Page 103 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

However, due to the inability of some countries to compute DYDs for bulls, the
deregressed proofs (DRP) of bulls became the variable of choice (Sigurdsson and
Banos, 1995) and the weighting factor became the effective daughter contributions
(EDC) of bulls (Fiske and Banos, 2001). The model in matrix notation is:

y = 1m + Z Qw + Z a + e (5.14)
i i i i i i i
where y is the vector of DRP from country i for one trait such as milk yield, m
i i
is a mean effect for country i, which reflects the definition of the genetic base for
that country, w is the vector of genetic group effects of phantom parents, a is
i i
the vector random sire proof for country i and e is the vector of random mean
i
residuals.The matrix Q relates sires to phantom groups (see Section 3.6) and Z
i i
relates DRP to sires. Given two countries, the variance–covariance matrix for w,
s and e is:
æ w ö æ ç A g A g A g A g 0 0 ö ÷
pn 11
pn 112
pp 12
pp 11
1
ç ÷ ç A g A g A g A g 0 0 ÷
ç w 2 ÷ ç pp 21 pp 22 pn 21 pn 22 ÷
ç s ÷ ç A g 11 A g 12 A nn g 11 A g 12 0 0 ÷
np
nn
np
varç 1 ÷ = ç ÷
ç s 2 ÷ ç A g A np 22 A nn 21 A nn 22 0 0 ÷
g
g
g
n np 21
ç e ÷ ç 2 ÷
ç ç 1 ÷ ÷ ç 0 0 0 0 D s e1 0 ÷
1
è e 2 ø ç ç è 0 0 0 0 0 D s 2 ÷
e2 ø
2
where n and p are the number of bulls and groups, respectively, g is the sire genetic
ij
(co)variance between countries i and j, and A is the additive genetic relationship for
all bulls and phantom parent groups based on the maternal grandsire (MGS) model
2
(see Section 3.6), s is the residual variance for country i, and D is the reciprocal of
ei i
the effective daughter contribution of the bull in the ith country.
The variable DRP, analysed in Eqn 5.14, are obtained by deregressing the national
breeding values of bulls such that they are independent of all country group effects and
additive genetic relationships among bulls, their sires and paternal grandsires, which
are included in the MACE analysis (Sigurdsson and Banos, 1995). DRP may therefore
contain additive genetic contributions from the maternal pedigree, which are included
at the national level but not in MACE. The deregression procedure involves solving the
MME associated with Eqn 5.14 for the right-hand side details. The details of the pro-
cedure are outlined in Appendix F. The computation of the EDC of bulls used as the
weighting factor for the analysis of DRP in Eqn 5.14 is dealt with in a subsequent
section.
The MME for the above model, which are modified such that sire solutions have
group solutions incorporated (see Section 3.6) are:
−
⎛ XR X X ′ R Z 0⎞ ⎛ c ⎞ ˆ ⎛ XR y⎞
′
−1
1
′
−1
⎜ −1 −1 −1 −1 −1 −1 ⎜ ⎟ ⎜ −1 ⎟
⎟
′
′
′
+
ˆ
⎜ Z R X Z RZ + A ⊗ G − AQ ⊗ G ⎟ ⎜ Qw a ˆ ⎟ = ⎜ ZR y ⎟ (5.15)
⎜ − 1 −1 −1 −1 ⎟ ⎝ w⎠ ˆ ⎜ ⎟
′
′
⎝ 0 − QA ⊗ G Q A Q ⊗ G ⎠ ⎝ 0 ⎠
Q
Genetic groups are defined for unknown sires and MGS on the basis of country of
origin and year of birth of their progeny. Also, maternal granddams (MGDs) are
always assumed unknown and assigned to phantom groups on the same basis.
Multivariate Animal Models 87