Page 45 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

Recall from Section 2.3 that A can be expressed as:
A = TDT′
If L = T D
A = LL′ (2.4)
where L is a lower triangular matrix and, since D is diagonal, D refers to a matrix
obtained by calculating the square root of the diagonal elements of D. Equation 2.4
implies that the diagonal element of A for animal i is:
i
ii ∑
2
a = l im (2.5)
m=1
Thus for a pedigree consisting of m animals:
a = l 2
11 11
a = l + l 2
2
22 21 22
a = l + l + l 2
2
2
33 31 32 33

a = l 2 + l 2 + l 2 + ... + l 2
mm m1 m2 m3 mm
From the above, all the diagonal elements of A can be computed by calculating L one
column at a time (Quaas, 1984). Only two vectors of dimension equal to the number
of animals for storage will be required: one to store the column of L being computed
and the second to accumulate the sum of squares of the elements of L for each animal.
The matrices L and A can be computed using the following procedure:
−1
From Eqn 2.4 the diagonal element of L for animal i is:
ii l = d i
ii l = [ 0.5 0.25( s F + F )]
-
d
ss a ) ; ]
ii l = [ 1.0 0.25( a + a dd with a = + Fss and a = + Fdd
-
1
1
dd
ss
Using equation [2.5]:
⎡ ⎛ s d ⎞ ⎤
l = ⎜ ∑ sm ∑ 2
+
2
ii ⎢ 10. − 025. l l dm ⎟ ⎥
⎣ ⎢ ⎝ m=1 m=1 ⎠ ⎥ ⎦
−1
−1
To set up A at the same time, calculate the diagonal element of D (a ) for animal i
i
−1
2
as a = 1/l . Then compute the contribution of animal i to A , applying the usual
i ii
−1
rules for computing A (see Section 2.4.1).
The off-diagonal elements of L to the left of the diagonal for animal i are calculated as:
l = 0.5(l + l ); s and d equal to or greater than j
ij sj dj
For the example pedigree used in Section 2.4.1 the L matrix is:
1 2 3 4 5 6
1 1.0 0.0 0.0 0.0 0.0 0.0
2 0.0 1.0 0.0 0.0 0.0 0.0
3 0.5 0.5 0.707 0.0 0.0 0.0
4 0.5 0.0 0.0 0.866 0.0 0.0
5 0.5 0.25 0.354 0.433 0.707 0.0
6 0.25 0.625 0.177 0.217 0.354 0.685
Genetic Covariance Between Relatives 29