Page 51 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

3.2 Brief Theoretical Background

Consider the following equation for a mixed linear model:
y = Xb + Za + e (3.1)

where:
y = n × 1 vector of observations; n = number of records
b = p × 1 vector of fixed effects; p = number of levels for fixed effects
a = q × 1 vector of random animal effects; q = number of levels for random effects
e = n × 1 vector of random residual effects
X = design matrix of order n × p, which relates records to fixed effects
Z = design matrix of order n × q, which relates records to random animal effects
Both X and Z are termed incidence matrices.
It is assumed that the expectations (E) of the variables are:
E(y) = Xb;E(a) = E(e) = 0
and it is assumed that residual effects, which include random environmental and non-
2
additive genetic effects, are independently distributed with variance s ; therefore,
e
2
2
var(e) = Is = R; var(a) = As = G and cov(a, e) = cov(e, a) = 0, where A is the
e a
numerator relationship matrix.
Then:
var(y) = V = var(Za + e)
= Zvar(a)Z′ + var(e) + cov(Za, e) + cov(e, Za)
= ZGZ′ + R + Zcov(a, e) + cov(e, a)Z′
Since cov(a, e) = cov(e, a) = 0, then:
V = ZGZ′ + R (3.2)
cov(y, a) = cov(Za + e, a)
= cov(Za, a) + cov(e, a)
= Zcov(a, a)
= ZG
and:
cov(y, e) = cov(Za + e, e)
= cov(Za, e) + cov(e, e)
= Zcov(a, e) + cov(e, e)
= R
The general problem with respect to Eqn 3.1 is to predict a linear function of
b and a, that is, k′b + a (predictand), using a linear function of y, say L′y (predictor),
given that k′b is estimable. The predictor L′y is chosen such that it is unbiased (i.e. its
expected value is equal to the expected value of the predictand) and PEV is minimized.
This minimization leads to the BLUP of a (Henderson, 1973) as:
ˆ
aˆ = BLUP(a) = GZ′V (y – Xb) (3.3)

−1
and:
ˆ
ˆ
L′y = k′b + GZ′V (y − Xb)
−1
Univariate Models with One Random Effect 35