Page 99 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

each daughter has one record per each trait, YD for the daughter i and trait j equals
ij
ˆ
(y − x b). Thus:
ij ij
æ YD ö æ201 - 175.5 ö æ 25 7 ö æ YD ö æ160 - 219.6 ö æ - 59 6ö
.
.
41
1
61
ç ÷ = ç ÷ = ç ÷ ; ç ÷ = ç ÷ = ç ÷
è YD 42 ø è 280 - 243.2 ø è 36 8 . ø è YD 62 ø è 190 - 243.2 ø è - 53 2 . ø
and:
⎛ YD ⎞ ⎛285 − 219.6 ⎞ ⎛ 65 4⎞
.
81
⎜ ⎝ YD ⎠ ⎟ = ⎜ ⎝ 300 − 240.6 ⎟ ⎠ = ⎜ ⎝ 59 4⎠ ⎟
.
82
For all three daughters, the dams are known, therefore W in Eqn 5.13 is the same
2prog
for all daughters and is:
−1
−1
W 2 prog =( Z RZ + 2′ −1 G ) ( Z RZ)′ −1
⎛ 0.2439 − 0.2176⎞ −1 ⎛ 0.0183 − 0.0071 ⎞ ⎛ 0.1713 0.0821⎞
= ⎜ ⎝ ⎝ − 0.2176 0.2802⎠ ⎟ ⎜ ⎝ 0.0071− 0.0170 ⎟ ⎠ = ⎜ ⎝ 0.1078 0.1244⎠ ⎟
8
The correction of the daughters’ YD for the breeding values of the mates of the sire
is follows:
⎛ 2YD − ˆ a 21 ⎞ ⎛ 51 4 −− ( 2 999 ⎞ ) ⎛54 399. ⎞
.
.
41
⎜ ⎝ 2YD 42 − ˆ a 22 ⎟ ⎠ = ⎜ ⎝ 73 6 −− ( 2 777 ⎠ ) ⎟ ⎟ = ⎜ ⎝76 377. ⎟ ⎠
.
.
⎛ 2YD − ˆ a 51 ⎞ ⎛ − 119 2 − −. ( 16 253⎞ ⎛ −102 947 ⎞
.
.
61
⎜ ⎝ 2YD − ˆ a ⎠ ⎟ = ⎜ ⎝ ⎝ − 106 4 − −. ( 15 824⎠ ⎟ = ⎜ ⎝ − 90 576 ⎟ ⎠
.
.
52
62
⎛2YD 81 − ˆ a 71 ⎞ ⎛ 130.8 8− .690⎞ ⎛ 122 .110⎞
.
⎜ ⎝2YD 82 − ˆ a 72 ⎟ ⎠ = ⎜ ⎝ 118 .8 8 .138⎠ ⎟ = ⎜ ⎝ 110 .662⎠ ⎟
−
equals 1 for all daughters of the bull, DYD for sire 1, using Eqn 5.13, is:
Since a prog
⎡ ⎛ 53.399⎞ ⎛ − 102.947⎞ ⎛ 122.110⎞ ⎤
DYD =(3 W 2 ) − 1 ⎢ W 2 prog ⎜ ⎟ + W 2 prog ⎜ ⎟ + W ⎜ ⎟ ⎥
⎣ ⎦
prog ⎝ 76.377⎠ ⎝ − 90..576⎠ 2 prog ⎝ 110.662⎠
⎛ 24.5207⎞
= ⎜ ⎝ 32.1543⎠ ⎟
3
Using Eqn 5.12, the breeding value of sire 1 can be calculated as:
⎛ ˆ a ⎞ ⎛ 24 .5207⎞ ⎛ . 7 439⎞
11
⎜ ⎝ ˆ a ⎠ ⎟ = M 3 ⎜ ⎝ 32 .1543⎠ ⎟ = ⎜ ⎝ . 7 387⎠ ⎟
12
where:
⎛ ⎞ − 1 ⎛ ⎞
.
M = 2G − 1 0.5 + 0.5G − 1 ∑ W 05G − 1 ∑ W prog ⎟
3 ⎜ ⎝ 3 2 prog a prog ⎟ ⎠ ⎜ ⎝ ⎝ 3 2 prog a ⎠
⎛ 0.1247 −0.1110 ⎞ −1 ⎛ 0.0120 −0.0058⎞ ⎛ 0.1937 0.0836⎞
M 3 = ⎜ ⎝ −0.1110 0.1432 ⎟ ⎠ ⎜ ⎝ −0..0058 0.0116⎠ ⎟ = ⎜ ⎝ 0.1099 0.1459⎠ ⎟
The vector of breeding value calculated for sire 1 using Eqn 5.12 is slightly lower than
that shown earlier in the table of results as contributions from the grand-progeny of
the sire are not included.
Multivariate Animal Models 83