Quantitative Genetics

419 Pages·2022·9.953 MB·English

by  Shizhong Xu

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
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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