Transcript Lecture 3: Resemblance Between Relatives

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
AncestralCollateral
relatives relatives,
e.g., parent
offspring
e.g.and
sibs
1
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Half-sibs
Full-sibs
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o* o*
o* o*
o. * o. *
o*
o*
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o* o*
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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)
Cov(x,y)
>=<0,
0,
negative
(linear)
(linear)
association
between
between
Cov(x,y)
Cov(x,y)
0,positive
no
= 0linear
DOES
association
NOTassociation
imply
between
no assocation
x & y x x&&y y
cov(X,Y)
cov(X,Y)
> 0=<00= 0
cov(X,Y)
cov(X,Y)
Y
Y Y
Y
X
X X
X
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) = p
Cov(x; y)
V a r (x) Va r (y)
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
yb = y + by j x ( x
x)
The slope of this linear regression is a function of Cov,
by j x
Cov(x; y)
=
Va r (x)
The fraction of the variation in y accounted for by knowing
x, i.e,Var(yhat - y), is r2
r2 = 0.9
0.6
Relationship between the correlation and the regression
slope:
s
r (x; y) = p
C ov(x; y)
Va r (x)Va r (y)
= by jx
Va r (x)
V a r (y)
If Var(x) = Var(y), then by|x = b x|y = r(x,y)
In the 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)
More generally
0
1
Xn
Xm
Xn Xm
C ov @
xi ;
yj A =
Cov(x i ; yj )
i= 1
j= 1
i= 1 j = 1
Var (x + y) = V ar (x) + Var (y) + 2Cov(x; y)
Hence, the variance of a sum equals the sum of the
Variances ONLY when the elements are uncorrelated
Genetic Covariance between
relatives
Sharing
meansarise
having
allelestwo
thatrelated
are
Genetic alleles
covariances
because
Father
Mother
identical
by are
descent
both
copies
of than
individuals
more(IBD):
likely to
share
alleles
can two
be traced
backindividuals.
to a single copy in a
are
unrelated
recent common ancestor.
One
allele IBD
IBD
No alleles Both
IBD alleles
Regressions and ANOVA
• Parent-offspring regression
– Single parent vs. midparent
– Parent-offspring covariance is a
intraclass (between class) variance
• Sibs
– Covariances between sibs is an interclass
(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
1
2
Var(B) = 2.5
Var(W) = 0.2
Var(T) = 2.7
3
Situation 2
4
t = 2.5/2.7 = 0.93
Var(B) = 0
Var(W) = 2.7
Var(T) = 2.7
t=0
Parent-offspring genetic covariance
Cov(Gp, Go) --- Parents and offspring share
EXACTLY one allele IBD
Denote this common allele by A1
G p = Ap + D p = Æ1 + Æx + D 1 x
G o = Ao + D o = Æ1 + Æy + D 1 y
IBD allele
Non-IBD alleles
C ov(G o ; G p ) = C ov(Æ1 + Æx + D 1 x ; Æ1 + Æy + D 1 y
= C ov(Æ1 ; Æ1 ) + C ov(Æ1 ; Æy ) + C ov(Æ1 ; D 1 y )
+ C ov(Æx ; Æ1 ) + C ov(Æx ; Æy ) + C ov(Æx ; D 1 y )
+ C ov(D 1 x ; Æ1 ) + C ov(D 1 x ; Æy ) + C ov(D 1 x ; D 1 y )
All red covariance terms are zero.
• 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
ž
C ov(Æx ; Æy ) =
0
Var (A)=2
if x 6
= y; i.e., not IBD
if x = y; i.e., IBD
Var (A)one
= Var
(ÆIBD
Hence, relatives sharing
allele
have
1 + Æ
2 ) = a2Var (Æ1 )
genetic covariance of Var(A)/2
so t hat
Var (Æ1 ) = C ov(Æ1 ; Æ1 ) = Var (A)=2
The resulting parent-offspring genetic covariance
becomes Cov(Gp,Go) = Var(A)/2
Half-sibs
Each sib gets exactly one
allele from common father,
different alleles from the
different mothers
2
1
o
1
o
2
Hence, the genetic
The half-sibs
covariance
share
of half-sibs
no
onealleles
alleleisIBD
just
(1/2)Var(A)/2 •= Var(A)/4
occurs with probability 1/2
Full-sibs
Father
Mother
Each sib gets
exact one allele
from each parent
Full Sibs
not IDB
[ Prob = 1/2
]
Paternal allele
[ Prob
Prob(exactly
oneIDB
allele
IDB)==1/2
1/2]
not IDB
[ Prob
= 1/2
[ Prob
= 1/2
] ]
= Maternal
1- Prob(0 allele
IBD) -IDB
Prob(2
IBD)
Prob(zero alleles IDB) = 1/2*1/2 = 1/4
-> Prob(both
Resulting Genetic Covariance between full-sibs
IBD alleles
IBD alleles
0
1
2
Probability
Probability
Contribution
Contribution
1/4
0
1
1/21/2
Var(A)/2
Var(A)/2
2
1/4
0
1/4
1/4
0
Var(A) + Var(D)
Var(A) + Var(D)
Cov(Full-sibs) = Var(A)/2 + Var(D)/4
Genetic Covariances for General Relatives
Let r = (1/2)Prob(1 allele IDB) + Prob(2 alleles IDB)
Let u = Prob(both alleles IDB)
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
E = Ec + Es
The Environmental variance can thus be written
in terms of variance components as
Total
environmental
value value experienced
Common
environmental
Specific
environmental
value,
by all any
members
of
a family, e.g.,effects
shared
unique
environmental
E
Ec
Es
maternal
effects by the individual
experienced
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.
V =V
+V
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