Íslenskar landbúnaðarrannsóknir - 01.03.1979, Page 98

96 ÍSLENZKAR LANDBÚNAÐARRANNSÓKNIR
From estimates ofgenotypic and phen-
otypic parameters of the characters inclu-
ded in the scoring index (Árnason, 1978),
a selection index will be developed so as to
maximize response from selection, in an
aggregate genotype, where the multiplic-
ation factors in the scoring index are taken
as direct indicators of economic impor-
tance of score of each point for each char-
acter.
H is not measureable and can not be sel-
ected for directly. Genetic improvement in
H is brought about by a single stage trun-
cation selection on an index:
I = b’X (2)
The genetic change in H, obtained by one
step of selection on I is:
AH = Bh.j ctj i (3)
Theory of selection indices
The theory of the selection index was or-
ginally developed by Smith (1936), and
extended by Hazel and Lush (1942). By
the work of Hazel (1943), the selection
index was introduced to animal breeding,
and is finding increasing use in recent ye-
ars due to the relative ease with which
indices can be calculated by computers.
The information required in con-
structing a selection index can be specified
in terms of the following 4 vectors and 2
matrices.
g = a vector of additive genetic values
(breeding values) for the traits
included in the aggregate genotype.
a = a vector of constants representing
the relative economic values of the
traits in g.
X = a vector of phenotypic variables to
be included in the index.
b = a vector of weighting factors to be
used in the index.
P = a variance-covariance matrix
appropriate to the vector X.
G = a variance-covariance matrix cor-
responding to g.
The aggregate genotype is defined as:
H = a’g (1)
where: BH . j = the regression coefficient
of H on the index.
erl = the standard deviation of the index.
i = the selection differential in standard
units of the normal distribution.
The aim is to find the b’s which max-
imize aH. This is equivalent to maximiz-
ing ®H • I°1 (since i is a constant). Using
Pij and Gij as the notation for the elements
in P and G respectively,
BH • I<U =
<rHj z^aibjGij W
°I (2 zbíbjPij)1'2
Bh . jcrj is maximized by differentiat-
ing the expression of log BH . jtrj with
respect to b; and equate the result to zero.
After scaling the weighting factors this
results in a set of simultaneous equations:
^bjPij = JajGij
lbjP2j = ZajG2j
ZbjPnj = ZajGnj
or: Pb = Ga (5)
to give:
b = p-'Ga
(6)