Page 47 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

MGS model, the relationship matrix A required pertains to males and can be approxi-
mated (Quaas, 1984) as:
a = 1 + 0.25a (2.6)
ii sk
a = 0.5a + 0.25a (2.7)
ij sj kj
where s and k are the sires and maternal grandsires, respectively, for sire i. When all
maternal granddams are unrelated (base animals) and there are no maternal half-sibs,
the above will yield the exact A.
The inverse of approximate A can be calculated from a list of sires and mater-
−1
nal grandsires, applying Eqn 2.3. In this case, T is a lower triangular matrix
with ones in the diagonal and the only non-zero elements to the left of the diagonal
in the row for the ith animal are −0.5 and −0.25 for the columns corresponding
to the sire and maternal grandsire, respectively. The elements of D and therefore
D can be calculated in a manner similar to that described in Sections 2.3
−1
and 2.4. The diagonal elements of D (d ) for animal i are calculated by the fol-
ii
lowing rules.
If both sire (s) and maternal grandsire (k) are known:
1 1 2
ii
2
s
4 k
i
d = [var(u ) − var( u + u )]/s u
where the u terms are breeding values. Following the same arguments as in
Section 2.3:
d = 11 - 1 4 F - 16 F k
1
ii
s
16
where F and F are inbreeding coefficients for sire and maternal grandsire,
s k
respectively.
When only the maternal grandsire is known:
1 2
ii i 4 k u
d = [var(u ) − var( u )]/s
d = 15 - 1 F
ii 16 16 k
When only the sire is known or no parents are known, d is as calculated in
ii
Section 2.3.
−1
−1
The elements of D are reciprocals of D, calculated above. Using Eqn 2.3, A can
−1
−1
be calculated on the basis of T and D , defined above, as follows:
−1
Initially, set A to zero.
If both sire (s) and maternal grandsire (k) of animal i are known, add:
−1
d to the (i,i) element
ii
−1
−d /2 to the (s,i) and (i,s) elements
ii
−d /4 to the (k,i) and (i,k) elements
−1
ii
−1
d /4 to the (s,s) element
ii
−1
d /8 to the (s,k) and (k,s) elements
ii
d /16 to the (k,k) element
−1
ii
Genetic Covariance Between Relatives 31