Page 333 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

D.2.3 Value of records on ancestors

In the third step, contributions from parents, ancestors and collateral relatives are
accumulated for each animal, processing the pedigree from oldest to youngest.
However, in step two, contributions from descendants were accumulated for all
animals, hence E for parent j of animal i includes the contribution from animal i.
j
The contributions of animal i have to be removed from E to avoid double counting.
j
The corrected block is:
E = 1 G -1 - 4 G (- E + F + 4 G ) -1 G -1
*
-1
-1
j 3 9 i j 3
where F is the sum of contributions from all sources of information for parent j. As
j
parents are processed in the pedigree before progeny, F is always computed before
j
the contribution of parent j to animal i is required. For animal i, F is:
j
i å
i å
F = t i E + D + n i E k
*
*
j
j=1 k=1
with t = 0, 1 or 2 denoting the number of parents of animal i that are known.
i
The matrix T of the approximate PEV and PEC for the genetic effects for animal
i
i is:
T = (F + G ) −1
i i
The approximate reliability for a linear function of EBVs for animal i then is:
2
r = 1 − k′T k/k′Gk
i i
with k calculated as described in Section 9.3.4.

Appendix D 317