Lecture 3: Resemblance Between Relatives

Heritability Estimates of VA require known collections of relatives Central concept in quantitative genetics Proportion of variation due to additive genetic values (Breeding values) h2 = VA/VP Phenotypes (and hence VP) can be directly measured Breeding values (and hence VA ) must be estimated Estimates of VA require known collections of relatives

Ancestral relatives e.g., parent and offspring Collateral relatives, e.g. sibs

Half-sibs Full-sibs

Key observations The amount of phenotypic resemblance among relatives for the trait provides an indication of the amount of genetic variation for the trait. If trait variation has a significant genetic basis, the closer the relatives, the more similar their appearance

Covariances Cov(x,y) = E [x*y] - E[x]*E[y] Cov(x,y) > 0, positive (linear) association between x & y Cov(x,y) < 0, negative (linear) association between x & y Cov(x,y) = 0, no linear association between x & y Cov(x,y) = 0 DOES NOT imply no assocation

Correlation r ( x ; y ) = C o v V a p Cov = 10 tells us nothing about the strength of an association What is needed is an absolute measure of association This is provided by the correlation, r(x,y) r ( x ; y ) = C o v V a p r = 1 implies a prefect (positive) linear association r = - 1 implies a prefect (negative) linear association

b y = + ( ) Regressions b = C o v ( ; ) V a r j x y j x Consider the best (linear) predictor of y given we know x, b y = + j x ( ) The slope of this linear regression is a function of Cov, b y j x = C o v ( ; ) V a r The fraction of the variation in y accounted for by knowing x, i.e,Var(yhat - y), is r2

s p r ( x ; y ) = C o v V a b j Relationship between the correlation and the regression slope: If Var(x) = Var(y), then by|x = b x|y = r(x,y) In this case, the fraction of variation accounted for by the regression is b2

Useful Properties of Variances and Covariances Symmetry, Cov(x,y) = Cov(y,x) The covariance of a variable with itself is the variance, Cov(x,x) = Var(x) If a is a constant, then Cov(ax,y) = a Cov(x,y) Var(a x) = a2 Var(x). Var(ax) = Cov(ax,ax) = a2 Cov(x,x) = a2Var(x) Cov(x+y,z) = Cov(x,z) + Cov(y,z)

V a r ( x + y ) = 2 C o v ; C o v @ X x ; y A ( ) More generally i = 1 x ; m j y A ( ) V a r ( x + y ) = 2 C o v ; Hence, the variance of a sum equals the sum of the Variances ONLY when the elements are uncorrelated

Genetic Covariance between relatives Sharing alleles means having alleles that are identical by descent (IBD): both copies of can be traced back to a single copy in a recent common ancestor. Genetic covariances arise because two related individuals are more likely to share alleles than are two unrelated individuals. Both alleles IBD One allele IBD No alleles IBD

Regressions and ANOVA Parent-offspring regression Sibs Single parent vs. midparent Parent-offspring covariance is a interclass (between class) variance Sibs Covariances between sibs is an intraclass (within class) variance

ANOVA Two key ANOVA identities Total variance = between-group variance + within-group variance Var(T) = Var(B) + Var(W) Variance(between groups) = covariance (within groups) Intraclass correlation, t = Var(B)/Var(T)

Situation 1 Situation 2 Var(B) = 0 Var(B) = 2.5 Var(W) = 2.7 Var(T) = 2.7 Var(B) = 2.5 Var(W) = 0.2 Var(T) = 2.7 t = 0 t = 2.5/2.7 = 0.93

Parent-offspring genetic covariance Cov(Gp, Go) --- Parents and offspring share EXACTLY one allele IBD Denote this common allele by A1 IBD allele G p = A + D Æ 1 x o y Non-IBD alleles

C o v ( G ; ) = Æ + D All white covariance terms are zero. p ) = Æ 1 + x D y • By construction, a and D are uncorrelated • By construction, a from non-IBD alleles are uncorrelated • By construction, D values are uncorrelated unless both alleles are IBD All white covariance terms are zero.

C o v ( Æ ; ) = Ω i f 6 . e , n t I B D V a r A 2 V a r ( A ) = Æ + s x ; y ) = Ω i f 6 . e , n t I B D V a r A 2 V a r ( A ) = Æ 1 + 2 s o t h C v ; Hence, relatives sharing one allele IBD have a genetic covariance of Var(A)/2 The resulting parent-offspring genetic covariance becomes Cov(Gp,Go) = Var(A)/2

Half-sibs The half-sibs share no alleles IBD • occurs with probability 1/2 The half-sibs share one allele IBD • occurs with probability 1/2 Each sib gets exactly one allele from common father, different alleles from the different mothers Hence, the genetic covariance of half-sibs is just (1/2)Var(A)/2 = Var(A)/4

Full-sibs Paternal allele IBD [ Prob = 1/2 ] Maternal allele IBD [ Prob = 1/2 ] -> Prob(both alleles IBD) = 1/2*1/2 = 1/4 Paternal allele not IBD [ Prob = 1/2 ] Maternal allele not IBD [ Prob = 1/2 ] -> Prob(zero alleles IBD) = 1/2*1/2 = 1/4 Each sib gets exact one allele from each parent Prob(exactly one allele IBD) = 1/2 = 1- Prob(0 IBD) - Prob(2 IBD)

Resulting Genetic Covariance between full-sibs BD al l el es P rob a bil i ty Co n tr but on 1/ 4 1 2 V r ( A ) / ) + Va r( D Cov(Full-sibs) = Var(A)/2 + Var(D)/4

Genetic Covariances for General Relatives Let r = (1/2)Prob(1 allele IBD) + Prob(2 alleles IBD) Let u = Prob(both alleles IBD) General genetic covariance between relatives Cov(G) = rVar(A) + uVar(D) When epistasis is present, additional terms appear r2Var(AA) + ruVar(AD) + u2Var(DD) + r3Var(AAA) +

Components of the Environmental Variance Total environmental value Common environmental value experienced by all members of a family, e.g., shared maternal effects E = Ec + Es Specific environmental value, any unique environmental effects experienced by the individual VE = VEc + VEs The Environmental variance can thus be written in terms of variance components as One can decompose the environmental further, if desired. For example, plant breeders have terms for the location variance, the year variance, and the location x year variance.

Shared Environmental Effects contribute to the phenotypic covariances of relatives Cov(P1,P2) = Cov(G1+E1,G2+E2) = Cov(G1,G2) + Cov(E1,E2) Shared environmental values are expected when sibs share the same mom, so that Cov(Full sibs) and Cov(Maternal half-sibs) not only contain a genetic covariance, but an environmental covariance as well, VEc