Page 141 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

Although the number of equations to be fitted for Eqn 8.6 is usually more than for
Eqn 8.7, the systems of equations for Eqn 8.7 are denser and more difficult to set up.
8.2.1 Illustration of a model with social interaction

Example 8.1
Table 8.1 contains the growth rate data of nine pigs housed in three pens during the
finishing period in groups of three. The pigs are from three different litters and the
aim is to estimate the direct and associative breeding values for all pigs, estimate sex
effect and common environment effect as some of the pigs are full-sibs. It is assumed
2
2
that genetic variances for direct and associative effects are 25.70 g and 3.60 g ,
respectively, with a covariance of 2.25 g between them. Also, it is assumed that the
2
variance for common environmental variance (s ) is 12.5 g and residual variances
2
c
2
2
2
2
for direct (s ) and associative (s ) effects are 40.6 g and 10.0 g , respectively, and
E D E S
the correlation among pigs in the same pen (r) is 0.2.
The MME in Eqn 8.6 are initially used to analyse the data. Based on the given
genetic parameters:
2 2 4 6 + ( −
(
6 6
)
1
var e () = s + n − s = 0. 3 1 1 ) 0 = 0.
E D E S
2
2
Since r = cov(e , e )/var(e) = 0.2, and cov(e , e ) = s in Eqn 8.6, then s = r var(e) =
i
j
g
i
g
j
*
0.2 60.6 = 12.12.
Therefore, the residual variance relevant to the analysis using Eqn 8.6 with groups
*
2
fitted is var(e ) = var(e) − s = 60.6 – 12.12 = 48.48 and:
g
⎡ a 1 a ⎤ − 1 2 ⎡ 1 9956 − 1 2472⎤
.
.
2
⎢ ⎥ in[ . ] s e* = ⎢ ⎥
86 = G
.
.
⎣ a 2 a 3⎦ ⎣ − 1 2472 14 24462 ⎦
Setting up the incidence matrices X, V, W and Z in Eqn 8.6 follows the pattern
D
already described for other models in previous chapters, with Z being a diagonal
D
matrix for animals with records and:
⎛1 1 1 0 0 0 0 0 ⎞ 0
⎜
V = 0 0 0 1 1 1 0 0 0 ⎟
⎜ ⎟
⎝0 0 0 0 0 0 1 1 ⎠ 1
relating records to pen (groups).
Table 8.1. The growth rate of a set of finishing pigs.
Animal Sire Dam Pen Sex Growth rate (g/day)*10
7 1 4 1 Male 5.50
8 1 4 1 Female 9.80
9 2 5 1 Female 4.90
10 1 4 2 Male 8.23
11 2 5 2 Female 7.50
12 3 6 2 Female 10.00
13 2 5 3 Male 4.50
14 3 6 3 Female 8.40
15 3 6 3 Male 6.40
Social Interaction Models 125