Page 139 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

Bijma et al. (2007a) presented this general formula for total genetic response per
generation (DG) to selection for traits with associative effects.
ΔG = {wt n − )( r 1 2 + (1 }
1
(
,
TBV − wt)s p TBV k
+ )s
s I
is the covariance between the phenotype of the individual and TBV, r meas-
p,TBV
where s
ures the degree of genetic relatedness, which is twice the coefficient of coancestry, k is the
selection intensity, s is the standard deviation of the index (I) that combines individual
I
phenotypes and phenotypes of group members, and wt defines the weights on individu-
als versus phenotypes of group members, such that:
n
I = wtP + −wt(1 ) ∑ p
i j
≠ ji
2 and the
Thus for a given r, n, wt and selection intensity, response is dependent on the s TBV
covariance between the phenotype of the individual and TBV. Therefore, response to selection
may not necessarily follow the same direction as the selection pressure as in classical quan-
titative theory. The interactions among individuals affect both the direction and magnitude
due to a large and
of selection response. Strong competition, for instance a negative s p,TBV
, will result in a response opposite in direction to the direction of selection.
negative s
A DS
8.2 Animal Model with Social Interaction Effects
Usually, data with associative effects tend to include animals that are full-sibs and
therefore there is the need to account for the common environmental effects in the
model. Thus the MME for a trait with social interaction effects could be written as:
y = Xb + Z u + Z u + Wc + e (8.2)
D D S S
where b is the vector of fixed effects, u and u are the vectors for direct and associa-
D S
tive genetic effects, respectively, c is the vector for common environmental effects and
e is the vector for residual error.
It is also assumed that:
u ⎡ D ⎤ ⎡g 11 A g 12 A ⎤
var ⎢ ⎥ = ⎢ ⎥
⎣ u S ⎦ ⎣ 21 A g 22 A ⎦
g
and if there are n animals in a group, then for the ith animal:
var e() = ( var(E + E ), j = 1 ,n −1 and i ≠ 2 + (n −1 2 (8.3)
i D ,i , S j ) j = s E D )s E S
Assuming that n = 3, with animals i, j and k in the group, then the residual covariance
between animal i and j in the same group or pen is:
cov = cov(e , e ) = cov(E + E + E ; E + E + E )
penmates i j D,i S,j S,k D,j S,i S,k
, (
(
(
= cov E , E ) + cov E , E ) + cov E , E ) = 2s E DS + n ( - 2 )s 2 E S (8.4)
Sk
,
S j
,
Sk
D j
,
,
,
D i
Si
s
Therefore, the correlation among animals in the same group (r) can be defined as:
2
ee
e
r = cov( , ) / var( ) = [2 s E DS + (n − s ) 2 E S ] / s [ 2 E D + (n − s ) 1 2 E S ]
j
i
Assuming that residual covariance among different groups is zero, the residual vari-
2
ance structure can then be defined as var(e) = R, with r = s , r = r s( 2 e ) for animals
ij
e
ii
i and j in the same group and r = 0 for animals i and j in different groups. Thus R is
ij
block diagonal and with n = 3, the block diagonal for one group is:
Social Interaction Models 123