1. Lecture 3: Resemblance Between Relatives

2. 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

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

4. Half-sibs
Full-sibs

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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)

11. V a r ( x + y ) = 2 C o v ; C o v @ X x ; y A ( ) More generallyi = 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

12. 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

13. 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

14. 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)

15. 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

16. 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

17. 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.

18. C o v ( Æ ; ) = Ω i f 6 . e , n t I B D V a r A 2 V a r ( A ) = Æ + sx ; 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

19. 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

20. 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)

21. Resulting Genetic Covariance between full-sibsBD 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

22. 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) +

23. 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.

24. 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