Page 56 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

The solutions indicate that male calves have a higher rate of gain up to weaning than
females calves, which is consistent with the raw averages for males and females. From
the first row in the MME (Eqn 3.4), the equations for sex effect are:
ˆ
(X′X)b= X′y − (X′Z)a ˆ
ˆ b= (X′X) X′(y − Zaˆ)
−1
Thus the solution for the ith level of sex effect may be written as:
⎛ ⎞
ij ∑
ˆ
b = ⎜∑ y − ˆ a diag (3.5)
i ij ⎟ i
⎝ j j ⎠
where y is the record and aˆ is the solution of the jth animals within the sex subclass
ij ij
i and diag is the sum of observations for the sex subclass i. For instance, the solution
i
for male calves is:
b = [(4.5 + 3.5 + 5.0) − (−0.009 + −0.249 + 0.183)]/3 = 4.358
1
The equations for animal effects from the second row of Eqn 3.4 are:
ˆ
(Z′Z + A a)aˆ = Z′y − (Z′X)bb
−1
ˆ
−1
(Z′Z + A a)aˆ = Z′(y − Xb )
(Z′Z + A a)aˆ = (Z′Z)YD (3.6)
−1
ˆ
with YD = (Z′Z) Z′(y − Xbb ), where YD is the vector of yield deviations (YDs) and
−1
represents the yields of the animal adjusted for all effects other than genetic merit
−1
and error. The matrix A has non-zero off-diagonals only for the animal’s parents,
progeny and mates (see Section 2.4), transferring off-diagonal terms to the right-hand
side of Eqn 3.6 gives the equation for animal i with k progeny as:
∑ u (aˆ − 0.5a )
ˆ
ii
s
d
i
ip
(Z′Z + u a)aˆ = au (aˆ + aˆ ) + (Z′Z)YD + a im anim m
k
where u is the element of the A between animal i and its parents with the sign reversed,
−1
ip
and u is the element of A between the animal and the dam of the kth progeny.
−1
im
Therefore:
∑ u (2a ˆ − a ) (3.7)
ˆ
ii i par prog anim m
(Z′Z + u a)aˆ = au (PA) + (Z′Z)YD + 0.5a
k
where PA is the parent average, u = 2(u ), with u equal to 1, ; or if both, one or
2
1
par ip ip 3 2
neither parents are known and u = u , with u equal to 1 when the mate of animal
prog im im
2
i is known or when the mate is not known.
3
−1
Multiplying both sides of the equation by (Z′Z + u a) (VanRaden and Wiggans,
ii
1991) gives:
a = n (PA) + n (YD) + n (PC) (3.8)
i 1 2 3
where:
PC = ∑u prog (2a ˆ anim − a )/∑u prog
ˆ
m
k k
is regarded as the progeny contribution and n , n and n are weights that sum to
1 2 3
one. The derivation of the equation for PC is given in Appendix C, Section C.3. The
numerators of n , n and n are au , Z′Z (number of records the animal has) and
1 2 3 par
0.5aΣ u , respectively. The denominator of all three n terms is the sum of the three
k prog
numerators.
40 Chapter 3