Page 77 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

4 Best Linear Unbiased

Prediction of Breeding Value:

Models with Random

Environmental Effects

4.1 Introduction

In some circumstances, environmental factors constitute an important component of
the covariance between individuals such as members of a family reared together
(common environmental effects) or between the records of an individual (permanent
environmental effects). Such environmental effects are usually accounted for in the
model to ensure accurate prediction of breeding values. This chapter deals with models
that account these two main types of environmental effects in genetic evaluations.

4.2 Repeatability Model

The repeatability model has been employed for the analysis of data when multiple
measurements on the same trait are recorded on an individual, such as litter size in
successive pregnancies or milk yield in successive lactations (Interbull, 2000). The
details of the assumptions and the components of the phenotypic variance have been
given in Section 1.3.2. Briefly, the phenotypic variance comprises the genetic (additive
and non-additive) variance, permanent environmental variance and temporary envi-
ronmental variance. For an animal, the model usually assumes a genetic correlation
of unity between all pairs of records, equal variance for all records and equal envi-
ronmental correlation between all pairs of records. In practice, some of these assump-
tions do not hold in the analysis of real data. A more appropriate way of handling
repeated measurements over time is by fitting a random regression model or a covari-
ance function, and this is discussed in Chapter 9. This section has therefore been
included to help illustrate the evolution of the model for the analysis of repeated
records over time. The phenotypic structure for three observations of an individual
under this model could be written (Quaas, 1984) as:
+
+
+
2
2
⎡ y ⎤ ⎡ s + 2 pe s 2 g s 2 pe s 2 g s 2 pe s ⎤ ⎤
g
1
t1 s
⎢ ⎥ ⎢ 2 2 2 2 2 2 2 ⎥
+
var y 2⎥ = ⎢ s pe s g s t2 + s pe + s g s pe + s g⎥
⎢
s
⎥
⎣ y ⎢ 3⎦ ⎢ ⎢ ⎣ s pe + s g 2 s pe + s g 2 s t3 + s pe + s g ⎦ ⎥ ⎥
2
2
2
2
2
2
2
with: s = temporary environmental variance specific to record i; s = covariance
ti pe
due to permanent environmental effects (variances and covariances are equal); and
2
s = genetic covariance (variances and covariances are equal). The correlation
g
© R.A. Mrode 2014. Linear Models for the Prediction of Animal Breeding Values, 61
3rd Edition (R.A. Mrode)