Page 170 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

é 0.5000 - 1.7320 1.4999ù
ê 0.5000 - ú
ê 0.9237 0.0999 ú
ê 0.5000 0.0000 - 1.44999ú
X s = ê ú
ê 0.5000 - 0.1154 0.0067 ú
ê 0.5000 0.8083 - 0.0999 ú
ê ú
ë ê 0.5000 1.7320 1.49999ú û
ˇ
Finally, delete from cˇ the elements for which C has elements i < j. The matrix cˇ is
ˇ
ˇ
ˇ
ˇ
now of the order k(k + 1)/2 by 1. For the example data, c′ = [C C C ]. The vector cˇ
00 10 11
can now be calculated by a weighted least-square procedure as:
-1
cˇ = (X′V ˆ -1 X ) X′ V ˆ -1 g ˘
s s s
For the example data, c calculated using the above equation is:
ˇ
cˇ = [341.8512 45.0421 24.5405]
ˇ
ˇ
The reduced coefficient matrix C is then constructed from the calculated c. Then a
row and column of zeros are inserted in positions corresponding to those polynomials
ˇ
ˇ
not included to obtain C. For the example data, C is now:
é341 8512. 45 0421 0 0 ù
.
.
C ˇ = ê ê 45 0421 24 5405 0 0 ú ú
.
.
.
ê ë 00 00 00 ú û
.
.
.
Kirkpatrick et al. (1990) presented the following chi-square statistic to test the good-
˘
ness of fit of the reduced covariance function to G:
χ 2 = (g - X cˆ)′ V (g˘ - X c)
˘
ˆ -1
ˆ
(m-p) s s
˘
where m = t(t + 1)/2 is the number of degrees of freedom in G and p = (k(k + 1)/2 is the
number of parameters being fitted. A significant result indicates that the model is
inconsistent with the data, and a higher order of fit may be needed. For the beef cattle
2
2
example, the value of c was 0.2231 with m = 6 and p = 3. This value of c was not
significant with three degrees of freedom and thus the reduced covariance function
˘
was not significantly different from G.
Another method of fitting a reduced-order CF, proposed by Mantysaari (1999),
involved eigenvalue decomposition of the coefficient matrix. The largest k eigenvalues of
ˆ
C in Eqn 9.9, for instance, are kept in a diagonal matrix (D ) and the matrix F replaced
a
ˆ
by the k corresponding eigenfunctions. Thus G in Eqn 9.7 can be approximated as:
ˆ
G » F[v v ... v ]D [v v ... v ]′F′ = TD T′
1 2 k a 1 2 k a
ˆ
where the v are the eigenvectors of C corresponding to eigenvalues in D .
i a
Similarly, if CF has been fitted to the environmental covariance matrix, a similar
reduction can be carried as follows:
R = FC F + Is e
2
p
= F[v v ... v ]D [v v ... v ]′F′ + Is e = QD Q′ + Is e (9.11)
2
2
1 2 k p 1 2 k p
154 Chapter 9