Page 81 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

The fixed-effect solutions for parity indicate that yield at second lactation is higher
than that at first, which is consistent with the raw averages. From the MME, the
solution for level i of the nth fixed effect can be calculated as:

diag in
b ˆ in = å y inf - å b ˆ inj - å a ˆ ink - å pe ˆ inl / diag in n (4.3)
f=1 j k l
where y is the record for animal f in level i of the nth fixed effect, diag is the number
inf in
of observations for level i of the nth fixed effect, b , aˆ and pe are solutions for
inj ink inl
levels j, k and l of any other fixed effect, random animal and permanent environmental
effects, respectively, within level i of the nth fixed effect. Thus the solution for level
two of HYS effect is:
ˆ
ˆ
b = [445 − (2b ) − (aˆ + aˆ ) − (p ˆe + p ˆe )]/2
21 12 6 8 6 8
= [445 − 2(175.472) − 5.807 − (0.118)]/2
= 44.065
Breeding values for animals with a repeatability model can also be calculated
using Eqn 3.8, except that YD is now yield corrected for the appropriate fixed effects,
permanent environmental effect and averaged. Thus for animal 4:
ˆ
ˆ
ˆ
ˆ
a = n [(aˆ + aˆ )/2] + n [((y − b − b − p ˆe ) + (y − b – b − p ˆe ))/2]
ˆ
4 1 1 2 2 41 1 5 4 42 3 6 4
+ n (2aˆ − aˆ )
3 7 3
where y is yield for cow j in lactation i, n = 2.8/5.5, n = 2/5.5 and n = 0.7/5.5 and
ji 1 2 3
5.5 = the sum of the numerator of n , n and n .
1 2 3
ˆ
a = n (3.532) + n [((201 − 0.0 − 175.472 − 8.417)
4 1 2
+ (280 − 0.0 − 241.893 – 8.147))/2] + n (18.656 − (−7.063))
3
= 13.581
The higher breeding value for sire 1 compared with sire 3 is due to the fact that
on average the daughters of sire 1 were of higher genetic merit after adjusting for the
breeding values of mates. The very high breeding value for cow 8 results from the
high parent average breeding value and she has the highest yield in the herd, resulting
in a large YD.
The estimate of pe for animal i could be calculated as:
æ é mi öù
ˆ pe = ç ê å Y - b ˆ - å ˆ a ÷ú ) (4.4)
i ç if å ij ik ÷ (m +a 2
i
ê è ë f j k øû ú
where m is the number of records for animal i a = s /s and other terms are as
2
2
i 2 e pe
defined in Eqn 4.3. Thus for animal 4:
p ˆe = [(201 − 0.0 − 175.472 − 13.581)
4
+ (280 − 0.0 − 241.893 − 13.581)]/(2 + 2.333)
= 8.417
The estimate of permanent environment effect for an animal represents environ-
mental influences and non-additive genetic effect, which are peculiar to the animal
and affect its performance for life. These environmental influences could either be

Models with Random Environmental Effects 65