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Lecture 4:
Heritability
Heritability
Narrow vs. board sense
Narrow sense: h2 = VA/VP
Slope of midparent-offspring regression
(sexual reproduction)
Board sense: H2 = VG/VP
Slope of a parent - cloned offspring regression
(asexual reproduction)
When one refers to heritability, the default is narrow-sense, h2
h2 is the measure of (easily) usable genetic variation under
sexual reproduction
Why h2 instead of h?
Blame Sewall Wright, who used h to denote the correlation
between phenotype and breeding value. Hence, h2 is the
total fraction of phenotypic variance due to breeding values
æ(A; P )
æ2A
æA
r (A; P ) =
=
=
= h
æA æP
æA æP
æP
Heritabilities are functions of populations
Heritability values only make sense in the content of the population
for which it was measured.
Heritability measures the standing genetic variation of a population,
A zero heritability DOES NOT imply that the trait is not genetically
determined
Heritabilities are functions of the distribution of
environmental values (i.e., the universe of E values)
Decreasing VP increases h2.
Heritability values measured one one environment
(or distribution of environments) may not be valid
under another
Measures of heritability for lab-reared individuals
may be very different from heritability in nature
Heritability and the prediction of breeding values
If P denotes an individual's phenotype, then best linear
predictor of their breeding value A is
æ(P ; A)
2
A=
(P
°
š
)
+
e
=
h
(P ° š p ) + e
p
2
æP
The residual variance is also a function of h2:
æ2e = (1 ° h 2 )æ2A
The larger the heritability, the tighter the distribution of true
breeding values around the value h2(P - mP) predicted by
an individual’s phenotype.
Heritability and population divergence
Heritability is a completely unreliable predictor of
long-term response
Measuring heritability values in two populations that
show a difference in their means provides no information
on whether the underlying difference is genetic
Sample heritabilities
People
hs
Height
0.65
Serum IG
0.45
Back-fat
0.70
Weight gain
0.30
Litter size
0.05
Abdominal Bristles
0.50
Body size
0.40
Ovary size
0.3
Egg production
0.20
Pigs
Fruit Flies
Traits more closely
associated with fitness
tend to have lower
heritabilities
Estimation: One-way ANOVA
Simple (balanced) full-sib design: N full-sib families,
each with n offspring
One-way ANOVA model:
zij = m + fi + wij
s2f = between-family variance = variance among family means
Trait value in sib j from
Deviation
Common
Effect
family
Mean
offor
i sibfamily
j fromi =the family mean
s2w = within-family variance
deviation of mean of i from
The common mean
s2P = Total phenotypic variance = s2f + s2w
Covariance between members of the same group
equals the variance among (between) groups
The variance
effects equals the
Cov(Full
Sibs) = among
æ( zi j ; zfamily
ik )
covariance =between
sibs
æ[ (š + ffull
i + wi j ); (š + f i + wi k ) ]
f i )+
+ æ(æf2i ; w
wi j ; f i ) + æ( wi j ; wi k )
i k ) + æ( 2
æ2f == æ(
æ22Af i ;=2
=4
+
æ
D
Ec
= æf
The within-family variance s2w = s2P - s2f,
æ2w ( F S ) = æP2 ° ( æA2 =2 + æ2D =4 + æE2 c )
= æA2 + æ2D + æE2 ° ( æA2 =2 + æ2D =4 + æE2 c )
= (1=2) æA2 + (3=4)æ2D + æE2 ° æ2E c
One-way Anova: N families with n
sibs, T = Nn
Factor
Degrees of
freedom, df
Sums of
Squares (SS)
Mean sum of
squares
(MS)
E[ MS ]
Among-family
N-1
SSf =
XN
SSf/(N-1)
s2w
SSw/(T-N)
s2w
n
(z i ° z)2
i= 1
Within-family
T-N
SSW =
XN Xn
i= 1 j = 1
(zi j ° z i )2
+n
s2f
Estimating the variance components:
Since
æ2f = æ2A =2 + æD2 =4 + æE2 c
2Var(f) is an upper bound for the additive variance
Assigning standard errors ( = square root of Var)
Nice fact: Under normality, the (large-sample) variance
for a mean-square is given by
2
2(MS
)
x
2
æ (MS x ) '
dfx + 2
(
(
)
•
•
MSf ° MS w
2(MSw ) 2
Var[ Var(f
) ] =S Var
Var(w(F
)) ] = Var(MSw ) '
n
T N + 2
µ
Ž
2
2
2 (MSf )
(MSw )
'
+
2
n
N + 1
T° N + 2
)
Estimating heritability
tF S
2 =4 + æ2
æ
Var(f )
1 2
D
Ec
=
= h +
Var(z)
2
æz2
Hence, h2 < 2 tFS
An approximate large-sample standard error
for h2 is given by
p
SE(h ) ' 2(1 ° t F S )[1 + (n °- 1)t F S ] 2=[N n(n °- 1)]
2
Worked example
10 full-sib families, each with 5 offspring are measured
Factor
Df
SS
MS
EMS
Among-familes
9
SSf = 405
45
Within-families
40
SSw = 800
20
s2f + 5 s2f
s2w
VA < 10
Var(w) = MSw = 20
Var(z) = Var(f ) + Var(w) = 25
SE(h 2 )
h2 < 2 (5/25) = 0.4
p
' 2(1 ° 0:4)[1 + (5 ° 1)0:4] 2=[50(5 ° 1)] = 0:312
Full sib-half sib design: Nested ANOVA
1
1
n
n
2
1
n
2
Full-sibs
o* o*
o* o*
o. * o. *
o*
o*
o. *
o
o*
1
1
2
2
3
3
..
..
Half-sibs
*
*
k
o
k
1
2
..
3
k
...
o* o*
o* o*
o. * o. *
o*
o*
o. *
o* o*
o*
..
1
1
2
2
3
3
k
..
k
1
2
..
3
k
Estimation: Nested ANOVA
Balanced full-sib / half-sib design: N males (sires)
are crossed to M dams each of which has n offspring
Nested ANOVA model:
zijk = m + si + dij+ wijk
s2s = between-sire variance = variance in sire family means
Effect
of sire
ofWithin-family
idam
= deviation
j of sire deviation
i = deviation
of kth
Value of the
kthEffect
offspring
Overall
mean
s2dfrom
= variance
dams
within
=sire
offor
mean
of
mean
ofi i’s
of
family
offspring
damsires
from
j from
from
theand
mean
overall
of the
the kth among
dam
sire
overall
mean
mean
ij-th
family
variance of dam means
for the
same
sire
s2w = within-family variance
s2T = s2s + s2d + s2w
Nested Anova: N sires crossed to M
dams, each with n sibs, T = NMn
Factor
Df
SS
MS
Sires
N-1
SSs =
Mi
XN X
Mn
EMS
SSs/(N-1)
æ2w + næ2d + M næ2s
(z i ° z) 2
i= 1 j = 1
Dams(Sires)
N(M-1)
SSs =
XN
SSd /(N[M-1])
XM
2
n
(z i j ° z i )
æw2 + næd2
i= 1 j= 1
Sibs(Dams)
T-NM
SSw =
XN
XM
SSw/(T-NM)
Xn
i= 1 j = 1 k= 1
(zi j k ° zi j )2
æw2
Estimation of sire, dam, and family variances:
MSs ° MSd
Var (s ) =
Mn
MSd ° MSw
Var(d) =
n
Var (e) = MSw
Translating these into the desired variance components
• Var(Total) = Var(between FS families) + Var(Within FS)
s2w = s2z - Cov(FS)
• Var(Sires) = Cov(Paternal half-sibs)
æd2 = æ2z ° æ2s ° æ2w
= æ(FS) ° æ(P HS)
Summarizing,
æ2s
= æ(PHS)
æw2 = æz2 ° æ(F S)
æd2 = æ2z ° æ2s ° æ2w
= æ(FS) ° æ(PHS)
Expressing these in terms of the genetic and environmental variances,
æs2 '
æd2 '
æ2w '
æA2
4
æA2
æD2
+
+ æE2 c
4
4
æA2
3æ2D
+
+ æ2E s
2
4
Intraclass correlations and estimating heritability
tP H S
tF S
Cov(P HS)
Var(s)
=
=
Var (z)
Var (z)
Cov(F S)
Var(s) + Var(d)
=
=
Var(z)
Var(z)
4tPHS = h2
h2 < 2tFS
Note that 4tPHS = 2tFS implies no dominance
or shared family environmental effects
Worked Example:
N=10 sires, M = 3 dams, n = 10 sibs/dam
Factor
Df
SS
MS
EMS
Sires
9
4,230
470
æw2 + 10æ2d + 30æ2s
Dams(Sires)
20
3,400
170
æw2 + 10æ2d
Within Dams
270
5,400
20
æw2
æ2w = M S w = 20
M Sd ° M Sw
170 ° 20
2
æd =
=
= 15
n
10
M Ss ° M Sd
470 ° 170
2
æs =
=
= 10
Nn
30
æP2 = æ2s + æ2d + æ2w = 45
æ2d = 15 = (1=4)æA2 + (1=4)æD2 + æ2E c
= 10 + (1=4)æ2D + æE2 c
æ2A = 4æ2s = 40
h2
æA2
40
= 2 =
= 0:89
æP
45
æD2 + 4æE2 c = 20
Parent-offspring regression
Single parent - offspring regression
zo i = š + bojp (zp i ° š ) + ei
The expected slope of this regression is:
æ(zo ; zp )
E (bojp ) =
'
2
æ (zp )
(æ2A =2) + æ(E o ; E p )
h2
æ(E o ; E p )
=
+
2
æz
2
æ2z
Residual error variance (spread around expected values)
Shared environmental values
Ž
2
h
2
2
æ
=
1
°
æ
z regressions are male-offspring, as
Toe avoid this term, typically
2
female-offspring more likely to share environmental values
µ
(
)
Midparent - offspring regression
µ
zoi = š + boj M P
(
zm i + zf i
° š
2
Ž
)+e
i
The expected slope of this regression is h2, as
Cov[zo ; (zm + zf )=2]
Residual error
(spread around expected values)
bokM Pvariance
=
Var[(zm + zf )=2]
Key: Var(MP) = (1/2)Var(P)
µ
Ž
2 o ; zm ) + Cov(zo ; zf )]=2
[Cov(z
h
2
æe2 = 1 °=
æ
[Var (z)
z + Var(z)]=4
(
2
)
2Cov(zo ; zp )
=
= 2bthere
ojp
Hence, even when heritability is one,
is considerable spread
Var(z)
of the true offspring values about their midparent values,
with Var = VA/2, the segregation variance
Standard errors
Single parent-offspring regression,
N parents, each with n offspring
Var(boj p ) '
n(t ° b2ojp ) + (1 ° t )
Nn
8
2
2 =4
t
=
h
for half-sib s
Var(h
)
=
Var(2b
H
S
<
Squared regression slope ojp ) = 4Var (bojp )
2 Total
2number of offspring
Sib correlation t =
æ
+
æ
D
E
c
: t F S = h 2 =2 +
for full sibs
Midparent-offspring regression,
2
æz
N sets of parents, each with n offspring
Var(h 2 ) = Var(boj M P ) '
2[n(t F S ° b2ojM P =2) + (1 ° t F S )])
Nn
• Midparent-offspring variance half that of single parent-offspring variance
Estimating Heritability in Natural Populations
A
lowersibs
bound
be placed
of heritability
using
Often,
arecan
reared
in a laboratory
environment,
parents
from nature and regressions
their lab-reared
offspring,
making parent-offspring
and sib
ANOVA
problematic for estimating heritability
h 2m i n
=
(b0oj M P
Var n (z)
)
Var l (A)
2
Why is this a lower bound?
•
•2
Covl ; n (A) Var n (z)
0
2 Var n (z)
(bojM P )
=
Var l (A)
Var n (z)
Var l (A)
)
(
= •2 h 2n
Cov l ; n (A)
where
•= p
Var n (A)Var
Covariance between breeding
value in
nature and BV in lab
l (A)
is the additive genetic covariance between
environments and hence
g2 < 1
Defining H2 for Plant Populations
Plant breeders often do not measure individual plants
(especially with pure lines), but instead measure a plot or
a block of individuals.
This can result in inconsistent measures of H2
even for otherwise identical populations.
zi j k ` = Gi + E j + GE i j + pi j k + ei j k `
If we set our unit of measurement as the average over all plots,
Genotype
Environment
i variance
Interaction
j
between
Genotype
I
the
phenotypic
becomes
Effect
of
deviations
plot
k for
ofGenotype
individual
I
2j
2
2
and environment
æ
æ
æ
in
environment
Plants
within
j
this plot
p
e
2
2
2
GE
æ (zi ) = æG + æE +
+
+
e
er
ern
Hence, VP, and hence H2, depends on our choice of e, r, and n
Number of Environments
Number of plots/environment
Number of individuals/plot