Page 138 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

non-additive components Q and Q and environmental components E and E .
D,i S,i D,i S,j
The non-additive components may be combined with the environmental components
such that the equation may be expressed:
n−1
+
P = A + E D i ∑ A + E
,
i D i , , S j , S j
ji ≠
Therefore the phenotypic variance can be derived as:
n−1
var(P) = var[A + E Di ∑ A + E ]
+
Di , , S j S j ,
,
ji ≠
Given that cov(E , E ) = 0 when i ≠ j, and cov(A, E) = 0 for all i, j, then:
D,i S,j
−
n−1 n−1 n −1
2 2 + ∑ ) + ∑ , ∑
Sj ,
var() P = s + s var ( A , Sj var( E , Sj ) + 2 cov(A Di , A )
A D E D
ji ≠ ji ≠ ≠ ji
with cov((E , E ) = 0
S,j S,j′
when:
⎛ n−1 ⎞
2
j ≠ j′, var ⎜ ⎝ ∑ E sj ,⎟ ⎠ = n ( − 1 )s E s
≠
ji
2
Also given that cov((A ,A ) = r s where r is the relatedness between animals j and j′, then:
, Sj , Sj′ jj′ A S jj′
⎛ n−1 ⎞
2
2
r )
var ⎜ ⎝ ∑ A sj⎟ ⎠ = n ( − 1 )s A S + n ( − 1 )( n − 2 s A S
,
≠
ji
with r equal to the mean relatedness within the groups. Finally:
⎛
n−1
Di ∑ ⎞ =
r )
cov(A , A sj ,⎟ n ( − 1 s
, ⎜
⎝ ji ⎠ A DS
≠
Collecting all the terms together gives the phenotypic variance as:
s = 2 + 2 + ( n 1)( 2 + s ) ( n 1 2 + ( n − 2) s ]
−
+
−
2
2
2
[
)
r
p s A D s E D s A S E S s A DS A S
However, the total breed value (TBV; Bijma et al., 2007a) for individual i is:
TBV = A + (n − 1) A (8.1)
i D,i s,j
Note that TBV is what the progeny of the individual i will inherit and is the relevant
breeding value in computing response for selection for traits affected by associative
effects. Therefore, the total heritable variance of the trait equals the variance of the
2 ) among individuals and is:
TBVs (s TBV
2
2 = s + - - 2 2
s TBV A D 2( n 1) s A DS + ( n 1) s A S
2 2
where s , s and s are the variance of direct breeding value (DBV), associative
A D A S A DS
breeding value (SBV) and the covariance between DBV and SBV, respectively. The
sign of this covariance provides a measure of the competition versus cooperation
among group members. Negative values may be interpreted as ‘heritable competition’
in the sense that animals’ positive DBV on the basis of their phenotype has a negative
heritable impact on the phenotypes of their associates. On the other hand, a positive
covariance may be interpreted as ‘heritable cooperation’ (Bijma et al., 2007b).
2
Thus the ratio of total heritable variance to the phenotypic variance (t ) for traits with
associative effects (Bergsma et al., 2008) can be expressed as t = s TBV s . A comparison of
2
2
2
p
2 2 2 2
t to the classical heritability (h = s A s p ) indicates the proportional contribution of
indirect additive effects to the total heritable variance for traits with associative effects.
122 Chapter 8