Page 180 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

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Applying these rules, the calculation of the inverse of G for the pedigree in
v
Example 10.1 is illustrated. For this pedigree, the matrix H and its inverse are:
H = diag(1111 0.18 0.18 0.18 0.18 0.1508 0.18) and
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H = diag(1111 5.556 5.556 5.556 5.556 6.630 5.556)
Note that in calculating the diagonal element for the paternal MQTL of animal 4 (d ),
4p,4p
an inbreeding coefficient of 0.162 (covariance between the maternal and paternal
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MQTL alleles of the sire and dam, respectively) has been accounted for. Set G with
v
ii,jj
elements represented as g to zero and the contribution from the first three animals
can be calculated as follows.
For animals 1 and 2, parents are unknown; the diagonal elements are equal to 1
for the MQTL alleles of these animals. Therefore, add 1 to g 1p,1p , g 1m,1m , g 2p,2p and
g 2m,2m , using the same coding as for the rows of G as in Section 10.3. For paternal
v
p
2 –1
MQTL allele of animal 3, r = 0.1 and d equals 5.556. Add (1 − 0.1) h = 4.50
o 3 p ,3 p 3 p ,3 p
2 –1
to g 1p1p , (1 − 0.1)0.1(h –1 ) = 0.5 to g 1p,1m , −(1 − 0.1)h –1 = −5.00 to g 1p,3p , (0.1) h = 0.056
3p,3p 3p3p 3p3p
to g 1m,1m , (−0.1)h –1 = 0.556 to g 1m,3p and h –1 to g 3p,3p . For the maternal allele of
3p3p 3p3p
animal 3, r = 0.9 and h –1 = 5.556. Add (1 − 0.9) h = 0.056 to g 2p,2p , (1 − 0.9)
p
2 –1
o 3m,3m 3m,3m
2 –1
0.9(h –1 ) = 0.5 to g 2p,2m , −(1 − 0.9)h –1 = −0.556 to g 2p,3m , (0.9) h = 4.50 to
3m,3m 3m3m 3m3m
g 2m,2m , (−0.9)h –1 = −0.500 to g 2m,3m and h –1 to g . Applying the rules to all
3m3m 3m3m 3m,3m
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animals in the pedigree gives G as:
v
1p 1m 2p 2m 3p 3m 4p 4m 5p 5m
1p 5.556 1.000 0.000 0.000 −5.000 0.000 −0.556 0.000 0.000 0.000
1m 1.000 5.556 0.000 0.000 −0.556 0.000 −5.000 0.000 0.000 0.000
2p 0.000 0.000 1.056 0.500 0.000 −0.556 0.000 0.000 0.000 0.000
2m 0.000 0.000 0.500 5.500 0.000 −5.000 0.000 0.000 0.000 0.000
3p −5.000 −0.556 0.000 0.000 14.556 1.000 0.000 −5.000 0.000 −5.000
3m 0.000 0.000 −0.556 −5.000 1.000 5.667 0.000 −0.556 0.000 −0.556
4p −0.556 −5.000 0.000 0.000 0.000 0.000 10.925 0.597 −5.967 0.000
4m 0.000 0.000 0.000 0.000 −5.000 −0.556 0.597 5.622 −0.663 0.000
5p 0.000 0.000 0.000 0.000 0.000 0.000 −5.967 −0.663 6.630 0.000
5m 0.000 0.000 0.000 0.000 −5.000 −0.556 0.000 0.000 0.000 5.556
Similarly, the inverse of G can be obtained using Eqn 10.11 (Van Arendonk
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v,i
et al., 1994) as:

⎡ G −1 ⎤ 0 −1 ss′ − s ⎤
⎡
G −1 , = ⎢ vi , −1 ⎥ +( ii − s′ i G v i , − ) ⎢ ii i ⎥ (10.16)
g
s
vi
1 i
⎣ ⎢0 ⎦ ⎥ 0 ⎣ −s′ i 1 1 ⎦
−1
The application of Eqn 10.16 for the calculation of G is briefly illustrated. It has
v
−1
been shown earlier that G for the MQTL alleles of the first two animals is an
v
164 Chapter 10