05)32.83(31.23,34.56)11.38(10.65,12.15)11.26(10.73,11.97)10.57(9.80,11.56)0.66(0.63,0.68)0.67(0.65,0.69)0.64(0.63,0.67) 71

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Fittedgrowthcurvesforthe9QTLgenotypesassumingtwoQTLarelocatedonmousechromosome7and10. Table3-4. EstimatedBayesfactorsonalogarithmicscalebetweendierentmodels ModelsbeingcomparedLogarithmicscaledBayesfactors 72

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AnF2populationwith600individualswassimulatedforachromosomesegmentwithlength100cMcoveredby11evenlyspacedmarkers.TwoQTLareplacedatboth24cMand72cMfromtherstmarkerontheleft-handside.Foraparticulardynamicphenotypictrait,saybodymass,tenmeasurementswithequidistanttimelagcanbesimulatedforeachindividual.Thesimulateddataisbasedonthefollowingassumptions.First,thesevectorsofmeasurementsareassumedtofollowmultivariatenormaldistributionswhosemean-vectorsaremodeledbythelogisticgrowthcurve 2{2 .Second,weassumethatbothQTLscontributetothisparticulardynamictraitbyaectingtheunderlyingparametersofgrowthcurves,hence9distinctsetsofcurvatureparametersj;j;j9j=1werepresetforeachofthe9QTLgenotypes.Finally,weassumethatthecovariancematrixiscommonacrossallindividualswithallgenotypes. Intherstsimulationscenario,truecurvatureparametersandthetruecovariancematrixwerepresetaccordingtothecorrespondingestimatesyieldedfromthemicedatainsection3.Inthissetting,theheritability(H2)wasestimatedtoberoughly0.1.SeveralpreliminaryrunsshowthatdespitefordierentinitialpointstheconstructedMarkovchaincanreachtoitsstationarydistributionafter10,000burn-incycles.Afterdiscarding 73

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Asmallscaledreplicatedsimulationshowedconsistentbehaviorofourapproach.Although10replicationsmightnotbesucientlylargetoallowaccurateestimationofstatisticpower,wecanstillgetageneralideaabouttheperformanceofourproposedmodel.Forall10replications,thetrueQTLlocationswerecapturedbytheircorresponding95%condenceintervals;andfor7times,theBayesfactorcomparingM1andM2wasinfavorofM2,i.e.theTwo-QTLmodel.Forthepurposeofillustration,resultsfromoneofthosesimulationsarepresentedhere.Therstlocus1wasestimatedat23:92cM,andthesecondlocus2wasestimatedat70:12cM.Resultsfromonesimulationfortheestimationsofallparametersaresummarized 3-5 .Wenoticedthatevenatsucharelativelylowheritabilitylevel,theprecisionofallestimatesfromourproposedmethodwasabletomaintainatasatisfactorylevel. 74

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Resultsfromasimulationstudy,wheres=2andH2=0:1.Numbersintheparenthesesarethe95%equal-tailcondenceintervals.Numbersintheboxbractsarethegivenvaluesoftheparameters. ParameterQTLgenotype 36.63[37.06](35.85,37.51)35.52[35.52](34.44,36.33)33.29[33.50](32.24,34.32)11.23[11.59](10.78,11.54)10.93[11.08](10.50,11.31)11.49[11.22](10.64,12.13)0.63[0.64](0.62,0.65)0.67[0.66](0.65,0.68)0.66[0.66](0.65,0.67) 35.66[35.57](34.53,36.61)35.53[35.46](34.92,36.16)32.15[32.84](31.09,33.04)11.49[11.22](11.20,12.02)11.01[11.14](10.71,11.32)11.19[11.38](10.75,11.60)0.66[0.66](0.64,0.67)0.64[0.65](0.65,0.65)0.66[0.65](0.65,0.68) 36.03[35.78](34.35,37.80)33.48[33.45](32.47,34.56)32.08[31.71](31.49,33.06)11.95[11.55](11.32,12.37)10.69[10.70](10.29,11.11)10.42[10.21](10.01,10.80)0.66[0.64](0.64,0.67)0.64[0.65](0.62,0.65)0.66[0.65](0.65,0.68)

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3-8 ,forall25simulatedindependentsamples.FortheQTLwithlargergeneticeects,about90%oftheseproleshadalargepeakwithin8cMaroundthegivenlocation.Ontheotherhand,fortheQTLwithsmallergeneticeects,noneoftheseprolesshowedpeakintheneighborhoodofthegivenlocation. Inanotherscenario,I'dliketoevaluatetheperformanceofourmodelwhentheheritabilityishigher.First,Ipresetthetruecovariancematrixasthecorrespondingestimatesyieldedfromthemicedatainsection3,thencurvatureparameterswerespeciedsothattheheritabilitywasestimatedtobe0:4.Again,sinceresultsfromseveralindependentreplicatedsimulationsarebroadlyconsistentwitheachother,resultsfromonlyoneofthesimulationwerepresented.PosteriordensitiesofthetwoQTLlocationaregiveningure 3-9 ,andtheresultingestimationsofallparametersaresummarizedTable 3-6 .Ingeneral,accuracyofallestimateswereimprovedwhenIincreasedtheheritability.Amongallparameters,locationsoftwoQTLshadthegreatestimprovement,theestimated95%condenceintervalswereaswideasonly1 4oftheestimated95%condenceintervalswhenheritabilitywas0.1.Finally,wecomparedttedgrowthcurvesandthegivencurvesforeachoftheninegenotypes.Asdisplayedin 3-10 ,ourmodelwasabletoprovideaperfectttothegrowthcurvesforallgenotypes. Insummary,ourproposedmethodprovidedmoreaccurateestimatestoallparametersofinterest,andamuchgreaterpowertoseparatecloselylinkedQTL.Mostimportantly,ourmodelnotonlyenableustoenumeratemultipleQTLalongthechromosome,butalsoawaytoevaluatethegeneticeectsofeachindividualQTLaswellastheinteractionbetweendierentQTL. 76

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Estimatedmarginalposteriordistributionsofthetwolocionasamechromosomeinasimulationstudy.Astothesettingsofthissimulation,weassumethattwoQTLcontrolbodymassgrowthtrajectoriesofmice,andtheheritabilityis0.1. 77

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TheproleoftheLRbetweenthefullandreduced(noQTL)modelestimatedfromtheSAD(1)modelforbodymassgrowthtrajectoriesinasimulationstudy.Astothesettingsofthissimulation,weassumethattwoQTLcontrolbodymassgrowthtrajectoriesofmice,andtheheritabilityis0.1. 78

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Estimatedmarginalposteriordistributionsofthetwolocionasamechromosomeinasimulationstudy.Astothesettingsofthissimulation,weassumethattwoQTLcontrolbodymassgrowthtrajectoriesofmice,andtheheritabilityis0.4. 79

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Fittedandgivengrowthcurvesforthe9QTLgenotypesinasimulationstudy,assumingtheheritabilityis0.4. 80

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Resultsfromasimulationstudy,wheres=2andH2=0:4.Numbersintheparenthesesarethe95%equal-tailcondenceintervals.Numbersintheboxbractsarethegivenvaluesoftheparameters. ParameterQTLgenotype 37.38[37.00](36.73,38.09)35.50[35.50](34.70,36.33)35.49[35.00](34.67,36.32)9.86[10.00](9.60,10.21)10.54[10.50](10.04,10.97)7.34[7.50](6.80,7.87)0.59[0.65](0.58,0.60)0.64[0.65](0.63,0.66)0.63[0.63](0.61,0.64) 35.34[35.50](34.68,36.01)34.39[34.00](33.75,34.80)34.75[35.00](34.09,35.27)8.19[8.00](7.87,8.52)12.14[12.00](11.87,12.42)9.9[10.00](9.63,10.27)0.67[0.65](0.65,0.68)0.55[0.55](0.54,0.56)0.72[0.71](0.71,0.73) 35.82[35.00](34.81,36.80)33.43[33.00](32.77,34.15)31.70[32.00](30.90,32.41)7.75[8.00](7.30,8.19)9.76[10.00](9.40,10.16)6.45[6.50](6.15,6.77)0.58[0.60](0.57,0.60)0.44[0.45](0.42,0.45)0.62[0.62](0.61,0.63)

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1 ][ 25 ][ 59 ][ 100 ].Throughanintegratedapproach,studiescanmovetocharacterizethedetailedandprecisepictureofthegeneticarchitectureofanybiologicalprocess.Byincorporatinggenetictestsintoagriculturalandbiomedicalresearchprograms,breedersmighthavemoredesirableopportunitiestoselectsuperiorgenotypesinpractice,andphysiciansmightimproveoutcomesforpatients. Inthischapter,IincorporatedBayesianapproachestomappinginteractiveorepistaticQTLthatgoverndynamicprocessesofaquantitativetrait.Thismodelhasbeenexaminedthroughsimulationstudies.Itsapplicationtoarealexampleindicatesthatthismodelwillbepracticallyuseful.Itcandetectsignicantgeneticvariantsthatcontrolbiologicalprocessesandprovidereasonablypreciseestimatesofthegeneticparameters. Thereareseveralwaysinwhichourmodelmaybeextended.Forsimplicity,ourpresentationisbasedontheinteractionbetweentwoQTLs.Itislikelythatthetwo-QTLinteractionmodelistoosimpletocharacterizegeneticvariantsforquantitativevariation.Withthefoundationforthetwo-QTLinteractionmodel,thismodelcanbeextendedtoincludethehigh-orderinteractionsamongmultipleQTLswhichareassociatedwiththephenotypicvariation.Ahigh-orderQTLepistaticmodelwillencountertheproblemofmanyparameterstobeestimated,butitcanbeexpectedtohavepotentialimplicationsforunderstandingtheoriginandevolutionofdevelopmentandthecontributionsofepistaticeectstoevolutionarychangesinthetheprocessofdevelopment. 82

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Inthelastchapterofthisdissertation,IwillpinpointseveralkeyareasinwhichBayesianapproachescanbeexpandedtobetterunderstandthegeneticsofdynamictraits.Someoftheseareasarebiological,somearestatisticalandmanyothersaremixed. a. Sigmoidcurvefortumorgrowth, b. Bi-exponentialequationsforHIVdynamics, c. SigmoidEmaxmodelsforpharmacodynamicresponse, d. Fourierseriesapproximationsforperiodiccellcycles, e. Biologicalthermaldynamics, d. Ordinaldierentialequationsforbiologicalclock, e. Fourierseriesapproximationforperiodiccellcycle. 83

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48 ][ 71 ].Statisticaltestsareperformedtomakeaninferenceaboutthesignicanceofmodelsforthemeanandcovariancestructures. Keystepsforconstructingageneralframeworkforthegeneticanalysisofbiologicalandbiomedicalprocesseshavebeenachievedmostlythroughparametricapproaches.Infact,wecanalsoconstructsuchaframeworkwithinnonparametricandsemiparametriccontextstomodelboththemeanandcovariancestructures. Whennofunctionalrelationshipexists,nonparametricanaloguesofparametricmodelingcanbeused.Nonparametricregressionmethodsusingkernelestimatorshavebeenconsideredforthemeanstructureofgrowthcurvedata.Allofthesenonparametricapproacheshaveincommonthattheunknownmeanresponsecurveovertimeisestimatedbysmoothingtherawdata,andtimeistheonlyexplanatoryvariable.[ 61 ]appliednonparametricregressionmethodstolongitudinaldatabutwithoutconsideringaserialcorrelationstructure. AnapproachbasedonB-splinebasisfunctionscanbeusedfornonparametricregressiontting.TheB-splineapproachconstructscurvesfrompiecesoflowerdegreepolynomialssmoothedatselectedpointed(knots).Brownetal.[ 7 ]extendedtheB-splinebasistomodelmultiplelongitudinalvariables.TheideaofB-splinecurvettingwillbeincorporatedintothefunctionalmappingmodel,aimedtoincreasethebreadthoftheuseoffunctionalmappinginsolvingpracticalgeneticproblems.Otherapplicabledimensionreductionmethodsincludefunctionalprincipalcomponentsanalysisthatisdata-drivenandletsdataspeakforthemselves. 84

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18 ]usedkernel-weightedlocallinearregressionsmoothingofsamplevariogramsordinatesandofsquaredresidualstoprovideanonparametricestimatorforthecovariancestructurewithoutassumingstationarity.Inaddition,theyusedthevalueoftheestimatorasadiagnostictoolbutdidnotstudytheuseoftheestimatorinmoreformalstatisticalinferenceconcerningthemeanproles.Wang[ 97 ]usedkernelestimatorstoestimatecovariancefunctionsinanonparametricway.Hisonlyassumptionwastohaveafullyunstructuredsmoothcovariancestructure,togetherwithaxedeectsmodel.Theproposedkernelestimatorwasconsistentwithcompletebutirregularlyspacedfollow-ups,orwhenthemissingmechanismisstronglyignorableMAR(missingatrandom). ZegerandDiggle[ 116 ]andMoyeedandDiggle[ 60 ]studiedasemiparametricmodelforlongitudinaldatainwhichthecovariatesenteredparametricallyandonlythetimeeectenterednonparametrically.Totthemodel,theyextendedtolongitudinaldatathebackttingalgorithmofHastieandTibshiraniforsemiparametricregression.Basedontheideausedinsemiparametricmeanresponsemodels,wecanextendthesemiparametricmodelsforfunctionalmappingtoallowformoreexibilitiesinbothmeanandcovariancestructures. 85

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4.4.1ModelSelection AfeasiblealternativewaytodeterminethenumberofQTLwouldbeanapplicationofthereversible-jumpalgorithm.Theoretically,ifweregardthenumberofQTLs,i.e., 86

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33 ]canbeutilizedtodeterminethevalueofs.Fora`deathmove'inareversiblejumpMCMC,aselectedQTLlociisremovedwithaparticularprobability(d)andthevalueofsisdeductedtos1;ontheotherhand,smayalsobeincreasedtos+1ina`birthmove'.Inabirthmove,anewlygeneratedcandidateQTLlociwillbeincludedinthemodelforacertainprobability(b),andanewsetofcurvatureparametersneedtobeproposed.However,adirectimplementationofreversiblejumpMCMCalgorithmmayencountergreatdiculties.Thisisdueto1)thedimensionofparameterspacechangegreatlywhenschanges,evenwhensisassmallas2)computingtimeisusuallylengthybecauseofslow-mixing3)convergenceofthealgorithmstronglydependedonagoodpropose.Hence,thereisalotofroomleftforimprovement.First,Descending-dimensionmethodscanbeexploredtodecreasethedimensionofparameterspaceespeciallywhensislarge.Second,theacceptanceprobabilityofabirthstepcanbeadjustedtopreventtoomuchjumpingbetweenmodels. 87

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42 ][ 50 ][ 65 ][ 69 ][ 70 ][ 107 ][ 115 ]itisnowpossibletodetailthegeneticarchitectureofacomplex,quantitativelyinheritedtrait.Inherseminalreview,Mackay[ 55 ]denedtheoverallpictureofthegeneticarchitectureofacomplextraitintermsofitscomposedelements,i.e.,thenumberofgenesinvolvedandtheirfrequenciesandpleiotropiceects,gene-gene(epistatic),gene-sexandgene-environmentinteractions.Inafollow-upreview,Mackayandcolleague[ 56 ]furtherdocumentedtheimportanceoftheseelementsincreatingandmaintainingthegeneticvariationofaspecicquantitativetraitinapopulation.Wuandcolleagues([ 54 ]and[ 107 ])pioneeredageneralstatisticalframeworkforfunctionalmappingthatcanbeusedtounravelthegeneticarchitectureofdynamiccomplextraits,asdenedbyMackay[ 56 ]. Inthisdissertation,IdevelopedageneralBayesianframeworkforfunctionalmappingofcomplexdynamictraits.Comparedwithtraditionalmaximumlikelihood-basedfunctionalmapping,theBayesianapproachdisplaysseveralfavorablepropertiesinparameterestimationandthecharacterizationofthenumberoftheunderlyinggenesorQTLsforthetrait.TheunderstandingoftheQTLnumberinvolvedisakeytodrawacomprehensivepictureofthegeneticnetworkandregulationthatdeterminethedevelopmentalshapeandprocessofacomplextrait.Withtheavailabilityofacompletereferencesequenceoftheentiregenomeforvariousspeciesandcontinuingadvancesin 88

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89

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TianLiuwasbornonApril1,1979,inShanghai,China.ShemajoredinappliedmathematicsatFudanUniversity(Shanghai,China),andobtainedherBachelorofSciencein2001.During2001and2002sheworkedasahighschoolmathteacherinKingsfordCommunitySchool,London,U.K.From2002topresent,shehasbeenstudyingintheDepartmentofStatisticsattheUniversityofFlorida.ShereceivedherMasterofScienceinstatisticsin2004.ShereceivedherPh.D.in2007.ShewillworkintheGenomeInstituteofSingaporeasaresearchscientist. 98