response to selection in variance
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29/01/2017
1
Response to selection in variance
Han Mulder and Piter Bijma
Contents
Effect of mass selection on Ve
Response to selection and breeding goal
● Use of information of relatives
● Breeding goal and deriving economic values
● Multi-trait selection on mean and variance
● What does nature say with respect to selection on variance?
● Selection experiments
Practical
● Mass selection
● Information of relatives and breeding goal
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Learning outcomes
To understand and calculate the response to mass selection in mean and variance
To calculate economic values for mean and variance
To understand and calculate the response to index selection in mean and variance using information of relatives
3
Effect of mass selection on Ve
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Response to selection in normal cases
Mass selection = selection on phenotype
∆𝑎 = 𝑏∆𝑥 = 𝑏𝑆
∆𝑎 = selection response
∆𝑥 = S= selection differential
𝑏 = regression coefficient of ∆𝑎 on ∆𝑥 =cov a,x
var x= ℎ2
∆𝑎 = 𝑏∆𝑥 = ℎ2𝑆
For directional truncation selection
∆𝑎 = 𝑏∆𝑥 = ℎ2𝑆 = 𝑖ℎ2𝜎𝑃 → breeders equation
General breeders equation: ∆𝑎 = i𝑟𝐼𝐻𝜎𝑎
𝑟𝐼𝐻 = accuracy of selection
5
Quantitative genetic model: the additive
model
AG,
0
0~ N
A
A
v
m
ivEimi AAEAP ,
2
,
2
2
cov
cov
vAmvA
mvAmA
G
1,0~ N
Am,i = breeding value of i for mean Av,i = breeding value of i for environmental variance
E2 = the mean environmental variance
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Single phenotype, stochastic simulations
When rA = 0, then
P relates nearly linearly to Am
P2 relates nearly linearly to Av
If we want to predict Am and Av from a single phenotypic observation, then it makes sense to use an index of P and P2
Mulder et al. 2007; Genetics
175:1895-1910
7
Estimation of breeding values and selection
responses
Start simple: only a phenotype is available, e.g. mass selection
E(Am|P)≈h2*(P-Pmean)
E(Av|P2)=b * (P2 - P2mean)
Mulder et al. 2007; Genetics
175:1895-1910
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Single phenotype, selection index with P
and P2
222,1,
222,1,
)()()(ˆ
)()()(ˆ
PPbPbA
PPbPbA
vvv
mmmIndex:
222
1
2221
)()(
)()(,
PP
P
x
x
PPxPx
x
Info:
xbx
xbx
'2,1,
'2,1,
ˆ
ˆ
vvvv
mmmm
bbA
bbA
Selection index in matrix-vector notation
xBa ')()(ˆ
ˆˆ
222,1,
2,1,
PP
P
bb
bb
A
A
vv
mm
v
m
Use both info sources for both traits, because Am and Av can be correlated
Mulder et al. 2007; Genetics 175:1895-1910 9
Single phenotype, selection index with P
and P2 = multiple regression
Solve the index weights
● B = P-1G, P = Var(x), G = Cov(x,a)
● Use moments of the normal distribution
24
2
323
3)(
vmv
mv
APA
APVar
xP
2
2
22
11
),(),(
),(),(),(
vmv
mvm
AA
AA
vm
vm
AxCovAxCov
AxCovAxCovCov
axG
- This allows you to estimate breeding values for mean and variance when you know the phenotype of the individual - From those selection index equations, you can also derive the accuracy for each trait
Mulder et al. 2007; Genetics 175:1895-1910
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Monte Carlo simulation
To evaluate goodness of fit
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
-3 -2 -1 0 1 2 3
P
Am
expectation
from MC
prediction
from MR
-0.2
-0.1
0
0.1
0.2
0.3
0.4
-3 -2 -1 0 1 2 3
P
Av
expectation
from MC
prediction
from MR
3.02 mh 12 P 05.02 Av 5.0Ar
Mulder et al. 2007; Genetics
175:1895-1910 11
Intermezzo, The Moment Generating
Function (MGF)
The MGF is a function that allows you to derive
“moments” of distributions
First moment: E(X) = mean
Second moment: E(X2)
● Variance = E(X2) – mean2
nth moment: E(Xn)
Really handy for derivations of variances and covariances
For the nth moment:
● Take the nth derivative of MX(t) with respect to t
● Calculate its value for t = 0
)0()0()( tdt
MdtMXE
n
Xn
nX
n
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Moments of the normal distribution
Number Raw moment Central moment
0 1 1
1 μ 0
2 μ2 + σ
2 σ
2
3 μ3 + 3μσ
2 0
4 μ4 + 6μ
2σ
2 + 3σ
4 3σ
4
5 μ5 + 10μ
3σ
2 + 15μσ
4 0
6 μ6 + 15μ
4σ
2 + 45μ
2σ
4 + 15σ
6 15σ
6
7 μ7 + 21μ
5σ
2 + 105μ
3σ
4 + 105μσ
6 0
8 μ8 + 28μ
6σ
2 + 210μ
4σ
4 + 420μ
2σ
6 + 105σ
8 105σ
8
Or, if you don’t like derivatives, look-up the moments at Wikipedia
Those things are used in the derivations in Mulder et al.
13
Single phenotype, selection index with P
and P2
Response to selection in mean and variance
● a = B’x a = B’x
● If you know the selection index weights in B, and the
selection differentials in x, you can calculate response
Directional truncation selection on P
● Selection differentials in mean and variance (e.g. Tallis,
1961)
222
2
1
)()( PP
P
ixPx
iPx
where x is the standardized truncation point
Selection intensity for the variance
Selection for the mean also generates a positive selection differential in the variance!
14
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Response to directional mass selection
p ∆Am ∆Av 𝜎𝐸2 Ratio
∆Av / ∆Am
20% 0.42 0.03 0.73 0.06
5% 0.60 0.08 0.78 0.13
1% 0.73 0.14 0.84 0.20
0
0.1
0.2
0.3
0.4
-4 -3 -2 -1 0 1 2 3 4
x
z
3.02 mh 12 P 0Ar
Animals with larger variance have larger probability of selection
→ Directional mass selection increases the environmental
variance
05.02 Av
Mulder et al. 2007; Genetics 175:1895-1910 15
Single phenotype, selection index with P
and P2
Can we increase uniformity using info on a single phenotype?
Try stabilizing selection
Again we can use a = B’x
2****22
1
)21/(21)(
0
PpxipPx
Px
where p* is the proportion in one of the tails
Doing those calculations tells us whether we can select for uniformity by taking a group of average animals
Mulder et al. 2007; Genetics 175:1895-1910
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Single phenotype, selection index with P
and P2
What selection differentials are feasible for P2 ?
You can select strongly for greater P2, but not for smaller P2
The selection differential limits the potential to reduce variability with mass selection. Stabilizing selection on own performance is not promising
The usual directional selection has the effect to increase variability! 17
Response of Av to mass selection
Directional selection
0
0.1
0.2
0.3
0.4
-3 -2 -1 0 1 2 3
x
z
0
0.1
0.2
0.3
0.4
-3 -2 -1 0 1 2 3
x
z
Stabilizing selection
0
0.1
0.2
0.3
0.4
-3 -2 -1 0 1 2 3
x
z
Disruptive selection
3.02 mh 12 P 05.02 Av 0Ar
p Directional Stabilizing Disruptive
20% 0.03 -0.02 0.05
5% 0.08 -0.02 0.11
1% 0.14 -0.02 0.17
70.02
0, E
Mulder et al. 2007; Genetics 175:1895-1910 18
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Heritability of environmental variance
In analogy of the normal heritability of the mean
Regression of Av on P2
Accuracy of BV =
24
2
2
222
22
,
2
32
)(/
)(/),(2
v
V
V
V
AP
A
v
Av
VPAv
h
PVarh
PVarPACovbh
2
vh
Mulder et al. 2007; Genetics 175:1895-1910 19
Conclusions mass selection
Opportunities to change the variance with info on single phenotypes is limited
● Accuracy is low ( ℎ𝑣2)
● The selection differential for stabilizing selection is
small
Traditional truncation selection has the tendency to increase variability
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11
Response to selection using information
of relatives
21
Selection based on relatives
Use a group of relatives of a single sire and/or dam
● E.g. progeny
Within family variance is a measure for Av
● HS-Progeny of a single sire
● VarW = ¾Var(A) + [ Var(E) + ½Av,sire ]
E.g. find the sire with the most uniform progeny, using 100
progeny per sire
This yields much higher accuracies than own performance info
The procedure is the same, but …
The mathematics becomes more tedious (Mulder et al., 2007)
Mulder et al. 2007; Genetics 175:1895-1910 22
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Accuracy of selection based on relatives
Selection based on the within family variance
● Variance within a group of sibs or progeny
Exact expression is complex approximation
Classical expression for selection on relatives
𝑟𝐼𝐻 = 𝑎ℎ
𝑛
1+ 𝑛−1 𝑡
a = additive genetic relationship between candidate and group of relatives t = intraclass correlation among the relatives t = awh2 , aw = additive genetic relationship among the group of relatives
Mulder et al. 2007; Genetics 175:1895-1910
Hill and Mulder, 2010; Genetics Res. 92:381-395 23
Accuracy of selection based on relatives
Expression for selection on the variance (vEBV)
𝑟𝐼𝐻,𝑣𝐸𝐵𝑉 ≈ 𝑎ℎ𝑛
1+ 𝑛−1 𝑡
● 𝑎 = additive genetic relationship between candidate and
group of relatives
● 𝑡 = intraclass correlation among the relatives
● 𝑡 = 𝑎𝑤ℎ𝑣2 , aw = additive genetic relationship among the
group of relatives
This is the same as for classical traits ● Limiting accuracies are 0.5 for HS, 0.71 for FS and
1 for progeny
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Comparison approximation with simulation
Table 6. Realized (MC) and predicted accuracy of vA for different numbers of half-sib
progeny per sire and 2
mA using either the exact prediction (MR exact) or the approximate
prediction (MR approx)a,b
.
Accuracy vA
Number of progeny
10 100
2
mA MC MR exact MR approx MC MR exact MR approx
0 0.235 0.235 0.235 0.607 0.607 0.607
0.1 0.236 0.236 0.235 0.615 0.615 0.607
0.3 0.243 0.244 0.235 0.633 0.633 0.607
0.6 0.251 0.260 0.235 0.648 0.663 0.607
Mulder et al. 2007; Genetics 175:1895-1910
25
Accuracy of vEBV
3.02 mh 12 P 023.02 vh 0Ar
Number of records Mass FS HS progeny
Own phenotype 0.15 - -
10 - 0.25 0.24
50 - 0.47 0.50
100 - 0.55 0.63
With phenotype only, accuracy is small and relies on the mean
when rA ≠ 0
Using 100 progeny yields meaningful accuracies
Mulder et al. 2007; Genetics 175:1895-1910 26
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Numerical results for accuracy
With phenotype only, accuracy is small and relies on the mean
when rA ≠ 0
Using 100 progeny yields meaningful accuracies
hv2 ≈ 0.5%, 2.5% and 5%
Mulder et al. 2007; Genetics 175:1895-1910 27
The accuracy of vEBV (approximation)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 200 400 600 800 1000
accu
racy
vEB
V
Number of relatives
heritability = 0.03
clones
full-sibs
half-sib progeny
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The accuracy of vEBV (approximation)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 200 400 600 800 1000
accu
racy
vEB
V
Number of relatives
heritability = 0.01
clones
full-sibs
half-sib progeny
Large number of clones or half-sib progeny needed to approach accuracy = 1.0 29
Selection to increase uniformity expressed in
standard deviations
Selected proportions: 5% in sires, 20% in dams
sires progeny tested, dams sib tested
%SDE = change (%) in environmental standard deviation %SDP = change (%) in phenotypic standard deviation
Large families needed!!!
n σAv
2 h2 acc sire acc dam %SDE %SDP
50 0.05 0.3 0.36 0.18 -5.45 -3.78
100 0.05 0.3 0.48 0.24 -7.18 -4.97
200 0.05 0.3 0.61 0.31 -9.06 -6.25
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Selection to improve uniformity expressed
in standard deviations
Selected proportions: 5% in sires, 20% in dams
sires progeny/sib tested, dams sib tested
- Progeny testing gives larger response
- More response for traits with low heritability
%SDE %SDP
n σAv
2 h2 sib progeny sib progeny
100 0.05 0.1 -5.40 -8.48 -4.85 -7.60
100 0.05 0.3 -4.56 -7.18 -3.17 -4.97
100 0.05 0.5 -2.58 -5.54 -1.28 -2.73
31
Summary response to selection
With info on relatives we can reach high accuracies despite low heritability
● Limiting accuracies are the same as in classical theory
● Clones would be ideal system to study genetics of Ve
Combined with the high GCV, this yields high potential response to selection
Selection to reduce Ve is promising, especially for traits with a low heritability of the phenotype
AIHir
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Some extensions
33
Some extensions
What is the heritability of the log-variance/standard deviation of repeated observations?
● Within-litter variance of birth weight
● Log-variance of repeated records for egg color/egg weight/milk yield
Response to selection with the exponential model
● Relationship response and GCV
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Heritability litter variance
We can use the classical equation
𝑟𝐼𝐻 =𝑛ℎ2
1+ 𝑛−1 𝑟
𝑟 = repeatability
If 𝑟 = ℎ𝑣2=0.01 and n=15 piglets:
ℎ𝑣,𝑙𝑖𝑡𝑡𝑒𝑟2 = 0.14
35
Heritability repeated observations
Egg color in purebred laying hens
ℎ𝑣2=0.01 and 𝑟=0.046
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
5 7 9 11 13 15
he
rita
bili
ty o
f w
ith
in-i
nd
ivid
ual
va
rian
ce
number of repeated observations
Series1
Mulder et al. 2016; GSE 48:39 36
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19
Response to selection with exponential
model
Remember: average Ve depends on genetic variance in Ve
𝜎𝐸2 = 𝜎𝐸,𝑒𝑥𝑝
2 exp 0.5𝜎𝐴𝑣2
∆𝑎𝑣 = 𝑖𝑟𝐼𝐻𝜎𝐴𝑣
∆𝜎𝐸2 = 𝜎𝐸,𝑡
2 (exp ∆𝑎𝑣 − 1)
𝜎𝐸,𝑡+12 = 𝜎𝐸,𝑡
2 + ∆𝜎𝐸2 = 𝜎𝐸,𝑡
2 (exp ∆𝑎𝑣 )
37
Response under additive and exponential
model
Response to selection approximately the same in the first generations
Additive model can be considered as a linear approximation of the exponential model
No biological evidence which model is better
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Response to selection: use of GCV
In exponential model: 𝐺𝐶𝑉𝜎𝐸2 ≈ 𝜎𝑎𝑣
∆𝑎𝑣
𝜎𝐸2 = 𝑖𝑟𝐼𝐻
𝜎𝑎𝑣
𝜎𝐸2 ≈ 𝑖𝑟𝐼𝐻𝜎𝑎𝑣
The response in the exponential model is the proportion of change in Ve
39
Selection response in standard deviations
𝐺𝐶𝑉𝜎𝑒≈
1
2𝜎𝑎𝑣
𝐺𝐶𝑉𝜎𝑃≈
1
2𝜎𝑎𝑣
𝜎𝐸2
𝜎𝑃2
∆𝑎𝑣
𝜎𝑃≈ 𝑖𝑟𝐼𝐻
1
2𝜎𝑎𝑣
𝜎𝐸2
𝜎𝑃2
Example: 𝑖 = 1.0, 𝑟𝐼𝐻 = 0.6; 𝜎𝑎𝑣2 = 0.05;
𝜎𝐸2
𝜎𝑃2 = 0.7
∆𝑎𝑣 = 1.0 ∗ 0.6 ∗ 0.05 = 13%
∆𝑎𝑣
𝜎𝐸= 1.0 ∗ 0.6 ∗ 0.05 ∗ 0.5 = 6.7%
∆𝑎𝑣
𝜎𝑃= 1.0 ∗ 0.6 ∗ 0.05 ∗ 0.5 ∗ 0.7 = 2.3%
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Summary
Classical equations can be used to calculate accuracy of selection using heritability of Ve
Many offspring/sibs needed to obtain reasonable accuracy
The additive model is easier for selection response than the exponential model
Genetic coefficient of variation parameters are easy parameters to know the change in standard deviation or variance
41
Breeding goal and deriving economic
values
29/01/2017
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When do we want to select on variance?
Optimum traits: bring population close to the optimum
How could we determine it more precisely?
Take the derivative with respect to the variance
● M=profit equation
Only non-zero economic value for non-linear profit equations!!!
02
PvA
d
Mdv
Mulder et al. Genet. Sel. Evol. 40:37-59.
43
Different cases of non-linear profit
Quadratic profit equation
● Age at first calving
● Some conformation traits in cattle
Animals between thresholds
● Carcass weight in pigs
● pH of meat in pigs
● Egg weight in laying hens
-3 -2 -1 0 1 2 3
P
profit=0 profit=1 profit=0
Optimum
range
Mulder et al. Genet. Sel. Evol. 40:37-59. 44
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Non-linear profit: Risk aversion
Risk aversion
Higher profit increases utility less than lower profit decreases utility
𝑈 = 1 − exp −𝑎𝑌
Y is yield
Eskridge, K. M. and B. E. Johnson. 1991. Expected utility maximization and selection of stable plant cultivars. Theor. Appl. Genet. 81:825-832.
45
Case non-linear profit: derivation of economic
values (quadratic profit)
Animal level
Population level
Economic values
2
2
1 )( aOPaM
2
12
2
11
2
1 2)( PaaOaOaadPPMfM
)(2 1 Oad
Mdv
mA
12a
d
Mdv
P
Av
Mulder et al. Genet. Sel. Evol. 40:37-59.
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Case non-linear profit
derivation of economic values (thresholds)
P
ulA
zzv
m
-3 -2 -1 0 1 2 3
P
profit=0 profit=1 profit=0
Optimum
range
2
21 )(
P
uullA
tztzv
v
-0.5
-0.3
-0.1
0.1
0.3
0.5
-4 -2 0 2 4
Population Mean
Ec
on
om
ic v
alu
e
Am
Av
threshold
Mulder et al. Genet. Sel. Evol. 40:37-59. 47
Case non-linear profit
mean and phenotypic variance
-2.0
-1.5
-1.0
-0.5
0.0
0 1 2 3 4 5
Generation
Me
an
quadratic
two thresholds
0.6
0.7
0.8
0.9
1.0
1.1
0 1 2 3 4 5
Generation
Ph
en
oty
pic
va
ria
nc
e
quadratic
two thresholds
3.02 mh 12 P023.02 vh 0Ar
Mulder et al. Genet. Sel. Evol. 40:37-59.
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Multi-trait selection on mean and variance
49
Multi-trait selection: selection index theory
𝐻 = 𝑣𝐴𝑚𝐴𝑚 + 𝑣𝐴𝑣𝐴𝑣 = 𝐯′𝐚 𝐯′ = 𝑣𝐴𝑚 𝑣𝐴𝑣 ; 𝐚 =𝐴𝑚
𝐴𝑣
Suppose we have information on the mean performance and the within-family variance of half-sibs (either as sibs or as offspring of selection candidate)
𝐼 = 𝑏1𝑃 + 𝑏2𝑣𝑎𝑟𝑊 = 𝐛′𝐱 𝐛′ = 𝑏1 𝑏2 ; 𝐱 = 𝑃
𝑣𝑎𝑟𝑊
𝐛 = 𝐏−𝟏𝐆𝐯 index weights to maximize genetic gain in H
𝐏 = var 𝐱 =var(𝑃 ) cov(𝑃 , 𝑣𝑎𝑟𝑊)
cov(𝑃 , 𝑣𝑎𝑟𝑊) var(𝑣𝑎𝑟𝑊)
𝐆 =𝑐𝑜𝑣(𝐴𝑚, 𝑃 ) 𝑐𝑜𝑣(𝐴𝑣 , 𝑃 )
𝑐𝑜𝑣(𝐴𝑚, 𝑣𝑎𝑟𝑊) 𝑐𝑜𝑣(𝐴𝑣 , 𝑣𝑎𝑟𝑊)
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26
Response to selection
∆𝐴 = ∆𝐴𝑚 ∆𝐴𝑣 =𝑖 𝐛′𝐆 𝐛′𝐏𝐛
𝑖 : selection intensity;
𝐛′𝐏𝐛 = standard deviation of index
Use classical equations for P and G and ℎ𝑣2 and additive
model for Ve (approximation)
𝐏 =var(𝑃 ) cov(𝑃 , 𝑣𝑎𝑟𝑊)
cov(𝑃 , 𝑣𝑎𝑟𝑊) var(𝑣𝑎𝑟𝑊)=
[𝜎𝑝2(1 + 𝑛 − 1 𝑎ℎ2)]/𝑛 3 + 𝑎 𝑛 − 3 𝑐𝑜𝑣𝑎𝑚𝑣 /𝑛
𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐 [(2𝜎𝑝4+3𝜎𝑎𝑣
2 )(1 + 𝑛 − 1 𝑎ℎ𝑣2)]/𝑛
a = additive genetic relationships between family members
51
Response to selection
𝐆 =𝑐𝑜𝑣(𝐴𝑚, 𝑃 ) 𝑐𝑜𝑣(𝐴𝑣 , 𝑃 )
𝑐𝑜𝑣(𝐴𝑚, 𝑣𝑎𝑟𝑊) 𝑐𝑜𝑣(𝐴𝑣 , 𝑣𝑎𝑟𝑊)=
𝑎𝑗𝜎𝑎𝑚2 𝑎𝑗𝑐𝑜𝑣𝑎𝑚𝑣
𝑎𝑗𝑐𝑜𝑣𝑎𝑚𝑣 𝑎𝑗𝜎𝑎𝑣2
𝑎𝑗 = additive genetic relationship between the animal to
be evaluated and the group of relatives
● 0.5 parent-offspring
● 0.25 when selection candidate is a half-sib
Exact elements in Mulder et al. Genet. Sel. Evol. 40:37-59.
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What does nature say with respect to
selection on variance?
What is the relationship between fitness
and Ve?
What is the best Ve from a fitness point of view?
Are there any trade-offs of a high or low variance?
What is the regression of fitness on trait values or on within-family variance?
We performed an analysis in Great Tits at the Veluwe, a nature reservation close to Wageningen
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Fitness and Ve in Great Tit
• Results show stabilizing selection on Ve • Ve is maintained
Mulder et al., Evolution 70:2004-2016 55
Selection experiments
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Selection experiments
Selection experiments with laboratory animals, e.g. Drosophila
Selection experiment in rabbits
Need for more selection experiments
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Selection experiment in rabbits
Garreau et al. 2008;
Livest. Sci. 119:55-62.
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Experiment in rabbits
Garreau et al. 2008;
Livest. Sci. 119:55-62.
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Summary
Selection index theory can be used to predict responses to selection
Selection to change the variance is important when the profit equation is non-linear
● Selection to improve uniformity is of importance when having an optimum
Need for knowledge on relationships between Ve and fitness (traits)
More selection experiments are needed to learn more about genetic architecture of Ve
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