Page 61 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

2
where r is the squared correlation between the true and EBVs. Thus:
2 2 2
i e a
d s = (1 − r )s
22
where d is the ith diagonal element of C .
i
2
2
d s /s = 1 − r 2
i e a
2
i
r = 1 − d a
and the accuracy (r) is just the square root of reliability.
From Eqn 3.14 the standard error of prediction (SEP) is:
SEP = var (a − ˆ) a
2
i
= d s e for animal i
Note also that:
2
2
2
r = 1 − (SEP /s )
a
The inverse of the coefficient matrix for Example 3.1 is:
⎡ 0.596 0.157 − 0.164 − 0.084 − 0.131 − 0.265 − 0.148 − 0.166 − 0.284 − 0.238⎤
⎢ 0.802 − 0.133 − 0.241 − 0.112 − 0.087 − 0.299 − 0.306 − 0.186 − ⎥
9
⎢ 0 0.157 0.199 ⎥
⎢ − 0.164 − 0.133 0.471 0.007 0.033 0.220 0.045 0.221 0.139 0.134⎥
⎢ ⎥
⎢ − 0.0884 − 0.241 0.007 0.492 − 0.010 0.020 0.237 0.245 0.120 0.111 ⎥
⎢ − 0.131 − 0..112 0.033 − 0.010 0.456 0.048 0.201 0.023 0.126 0.218 ⎥
⎢ ⎥
⎢ − 0.265 − 0.087 0 0.220 0.020 0.048 0.428 0.047 0.128 0.243 0.123⎥
⎢ − 0.148 − ⎥
⎢ 0.299 0.045 0 0.237 0.201 0.047 0.428 0.170 0.220 0.178 ⎥
− ⎢ 0.166 − 0.306 0.221 0.245 0 0.023 0.128 0.170 0.442 0.152 0.219⎥
⎢ ⎥
⎢ − 0.284 − 0.186 0.139 0.120 0.126 0 0.243 0.220 0.152 0.442 0.168 ⎥
⎢ ⎣ − 0.238 − 0.199 0.134 0.111 0.218 0.123 0 0.178 0.219 0.168 0.422 ⎦ ⎥
2
The r , r and SEP for animals in Example 3.1 are:

Animal Diagonals of inverse r 2 r SEP
1 0.471 0.058 0.241 4.341
2 0.492 0.016 0.126 4.436
3 0.456 0.088 0.297 4.271
4 0.428 0.144 0.379 4.138
5 0.428 0.144 0.379 4.138
6 0.442 0.116 0.341 4.205
7 0.442 0.116 0.341 4.205
8 0.422 0.156 0.395 4.109

In the example, the reliabilities of animals with records are generally higher than
those of ancestors since each has only two progeny. The two calves in the female sex
subclass are progeny of dam 2 and this may explain the very low reliability for this
ancestor as the effective number of daughters is reduced. The amount of information

Univariate Models with One Random Effect 45