Page 332 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

D.2 Computing Approximate Reliabilities for Random
Regression Models

Meyer and Tier (2003) extended the method in Appendix D.1 to estimate reliabilities
for multivariate and random regression models. They outlined several steps.

D.2.1 Determine value of observation for an animal

Compute the diagonal block (D ) for animal i in the MME, based on the information
i
from the data, as:
−1
D = Z′R Z
i i i i
However, to account for the limited subclass sizes of contemporary group effect, such
as HTD in dairy cattle, D can better be calculated as:
i
−1
−1
−1
−1
D = Z′(R − R (S )R )Z
i i i i i i i
−1
−1
where Z and R are submatrices of Z and R for animal i, and S is the block of
i i i
coefficient matrix pertaining to the contemporary groups of which animal i is a member.
Then the permanent environmental (pe) effects are also absorbed into the block
corresponding to animal genetic effects:
−1
*
−1
D = D − Z′R −1 Q (Q′R −1 Q + P )Q′R Z
i i i i i i i i i i i
where Q is a submatrix of the matrix Q defined in Section 9.3. Limited subclass
i
effects of pe can be accounted for by using weights w = (n − k)/n ≤ 1, for the mth
m m m
record, with n the size of the subclass to which the record belongs and k the number
m
of ‘repeated’ records it has in that subclass. Then R in the above equation is replaced
i
2
*
with R = Diag(w s ).
i m e
D.2.2 Value of records on descendants
In this second step, the contributions from progeny and other descendants are accu-
mulated for each animal, processing the pedigree from youngest to the oldest. Let E
i
be the block of contributions for animal i that has n progeny. Then:
i
æ n i ö -1
i å
E = 1 G -1 - 4 G ç D + E + 4 G ÷ G -1
-1
*
-1
i 3 9 ç k 3 ÷
è k=1 ø
This block is accumulated for both sire and dam of the ith animal. This equation can
be derived by assuming each progeny has only one parent known and that the parent
has no other information; then the MME are set up for the animal and the parent and
the equations for the animal are absorbed into those of the parent. The above equation
will give an overestimate of the individual’s contribution to its parents if it were in a
contemporary group with many of its half-sibs. This can be discounted by weighting
contributions with a factor dependent on the proportion of sibs in a subclass. Let H i
be a diagonal matrix of weights w < 1, with w = (n − s )/n , where n is the
m m m m m m
total number of records in the subclass for trait m and s the total number of sibs of
m
*
**
animal i in the subclass. Calculate D = H D H and then replace D with D .
**
*
i i i i i i
316 Appendix D