Page 50 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

3 Best Linear Unbiased

Prediction of Breeding Value:

Univariate Models with One

Random Effect

3.1 Introduction

In Chapter 1, the use of the selection index (best linear prediction) for genetic
evaluation was examined; however, it is associated with some major disadvan-
tages. First, records may have to be pre-adjusted for fixed or environmental factors
and these are assumed to be known. These are not usually known, especially when
no prior data exist for new sub-classes of fixed effect or new environmental fac-
tors. Second, solutions to the index equations require the inverse of the covariance
matrix for observations and this may not be computationally feasible for large
data sets.
Henderson (1949) developed a methodology called best linear unbiased predic-
tion (BLUP), by which fixed effects and breeding values can be simultaneously
estimated. The properties of the methodology are similar to those of a selection
index and the methodology reduces to selection indices when no adjustments for
environmental factors are needed. The properties of BLUP are more or less incor-
porated in the name:
● Best – means it maximizes the correlation between true (a) and predicted breeding
value (aˆ) or minimizes prediction error variance (PEV) (var(a − aˆ)).
● Linear – predictors are linear functions of observations.
● Unbiased – estimation of realized values for a random variable such as ani-
mal breeding values, and of estimable functions of fixed effects are unbiased
(E(a = a)).
ˆ
● Prediction – involves prediction of true breeding value.
BLUP has found widespread usage in genetic evaluation of domestic animals
because of its desirable statistical properties. This has been enhanced by the steady
increase in computing power and has evolved in terms of its application to simple
models, such as the sire model, in its early years, to more complex models such as the
animal, maternal, multivariate and random regression models, in recent years. Several
general purpose computer packages for BLUP evaluations such as PEST (Groeneveld
et al., 1990), BREEDPLAN, Mix 99 (Lidauer et al., 2011) and a host of others have
been written and made available. In this chapter, BLUP’s theoretical background is
briefly presented, considering a univariate animal model, and its application to several
univariate models in genetic evaluation is illustrated.

34 © R.A. Mrode 2014. Linear Models for the Prediction of Animal Breeding Values,
3rd Edition (R.A. Mrode)