Page 42 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

−1
i are unknown, all elements of row i are zero. For the pedigree in Table 2.1, T can
be calculated as:
é 1 00000ù é 0.0 0.0 0.0 0.0 0.0 0.0ù
ê ú ê ê ú
ê 0 1 0000 ú ê 0.0 0.0 0.0 0.0 0.0 0.0 ú
ê 00 1 000ú ê 0.5 0.5 0.0 0.0 0.0 0.0ú
0
ê ú - ê ú
ê 000 1 0 0 ú ê 0.5 0.0 0.0 0.0 0.0 0.0 ú
ê 0000 1 0 ú ê 0.0 0.0 0.5 0.5 0.0 0.0 ú
ê ú ê ú
ë ê 00000 1ú û ë ê 0.0 0.5 0.0 0.0 0.5 0.0ú û
M
(I) ) ()
é ù
ê 1.0 0.0 0.0 0.0 0.0 0.0ú
ê ú
ê 0.0 1.0 0.0 0.0 0.0 00.0 ú
ê - 0.5 - 0.5 1.0 0.0 0.0 0.0 ú
= ê ú
- ê 0.5 0.0 0.0 1.0 0.0 0.0ú
ê 0.0 - 0.5 - ú
ê 0.0 0.5 1..0 0.0 ú
ê 0.0 - 0.5 0.0 0.0 - 0.5 1.0ú
ê ë ( T - 1 ) ú û

and:
−1
D = diag(1, 1, 2, 1.333, 2, 2.133)

2.4.1 Inverse of the numerator relationship matrix ignoring inbreeding

The relationship shown in Eqn 2.3 was used by Henderson (1976) to derive simple
−1
rules for obtaining A without accounting for inbreeding. With inbreeding
−1
4
ignored, the diagonal elements of D are either 2, or or 1 if both or one or no
3
−1
parents are known, respectively. Let a represent the diagonal element of D for
i
−1
animal i. Initially set A to zero and apply the following rules.
If both parents of the ith animal are known, add:
a to the (i,i) element
i
−a /2 to the (s,i), (i,s), (d,i) and (i,d) elements
i
a /4 to the (s,s), (s,d), (d,s) and (d,d) elements
i
If only one parent (s) of the ith animal is known, add:
a to the (i,i) element
i
−a /2 to the (s,i) and (i,s) elements
i
a /4 to the (s,s) element
i
If neither parent of the ith animal is known, add:
a to the (i,i) element
i

26 Chapter 2