Page 136 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

This tends to be more common for calving traits such as calving ease or stillbirth
(Wiggans et al., 2003). The calving event is regarded as a direct effect of the service
sire (direct effect). This predicts how easily his progeny are born and is computed by
fitting the service sire. The maternal effect, which predicts how easily the bull daugh-
ters calve, is computed by fitting the MGS, hence the name sire-maternal grandsire
(S-MGS) model.
The model then is similar to Eqn 7.1 and can be written as:
y = Xb + Zs + Wmgs + Spe + e (7.11)

where y = vector of observations, s = vector of random service sire (direct) effects,
mgs = vector of random MGS (indirect) genetic effects and other terms defined as in
Eqn 7.1, but Z and W are now incidence matrices relating records to service sire and
MGS genetic effects, respectively. Note that if only first lactation data is being ana-
lysed, then the pe can be omitted from the model.
It is assumed that:
2
2
2 2 2 2 2 =( 1 s + 1 s + 1 ),
,
s
s
u
var(s) = As , with s = 0.25s ; var(mgs) = As mgs with s mgs 16 u 4 m 16 s u m
2
where s and s are the additive genetic variance and maternal genetic variance,
2
u m
respectively.
2
, = 1 s + 1
,
,
cov(s, mgs) = As s,mgs with s smgs 8 u 4 s u m
pe 2 = 3 2 3 2 3 2 ) and
4
16
4
var( ) = s pe ( s u + s m + s , u m + s pe
2 = 1 2 2 1 2
te
u
2
4
var( ) e = s e ( s + s )
The same principles described in Section 7.2 can be used in the application of
Eqn 7.11 to estimate breeding values and solutions for fixed effects. Note, however,
that MME from such an analysis will produce predicted transmitting abilities (PTAs)
(which is half of the EBV) for the service sire (direct effect) and PTAs for maternal
effect are computed as:
*
PTA maternal effect = PTA for MGS from MME − 0.25 (PTA for direct effect)
The variance components for a S-MGS model can be converted to variances for an
animal model direct and maternal effects from the details of the components of
2
2
the variances defined above. Thus the direct genetic variance component (s ) = 4s , the
u
s
* 2
u,m s,mgs u
covariance between direct and maternal component (s ) = 4 (s ) – 0.5s and the
2 2 2
m mgs u u,m
maternal genetic variance component (s ) = 4s – 0.25s − s . The computation
2
of maternal genetic component (s ) can be illustrated as:
m
2
2
2
2
2 = 4 ( 2 − 0 25 s − ) = 4( 1 s + 1 s + 1 ) − 0 25 s − 2
.
.
,
,
,
s m s mgs u s u m 16 u 4 m 4 s u m u s u um = s m
120 Chapter 7