Page 181 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

−1
identity matrix of order 4. The matrix G with the paternal MQTL allele of animal
v
3 added can be computed as:
⎡ 0.81 0.09 0 0 −0.9⎤
⎢ 0..09 0.01 0 0 − 0.1 ⎥
⎡ G − 1 ⎤ 0 ⎢ ⎥
−1
G − 1 = ⎢ v, 2 m ⎥ + (1 − 0.82) ⎢ 0 0 0 0 0 ⎥
v,3 p
⎣ ⎢ 0 ⎦ ⎥ 0 ⎢ ⎥
⎢ 0 0 0 0 0 ⎥
⎢ ⎣ − 0.9 − 0.1 0 0 1 ⎥ ⎦
⎡ 5.500 0.500 0.0 0.0 − 5.000⎤
0
⎢ 1.056 0.0 0.0 − ⎥
⎢ 0.500 0.556 ⎥
= ⎢ 0.0 0.0 0.0 0.0 0.0 ⎥
⎢ ⎥
0
⎢ 0.0 0.0 0.0 0.0 0.0 ⎥
⎢ ⎣ − 0.500 − 0.556 0.0 0.0 5.556 ⎥ ⎦

10.6 Prediction of Breeding Values with Marker Information

The model in Eqn 10.2 for breeding value prediction with marker information can be
written in matrix notation as:
y = Xb + Zu + Wv + e (10.17)
where y is the vector of observation, b is the vector of fixed effects, u is the random
vector for additive genetic effects due to loci not linked to ML, v is the random vector
with allelic effects at the MQTL and e is random residual effects. The matrices X, Z
2
2
2
and W are incidence matrices. Var(u) = A s , var(v) = G s , var(e) = Is and cov(u, v) =
u u v v e
cov(u, e) = cov(v, e) = 0.
The MME for the above linear model are:
′
′
′
b
⎡XX XZ XW ⎤ ⎡ ˆ ⎤ ⎡Xy
′ ⎤
⎢ −1 ⎥ ⎢ ⎥ ⎢ ⎥
′
′
′
ˆ a
⎢ ZX ZZ a 1 Z W ⎥ ⎢ ⎥ = Zy ⎥ (10.18)
⎢
′ + A a
⎢ −1 ⎥ ⎢ ⎥ ′ ⎥
′
v ˆ
′
′
⎣ WX WZ WW + G a 2 ⎦ ⎣ ⎦ ⎣ ⎢Wy ⎦
⎣
v
where:
2 2 2 2
1 e u 2 e v
a = s /s and a = s /s
10.6.1 An illustration
Example 10.2
Using the data for Example 10.1, the breeding value of animals for QTL not linked
to ML (simply referred to subsequently as breeding values), additive MQTL effects
are predicted for the beef calves and sex effects are estimated. It is assumed that
2
2
s = 0.3, s = 0.05 and s = 0.6. Therefore, a = 0.6/0.3 = 2 and a = 0.6/0.05 = 12.
2
u v e 1 2
The parameters are expressed as a proportion of the phenotypic variance. Note that
2
the total genetic variance s = (s + 2s ) = 0.3 + 2(0.05) = 0.40. Thus 40% of the
2
2
a u v
phenotypic variance is due to additive genetic variance, of which 25% can be
explained by the MQTL.
Use of Genetic Markers in Breeding Value Prediction 165