Page 112 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

Let y be vectors of observations:
var(y) = G + R (6.1)
where G and R are variance and covariance matrices for the additive genetic and
residual effects, respectively. Assuming G and R are positive definite matrices, then
there exists a matrix Q, such that:
QRQ′ = I and QGQ′ = W
where I is an identity matrix and W is a diagonal matrix (Anderson, 1958). This
implies that pre- and post-multiplication of R by the transformation matrix (Q)
reduces it to an identity matrix and G to a diagonal matrix. The multiplication of y
*
by Q yields a new vector of observations y that are uncorrelated:
*
y = Qy
*
var(y ) = W + I; which is a diagonal matrix.
Since there are no covariances between the transformed traits, they can be inde-
pendently evaluated. The procedure for calculating the transformation matrix Q is
given in Appendix E, Section E.1.

6.2.1 The model

A single trait analysis is usually carried out on each of the transformed variables.
The model for the ith transformed variable can be written as:
*
*
*
y = Xb + Za + e * (6.2)
i i i i
*
*
where y = vector of transformed variables for the ith transformed trait; b = vector of
i i
*
fixed effects for the ith transformed variable i; a = vector of random animal effects
i
*
for transformed trait i; e = vector of random residual errors for the ith transformed
i
trait; and X and Z are incidence matrices relating records to fixed and random effects,
respectively.
*
*
The MME to be solved to obtain the BLUE of b and the BLUP of a are the same
i i
as those presented in Section 3.2 for the univariate model. These equations are:
′ ⎤ ⎡ ⎤
′ ⎤
′
⎡XX XZ ˆ * ⎡ Xy * i
⎢ ′ − 1 ⎥ ⎢ ⎥ = ⎢ ⎥
b i
′ ⎥
′ +
⎣ ZX ZZ A a i ⎦ ⎣ ⎢ ⎥ ⎦ ⎢Zy * i ⎥ ⎦
*
⎣
a i ˆ
As explained earlier, it is assumed for the ith trait that:
*
*
*
var(a ) = Aw ; var(e ) = I and var(y ) = ZAZ′w + I
i ii i i ii
where w refers to the ith element of the diagonal matrix W.
ii
*
*
The MME are solved for b and a and the transformation back to the original
i i
scale is achieved as:
−1 *
b = Q b (6.3)
i i
a = Q a (6.4)
−1 *
i i
Thus the multivariate analysis is simplified to i single trait evaluations.
96 Chapter 6