Page 143 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

Although solutions for sex of pig and common environmental effects were gener-
ally in the same direction in models with or without associative effects, there was a
major re-ranking of animals based on the EBVs. Griffing (1967) has indicated that
selection schemes that ignore this social effect of an individual on the phenotypes of
its group members could result in less optimum response, while Bijma et al. (2007a)
observed that the presence of social interaction among individuals may increase the
total heritable variance in a trait.

8.3 Partitioning Evaluations from Associative Models

The equations for DBV and SBV for animal i can be written as:
ˆ
′
−
⎡ZZ i D + A −1 Z i ′ Z S i + A −1 ⎤ ⎡ ˆ u ⎤ ⎡ Z i ′ (y i − Xb Z u ˆ ) ⎤
D
D
D
S j
S j
i
⎢ a 1 a 2 ⎥ ⎥ ⎢ D ⎥ = ⎢ ⎥
′
ˆ
−1
−1
−
⎢ZZ i D + A a 2 Z ′ Z S i + A ⎥ ⎣ ˆ u S ⎦ ⎢ ⎣ Z ′ (y j − Xb Z u S j − Z u ˆ ) ⎥
ˆ
⎣
⎦
D ⎦
S i
S i
S i
jD
D
S j
j
a 3
⎛ ZZ 0 ⎞ ⎛ yd ⎞
′
= ⎜ i D i D ⎟ ⎜ 1 ⎟
′
⎝0 ZZ )⎠ ⎝ yd ⎠ (8.8)
2
S i
S i
with i ≠ j and j = (1, n − 1), where n is the number of animals in the same group and:
yd = Z′ ( Z ) −1 Z′ ( y − ˆ −
s s j ) and
j
1 i D i D i D i Xb Z ˆ u
yd 2 = ( Z′ Z S i ) −1 Z′ (y − Xb − Z u ˆ jS − Z u ˆ )
y
S i
jD
S i
i
jD
jS
Thus yd is the yield record of animal i corrected for all fixed effects and the SBVs of
1
all other members in the same group, and yd is the average of the yield records of
2
all animals in the same group apart from animal i corrected for all fixed effects, the
DBVs and SBVs of the members of the group. Transferring the left non-diagonal
−1
terms of A in Eqn 8.8 to the right side of the equation gives:
′
′
ii
ii
⎡ZZ i D + a a 1 ZZ S i + a a 2 ⎤ ⎡ ˆ u D ⎤ ⎛a 1 a 2 ⎞ ⎛ PA 1 ⎞
D
i
D
i
⎢ ii ii ⎥ ⎥ ⎢ ⎥ = 2a par ⎜ ⎟ ⎜ ⎟
′
′
⎣ ⎢ZZ i D + a a 2 ZZ S i + a a ⎥ ⎣ ˆ u S ⎦ ⎝a 2 a 3 ⎠ ⎝PA 2 ⎠
⎦
3
S i
S i
⎛ ZZ 0⎞ ⎛ yd ⎞ ⎞ ⎛ u ˆ 2 − ˆ u ˆ Dmate ⎞
′
+ ⎜ iD i D ⎟ ⎜ 1 ⎟ + . 05 a prog ⎜ ⎛a 1 a 2 ⎟ ⎜ Dprog ⎟
′
⎝ 0 Z ZZ )⎠ ⎝ yd ⎠ ⎝a 2 a 3 ⎠ ⎝ 2 ˆ u Sprog − ˆ u Smate ⎠
2
iS
S i
2
where PA and PA are the parent averages for DBV and SBV for animal i; a = 1, 3
1 2 par
1
or if both, one or neither parents are known, respectively; and a = 1 if the animal’s
2 prog
2 ii
mate is known and 3 if unknown. Note that a = 2a + 0.5a , therefore pre-
par prog
multiplying both sides of the above equation by the inverse of DIAG, with:
⎡ ′ ZZ ii ′ ZZ + a a ⎤
ii
DIAG = ⎢ i D i D + a a 1 i D S i 2 ⎥
⎢ ⎣ ⎣ ′ ZZ i D + a a 2 ′ ZZ S i + a a ⎥
ii
ii
3 ⎦
S i
S i
gives:
ˆ u ⎡ D ⎤ ⎛ PA ⎞ ⎛ yd ⎞ ⎛ PC1 ⎞
1
1
⎢ ⎥ = WT 1 ⎜ ⎟ + WT 2 ⎜ ⎟ + WT 3 ⎜ ⎟ (8.9)
⎣ ˆ u S ⎦ ⎝ PA ⎠ ⎝ yd ⎠ ⎝PC2⎠ ⎠
2
2
Social Interaction Models 127