Page 135 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

3
1
where d is either if both parents are known or if only one parent is known. For
2 4
the example data, both parents of the non-parent individuals are known,
therefore:
k = 1/(1 + 2(2.333)) = 0.17647
i
For animal 10, for instance, the breeding value is:
ˆ
ˆ
u = 0.5(uˆ + uˆ ) + k(y − b − b − mˆ − p ˆ − 0.5(uˆ + uˆ ))
ˆ
10 3 2 10 2 5 2 2 3 2
= 0.5(1.165 + −1.244) + 0.17647(22 − 3.386 − 27.691 − (−1.583)
− (−1.701) − 0.5(1.165 + −1.244))
= −1.055
BACK-SOLVING FOR MATERNAL EFFECTS
The equation for obtaining genetic maternal effects for non-parents can be derived
as follows. From the MME, the equation for direct and genetic maternal effects for
non-parent i is:
r ⎡ −1 + n g n g ⎤ ⎡ ˆ ⎡ u s + ˆ d ⎤ ⎡ i y − ˆ i b − m ˆ dam − pe ˆ dam ⎤
u i ⎤
−1 11
−1 12
u
ˆ
d
−1
−1
⎢ −121 −1 22 ⎥ ⎢ ⎥ = Gk 2 ⎢ ⎥ + r ⎢ ⎥ (7.9)
m i⎦
⎣ ng ng ⎦ ⎣ ˆ ⎣ m s ˆ + m d ˆ ⎦ ⎣ ⎢ ⎦ ⎥ 0
where n is defined in Eqn 7.8 and other terms are as defined in Eqn 7.4.
From the above equations:
22
m = [g (mˆ + mˆ ) + g (uˆ + uˆ ) − n g (uˆ )]/n g
21
ˆ
−1 22
−1 21
i s d s d i
ˆ
22
ˆ
21
ˆ
ˆ
21
m = n(m + m ) + [(g n(u + u ) − g u)/g ]
ˆ
ˆ
i s d s d i
21
m = n(mˆ + mˆ ) + g /g (n(uˆ + uˆ ) − uˆ )
22
ˆ
i s d s d i
Note that:
22
21
g /g = {−g /(g g − g g )}{(g g − g g )/g }
12 11 22 12 21 11 22 12 21 11
= −g /g
12 11
Therefore:
m = n(mˆ + mˆ ) + g /g (uˆ − d(uˆ + uˆ )) (7.10)
ˆ
i s d 12 11 i s d
When both parents are known:
ˆ
m = 0.5(mˆ + mˆ ) + (g /g )(uˆ − 0.5(uˆ + uˆ ))
i s d 12 11 i s d
For instance, for animal 10:
m = 0.5(mˆ + mˆ ) + (g /g )(uˆ − 0.5(uˆ + uˆ ))
ˆ
10 3 2 12 11 5 3 2
= 0.5(0.736 + (−1.583)) + (−40/150)(−1.055 − 0.5(1.165 + −1.244))
= −0.153
The solutions for direct and maternal effects of all non-parents in the example data
(animals 10 to 14) applying Eqns 7.7 and 7.9 are exactly the same as obtained for
these animals in the animal model.
7.4 Sire and Maternal Grandsire Model
In some cases, due to the structure of the available data, a sire and maternal grand-
sire model may be fitted for traits affected by direct and maternal genetic effects.
Maternal Trait Models: Animal and Reduced Animal Models 119