Íslenskar landbúnaðarrannsóknir - 01.03.1970, Page 59

COLOUR INHERITANCE IN ICF.LANDIC SHEEP 57
TABLE 16
Genotypic probabilities in terms of redefined parameters
Allcle from sire Allele from dam Sum
Ak Ai
A, ai bk ai bi ai
^ ai + aj bj, + bj ai + aj bk + bi ai + aj
A- ai bk ai bi aJ
J g,i “ as + aj bk + bj S‘ “ ai + aj bk + bi ai + aj
b. b.
1
bk + bi bk + bi
and b,. 'l’his makes possible the assessment
of the overall representation of an allele
among observed progeny, irrespective of
the genotype it occurs in, and from the
aj’s, aj’s, bk’s and b/s the probability to
be associated with each allele in all indi-
vidual genotypes can be calculated.
The new parameters a^, a^, bk and b,
have been estimated by the method of
maximum likelihood, on he basis of the
data in table 9.
In the estimation, allele Aö was exclud-
ed because of the low numbers of progeny
in the matings where it occurred. This
gave 10 parameters, a4 to a5 and bx to b5
to be estimated. Because only the relative
magnitude of the parameters can be assess-
ed and not their absolute magnitude, all
the parameters were redefined, by dividing
through all the a’s by a5, and all the b’s
by b5. This gives a solution for a4 to a4
and b4 to b4 as a ratio to a5 and b5 re-
spectively, while a5 and b5 will occur as
constants of value 1 in the equations. The
number of parameters actually solved for
was thus reduced to 8.
The probabilities for the 4 genotypic
classes in table 14, when expressed in terms
of the redefined parameters are shown in
table 16.
The maximum likelihood estimation was
based on the probabilities for the pheno-
typic classes in situation 1—7, as shown in
table 15, where these probabilities were ex-
pressed in terms of the redfined express-
ions for the g’s shown in table 16.
Table 17 shows the number of pheno-
typic classes in each situation, and the
probabilities within each class. The likeli-
hood for a single mating within each
situation was then:
eL oc npi"i,
i
ancl the complete likelihood was obtained
as the product of tlie likelihoods from all
situations and all matings.
The log-likelihood and its derivatives
consist of additive terms. For ease of
handling, the terms relating to each situa-
tion were differentiated separately and the
expectations of the 2nd derivatives found.
A computer program was then developed
which read in as data the indices of the