Page 177 - Linear Models for the Prediction of Animal Breeding Values

1
When both sire (f) and dam (d) of i are known, s′A s = (a + a + a + a ) where
i i−1 4 ff fd df dd
−1
a are the elements of A for f and d. Since a = (1 + a ), then (a − s′A s) can be writ-
jj i−1 ii ff ii i i−1
−1
−1
ten as (1 − (a + a )) . The application of Eqn 10.11 to calculate A for the pedi-
1
4 ff dd
gree in Example 10.1 is straightforward. For instance, for the first two animals with
−1
−1
parents unknown, A is an identity matrix of order 2. Then A can then be calcu-
2 3
−1
lated using Eqn 10.11. Given that A has been calculated, the inverse of A for all five
4
animals can be illustrated as follows:
−1
−1
−1
For animal 5, (a − s′ A s ) = (1 − (a + a )) = (1 − (1 + 1.25)) = 2.286.
1
1
55 5 4 5 4 33 44 4
Then Eqn 10.11 is:
⎡ 2.000 0.500 −0.500 −1.000 0.000 ⎤ ⎛ 0.0 0.0 0.0 0.0 0.0 ⎞
0
⎢ ⎥ ⎜ ⎟
⎢ 0.500 1.500 −1.000 0.000 0.0000 ⎥ ⎜ 0.0 0.0 0.0 0.0 0.0 ⎟
⎢ ⎥ ⎜ ⎟
⎢
⎜
A 5 = 0.500− − 1.000 2.500 − 1.000 0.000 ⎥ ⎥ + (2.286) 0.0 0.0 0.25 0.25 − 0.5 ⎟ ⎟
−1
⎢
⎜
⎢ − 0.000 − ⎥ ⎜ 0..25 − ⎟
⎢ 1.000 1.000 2.000 0.000⎥ ⎜ 0.0 0.0 0.25 0.5 ⎟
⎢ ⎥ ⎜ ⎟
⎢ ⎣ 0 0.000 0.000 0.000 0.000 0.000 ⎦ ⎥ ⎝ 0.0 0.0 − 0.5 − 0.5 1.0⎠
⎡ 2.000 0.500 − 0..500 − 1.000 0.000 ⎤
⎢ ⎥
⎢ 0.500 1.500 − 1.000 0.000 0.000 ⎥
⎢ ⎥
⎢
= − 0.500 − 1.000 3.071 − 0.429 − 1.143 ⎥ ⎥
1
⎢
⎢ ⎥
⎢ − 1.000 0.000 − 0.429 2.571 − 1.143⎥
⎢ ⎥
−
⎣ ⎢ 0.000 0.000 − 1.143 −1.143 2.28 ⎦ ⎥
where s′ = (0 0 0.5 0.5).
5
Applying Eqns 10.9, Van Arendonk et al. (1994) showed that when alleles are
ordered chronologically, G can be calculated as:
v,i
é G vi-1 G vi-1 s i ù
,
,
G vi = ê s¢ i ë ê G v i-1 g ii ú û ú (10.12)
,
,
where s is the column vector of i − 1 elements containing non-zero elements relating
i
allele i to paternal and maternal alleles of parent (if known) and zeros elsewhere;
G is the covariance matrix for MQTL for alleles 1 to (i − 1); and g is the diagonal
v,i−1 ii
element of G for the k allele, which is equal to 1. Using the same notation for the rows
v
in G shown in Section 10.3.1, s for animals 3, 4 and 5 are: s′ =[1 – rr 0 0], s′ =
v i 3p 3m
[0 0 r (1 − r) 0], s′ = [r 1 – r 0 0 0 0], s′ = [0 0 0 0 (1 − r) r 0], s′ = [0 0 0 0 0 0
4p 4m 5p
(1 − r) r] and s′ = [0 0 0 0 (1 − r) r 0 0 0]. Thus G can easily be constructed using
5m v
Eqn 10.12.
10.5 Calculating the Inverse of G
v
Fernando and Grossman (1989) used an approach similar to that for setting up A −1
in calculating the inverse of G . They showed that G could be expressed as:
v v
G = (Q )′ HQ −1
−1
v
−1
Therefore, G can be written as:
v
G = QH Q′ (10.13)
−1
−1
v
Use of Genetic Markers in Breeding Value Prediction 161