Introduction to Quantitative Genetics II: …

1 ** ** ** Introduction to Quantitative Genetics II:Resemblance Between RelativesBruce Walsh. 28 November 2006. EEB 600 ATheheritabilityof a trait, a central concept in Quantitative Genetics , is the proportion of variationamong individuals in a population that is due to variation in the additive genetic ( , breeding)values of individuals:h2=VAVP=Variance of breeding valuesPhenotypic VarianceSince an individual s phenotype can be directly scored, the phenotypic varianceVPcan be estimatedfrom measurements made directly on the contrast, an individual s breeding value cannot be observed directly, but rather must be inferredfrom the mean value of its offspring (or more generally using the phenotypic values of other knownrelatives).

2 Thus estimates ofVArequire known collections of relatives. The most common situations(which we focus on here) are comparisons between parents and their offspring or comparisonsamong sibs. We can classify relatives as eitherancestralorcollateral, and we focus here ondesigns with just one type of relative. In a more general pedigree, information from both kinds ofrelatives is : , parent and offspring* Measure phenotypes of one or both parents, andkoffspringCollateralrelatives:Full Sibs have both parents in common*Measurekoffspring in each family, but not the to Quantitative Genetics , II . ** ** ** ** ** **n..Half Sibs have one parent in common* Measure phenotype of k progeny of each family, but not the parents.

3 Note that ifk>1, this designinvolves both full- (within any column) and half-sibs (between columns from the same sire) and isreferred to as anested half-sib/full-sib observation:The amount of phenotypic resemblance among relatives for the trait provides an indicationof the amount of genetic variation for that trait. If trait variation has a significant genetic basis, the closer therelatives, the more similar their Resemblance Between RelativesQuantitative Genetics as a field traces back to R. A. Fisher s 1918 paper showing how the phenotypiccovariance between relatives can be expressed in terms of genetic variances, as we detail Parent-offspring regressionsThere are three types of parent-offspring regressions: twosingle parent - offspring regressions(plotting offspring mean versus either the trait value in their fatherPfor their motherPm), and themidparent-offspring regression(the offspring mean regressed on the mean of their parents, themidparentMP=(Pf+Pm)=2).

4 The slope of the (single) parent-offspring regression is estimated bybOjP=Cov(O;P)Var(P);whereCov(O;P)=1n 1 nXi=1 OiPi nO P!whereOiis the mean trait value in the offspring of parenti,Othe offspring mean over all familes (alsocaled thegrand meanof all offspring),Pthe mean of all parents, and we examinenparent-offspringfamilies. One could compute separate regressions using males (Pm) and females (Pf), althoughthe later potentially includes maternal effect contributions and hence single-parent regressions areusually restricted to midparent-offspring regression slope is estimated bybOjMP=Cov(O;MP)Var(MP);whereCov(O;MP)= 1n 1 nXi=1 OiPmp;i nO MP!

5 WhereOiis the mean trait value in the offspring of parents in pairi, where these parents have anaverage trait valueMPiand we examinenparent-offspring pairs .Notice that all of the three regressions involve the covariance between parents and their to Quantitative Genetics , II . 22. Collateral relationships: ANOVAWith collateral relatives, the above formulae for the sample covariance is not appropriate, for tworeasons. First, there are usually more than two collateral relatives per family. Second, even if familiesconsist of only two relatives, the order of the two is arbitrary , there is no natural distinctionbetween X and Y , as exists in the case of parents and way of stating the second point is that collateral relatives belong to the same class orcategory.

6 In contrast, parents and offspring belong to different classes. The covariance betweenparents and offspring is aninterclass(between-class) covariance, while the covariance betweencollateral relatives is anintraclass(within-class) covariance. The analysis of variance (ANOVA),first proposed in Fisher s 1918 paper, is used to estimate intraclass the simplest ANOVA framework, we can consider the total variance of a trait to consist of twocomponents: abetween-group(also called theamong-group) component (for example, differencesin the mean values of different families) and awithin-groupcomponent (the variation in trait valuewithin each family).

7 The total variance is the sum of the between and within group variances,Var(T)=Var(B)+Var(W)( )A key feature of ANOVA is thatthe between-group variance equals the within-group covariance. Thus,the larger the covariance between members of a family, the larger the fraction of total variation thatis attributed to differences between family see this point, consider the following extreme patterns of phenotypes in full sib families:Situation 1 Suppose the between group varianceVar(B)=2:5, and the within-group varianceVar(W)=0 gives a total phenotypic variance ofVP=Var(T)=Var(B)+Var(W)=2:7. Here: members of a family resemble each other more closely than they do membersof other families there are large differences in average phenotype between familiesIntro to Quantitative Genetics , II.

8 3 The resulting intraclass correlationtist=Cov(full sibs)VP=Var(B)VP=0:93where we have used the above-mentioned ANOVA identity that the between-group variance equalsthe within-group covariance (here, the covariance between full sibs). Since elements of the sameclass are full-sibs, this is often denoted bytFSto distinguish it from other intraclass 2 Suppose the total (phenotypic) variance is the same as in situation 1, withVar(T)=VP=2 , suppose there is no between-group variance (Var(B)=0), implying thatVar(W)=2:7and the intraclass correlation ist=0. Here: members of a family resemble each other no more than they do members ofother families there are no significant differences in average phenotype between families ( ,all families have the same mean value) phenotypic resemblance is low, so genetic variation is lowNote that phenotypic resemblance among relatives can equivalently be considered as a measureof thesimiliaryamong a group of relatives for the phenotype of a Quantitative trait (the covarianceof family members), or thedifferencein phenotype between different families (the between-groupvariance = variance of family means).

9 Causes of Phenotypic Covariance Among RelativesRelatives resemble each other for Quantitative traits more than they do unrelated members of thepopulation for two potential reasons: relatives share genes. The closer the relationship, the higher the proportion ofshared genes relatives share the same environmentThe genetic Covariance Between RelativesThe genetic CovarianceCov(Gx;Gy)= covariance of the genotypic values (Gx,Gy) of the will first show how the genetic covariances between parent and offspring, full sibs, and halfsibs depend on the genetic variancesVAandVD. We will then discuss how these covariances areestimated in covariances arise because two related individuals are more likely to share alleles than aretwo unrelated individuals.

10 Sharing alleles means having alleles that areidentical by descent(IBD):namely that both copies of an allele can be traced back to a single copy in a recent common can also beidentical in statebut not identical by descent. For example, both alleles in anA1A1individual are the same type (identical in state), but they are only identical by descent if bothcopies trace back to (descend from) a single copy in a recent example, consider the offspring of two parents and label the four allelic copies in the parents by1 - 4, independent of whether or not any are identical in to Quantitative Genetics , II . 41o12o2 Parents:A1A2 A3A4 Offspring:o1=A1A3o2=A1A4o3=A1A3o4=A2A4 Here,o1ando2share one allele IBD,o1ando3share two alleles IBD,o1ando4share no alleles Offspring and one is the covariance of genotypic values between an offspring (Go) and its parent (Gp)?