Íslenskar landbúnaðarrannsóknir - 01.03.1979, Page 103

TRAITS IN ICEL. TOELTER HORSES II. 101
ons of more than one variable been studi-
ed.
The solution to the matrix equation (6)
for progeny tests involves use of the in-
verse ofthe matrix [G — (P — G)n~‘]. As n
increases this matrix obviously approac-
hes G. In the case of G being a non-posit-
ive definite matrix this may result in an
invertion of a singular matrix, which is the
most probable explanation of the erran-
eous solutions obtained in table 2. The
influence of a non-positive deflnite G on a
selection index for individual selection is
probably not very serious in most cases, as
only P is inverted, P being a positive def-
inite matrix. Still the index is not correct
and could possibly be improved by reg-
ressing the genetic correlations iteratively
until positive definite G is obtained. This
solution to the problem is however quite
arbitrary and might be substituted later
by other methods, better established and
more satisfactory from theoretical view-
point.
After this paper had been written an
article on the probabilities of non-positive
definite genetic covariance matrices was
presented by W.G. Hill and R. Thompson
in Edinburgh (Hill and Thompson,
1978). Their work, which was partly
stimulated by previous result of the pres-
ent author (Árnason, 1976), indicates
high probability that estimated between-
group covariance matrix of many variab-
les is not positive deflnite, even for large
experiments.
ACKNOWLEDGEMENTS
I wish to express my appreciation to A.
Robertson and W.G. Hill at the Institute
of Animal Genetics, Edinburgh, for their
constructive criticisnr and assistance
throughout the preparadon of this study.
I am grateful to Halldór Pálsson and
Porkell Bjarnason for giving me an access
to the data and to-Gísli Sverrisson for its
preparation.
I am also indebted to Jenny Smith and
Marjorie McEwan for computational
assistance, and to Olafur R. Dýrmunds-
son and the referee for their valuable
comments on the manuscript.