Citation
Statistical Model for Mapping Quantitative Trait Loci in Autotetraploid

Material Information

Title:
Statistical Model for Mapping Quantitative Trait Loci in Autotetraploid
Creator:
Li, Jiahan
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
Publication Date:
Language:
english
Physical Description:
1 online resource (28 p.)

Thesis/Dissertation Information

Degree:
Master's ( M.S.Stat.)
Degree Grantor:
University of Florida
Degree Disciplines:
Statistics
Committee Chair:
Wu, Rongling
Committee Members:
Berg, Arthur
Graduation Date:
12/19/2008

Subjects

Subjects / Keywords:
Additive effects ( jstor )
Chromosomes ( jstor )
Gametes ( jstor )
Genotypes ( jstor )
Modeling ( jstor )
Polyploidy ( jstor )
Quantitative trait loci ( jstor )
Sample size ( jstor )
Statistical models ( jstor )
Tetraploidy ( jstor )
Statistics -- Dissertations, Academic -- UF
autotetraploid, qtl
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Statistics thesis, M.S.Stat.

Notes

Abstract:
Autotetraploids that include many plant species, such as potato, sugarcane and rose, are of paramount importance to agricultural production and biological research. Quantitative trait locus (QTL) mapping in autotetraploids is challenged by their unique cytogenetic properties, such as preferential pairing and double reduction. In this article, we present a statistical model for mapping autotetraploid QTLs by considering these cytogenetic properties. Our model is built in the mixture model-based framework and implemented with the EM algorithm. The model allows simultaneous estimation of QTL positions, QTL e?ects, the chromosomal pairing factor and the degree of double reduction as well as the assessment of the estimation precision of these parameters. Computer simulation is used to examine the statistical properties of the model. Our model will provide a useful tool for QTL mapping in autotetraploids that undergo double reduction. ( en )
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.
Thesis:
Thesis (M.S.Stat.)--University of Florida, 2008.
Local:
Adviser: Wu, Rongling.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-12-31
Statement of Responsibility:
by Jiahan Li.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright Li, Jiahan. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Embargo Date:
12/31/2010
Resource Identifier:
430115406 ( OCLC )
Classification:
LD1780 2008 ( lcc )

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IamgratefultomyadvisorDr.RonglingWu.Withouthim,thisthesiswouldnothavebeenpossible.Ithankhimforhispatienceandencouragementthatcarriedmeonthroughdiculttimes,andforhisinsightsandsuggestionsthathelpedtoshapemyresearchskills.Ialsothanktherestofmythesiscommitteemembers:Dr.ArthurBergandDr.MyronChang.Theirvaluablefeedbackhelpedmetoimprovethethesisinmanyways. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 6 ABSTRACT ........................................ 7 CHAPTER 1INTRODUCTION .................................. 8 2THEMODEL ..................................... 10 2.1GeneticDesign ................................. 10 2.2ExtensiontoPartiallyInformativeMarkers .................. 13 2.3QuantitativeGeneticModel .......................... 13 2.4Mixture-BasedLikelihoodandEMAlgorithm ................ 15 2.5HypothesisTesting ............................... 19 3COMPUTERSIMULATION ............................ 20 4DISCUSSION ..................................... 25 REFERENCES ....................................... 26 BIOGRAPHICALSKETCH ................................ 28 5

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Table page 3-1SampleSizen=100.EstimatesofDoubleReduction,RecombinationFraction,OverallMean,AdditiveEects,andDominantEectsfromFullyInformativeMarkers. ........................................ 21 3-2SampleSizen=200.EstimatesofDoubleReduction,RecombinationFraction,OverallMean,AdditiveEects,andDominantEectsfromFullyInformativeMarkers. ........................................ 22 3-3SampleSizen=400.EstimatesofDoubleReduction,RecombinationFraction,OverallMean,AdditiveEects,andDominantEectsfromFullyInformativeMarkers. ........................................ 23 3-4Samplesizen=400.EstimatesofDoubleReduction,RecombinationFraction,OverallMean,AdditiveEects,andDominantEectswithPartiallyInformativemarkers. ........................................ 24 6

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1995 )( 1998 )( 2001 )( 1998 )Statisticalmodelsforlinkageanalysisandmapconstructionthatconsideruniquebiologicalpropertiesofpolyploidshavebeendeveloped.( 1998 )( 1999 )( 2001 )Forbivalentpolyploids,Wuetal.( 2002 )incorporatedtheso-calledchromosomalpairingpreference( 1994 )intothelinkageanalysisframework,aimedtoincreasethebiologicalmeaningoflinkagemappingmodels.Therehavebeenseveralstatisticalmodelsdevelopedtomapquantitativetraitloci(QTLs)inbivalentpolyploids.( 2002 )Thereisalsoagroupofpolyploids,calledmultivalentpolyploids,inwhichchromosomespairamongmorethantwohomologouscopiesatmeiosis,ratherthanonlytwocopiesaslikeinbivalentpolyploids.Theconsequenceofthismultivalentpairingistheformationofdoublereduction,i.e.,twosisterchromatidsofachromosomesortintothesamegamete.( 1929 )Fisher( 1947 )proposedaconceptualmodelforcharacterizingtheprobabilitiesof11dierentmodesofgameteformationforaquadrivalentpolyploidintermsoftherecombinationfractionbetweentwodierentlociandtheirdoublereductions.Wuetal.( 2001a )usedFisher'smodeltoderivetheEMalgorithmfortheestimationofthelinkagebetweenfullyinformativemarkers.WuandMa( 2004 )extendedthismodelintoanalyzeanytypeofmarkers,regardlessoftheirinformativenessanddominantorcodominantnature.ThesignicantadvantageofthemodelsbyWuandcolleaguesdirectlyliesintheirgenerality,exibilityandrobustness. 8

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1947 )11classicationsofgameteformation.ThemodelallowstheestimationandtestofnotonlytheQTL-markerlinkage,butalsotheextentofdoublereductionoftheQTL.Becauseoftheinherentcomplexityofclassicationanalysesofgameteformation,wewillfocusonthemodelingandanalysisofone-marker/one-QTLassociations,butbothfullyinformativemarkersandpartiallyinformativemarkersareconsidered.Atwo-stagehierarchicalmodelisderivedtoestimatetheprobabilitiesofgameteformationmodesandthereforedoublereductionintheupperhierarchyandestimatethemarker-QTLrecombinationfractioninthelowerhierarchywithinthemaximumlikelihoodcontextimplementedwiththeEMalgorithm.Computersimulationstudiesareperformedtoinvestigatestatisticalpropertiesofourmodelanditsanalyticalandbiologicalmerits. 9

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1947 )classiedthese136formationmechanismsinto11gametemodes.Ofthese11gametemodes,however, 10

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2001b )inmatrixformexpressedasQ1Q1Q2Q2Q3Q3Q4Q4Q1Q2Q1Q3Q1Q4Q2Q3Q2Q4Q3Q4g=M1M1M2M2M3M3M4M4M1M2=M2M1M1M3=M3M1M1M4=M4M1M2M3=M3M2M2M4=M4M2M3M4=M4M3266666666666666666666666666666641 4g11 12g21 12g21 12g21 12g51 12g51 12g51 12g61 12g61 12g61 12g21 4g11 12g21 12g21 12g51 12g61 12g61 12g51 12g51 12g61 12g21 12g21 4g11 12g21 12g61 12g51 12g61 12g51 12g61 12g51 12g21 12g21 12g21 4g11 12g61 12g61 12g51 12g61 12g51 12g51 12g31 12g31 12g41 12g41 6g71 24g81 24g81 24g81 24g81 6g91 12g31 12g41 12g31 12g41 24g81 6g71 24g81 24g81 6g91 24g81 12g31 12g41 12g41 12g31 24g81 24g81 6g71 6g91 24g81 24g81 12g41 12g31 12g31 12g41 24g81 24g81 6g91 6g71 24g81 24g81 12g41 12g31 12g41 12g31 24g81 6g91 24g81 24g81 6g71 24g81 12g41 12g41 12g31 12g31 6g91 24g81 24g81 24g81 24g81 6g737777777777777777777777777777775; 2{8 ),weseethatthereisno,oneandtworecombinanteventsinthecells(g1),(g3;g5)and(g2;g4;g6;g9),respectively.Thecells(g7)and(g8)areeachamixtureoftwodierentgameteformationmechanismsorcongurations(AandB),i.e.,g7=g7A+g7Bandg8=g8A+g8B,withrelativeproportionsdeterminedbyr.Becausedierentcongurationscontaindierentnumbersofrecombinationevents,theexpectednumberofrecombinationeventsineachcell,i.e.,anobservablegametegenotype,shouldbetheweightedaverageofthenumberofrecombinationeventsforeachconguration.Wuetal.( 2001b )usedamatrixformtocounttheexpectednumberofrecombinationeventsforeachobservablegamete 11

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Basedonmatrices( 2{8 )and( 2{2 ),theexpressionsforthefrequenciesofdoublereduction(and)andtherecombinationfractionrcanbeexpressedintermsofgias (2{5) (2{6) 2[g3+g5+2(g2+g4+g6+g9)+2g7+(1+)g8] (2{7) 12

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4g11 12g21 12g21 12g21 12g51 12g51 12g51 12g61 12g61 12g6g2 13

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whereistheoverallmean,a1,a2,anda3aretheadditivegeneticeectsofallelesQ1,Q2,andQ3relativetoalleleQ4,andd12,d13,d14,d23,d24,andd34arethedominantgeneticeectsduetointeractionsbetweendierentallelesQ1andQ2,Q1andQ3,Q1andQ4,Q2andQ3,Q2andQ4,andQ3andQ4,respectively. 14

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4(11+22+33+44); 4(311223344); 4(322113344); 4(333112244); 4(333+3441122); 4(322+3441133); 4(322+3331144); 4(311+3442233); 4(311+3332244); 4(311+3223344): wherej1j4jiistheindicatorvariablethatisdenedas1ifindividualihasaQTLgenotypej1j2(j1j2=Q1;Q2;Q3;Q4),and0otherwise,j1j2isthegenotypicvalueofQTLgenotypej1j2asdenedinequation( 2{9 ),andeiistheresidualerrorassumedtobenormallydistributedwithmeanzeroandvariance2.Weusetodenotetheunknownvector(11;22;33;44;12;13;14;23;24;34;2).ForaQTLmappingexperiment,markergenotypesareobservable.Letnl1l2betheobservationofmarkergenotypel1l2(l1l2=M1;M2;M3;M4).Thelikelihoodofthephenotypic(y)andmarkerdata(M)isconstructed,withinthemixturemodelframework, 15

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wherej1j2jl1l2istheconditionalprobabilityofQTLgenotypej1j2givenmarkergenotypel1l2,andfj1j2(yi)isassumedtofollowanormaldistributionwithmeanj1j2andvariance2.Priorconditionalprobabilityj1j2jl1l2iscalculatedasthefrequencyofjointmarker-QTLgenotypel1l2j1j2,expressedintermsofninegprobabilitiesinmatrix( 2{2 ),dividedbythefrequencyofmarkergenotypel1l2.Markergenotypefrequenciesare=4foreachofdoublereductiongametesM1M1,M2M2,M3M3,andM4M4,and(1)=6foreachofnon-doublereductiongametesM1M2,M1M3,M1M4,M2M3,M2M4,andM3M4.Theestimatesofunknownparametersthatmaximizethelikelihood( 2{21 )canbeobtainedbyimplementingtheEMalgorithm.InstepE,wecalculatetheposteriorprobabilityofaQTLgenotypegivenaspecicmarkergenotypeofindividualiby j1j2jl1l2i=j1j2jl1l2fj1j2(yi) 16

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Basedon 2{8 ,theMLEsofg1;g2;;g9intheproblemofpartiallyinformativemarkerscanbeexpressedinasimilarway.Theseleadtotheestimatesofthefrequenciesofdoublereduction ^=^g1+^g2+^g3+^g4=1 ^=^g1+^g2+^g5+^g6; ^r=1 2[^g3+^g5+2(^g2+^g4+^g6+^g9)+2^g7+(1+)^g8]: ThegenotypicvalueofQTLgenotypej1j2andresidualvarianceareestimatedby ^j1j2=Pni=1PM4l1=M1PM4l2=M1j1j1jl1l2iyi ^2=1 TheiterationisrepeatedbetweentheEstep,equations( 2{3 ),( 2{4 ),and( 2{22 ),andMstep,equations( 2{23 ){( 2{28 ),untilstableestimatesareobtained.Thestableestimatesarethemaximumlikelihoodestimates(MLEs)ofparameters. 18

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Thedierencebetweenthelog-likelihoodfunctionsunderthenullandalternativehypothesesarecalculated.Butthedistributionofthislog-likelihoodratio(LR)isnotknownbecauseoftheviolationofregularityconditionsforthemixturemodel.Forthisreason,acommonlyusedempiricalapproachbasedonpermutationtestsbyreshuingtherelationshipsbetweenthemarkergenotypesandphenotypes( 1994 )isusedtodeterminethecriticalthreshold,inordertojudgewhetherthereisaQTLforthetrait.AfterasignicantQTLisdetected,thenexthypothesisisabouttheadditivegeneticeectoftheQTL.Thiscanbetestedbyformulatingthenullhypothesis,H0:a1=a2=a3=0;underwhichtheestimatesofgenotypicvaluesofQTLgenotypescanbeobtainedwiththeEMalgorithmasdescribedabove,butposingthreeconstraintsderivedfromequations( 2{11 ),( 2{12 )and( 2{13 ).Similarly,thedominantgeneticeectscanbetestedwiththenullhypothesis,H0:d12=d13=d14=d23=d24=d34=0;withestimatesofgenotypicvaluesundertheconstraintsderivedfromequations( 2{14 ){( 2{19 ).Allthesegeneticeectscanbetestedindividually. 19

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2{9 )whentheoverallmeanisassignedas1,andtheadditiveanddominanteectsassignedasa1=a2=a3=0:6andd12=d13=d14=d23=d24=d34=0:5.TheerrorvarianceisdeterminedaccordingtotheheritabilityofH2=0:1and0.4,respectively.ThemeansoftheMLEsofunknownparametersandtheirstandarderrorsbasedon1000simulationreplicatesareillustratedintable 3-1 ,table 3-2 ,table 3-3 andtable 3-4 .Ourmodelprovidesreasonableestimatesofallthemodelparameters.TheprecisionoftheMLEs,asassessedbythestandarderrors,increasesforalltheparameterswithincreasedsamplesize,increaseddegreeoflinkage,andincreasedheritability.Dierentdegreesofdoublereductioncanbepreciselyestimated. 20

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SampleSizen=100.EstimatesofDoubleReduction,RecombinationFraction,OverallMean,AdditiveEects,andDominantEectsfromFullyInformativeMarkers. 0:30:250:10.2950.3140.0470.1160.0372.2283.007-1.9532.375-2.661-2.107(SD)(SD)(0.047)(0.120)(2.339)(2.651)(2.496)(4.013)(4.237)(4.499)(4.349)(4.453)(4.407)0:30:250:40.3010.2510.5560.4960.5460.6770.6250.3130.4820.1810.138(SD)(SD)(0.048)(0.098)(0.986)(1.003)(1.133)(1.744)(1.861)(2.143)(1.947)(2.103)(1.931)0:30:050:10.2970.1420.6590.5800.5121.2161.130-0.0361.2860.119-0.084(SD)(SD)(0.041)(0.114)(1.330)(1.293)(1.303)(1.967)(2.208)(2.301)(2.129)(2.193)(2.102)0:30:050:40.2960.0570.6420.6430.5330.4450.5620.4280.6010.4880.594(SD)(SD)(0.045)(0.051)(0.461)(0.581)(0.497)(0.868)(0.787)(0.851)(0.973)(0.762)(0.687)0:150:250:10.1500.3050.2440.8210.9451.2321.3340.1670.660-0.150-0.373(SD)(SD)(0.034)(0.155)(1.800)(1.794)(1.965)(2.707)(2.645)(2.819)(2.820)(2.658)(2.614)0:150:250:40.1520.2390.5220.5480.5980.6450.4840.4490.4920.3820.230(SD)(SD)(0.031)(0.097)(0.777)(0.812)(0.794)(1.082)(1.166)(1.100)(1.203)(1.227)(1.238)0:150:050:10.1440.1680.4110.5400.9301.6161.2420.2711.033-0.037-0.313(SD)(SD)(0.035)(0.127)(2.103)(1.962)(1.635)(2.854)(2.698)(2.488)(2.452)(3.147)(2.700)0:150:050:40.1430.0570.3720.7360.5450.6850.9650.6210.6770.2820.453(SD)(SD)(0.035)(0.058)(0.877)(0.759)(0.775)(1.192)(1.148)(1.085)(1.085)(1.167)(1.147)0:050:250:10.0520.3710.3560.3000.4552.2392.178-0.6912.031-0.426-0.588(SD)(SD)(0.022)(0.180)(2.122)(2.576)(2.392)(3.425)(3.478)(3.104)(2.787)(2.998)(3.563)0:050:250:40.0510.2530.4400.4060.4581.2431.2020.5001.2240.5530.362(SD)(SD)(0.022)(0.111)(1.067)(1.003)(1.105)(1.522)(1.605)(1.341)(1.541)(1.483)(1.482)0:050:050:10.0520.1930.0561.0120.3621.5521.977-0.0651.068-1.077-0.438(SD)(SD)(0.025)(0.136)(2.119)(2.537)(2.121)(3.110)(2.718)(3.191)(3.427)(3.354)(3.391)0:050:050:40.0520.0640.5250.3610.3371.1191.1750.1931.2550.3030.351(SD)(SD)(0.024)(0.059)(1.005)(0.959)(1.076)(1.480)(1.482)(1.542)(1.636)(1.473)(1.285)

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SampleSizen=200.EstimatesofDoubleReduction,RecombinationFraction,OverallMean,AdditiveEects,andDominantEectsfromFullyInformativeMarkers. 0:30:250:10.3020.3130.3970.2180.1262.3032.553-1.6492.272-1.471-1.268(SD)(SD)(0.031)(0.093)(1.708)(1.697)(1.443)(2.844)(3.144)(3.576)(3.048)(3.454)(3.526)0:30:250:40.3010.2340.5820.5530.6770.4930.3170.3400.6460.4590.262(SD)(SD)(0.034)(0.085)(0.781)(0.730)(0.727)(1.432)(1.402)(1.438)(1.416)(1.435)(1.362)0:30:050:10.3020.0930.5450.5510.6600.7390.7030.2060.9240.1140.089(SD)(SD)(0.032)(0.093)(0.789)(0.829)(0.753)(1.432)(1.561)(1.309)(1.355)(1.453)(1.486)0:30:050:40.2970.0470.5960.5850.5740.5220.5310.4640.5090.4200.530(SD)(SD)(0.032)(0.037)(0.362)(0.346)(0.342)(0.570)(0.645)(0.666)(0.611)(0.577)(0.569)0:150:250:10.1520.2770.6920.5630.5870.9530.968-0.2321.224-0.034-0.033(SD)(SD)(0.028)(0.117)(1.168)(1.225)(1.215)(1.866)(1.930)(1.862)(1.900)(1.780)(1.692)0:150:250:40.1470.2340.5810.6750.5800.4200.5510.5610.3860.4840.617(SD)(SD)(0.024)(0.078)(0.520)(0.471)(0.530)(0.838)(0.724)(0.775)(0.822)(0.745)(0.736)0:150:050:10.1490.1400.4750.5480.7231.2320.9850.2460.8630.048-0.325(SD)(SD)(0.024)(0.106)(1.251)(1.257)(1.479)(1.814)(1.719)(1.990)(1.968)(1.868)(1.824)0:150:050:40.1490.0550.6910.5730.5580.3940.3800.4780.5360.4850.523(SD)(SD)(0.026)(0.047)(0.558)(0.511)(0.585)(0.684)(0.799)(0.800)(0.862)(0.754)(0.745)0:050:250:10.0490.3260.7500.3250.5571.2351.190-0.4201.392-0.127-0.331(SD)(SD)(0.015)(0.143)(2.486)(2.232)(1.868)(2.857)(2.619)(3.332)(3.113)(3.023)(2.775)0:050:250:40.0480.2410.6190.3450.5280.8850.6580.2340.8790.4980.242(SD)(SD)(0.013)(0.075)(1.038)(1.037)(0.932)(1.365)(1.305)(1.282)(1.317)(1.190)(1.276)0:050:050:10.0500.1960.4900.2940.6121.7551.435-0.4871.713-0.146-0.379(SD)(SD)(0.015)(0.127)(1.941)(2.134)(2.157)(2.902)(2.792)(2.747)(2.872)(2.758)(2.855)0:050:050:40.0490.0590.6090.3510.6970.9520.6460.5480.9080.7370.412(SD)(SD)(0.017)(0.049)(0.978)(0.988)(0.967)(1.130)(1.527)(1.419)(1.285)(1.279)(1.001)

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SampleSizen=400.EstimatesofDoubleReduction,RecombinationFraction,OverallMean,AdditiveEects,andDominantEectsfromFullyInformativeMarkers. 0:30:250:10.3000.3000.1820.4350.2081.8002.002-1.2361.546-1.574-1.318(SD)(SD)(0.023)(0.082)(1.405)(1.400)(1.074)(2.357)(2.429)(2.829)(2.130)(2.744)(2.906)0:30:250:40.3010.2310.5850.5570.6030.5080.4510.3610.4940.4630.309(SD)(SD)(0.023)(0.078)(0.552)(0.497)(0.532)(1.050)(0.984)(0.977)(1.115)(1.060)(0.964)0:30:050:10.2980.0970.5770.6460.6290.7040.8180.2450.6890.2020.208(SD)(SD)(0.023)(0.077)(0.603)(0.637)(0.580)(1.067)(1.138)(1.206)(1.028)(1.120)(1.071)0:30:050:40.2990.0500.6260.5530.6390.5250.4420.4580.5070.5750.414(SD)(SD)(0.025)(0.027)(0.245)(0.253)(0.250)(0.388)(0.435)(0.405)(0.403)(0.388)(0.350)0:150:250:10.1500.2610.6410.4630.6110.9750.8600.1920.9950.2650.202(SD)(SD)(0.016)(0.135)(0.684)(0.904)(0.737)(1.372)(1.476)(1.423)(1.286)(1.458)(1.353)0:150:250:40.1500.2500.5840.6240.5650.4950.5520.4790.5160.4520.424(SD)(SD)(0.017)(0.060)(0.329)(0.358)(0.326)(0.511)(0.526)(0.510)(0.558)(0.478)(0.419)0:150:050:10.1500.1130.5830.5830.7060.7400.7050.2180.6670.1700.058(SD)(SD)(0.017)(0.095)(0.866)(0.846)(0.838)(1.341)(1.254)(1.322)(1.527)(1.272)(1.218)0:150:050:40.1500.0490.5260.6540.6070.5310.5720.5470.4060.4250.482(SD)(SD)(0.017)(0.034)(0.307)(0.382)(0.366)(0.557)(0.511)(0.551)(0.516)(0.506)(0.483)0:050:250:10.0500.3220.5160.7260.6291.2561.3270.1871.026-0.191-0.037(SD)(SD)(0.010)(0.113)(1.819)(1.626)(1.773)(2.170)(2.180)(2.155)(2.213)(2.437)(2.459)0:050:250:40.0500.2500.6100.6520.5510.4040.5620.4810.4800.4570.545(SD)(SD)(0.012)(0.057)(0.664)(0.680)(0.624)(0.873)(0.789)(0.867)(0.916)(0.909)(0.991)0:050:050:10.0500.1670.6560.3430.7231.1560.796-0.1641.1470.111-0.211(SD)(SD)(0.011)(0.109)(1.548)(1.553)(1.581)(2.032)(2.084)(2.157)(2.213)(1.997)(2.140)0:050:050:40.0510.0540.5910.6080.6350.5120.4590.5180.4690.4770.450(SD)(SD)(0.011)(0.037)(0.637)(0.624)(0.698)(0.771)(0.895)(0.894)(0.931)(0.800)(0.770)

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Samplesizen=400.EstimatesofDoubleReduction,RecombinationFraction,OverallMean,AdditiveEects,andDominantEectswithPartiallyInformativemarkers. 0:30:250:10.30.2180.8860.450.450.2550.2550.2590.880.7740.774(SD)(SD)(0.038)(0.097)(0.677)(0.49)(0.49)(0.763)(0.763)(1.094)(1.394)(0.852)(0.852)0:30:250:40.30.2380.5620.5780.5780.3460.3460.5190.1820.3040.304(SD)(SD)(0.035)(0.111)(0.434)(0.296)(0.296)(0.47)(0.47)(0.588)(0.816)(0.503)(0.503)0:30:050:10.2970.0980.9211.2431.2430.550.550.8380.8781.4991.499(SD)(SD)(0.039)(0.089)(0.608)(0.673)(0.673)(1.03)(1.03)(1.2)(2.467)(1.124)(1.124)0:30:050:40.2990.070.6110.5540.5540.6230.6230.4840.5950.5770.577(SD)(SD)(0.04)(0.081)(0.298)(0.27)(0.27)(0.402)(0.402)(0.57)(0.97)(0.625)(0.625)0:150:250:10.1460.250.5610.6420.6420.5420.5420.5430.3680.520.52(SD)(SD)(0.032)(0.111)(0.547)(0.508)(0.508)(0.64)(0.64)(1.02)(1.395)(0.805)(0.805)0:150:250:40.150.2590.5940.5540.5540.5720.5720.5190.3860.4010.401(SD)(SD)(0.031)(0.099)(0.401)(0.325)(0.325)(0.446)(0.446)(0.7)(1.054)(0.651)(0.651)0:150:050:10.1540.0880.6350.6450.6450.5590.5590.6490.320.6020.602(SD)(SD)(0.03)(0.093)(0.479)(0.363)(0.363)(0.474)(0.474)(0.768)(1.204)(0.753)(0.753)0:150:050:40.1510.0810.6130.4570.4570.650.650.4280.550.3950.395(SD)(SD)(0.031)(0.092)(0.341)(0.327)(0.327)(0.372)(0.372)(0.711)(0.955)(0.516)(0.516)0:050:250:10.0540.2831.2581.4731.4730.4860.4861.8680.9232.5342.534(SD)(SD)(0.019)(0.197)(1.722)(1.423)(1.423)(1.963)(1.963)(2.848)(3.577)(2.291)(2.291)0:050:250:40.0510.2510.7130.7060.7060.4840.4840.8350.3530.9490.949(SD)(SD)(0.021)(0.13)(0.874)(0.655)(0.655)(0.707)(0.707)(1.357)(1.813)(1.098)(1.098)0:050:050:10.0530.1531.21.6591.6590.3270.3272.2270.0652.2842.284(SD)(SD)(0.017)(0.141)(1.359)(1.042)(1.042)(1.533)(1.533)(2.147)(3.063)(2.018)(2.018)0:050:050:40.0520.0810.7150.5680.5680.5850.5850.570.5750.6470.647(SD)(SD)(0.018)(0.088)(0.662)(0.531)(0.531)(0.741)(0.741)(1.057)(1.374)(0.829)(0.829)

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2001b )( 2004 )itisimportanttoincorporatedoublereductionintoaQTLmappingframework.OurmodelprovideapowerfultoolforQTLmappingandunderstandingthegeneticcontrolofaquantitativetraitinanautotetraploid.Ourmodelmadeuseof11dierentclassicationsoftwo-locusgameteformations,derivedbyFisher,( 1947 )duringtetraploidmeiosisandhasproventobepowerfulforsimultaneousestimationofthefrequenciesofdoublereductionandtherecombinationfractionbetweendierentloci.Wewillbeinabetterpositiontounderstandthegeneticdierentiationamongpolyploidgenomesandcharacterizethegeneticarchitectureofquantitativelyinheritedtraitsforthisuniquegroupofspecies. 25

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MynameisJiahanLi.IwasborninLuoyang,China.AftergraduatingfromShanghaiJiaotongUniversity,IcametoDepartmentofStatistics,UniversityofFlorida. 28