Page 83 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

contribute to the resemblance between relatives. When members of a family are
reared together, such as litters of pigs, they share a common environment and this
contributes to the similarity between members of the family. Thus there is an addi-
tional covariance between members of a family due to the common environment
they share and this increases the variance between different families. The environ-
mental variance may be partitioned therefore into the between-family or group
2
component (s ), usually termed the common environment, which causes resem-
c
blance between members of a family, and the within-family or within-group vari-
2
ance (s ). Sources of common environmental variance between families may be due
e
to factors such as nutrition and/or climatic conditions. All sorts of relatives are
subject to an environmental source of resemblance, but most analyses concerned with
this type of variation in animal breeding tend to account for the common environ-
ment effects associated with full-sibs or maternal half-sibs, especially in pig and
chicken studies.

4.3.1 Defining the model

Genetic evaluation under this model is concerned with prediction of breeding val-
ues and common environmental effects and the phenotypic variance may be parti-
tioned into:
1. Additive genetic effects resulting from additive genes from parents.
2. Common environmental effects affecting full-sibs or all offspring of the same dam.
In the case of full-sibs, it may be confounded with dominance effects peculiar to off-
spring of the same parents. Further explanation is given later on the components of
the common environmental effect.
3. Random environmental effects.

In matrix notation, the model, which is exactly the same as in Eqn 4.1, is:
y = Xb + Za + Wc + e
where all terms are as given in Eqn 4.1 except c, which is the vector of common
environmental effects and W now relates records to common environmental
effects.
It is assumed that common environmental and residual effects are independently
2
2
2
distributed with means of zero and variance s and s , respectively. Thus var(c) = Is ,
c e c
2
2
var(e) = Is and var(a) = As .
e a
The MME for the BLUP of a and c and BLUE of estimable functions of b are
2
2
2
2
exactly the same as Eqn 4.2 but with a = s /s and a = s /s .
1 e a 2 e c
4.3.2 An illustration
Example 4.2
Consider the following data set on the weaning weight of piglets, which are progeny
of three sows mated to two boars:
Models with Random Environmental Effects 67