Page 231 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

which is the same as the D (Section 12.3.1) calculated from the pedigree using Eqn 12.1.
−1
Let D be partitioned as:
*
⎡ − 1 − 1 ⎤
− 1 D 11 D * 12
D = ⎢ * 1 ⎥
⎣ D * 21 D * 22⎦
* − 1 −
−1
−1
where D 11 is the top 12 by 12 block for dominance effects for animals, D 22 is
* −1 *
the bottom 6 by 6 block for subclass effects and D 12 is the block for dominance
* −1
by subclass effects. For the example data using Eqn 12.15, the submatrices of D
*
are:
(
− 1 =diag 43 43 4 3 4 3 43 43 43 43 43 43 43 4//3)
D * 11 /, /, /, /, /, /, /, /, /, /, /,
é 0 0 0 0 0 0 0 0 0 0 - 1.333 - 1.333ù
ê ú
ê 0000 0 0 - 1.333 0 0 0 0 0 ú
ê 0000 0 0 0 0 - 1..333 - 1.333 0 0 ú
D -1 = ê ú
* 12 ê 0000 0 - 1.333 0 0 0 0 0 0 ú
ê 0000 - 1.333 0 0 0 0 0 0 0 ú
ê ú
ë ê 0 0 0 0 0 0 0 0 0 0 0 0 0 û ú
−1
−1
D 12 is the transpose of D 21, and:
* *
⎡ 7.111 − 3.556 − 3.556 0 0 1.778⎤
⎢ ⎥
⎢ − 3.556 7.111 1.778 0 0 − 3.556 ⎥
⎢ −33.556 1.778 7.111 0 0 − 3.556⎥
− 1 = ⎢
D * 22 ⎥
⎢ 0 0 0 4.0 0 0 ⎥
⎢ 0 0 0 0 4.0 0 ⎥
⎢ ⎥
⎣ ⎢ 1.778 − 3.556 − 3.556 0 0 0 7.111⎥ ⎦
−1
The matrix D can be included in the usual MME for the prediction of domi-
*
nance and subclass effects.

12.5 Epistasis

Epistasis refers to the interaction among additive and dominance genetic effects; for
instance, additive by additive, additive by dominance, additive by additive by domi-
nance, etc. The epistasis relationship matrix can be derived from A and D as:

A#A for additive by additive
D#D for dominance by dominance
AA#D for additive by additive by dominance
where # represents the Hadamard product of the two matrices. The ij element of the
Hadamard product of the two matrices is the product of the ij elements of the two
matrices. Thus if M = A#B, then m = (a )(b ) where the matrices A and B should be
ij ij ij
of the same order.

Non-additive Animal Models 215