Íslenskar landbúnaðarrannsóknir - 01.03.1979, Page 102

100 ÍSLENZKAR LANDBÚNAÐARRANNSÓKNIR
TABLE 2.
Obtained values (bí) of selection indices aimed at maximizing the correlation between a progeny test and
the breeding value of the parent tested.
No. of
Offspring Traits
n Xi Xa X3 X4 X5 Xe X7 Xs X9 Xio
i 0.48 0.10 0.60 0.59 0.46 0.76 0.20 -0.16 -0.06 -0.54
20 6.58 5.98 0.73 8.33 6.47 4.92 -3.44 0.19 -3.14 -13.90
40 -7.44 25.31 -15.24 -6.71 7.43 9.11 2.05 -0.10 8.45 -0.87
100 -3.07 7.25 10.89 5.94 -1.89 2.87 -4.57 9.35 0.11 7.43
oo 8.00 4.00 8.00 3.33 2.67 3.33 2.00 4.00 2.67 2.00
eny groups of size 1 and oo were the same
as the bi in the individual selection index
and four times the value of the „base“
index (4Xai) respectively. This was actu-
ally obtained, apart from rounding error,
but intermediate progeny group sizes res-
ulted in quite suspicious bi values, e.g. for
n = 20, n = 40, n = 100, very fluctuating
relative values given to the traits were
obtained, and no directional trend tow-
ards the limits of infínite progeny group
size noted. See table 2.
This lead to a closer check of the mat-
rices. Negative eigenvalues (latent roots)
were found for the genotypic matrix (G),
thus indicating that G was not a positive
defmite matrix. This means that the con-
ditions for a set ofvariables, including the
multivariate normal distribution, did not
hold. If yi, yi. . . ,yP are p random var-
iables, and if Y is the p X 1 vector of these
variables, one of the requirements for the
function
-Ví(Y -M)’ R(Y-M)
f(yl y2 . .. .yp) = Ke
oo<yp< oo
to be a multivariate normal frequency
function is that R is a positive definite
matrix where elements (rij) are constants.
(Graybill, 1961; p 48).
The fact that G is not a positive definite
matrix corresponds to some partial gen-
etic correlations exceeding their defined
limits (+1 to —1). The matrix was thor-
oughly checked for errors in calculations,
so this is probably only a result of sampl-
ing variance. This should be a warning to
people constructing selection indices for
practical use, including many traits. It
can easily be seen (Árnason, 1978; table
5) that all the parameter estimates are
within defined limits of uni- and bivariate
distributions. The partial correlations
which exist between traits quickly become
very complicated as the number of traits
increases, and undefined parameter est-
imates are therefore easily hidden to vis-
ual inspection. Only a thorough check and
evaluation of eigenvalues of the matrices
can prohibit the possibilities of this kind of
error in construction of selection indices.
To the author’s knowledge, the effect of
these kind of errors on effíciency of selecti-
on indices has not been investigated, nor
has the actual probability of getting im-
possible estimates of fixed genetic param-
eters following defined frequency functi-