Lecture 3: Resemblance Between

Relatives

Heritability• 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

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Ancestral relatives e.g., parent and offspringCollateral relatives, e.g. sibs

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Full-sibsHalf-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]

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cov(X,Y) > 0

Cov(x,y) > 0, positive (linear) association between x & yCov(x,y) < 0, negative (linear) association between x & y

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cov(X,Y) < 0

Cov(x,y) = 0, no linear association between x & y

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cov(X,Y) = 0

Cov(x,y) = 0 DOES NOT imply no assocation

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cov(X,Y) = 0

CorrelationCov = 10 tells us nothing about the strength of anassociation

What is needed is an absolute measure of association

This is provided by the correlation, r(x,y)

r(x;y) =Cov(x;y)

V ar(x)Var(y)p

r = 1 implies a prefect (positive) linear association

r = - 1 implies a prefect (negative) linear association

RegressionsConsider the best (linear) predictor of y given we know x,

by = y+by j x( x x)The slope by|x of this linear regression is a function of Cov,

by j x =Cov(x;y)Var(x)

The fraction of the variation in y accounted for by knowing x, i.e,Var(yhat - y), is r2

s

pr(x;y) =Cov(x;y)

Var(x)Var(y)=byjx

Var(x)V ar(y)

Relationship between the correlation and the regressionslope:

r2 = 0.6r2 = 0.9r2 = 1.0r2 = 0.3

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)

Cov@nX

i=1

xi;mX

j=1

yj A =nX

i=1

mX

j=1

Cov(xi;yj )

0 1

Var(x +y) =V ar(x) +Var(y) +2Cov(x;y)

Hence, the variance of a sum equals the sum of theVariances 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.

Father Mother

No alleles IBD One allele IBDBoth 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)

4321 4321

Situation 1

Var(B) = 2.5Var(W) = 0.2Var(T) = 2.7

Situation 2

Var(B) = 0Var(W) = 2.7Var(T) = 2.7

t = 2.5/2.7 = 0.93 t = 0

Parent-offspring genetic covariance

Cov(Gp, Go) --- Parents and offspring share EXACTLY one allele IBD

Denote this common allele by A1

Gp = Ap + Dp =Æ1 +Æx + D1x

Go = Ao + Do =Æ1 +Æy + D1y

IBD alleleNon-IBD alleles

Cov(Go;Gp) = Cov(Æ1 +Æx + D1x;Æ1 +Æy + D1y

= Cov(Æ1;Æ1) + Cov(Æ1;Æy) +Cov(Æ1; D1y)+ Cov(Æx;Æ1) +Cov(Æx;Æy) +Cov(Æx; D1y)

+ Cov(D1x;Æ1) + Cov(D1x;Æy) + Cov(D1x; D1y)

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(Æx;Æy) =Ω0 if x6=y; i.e., not IBD

Var(A)=2 if x =y; i.e., IBD

Var(A) = Var(Æ1 +Æ2) = 2Var(Æ1)

so thatVar(Æ1) = Cov(Æ1;Æ1) = Var(A)=2

Hence, relatives sharing one allele IBD have agenetic covariance of Var(A)/2

The resulting parent-offspring genetic covariance becomes Cov(Gp,Go) = Var(A)/2

Half-sibs

The half-sibs share one allele IBD • occurs with probability 1/2

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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-sibsFather Mother

Full SibsPaternal 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 getsexact one allelefrom each parent

IB D alleles Probability Contr ibution

0 1/ 4 0

1 1/ 2 Var(A)/ 2

2 1/ 4 Var(A) + Var(D)

IBD alleles Probability Contribution

0 1/4 0

1 1/2 Var(A)/2

2 1/4 Var(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 relativesCov(G) = rVar(A) + uVar(D)

When epistasis is present, additional terms appearr2Var(AA) + ruVar(AD) + u2Var(DD) + r3Var(AAA) +

Components of the Environmental Variance

E = Ec + Es

Total environmental valueCommon environmental value experiencedby all members of a family, e.g., shared maternal effects

Specific environmental value,any unique environmental effectsexperienced by the individual VE = VEc + VEs

The Environmental variance can thus be writtenin terms of variance components as

One can decompose the environmental further, ifdesired. For example, plant breeders have termsfor the location variance, the year variance, and the location x year variance.

Shared Environmental Effects contributeto the phenotypic covariances of relatives

Cov(P1,P2) = Cov(G1+E1,G2+E2) = Cov(G1,G2) + Cov(E1,E2)

Shared environmental values are expectedwhen sibs share the same mom, so thatCov(Full sibs) and Cov(Maternal half-sibs)not only contain a genetic covariance, butan environmental covariance as well, VEc