Lecture 3: Resemblance Between Relatives

Heritability Central concept in quantitative genetics Proportion of variation due to additive genetic values (Breeding values) –h 2 = V A /V P –Phenotypes (and hence V P ) can be directly measured –Breeding values (and hence V A ) must be estimated Estimates of V A require known collections of relatives

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

Full-sibs Half-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 & yCov(x,y) < 0, negative (linear) association between x & y Cov(x,y) = 0, no linear association between x & yCov(x,y) = 0 DOES NOT imply no assocation

Correlation 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)= Cov(x;y) Var(x)Var(y) p r = 1 implies a prefect (positive) linear association r = - 1 implies a prefect (negative) linear association

Regressions Consider the best (linear) predictor of y given we know x, b y=y+b yjx (xx) The slope b y|x of this linear regression is a function of Cov, b yjx = Cov(x;y) Var(x) The fraction of the variation in y accounted for by knowing x, i.e,Var(yhat - y), is r 2

s p r(x;y)= Cov(x;y) Var(x)Var(y) =b yjx Var(x) Var(y) Relationship between the correlation and the regression slope: If Var(x) = Var(y), then b y|x = b x|y = r(x,y) In this case, the fraction of variation accounted for by the regression is b 2

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) = a 2 Var(x). –Var(ax) = Cov(ax,ax) = a 2 Cov(x,x) = a 2 Var(x) Cov(x+y,z) = Cov(x,z) + Cov(y,z)

n X i=1 x i ; m X j=1 y j A = n X i=1 m X j=1 Cov(x i ;y j ) 01 Var(x+y)=Var(x)+Var(y)+2Cov(x;y) Hence, the variance of a sum equals the sum of the Variances ONLY when the elements are uncorrelated More generally

Genetic Covariance between relatives Genetic covariances arise because two related individuals are more likely to share alleles than are two unrelated individuals. 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. No alleles IBD One allele IBD Both alleles IBD

Regressions and ANOVA Parent-offspring regression –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 Var(B) = 2.5 Var(W) = 0.2 Var(T) = 2.7 Situation 2 Var(B) = 0 Var(W) = 2.7 Var(T) = 2.7 t = 2.5/2.7 = 0.93 t = 0

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

All white covariance terms are zero. By construction,  and D are uncorrelated By construction,  from non-IBD alleles are uncorrelated By construction, D values are uncorrelated unless both alleles are IBD

Cov(Æ x ;Æ y )= Ω 0ifx6=y;i.e.,notIBD Var(A)=2ifx=y;i.e.,IBD Var(A)=Var(Æ 1 +Æ 2 )=2Var(Æ 1 ) sothat Var(Æ 1 )=Cov(Æ 1 ;Æ 1 )=Var(A)=2 Hence, relatives sharing one allele IBD have a genetic covariance of Var(A)/2 The resulting parent-offspring genetic covariance becomes Cov(G p,G o ) = Var(A)/2

Half-sibs The half-sibs share one allele IBD occurs with probability 1/2 The half-sibs share no alleles 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 not IBD [ Prob = 1/2 ] Maternal allele not IBD [ Prob = 1/2 ] -> Prob(zero alleles IBD) = 1/2*1/2 = 1/4 Paternal allele IBD [ Prob = 1/2 ] Maternal allele IBD [ Prob = 1/2 ] -> Prob(both alleles IBD) = 1/2*1/2 = 1/4 Prob(exactly one allele IBD) = 1/2 = 1- Prob(0 IBD) - Prob(2 IBD) Each sib gets exact one allele from each parent

IBD alleles ProbabilityContribution 01/40 1 2Var(A)/2 2 4Var(A) +Var(D) Resulting Genetic Covariance between full-sibs 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 r 2 Var(AA) + ruVar(AD) + u 2 Var(DD) + r 3 Var(AAA) +

Components of the Environmental Variance E = E c + E s Total environmental value Common environmental value experienced by all members of a family, e.g., shared maternal effects Specific environmental value, any unique environmental effects experienced by the individual V E = V Ec + V Es 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(P 1,P 2 ) = Cov(G 1 +E 1,G 2 +E 2 ) = Cov(G 1,G 2 ) + Cov(E 1,E 2 ) 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, V Ec