Page 168 - Linear Models for the Prediction of Animal Breeding Values

˘
Thus for the example G:
˘
g′= [132.3 127.0 136.6 172.8 200.8 288.0]
Define F of order t by k, obtained by deleting (t − k) columns of F correspond-
r
ing to those f not in the reduced-order fit. The relationship between the observed covari-
j
˘
ance matrix, g, and the coefficient matrix of the reduced fit to be estimated is given
by the following regression equation:
g˘ = X cˇ + e (9.10)
s
where e is the vector of the difference between observed covariances and those pre-
2
dicted by the covariance function, c is a vector of dimension k , containing the
ˇ
ˇ
ˇ
elements of the coefficient matrix of the reduced fit (C). The order of elements of C
ˇ
ˇ
ˇ
ˇ
in cˇ is the same as in g: that is, cˇ = (C ,..., C ,..., C ,..., C ). X is the Kronecker
˘
00 k0 k1 kk s
product of F with itself (X = F × F ) and is of the order t by k . Since only the
2
2
r
r
r
s
first two polynomials are fitted, the matrix F can be derived by deleting from F
r
the third column, corresponding to the missing second-degree polynomial. Thus for
the beef cattle example:
é0 7071 -1 2247 ù
.
.
ê
.
F = 0 7071 - 0 0816 ú
.
r ê ú
ê0 7071 1 2247 ú û
.
.
ë
and X is:
s
⎡ 0.5000 − 0.8660 − 0.8660 1.4999⎤
⎢ 0.5000 − − ⎥
⎢ 0.0577 0.8660 0.0999 ⎥
⎢ 0..5000 0.8660 − 0.8660 − 1.4999⎥
⎢ ⎥
⎢ 0.5000 − 0.8660 − 0.0577 0.0999 ⎥
⎢
−
X s = 0.5000 −0.0577 −0.0577 0.0067 ⎥
⎢ ⎥
⎢ 0.5000 0.8660 −0.0577 −0.0999 ⎥
⎢ 0.8660 − ⎥
⎢ 0.5000 −0.86660 1.4999 ⎥
⎢ 0.5000 − 0.0577 0.8660 − 0.0999⎥
⎢ ⎥
⎣ 0.5000 0.8660 0.86660 1.4999 ⎦
ˇ
The application of weighted least squares to obtain solutions for c in Eqn 9.10
˘
requires the covariance matrix (V) of sampling errors of g. Kirkpatrick et al. (1990) pre-
sented several methods for estimating V, examining three different experimental designs.
˘
However, in animal breeding, most estimates of G are from field data and may not
fit strictly to the designs they described, but estimates of sampling variances from
˘
REML analysis could be used. For the example G for the beef cattle data, V has been
estimated using the formula given by Kirkpatrick et al. (1990) for a half-sib design, assum-
ing that 60 sires were each mated to 20 dams. The mean cross-product for the residual
ˆ
ˆ
ˆ
ˆ
effect (W) was estimated as W = P − 0.25G ˘ and that among sires (W) as W =
e e,ij ij ij a a,ij
(n − 1/4)G ˘ + P , where P is the phenotypic variance and n is the number of dams.
ij ij ij
ˆ
ˆ
ˆ
2
˘
Sampling variance for g was then calculated as: V = (16/n )[cov(W , W ) + cov(W ,
a,ij
e,ij
a,kl
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
W )], where cov(W , W ) = (W W + W W )/ df, with df = number of degrees of freedom
e,kl ij kl ik jl il jk
152 Chapter 9