Page 46 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

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and A with inbreeding accounted for is:
1 2 3 4 5 6
1 1.833 0.5 −1.0 −0.667 0.0 0.0
2 0.5 2.033 −1.0 0.0 0.533 −1.067
3 −1.0 −1.0 2.5 0.5 −1.0 0.0
4 −0.667 0.0 0.5 1.833 −1.0 0.0
5 0.0 0.533 −1.0 −1.0 2.533 −1.067
6 0.0 1.067 0.0 0.0 −1.067 2.133
The calculation columns of L and a for the first three animals are illustrated below:
i
+
-
)
]
l 11 = [ 10 25 0 0 = 1
.
(
−1
a = 1 and its contribution to A is computed using the rules in Section 2.4.1
1
l = 0
21
l = 0.5(l + l ) = 0.5(1 + 0) = 0.5
31 11 21
l = 0.5(l ) = 0.5
41 11
l = 0.5(l + l ) = 0.5(0.5 + 0.5) = 0.5
51 41 31
l = 0.5(l + l ) = 0.5(0.5 + 0) = 0.25
61 51 21
-
+
.
)
(
]
l = [ 10 25 0 0 = 1
22
−1
a = 1 and its contribution to A is computed using the rules in Section 2.4.1
2
l = 0.5(l + l ) = 0.5(0 + 1) = 0.5
32 12 22
l = 0.5(l ) = 0.5(0) = 0
42 12
l = 0.5(l + l ) = 0.5(0 + 0.5) = 0.25
52 42 32
l = 0.5(l + l ) = 0.5(0.25 + 1.0) = 0.625
62 52 22
-
2
2
l = [ 10 25( l ) - 0 25( l 2 + l )]
.
.
33 11 21 22
+
-
(
)
]
(
)
.
.
= [ 1 0 25 1 - 0 25 0 1 = 0..707
a = 1/(0.707) = 2.0 and its contribution to A is computed using
−1
2
3
the usual rules
l = 0.5(l ) = 0.5(0) = 0
43 13
l = 0.5(l + l ) = 0.5(0 + 0.707) = 0.354
53 43 33
l = 0.5(l + l ) = 0.5(0.354 + 0) = 0.177
63 53 23
Faster algorithms for computing the inverse of A accounting for inbreeding based
on the L matrix have been published by Meuwissen and Luo (1992) and Quaas
(unpublished note, 1995), and these are presented in Appendix B.
2.5 Inverse of the Relationship Matrix for Sires and Maternal
Grandsires
In some cases, the prediction of breeding value is only for sires and maternal grandsires,
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the so-called sire and maternal grandsire (MGS) model. In such cases, the A to be
incorporated in the mixed model equations (MME) involves only sire and maternal
−1
grandsires and the rules for calculating A are different from those discussed in the
previous sections relating to pedigrees with individuals, sires and dams. With the
30 Chapter 2