Page 130 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

where n is the number of records for animal i; n is the number of progeny records with
1 2
4
animal i as the dam; d = 2, or 1 when both, one or no parents of animal i are known;
3
2
1
k = 1 or when both or one parent of animal i are known; k = and k = 1 when the
2 3 1 2 3
1
2
mate of animal i is known or k = and k = with the mate unknown and:
1 3 3 3
⎡ 1 a ⎤
H = ⎢ a 2 ⎥
⎣ a 2 a 3⎦
For instance, the solutions for direct and genetic maternal effects for animal 5 are:
⎡ ˆ u 5 ⎤ ⎡ 1+(2+0.5)2.647 (2+0.5)1.176⎤ − 1 ⎡ u 1 +u 2 ⎤ ⎡ y 5 − b ˆ 1 − m ˆ 2 − ˆ p ⎤
ˆ
2
=
+
⎢ ⎥ ⎢ ⎥ Hk 2 ⎢ ⎥ ⎢ ˆ ⎥
2
⎣ ˆ m 5⎦ ⎣ (2+0.5)1.176 1+(2+0.5)4.412 ⎦ ⎣ m + m 2 ⎦ ⎣ y − b 1 − ˆ u − ˆ p 5⎦
8
8
2
⎡ ˆ a − 0.5()⎤
ˆ a
+ H k ⎢ 8 3 ⎥
3
⎣ ˆ m − 0.5 ( ˆ m ⎦
(
)
8
3
−
⎡ ˆ u 5 ⎤ ⎡ 1+(2+0.5)2.647 (2+0.5)1.176⎤ − 1 ⎡ 0.564+( 1.244)⎤
=
⎢ ⎥ ⎢ ⎥ H (1) ⎢ ⎥
−
⎣ ˆ m 5⎦ ⎣ (2+00.5)1.176 1+(2+0.5)4.412 ⎦ ⎣ 0.262+ (1.5583) ⎦
− −
−−
−−
⎡ 35 0 34.54 ( 1.583) ( 1.701)⎤
+ ⎢ ⎥
−
− −
⎣ 40 0 34.54 1.917 − 00.415 ⎦
−
⎡ 1.917 0.5(1.165)⎤ ⎡ 0 630 ⎤
.
+ H (1) ⎢ ⎥ = = ⎢ ⎥
−
⎣ − 0.158 0.5(0.736) ⎦ ⎣ −0 507. ⎦
The solution for the permanent environmental effect for dam j from the MME is:
ˆ
p ˆ e = (y − b − uˆ − mˆ )/(n + a ) (7.5)
j o o o j 2 3
where all terms are as defined in Eqn 7.4. For animal 5, the solution for the perma-
nent environmental effect is:
p ˆ e = 40 − 0 − 34.54 − 1.917 − (−0.507)/(1 + 8.75) = 0.415
5
Additive genetic maternal effects represent good mothering ability, which is
passed on from dams to progeny, while permanent environment effects refer to per-
manent environmental and maternal non-additive genetic influences on the mother-
ing ability of the dam. Thus selection of dams for the next generation in a maternal
line would place emphasis on good genetic maternal effects in addition to a good
estimate of breeding value. If equal emphasis is placed on both effects, dams 7 and 5
would be the top two dams in the example while dam 2 ranks lowest. However, if the
main interest is the performance of the future dams in the same herd, then selection
of dams would be based on some combination of the solutions for direct, maternal
genetic and permanent environmental effects for the dams. Again, in the example
data, dam 2 ranks lowest while the best two dams are dams 6 and 7 if equal emphasis
is placed on the three components.
In the case of males, the selection of sires for a maternal line, for instance, would
be based on a combination of solutions for direct and maternal genetic effects.
Obviously, sires 3 and 1 would be the top two bulls for such a purpose. However, if
the emphasis is only on direct genetic effects, probably to breed a bull, then sire 8 in
the example would be the bull of choice.
114 Chapter 7