Table Of Content

Shizhong Xu Quantitative Genetics Quantitative Genetics Shizhong Xu Quantitative Genetics ShizhongXu DepartmentofBotanyandPlantSciences UniversityofCalifornia,Riverside Riverside,CA,USA ISBN978-3-030-83939-0 ISBN978-3-030-83940-6 (eBook) https://doi.org/10.1007/978-3-030-83940-6 #SpringerNatureSwitzerlandAG2022 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,reproduction onmicrofilmsorinanyotherphysicalway,andtransmissionorinformationstorageandretrieval,electronicadaptation, computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublicationdoesnot imply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelawsand regulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbookarebelievedto betrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsortheeditorsgiveawarranty, expressedorimplied,withrespecttothematerialcontainedhereinorforanyerrorsoromissionsthatmayhavebeen made.Thepublisherremainsneutralwithregardtojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface Polygenic traits are those controlled by multiple loci. Most economically important traits in plants and animals are polygenic in nature. Many disease traits are also polygenic although some disease phenotypes may appear to be categorical. Quantitative genetics is an area of geneticsthatstudiesthegeneticbasisofpolygenictraits.Becausethevariationofapolygenic trait,bydefinition,iscollectivelycontrolledbymultipleloci,toolscommonlyusedinclassical Mendeliangeneticsarenotsufficientforanalysisofpolygenictraits.Advancedstatisticaltools playanimportantroleinquantitativegenetics.Althoughtheterm“expectation”instatisticsis easytounderstand,terminologysuchas“variance”and“covariance”arethefociofquantitative genetics.Therefore, theproperties ofvariance andcovariance arefundamentallyimportantin understandingallequationspresentedinthetextbook.Awholechapter(Chap.4)isdedicatedto statisticalreview,especiallyvariance(covariance)anditsproperties. I have been teaching Quantitative Genetics (BPSC 148) as an undergraduate course at the UniversityofCalifornia,Riverside,since1995.Thistextbookisacollectionoflecturenotesfor thiscourse.MaterialandmethodsaremainlyextractedfromFalconerandMackay(1996)and Lynch and Walsh (1998). Some topics are adopted from the 1987 handouts of Quantitative Genetics (AGR/ANSC 611) by Professor Wyman E. Nyquist at Purdue University. The Quantitative Genetics course at UCR is offered for one quarter (10 weeks) annually. The Lynch and Walsh (1998) book is too much as a textbook for this one-quarter course. The FalconerandMackay(1996)bookisabouttherightsizebutderivationsofimportantequations areoftennotprovided.Iselectivelyadoptedtopicsfrombothbooksandextendedthesetopics in detail by providing derivations for some fundamentally important equations. More impor- tantly,thebookprovidessampledataandSAScodeforanalysesofthesedata.Thestatistical analysissystem(SAS)isasuiteofanalyticssoftware.Itisthelargestprivatesoftwarecompany intheworldwiththebesttechnical supportsystem. TheSASproductsaremorereliablethan products from all other software companies. This is why I chose SAS as the analytic tool for dataanalysis. Unliketheintroductionsofotherquantitativegeneticstextbooks,theintroduction(Chap.1) of this book presents two examples about breeding values and their applications to plant and animal breeding. In addition, the chapter describes the relationship between quantitative genetics and statistics. Major statistical methods that are commonly used in quantitative genetics are mentioned briefly in the introduction chapter. Chapters 2–4 review Mendelian genetics,populationgenetics,andstatistics.TheactualquantitativegeneticsstartsatChap.5by defining various kinds of genetic effects followed by definitions of genetic variance and its variancecomponentsinChap.6.Genotypebyenvironment(G(cid:1)E)interactionisdiscussedin Chap. 7 where environmental variation is classified into systematic error and random error. MuchofthecontentofG(cid:1)EinteractionisadoptedfromProfessorNyquist’s1987lecturenotes for Quantitative Genetics. Chapter 8 introduces the concept of major genes and describes methodsofmajorgenedetectionandsegregationanalysis,atopicnotoftenseeninquantitative genetics textbooks. Chaps. 9–11 cover theory and methods for estimation of heritability, primarily adopted from Falconer and Mackay (1996). Methods of kinship matrix calculation areaddedtoChap.11wheretheInbreedProcedureinSASisusedtocalculatekinshipmatrices v vi Preface forarbitrarilycomplicatedpedigrees.Linearmixedmodels(LMM)areintroducedinChap.12 for estimation of variance components from pedigree data. PROC MIXED in SAS is used to implementthemixedmodelanalysis.Multipletraitsandgeneticcorrelationbetweenmultiple traits are discussed in Chap. 13. Theory and methods of artificial selection are covered by Chaps. 14–16, followed by methods of multiple trait selection by Chap. 17. The last three chapters cover quantitative trait locus (QTL) mapping, genome-wide association studies (GWAS), and genomic selection (GS). The three topics belong to molecular quantitative geneticsandmolecularbreedingandareoftenignoredinclassicalquantitativegenetics. Thisbookcanbeusedasatextbookforquantitativegeneticscoursesforbothundergraduate andgraduateteaching.Forundergraduateteaching,instructorscanskipthederivationstepsand directlypresentthefinalformsofequations.Somecontentscanalsobeignoredforundergrad- uate teaching, e.g., calculation of the standard error of an estimated heritability from family data.Forgraduateteaching,derivationsareimportantpartoftheteachingandcanimprovethe depth of understanding of thecourse material, and thus should not be skipped. The book can also be used as a reference book for experimental quantitative geneticists, plant and animal breeders, theoretical quantitative geneticists, population geneticists, and even biologists who are interested in quantitative methodology and statisticians who are looking for examples of statistical methodsthatareoftenappliedtobiology.Thebookmaybeparticularlyusefulasa referencebookforjuniorfacultyasinstructorsofquantitativegenetics,evenifitisnotusedas thetextbookforstudents. I sincerely thank my graduate advisor for my MS degree, Professor Zhilian Sheng, who introduced me to this wonderful area of genetics (quantitative genetics). I learned his pooled analysisofsibcorrelationfrommultiplefarmsforestimatingheritabilityandgeneticcorrelation when I was a graduate student at Northeast Agricultural University under his supervision. Professor Sheng called his method “the within-unit sib analysis.” After I started my PhD program in the USA, I learned the method of restricted maximum likelihood method (REML) for estimation of variance components and immediately realized that the pooled analysisofsibcorrelationisidenticaltotheREMLmethod.TheREMLmethodwasformally publishedin1971byPattersonandThompsonwhileProfessorShengalreadyusedtheideaof REMLforestimatinggeneticparametersinthemid-1960s.ProfessorShengformallypublished his“within-unitmethodforsibcorrelationanalysis”in1980whenREMLwasnotwellknown tothemajorityofstatisticians.Mywholequantitativegeneticscareerbenefitedfromthe3-year trainingunderProfessorSheng’ssupervision.Icontinuedmytraininginquantitativegenetics under the supervision of Professor William M. Muir at Purdue University. Not only did Professor Muir teach me all the analytical skills of quantitative genetics, but he also taught metheSASprogramming,whichallowedmetosuccessfullycompeteforastudentstatistical consultingpositionatPurduetosupportmyfamilyduringthedifficultyearsasaPhDstudent.I am extremely grateful to Professor Muir for his continued supervision during my whole quantitative genetics career. The third person who was important to my quantitative genetics researchandteachingcareerwasProfessorWymanNyquistatPurdueUniversity.Hetaughtme Quantitative Genetics (AGR/ANSC 611) and Statistical Genetics (AGR/ANSC 615). He was thegreatesteducatorIhaveevermetandhisinfluenceonmyacademiccareerwaslifelong.My writtenandteachingstylesareprimarilyinheritedfromhis.Withouttheabovethreeinfluential mentors,Imighthavetakenadifferentcareerpathotherthanquantitativegenetics.Withouta careerinquantitativegenetics,Iwouldnothavetheopportunitytoteachquantitativegenetics andtowritethistextbook. TwoofmyPhDdissertationcommitteemembersalsodeservespecialappreciation:Profes- sor Thomas Kuczek in Statistics and Professor Truman Martin in Animal Sciences at Purdue University.WhenIlearnedtheBulmereffect(selectiononthereductionofgeneticvariance),I wastryingtoextendthemethodtoseehowselectionaffectthegeneticcovariancebetweentwo traitsthatarerelatedtothetargettraitofselection.IconsultedwithProfessorThomasKuczek, whohadanofficenexttoBillMuir’slaboratoryinLillyHall.Althoughhedidnottellmethe Preface vii answer immediately, he pointed me to a direction where we can express both traits as linear functions of the target trait of selection. Following his direction, I successfully derived the formulathenextday.IcouldnotwaittotellProfessorThomasKuczekabouttheanswerand howgratefulIwastohimforthevaluablesuggestion.Afewdayslater,Ihadachancetomeet with Professor Truman Martin. I told him about my derivation on the change of the genetic covariancebetweentwotraitsrelatedtothetargettrait.Hedidnotsayawordbutopenedadraw of his filecabinets, pulled a paper published by Alan Robertson in 1977 and told me that the formulawasalreadypresentedadecadeago.IwassadinthebeginningthatIwasnotthefirst persontoderivetheformula,butsoamazedbyProfessorTrumanMartin’squickresponseand clearmemoryabouttheformula.Mylessonslearnedfromthiseventare(1)alwayssearching literature thoroughly for a project before trying to investigate the project by yourself and (2)consultingwithyouradvisors,colleagues,classmates,evenroommatesforaprojectbefore actuallystartingtheproject. AllUCRstudentstakingthiscourse(BPSC148)forthelast25yearsorsohavecontributed to the book in one way or another. Errors in the lecture notes were periodically identified by studentsandthencorrectedduringthequarterswhenthecoursewasoffered.Specialapprecia- tion goes to all my former students who contributed to my research in quantitative genetics; some of their work was eventually integrated into the textbook. I particularly thank Meiyue Wang,arecentlygraduatedstudentfrommylab,forherhelpduringthebookwritingprocess bydrawingallthefigures,communicatingwiththeSpringerPublisherforcopyrightsissuesand making sure the book production process was smooth. Appreciation also goes to two more graduate students in theUCR Quantitative Genetics Program, Fangjie Xie from Xu’s lab and HanQufromJia’slab,whoofferedhelpinthebookpreparationprocess.Dr.XiangyuWang,a visiting scientist from The People’s Republic of China, audited the class (BPSC 148) in the 2020Winterquarterandidentifiedafewerrorsinthetextbook.Theerrorshavebeenfixedand shedeservesspecialappreciation. BeforethemanuscriptwassubmittedtotheSpringerPublisher,Idistributedeachchapterto one current or former member of the big Xu & Jia laboratory for proof reading. They all put significant effort to correct any errors occurring in an earlier version of the book manuscript. These contributors include John Chater, Ruidong Li, Meiyue Wang, Shibo Wang, Qiong Jia, TiantianZhu,FupingZhao,XuesongWang,XiangyuWang,JulongWei,HanQu,ChenLin, Saka Sigdel, Arthur Jia, Ryan Traband, Yanru Cui, Lei Yu, Huijiang Gao, Fangjie Xie, and YangXu.Theyalldeservespecialappreciation. Finally, I thank my wife (Yuhuan Wang), daughters (Mumu Xu and Nicole Xu), and grandson (Max Borson) for their support and patience. My wife originally thought that I could make a lot extra money by writing a book and only realized that the return was not worthforthesignificanteffortIputforth. Riverside,CA,USA ShizhongXu Contents 1 IntroductiontoQuantitativeGenetics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 BreedingValueandItsApplication. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 GeneticGainviaArtificialSelection(Example1). . . . . . . . . . . . 1 1.1.2 PredictingHumanHeight(Example2). . . . . . . . . . . . . . . . . . . 1 1.1.3 BreedingValue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 ComplicatedBehaviorTraits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 QualitativeandQuantitativeTraits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 TheRelationshipBetweenStatisticsandGenetics. . . . . . . . . . . . . . . . . . 6 1.5 FundamentalStatisticalMethodsinQuantitativeGenetics. . . . . . . . . . . . 8 1.6 StatisticalSoftwarePackages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.7 ClassicalQuantitativeGeneticsandModernQuantitativeGenetics. . . . . . 10 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 ReviewofMendelianGenetics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Mendel’sExperiments. . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . 13 2.2 Mendel’sLawsofInheritance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Vocabulary. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . .. . . . . . . . . . .. . . 19 2.4 DeparturefromMendelianRatio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5 TwoLociWithoutEnvironmentalEffects. . . . . . . . . . . . . . . . . . . . . . . . 20 2.6 MultipleLociwithorWithoutEnvironmentalEffects. . . . . . . . . . . . . . . 21 2.7 EnvironmentalErrorsCanLeadtoaNormalDistribution. . . . . . . . . . . . 22 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3 BasicConceptofPopulationGenetics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1 GeneFrequenciesandGenotypeFrequencies. . . . . . . . . . . . . . . . . . . . . 25 3.2 Hardy-WeinbergEquilibrium. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 25 3.2.1 ProofoftheH-WLaw. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2.2 ApplicationsoftheH-WLaw. . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2.3 TestofHardy-WeinbergEquilibrium. . . . . . . . . . . . . . . . . . . . . 28 3.3 GeneticDrift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.4 Wright’sFStatistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.5 EstimationofFStatistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4 ReviewofElementaryStatistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.1 Expectation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.1.1 Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.1.2 PropertiesofExpectation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.1.3 EstimatingtheMean. . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . 40 4.2 Variance. . . .. . . . .. . . .. . . .. . . .. . . .. . . .. . . . .. . . .. . . .. . . .. . 40 4.2.1 Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.2.2 PropertiesofVariance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ix x Contents 4.2.3 NormalDistribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2.4 EstimatingVariancefromaSample. . . . . . . . . . . . . . . . . . . . . . 41 4.2.5 AnApplicationoftheVarianceProperty. . . . . . . . . . . . . . . . . . 42 4.3 Covariance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.3.1 Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.3.2 PropertiesofCovariance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.3.3 EstimatingCovariancefromaSample. . . . . . . . . . . . . . . . . . . . 44 4.3.4 ConditionalExpectationandConditionalVariance. . . . . . . . . . . 45 4.4 SampleEstimatesofVarianceandCovariance. . . . . . . . . . . . . . . . . . . . 47 4.5 LinearModel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.5.1 Regression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.5.2 Correlation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.5.3 EstimationofRegressionandCorrelationCoefficients. . . . . . . . 49 4.6 MatrixAlgebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.6.1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.6.2 MatrixAdditionandSubtraction. . . . . . . . . . . . . . . . . . . . . . . . 52 4.6.3 MatrixMultiplication. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.6.4 MatrixTranspose. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.6.5 MatrixInverse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.6.6 GeneralizedInverse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.6.7 DeterminantofaMatrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.6.8 TraceofaMatrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.6.9 OrthogonalMatrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.6.10 EigenvaluesandEigenvectors. . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.7 LinearCombination,QuadraticForm,andCovarianceMatrix. . . . . . . . . 59 5 GeneticEffectsofQuantitativeTraits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.1 Phenotype,Genotype,andEnvironmentalError. . . . . . . . . . . . . . . . . . . 63 5.2 TheFirstGeneticModel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.3 PopulationMean. . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . 66 5.4 AverageEffectofGeneorAverageEffectofAllele. . . . . . . . . . . . . . . . 67 5.5 AverageEffectofGeneSubstitution. . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.6 AlternativeDefinitionofAverageEffectofGeneSubstitution. . . . . . . . . 67 5.7 BreedingValue. . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . . .. . . 68 5.8 DominanceDeviation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.9 EpistaticEffectsInvolvingTwoLoci. . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.10 AnExampleofEpistaticEffects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.11 PopulationMeanofMultipleLoci. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6 GeneticVariancesofQuantitativeTraits. . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6.1 TotalGeneticVariance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6.2 AdditiveandDominanceVariances. . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 6.3 EpistaticVariances(GeneralDefinition). . . . . . . . . . . . . . . . . . . . . . . . . 79 6.4 EpistaticVarianceBetweenTwoLoci. . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.5 AverageEffectofGeneSubstitutionandRegressionCoefficient. . . . . . . 81 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 7 EnvironmentalEffectsandEnvironmentalErrors. . . . . . . . . . . . . . . . . . . . . 85 7.1 EnvironmentalEffects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7.2 EnvironmentalErrors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7.3 Repeatability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 7.3.1 EstimationofRepeatability. . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 7.3.2 ProofoftheIntra-ClassCorrelationCoefficient. . . . . . . . . . . . . 89 Contents xi 7.3.3 EstimationofRepeatabilitywithVariableNumbers ofRepeats. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 7.3.4 AnExampleforEstimatingRepeatability. . . . . . . . . . . . . . . . . 91 7.3.5 ApplicationofRepeatability. . . . . . . . . . . . . . . . . . . . . . . . . . . 94 7.4 GenotypebyEnvironment(G(cid:1)E)Interaction. . . . . . . . . . . . . . . . . . . . 95 7.4.1 DefinitionofG(cid:1)EInteraction. . . . . . . . . . . . . . . . . . . . . . . . . 95 7.4.2 TheoreticalEvaluationofG(cid:1)EInteraction. . .. . . . . .. . . . .. . 97 7.4.3 SignificanceTestofG(cid:1)EInteraction. . . . . . . . . . . . . . . . . . . 98 7.4.4 PartitioningofPhenotypicVariance. . . . . . . . . . . . . . . . . . . . . 100 7.4.5 Tukey’sOneDegreeofFreedomG(cid:1)EInteractionTest. . . . . . 102 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 8 MajorGeneDetectionandSegregationAnalysis. . . . . . . . . . . . . . . . . . . . . . . 107 8.1 TwoSamplet-TestorF-Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 8.2 F-TestforMultipleSamples(ANOVA). . . . . . . . . . . . . . . . . . . . . . . . . 110 8.3 RegressionAnalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 8.3.1 TwoGenotypes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 8.3.2 ThreeGenotypes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 8.4 MajorGeneDetectionInvolvingEpistaticEffects. . . . . . . . . . . . . . . . . . 118 8.4.1 TestofEpistasis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 8.4.2 EpistaticVarianceComponentsandSignificanceTestfor eachTypeofEffects. . .. . . . . .. . . . .. . . . . .. . . . .. . . . .. . . 119 8.5 SegregationAnalysis. . .. . . . . . . . . . .. . . . . . . . . .. . . . . . . . . . .. . . . 122 8.5.1 QualitativeTraits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 8.5.2 QuantitativeTraits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 9 ResemblancebetweenRelatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 9.1 GeneticCovarianceBetweenOffspringandOneParent. . . . . . . . . . . . . . 136 9.1.1 ShortDerivation. .. . . . .. . . .. . . . .. . . .. . . . .. . . . .. . . .. . 136 9.1.2 LongDerivation. . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 9.2 GeneticCovariancebetweenOffspringandMid-Parent. . . . . . . . . . . . . . 139 9.3 GeneticCovariancebetweenHalf-Sibs. . . . . . . . . . . . . . . . . . . . . . . . . . 139 9.4 GeneticCovariancebetweenFull-Sibs. . . . . . . . . . . . . . . . . . . . . . . . . . 140 9.4.1 ShortWayofDerivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 9.4.2 LongWayofDerivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 9.5 GeneticCovariancebetweenMonozygoticTwins(IdenticalTwins). . . . . 142 9.6 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 9.7 EnvironmentalCovariance. . . .. . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . 143 9.8 PhenotypicResemblance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 9.9 DerivationofwithinFamilySegregationVariance. . . . . . . . . . . . . . . . . 143 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 10 EstimationofHeritability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 10.1 F DerivedfromaCrossofTwoInbredParents. . . . . . . . . . . . . . . . . . . 148 2 10.2 MultipleInbredLinesorMultipleHybrids. . . . . . . . . . . . . . . . . . . . . . . 148 10.2.1 WithReplications. . . .. . . .. . . . .. . . .. . . .. . . .. . . .. . . .. . 149 10.2.2 WithoutReplications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 10.3 Parent-OffspringRegression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 10.3.1 SingleParentVs.SingleOffspring. . . . . . . . . . . . . . . . . . . . . . 152 10.3.2 MiddleParentVs.SingleOffspring. . . . . . . . . . . . . . . . . . . . . . 153 10.3.3 SingleParentVs.MeanOffspring. . . . . . . . . . . . . . . . . . . . . . . 154