Selection Theory – Change of variance
Selection intensity
selint.xls
Superiority of selected group (in SD units)
proportion selected
p
10.00%
Selection threshold
Height at threshold
Selection intensity
x
z
i
1.282
z
0.1755
1.755
x i
X: cumulative density function(x > ∞) = p
x= -NORMSINV(p) in excel
𝑧=
2
−0.5𝑥
𝑒
/√2𝜋
i = z/p
Loss of variance due to selection
“Bulmer Effect”
Variance among selected individuals is reduced
Selected parents have a reduced variance
select of phenotype: VPs = (1-k)VP
k = i (i-x)
Both Environmental en Genetic variance are reduced
Genetic Variance in selected group is reduced to proportion (1-r2k)
where r is the correlation(selection criterion, breeding value)
i.e. r = selection accuracy
select of phenotype: VAs = (1-h2k)VA
Why is genetic variance reduced?
Bulmer 1971: Due to LD between loci (even if unlinked)
Example: Selection on the sum of two unlinked loci
+
Selected individuals
Locus 1
+
-
All individuals
prior to selection
--
Line representing
minimum requirement
for selection
Locus 2
In the selected group, there will be a negative covariance between loci
var(x1+x2) = car(x1) + var(x2) + 2cov(x1,x2) = reduced
This reduction will disappear if you stop selecting, i.e. no loss due to allele fixation
Reduction in genetic variance after selection
Variance among parents is reduced
What of this reduction do we find back in progeny?
we find again full residual variance
….but genetic variance is still reduced
Only the part coming from the parents is reduced
2
2
A

(
1

r
2 ks ) A
s _t
t
2
2
A
 (1  r 2 kd ) A
d _t
t
New variance is generated due to Mendelian Sampling
This is NOT affected by selection
2
A

t 1
1
2

4
As t 
1
2

4
Ad t 
1
2

2
A0
Genetic variance stabilizes after a few generations
Change of variance over time
(P-males=10%, P-females = 50%, h2 = 0.5)
Generation
0
1
2
3
4
5
6
7
8
9
10
Phenotypic
Variance
200.0
181.7
177.4
176.3
176.0
176.0
176.0
175.9
175.9
175.9
175.9
Mendelian Variance among Variance among
Sampling selected sires selected dams
Genetic Variance Variance
(% of
(% of
Heritability Genetic mean
unselected)
unselected)
100.0
50.0
58.5
68.2
0.500
0.0
81.7
50.0
51.2
58.3
0.450
9.0
77.4
50.0
49.3
55.9
0.436
16.8
76.3
50.0
48.9
55.3
0.433
24.2
76.0
50.0
48.7
55.1
0.432
31.5
76.0
50.0
48.7
55.1
0.432
38.8
76.0
50.0
48.7
55.1
0.432
46.1
75.9
50.0
48.7
55.1
0.432
53.4
75.9
50.0
48.7
55.1
0.432
60.8
75.9
50.0
48.7
55.1
0.432
68.1
75.9
50.0
48.7
55.1
0.432
75.4
• Variance, heritability, and response decline rapidly (about 15-25%)
• Stabilize after few generations
= asymptotic response
Response to
selection
9.03
7.73
7.41
7.34
7.32
7.31
7.31
7.31
7.31
7.31
7.31
Less response over time due to Bulmer effect (here 16%)
(P-males=10%, P-females = 50%, h2 = 0.5)
100.0
90.0
80.0
genetic mean
70.0
60.0
50.0
not accounting ofr Bulmer
40.0
accounting ofr Bulmer
30.0
20.0
10.0
0.0
0
2
4
6
8
10
12
Generation
Conclusion:
The Bulmer effect accounts for a reduction in
genetic variance, heritability, selection response
in populations under selection
Another consequence of the Bulmer effect
Estimation of breeding values is based on information sources that
combine
• Information about between family
• All pedigree and collateral sib information
• Information about Mendelian sampling variation
• Own performance and progeny information
Under selection, the variation between family is reduced
But the Mendelian sampling variance is not
 Under selection, the family (pedigree) information becomes less important
 The information from own performance and progeny becomes more important
Incorporating the Bulmer in Selection Index Calculations
Bulmer effect affects elements of P and G
Example:
x1 = individual's performance
x2 = mean performance of that individual's m full sibs
h2 = 0.5,  2g =25 ,  2p = 50, m=5
E.g.
(0)
 50 12.5
P(0) = 

12.5 20 
(0)
25 
G(0) =  
12.5
'G
b (0)
(0)
r(0) =
 g2
-1
.4074 
b(0) = P(0) G(0) = 

.3704 
and
= 0.77
(0)
ps = pd = 5%  k = 0.863
2
 g* =  g* = (1- k r 2 )  2g = (1-0.863x0.77 )25 = 12.21
2
s ( 1)
2
d (1)
 2g = ¼ 
(t )
 43.61 6.11
P(1) = 

 6.11 13.61
(0)
*2
g s ( t 1)
+ ¼
(0)
*2
g d ( t 1)
+  g2m = 18.61
18.61
G(1) = 

6.11 
r(1) =
'G
b (1)
(1)
 g2
(1)
-1
.3883 
b(1) = P(1) G(1) = 

.2746 
= 0.69
How accurate is the parent average EBV?
selected proportion males
selected proportion females
Information used
5%
5%
Nr.Records
Index
weight
0
-
-
-
0
0
0
0.500
0.500
-
EBV
EBV
-
dam
sire
-
nr of own records
nr. of dams per sire
nr of progeny per dam
nr. of progeny
Equilibrium Va
Equilibrium h2
0.500
0.500
SD of EBV
0.383
Accuracy of EBV
correlation EBV FS
correlation EBV HS
No Bulmer Correction
STEBVaccurcay.xls
Run with Bulmer Correction
0.5412
1.000
0.500
Equilibrium Va
Equilibrium h2
0.364
0.422
SD of EBV
0.104
Accuracy of EBV
correlation EBV FS
correlation EBV HS
With Bulmer Correction
0.1730
1.000
0.500
Parameters
Heritability
Repeatability of subsequent records
c-squared (among full sibs)
selected proportion males
selected proportion females
Information used
0.5
0.5
0
5%
5%
Nr.Records
Index
weight
1
0.381
1
own
0.189
0.189
0.240
-
EBV
EBV
5
-
dam
nr of own records
nr. of dams per sire
0
nr of progeny per dam
6
nr. of progeny
0
recorded on both s exes
Equilibrium Va
Equilibrium h2
0.500
0.500
SD of EBV
0.556
Accuracy of EBV
correlation EBV FS
correlation EBV HS
No Bulmer Correction
STEBVaccurcay.xls
0.7865
0.500
0.296
sire
FS
-
Parameters
Heritability
Repeatability of subsequent records
c-squared (among full sibs)
selected proportion males
selected proportion females
Information used
0.5
0.5
0
5%
5%
Nr.Records
Index
weight
1
0.382
1
own
0.188
0.188
0.242
-
EBV
EBV
5
-
dam
nr of own records
nr. of dams per sire
0
nr of progeny per dam
6
nr. of progeny
0
recorded on both s exes
Equilibrium Va
Equilibrium h2
0.356
0.416
SD of EBV
0.409
Accuracy of EBV
correlation EBV FS
correlation EBV HS
With Bulmer Correction
STEBVaccurcay.xls
0.6851
0.229
0.106
sire
FS
-
Bulmer effect and BLUP selection (Dekkers, 1992)
NOTE: Incorporating Bulmer effect into pseudo-BLUP index does
NOT affect index weights.
BLUP EBV can be derived without considering the Bulmer effect.
However, the accuracy of BLUP EBV is affected by the Bulmer effect
and needs to be derived.
Important Henderson (1975) result:
Prediction error variance (PEV) of BLUP EBV does not depend on
selection, but only on the amount of effective information used:
PEV unaffected by selection 
 2
(0)
= (1- r( 0 ) ) 
2
2
g( o )
=
 2 = (1- r 2 )  g2
(t )
(t )
(t )
As a result, can derive Asymptotic or Steady State Genetic Variance
and Response
Select on BLUP EBV
Accuracy at the limit:
Equal selection in males and females:
r(2L ) = 1 - (1- r(20 ) )  g2( o ) /  g2( L )
Genetic var. at limit:
2
2
 2g = ½(1- k r( L ) )  2g + ½  g( o )
( L)
( L)
 2g = [1+k(1- r( 0 ) )] 

2
( L)
2
g( o )
2
2
 2g =  g( o ) /(1- k r( L ) )
( L)
/(1+k)
R(L)/R(0) = r( L )  g( L ) / r( 0 )  g( o ) =
Response at the limit:

1
1 k
Unequal selection in males and females:
is 2
R(L)/R(0) =
rs2( 0 )
rd2( 0 )
 kd (
(is
rs2( 0 )
rd2( 0 )
rs( 0 )
rd ( 0 )
 1)  id 2  k s (1 
 id ) 2  k s  k d
rs2( 0 )
rd2( 0 )
)
IF rs = rd R(L)/R(0) =
2
2  ks  kd
So with BLUP selection, the reduction in response does
not depend on the initial selection accuracy
Summary
In populations under selection, the variance decreases rapidly by about
20% due to LD between loci under selection
 Bulmer effect
Predicted responses will be reduced by a similar amount (compared with
assuming no reduced variance
As a result, the value of ‘family information’ (ancestors and collateral sibs),
will be reduced, as the variation between families is reduced
(e.g. parental mean)
 Effectively less accuracy as predicted by selection index