Page 226 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

If the nine subclasses in Eqn 12.6 are identified by 1, 2, 3, 4, 5, 6, 7, 8 and 9 (i.e. f = 1,
SD
2
f = 2, etc.), the covariances between f and its parent subclasses (cov(f , f )/s )
S,SD SD SD par f
using Eqn 12.9 are:
2 3 4 5 6 7 8 9
1 [0.5 0.5 0.5 0.5 0.25 0.25 0.25 0.25] (12.10)
2
and the relationship matrix among parent subclasses (var(f )/s ) using Eqn 12.9 is:
par f
⎡ 1.0 0.0 0.25 0.25 0.5 0.0 0.5 0.0⎤
⎢ ⎥
⎢ 0.0 1.0 0.25 0.25 0.0 0.5 0.0 0.5 ⎥
⎢ 0.25 0.225 1.0 0.0 0.5 0.5 0.0 0.0⎥
⎢ ⎥
⎢ 0.25 0.25 0.0 1.0 0.0 0.0 0.5 0.5 ⎥
⎢ 0.5 0.0 0.5 0.00 1.0 0.0 0.0 0.0 ⎥ (12.11)
⎢ ⎥
⎢ 0.0 0.5 0.5 0.0 0.0 1.0 0.0 0.0 ⎥
⎢ ⎥
⎢ 0.5 0.0 0.0 0.5 0.0 0.0 1.000.0 ⎥
⎣ ⎢ 0.0 0.5 0.0 0.5 0.0 0.0 0.0 1.0⎥ ⎦
From the two matrices above (Eqns 12.10 and 12.11) the regression coefficients
(Eqn 12.7) are:

b′ = [0.5 0.5 0.5 0.5 −0.25 −0.25 −0.25 −0.25] (12.12)

which are identical to the coefficients in Eqn 12.6. It should be noted that there is
no need to add more remote ancestors of S and D as the partial regression of these
are zero.

12.4.1 Inverse of the relationship matrix of subclass effects

The recurrences in Eqn 12.6 could be represented as:
f = Qf + e (12.13)

where f is the vector of sire by dam subclasses and the row i of Q contains the elements
of b from Eqn 12.7 in columns pertaining to identified parent subclasses of subclass i.
2
The relationship matrix for subclasses in f is F = var(f)/s . From Eqn (12.13):
f
f = (I − Q) e
−1
The variance–covariance of f is:
2 −1 −1 2
f f
var(f) = Fs = (I − Q′) R(I − Q) s
with:
Rs = var(e)
2
f
Therefore:
F = (I − Q′) R (I − Q) (12.14)
−1
−1
210 Chapter 12