Page 339 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

Appendix F: Procedure for

Computing Deregressed

Breeding Values

The deregressed breeding values (DRB) of bulls used in multi-trait across-country
evaluations (MACE) are obtained by solving Eqn 5.15 for y considering data from
only one country at a time. Jairath et al. (1998) presented an algorithm for calculating
DRP. For instance, Eqn 5.15 for country i can be written as:
æ 1R 1 1 R -1 0 0 ö -1
′
′
-1
1R y ö
ç i i ÷ æ ˆ m i ö æ ′ i i ÷
ç
-1
-1
-1
-1
ç R1 R -1 + A a i A a i A a ÷ ç Qg + ˆ s ii ÷ ÷ ç -1 ÷
i
np
ng
i
i
nn
ç ÷ ç i = ç Ry i ÷ (f.1)
i
ç 0 A a i A a i A -1 a ÷ p ˆ ç i ÷ ç 0 ÷
1
-1
-1
i
pp
pg
pn
ç ÷ç ç ÷ ÷ ç ÷
ç -1 -1 -1 ÷è ˆ g i ø è 0 ø
è 0 A gn a i A gp a i A gg a i ø
−1
where p is the vector of identified parents without EBV and A are blocks of the
i jj
inverse of the relationship (see Chapter 3, Section 3.6) with j = n, p and g for animals
2
2
with records, ancestors and genetic groups, respectively, and a = (4 − h )/h , the ratio
i i i
of residual variance to sire variance for the ith country. The deregression of EBV
involves solving Eqn f.1 for y . The constant m and vectors s , p , g and y are
i i i i i i
unknown but a , the vector of genetic evaluations for sires, is known, as well as matri-
i
ces Q, R and A . Let a = 1m + Qg + s . The following iterative procedure can be
−1
−1
i jj i i i i
used to compute the vector of DRB, y :
i
1. Set 1m , p , s and g to 0.
i i i i
2. Calculate Qg + s = a − 1m .
i i i i
3. Compute:
-1
-1
-1
æ ˆ p ö æ A -1 A ö æ A ö ö
pg
pn
pp
i
s ˆ
ç ÷ =-ç ÷ ç ÷ (Qg ˆ i + )
i
è ˆ g i ø ç è A -1 A -1 ÷ ç A -1 ÷
ø è
gn ø
gp
gg
4. Generate:
Ry = R 1m i + ( R -1 + A )( Q +gˆ i s + A p a i + A gˆ a i
-1
-1
-1
-1
-1
)
pn i
i
nn
i
i
gn i
i
i
−1
and 1′R y
i i.
5. Now calculate:
−1
−1
−1
m = (1′R 1) 1′R y
i i i i
6. Continue at step 2 until convergence is achieved.
−1
7. Then compute DRB as y = R (R y ).
i i i i
Using the data for country 1 in Example 5.5, the deregression steps above are
illustrated in the first iteration. For country 1, a = 206.50/20.5 = 10.0732 and, con-
1
sidering only the bulls with evaluations, R = diag(0.0172, 0.0067, 0.0500, 0.0400).
1
© R.A. Mrode 2014. Linear Models for the Prediction of Animal Breeding Values, 323
3rd Edition (R.A. Mrode)