Agriculture Reference
In-Depth Information
Now to consider the mean value of A, this is given by:
as the
joint scaling test
. This test effectively combines
the whole set of scaling tests into one and thus offers
a more general, more convenient, more adaptable and
more informative approach.
The joint scaling test consists of estimating the
model's parameters,
m
,
2 B
1
−
F
1
−
P
1
=
A
=
(
)
−
−
=
2
107.3
105.5
109.1
2.0
Finally:
[
]
[
]
from the means
of all types of families available, followed by a compari-
son of these observed means with their expected values
derived from the estimates of the three parameters. This
makes it clear at once that at least three types of fam-
ily are necessary if the parameters of the model are to
be estimated. However, with only three types of fam-
ily available no test can be made of the goodness of fit
of the model since in such a case a perfect fit
must
be
obtained between the observed means and their expec-
tations from the estimates of the three parameters. So
to provide such a test, at least four types of family must
be raised.
The procedure for the joint scaling test is illus-
trated by considering the example given by Mather
and Jinks (
Introduction to Biometrical Genetics
). The
data they presented have been truncated for simplicity
and so differences due to rounding errors may occur.
Their example consists of a cross between two pure-
breeding varieties of rough tobacco (
Nicotiana rustica
).
The means and variances of the means for plant height
of the parental, F
1
,F
2
and first back-cross families
(B
1
and B
2
)
a
and
d
=
/σ
A
=
/
=
0.142.
So, a value of
t
has been calculated for these data. This
is based on both the deviation of A from its expected
value of zero (i.e. 2.0) and the variability found in the
P
1
,F
1
and B
1
plants measured, all the variability being
summarized in the standard error of A (i.e. 14.1). The
question now is the deviation statistically significant?
In order to decide this, it is necessary to account
for the number of plants measured in each generation
on which the values of A and
t
A
2.0
14.1
σ
A
are based. In fact, it
is not the number of plants as such that is used but
the relevant numbers of degrees of freedom, where the
degrees of freedom of A
=
the degrees of freedom of
B
1
+
F
1
+
the degrees of freedom of
the degrees of
freedom of P
1
.
Degrees of freedom have been previously mentioned
in connection with the
2
test. Generally, the number
of degrees of freedom associated with a generation is
one fewer than the number of plants representing that
generation. Thus, if 11 plants of each of generations B
1
,
F
1
and P
1
were measured then the degrees of freedom
of A
χ
derived from this cross are shown in
=
(
−
)
+
(
−
)
+
(
−
)
=
30 df.
It is necessary to look up the value of
t
(i.e. 0.142)
for 30 degrees of freedom in a table of probabilities
for
t
. As the
t
value we obtained is smaller in magnitude
compared to the table value with 30 degrees of freedom
so there is no reason to reject the additive-dominant
model in this instance, and so it is provisionally accepted
as an adequate explanation of the inheritance of the
character in question.
11
1
11
1
11
1
Table 5.7.
Also shown in this table is the number of plants
that were evaluated from each generation. Family size
was deliberately varied with the kind of family. It was
set at as low as 20 for the genetically uniform parents
and in excess of 100 for the F
2
and back-crosses, to
compensate for the greater variation expected in these
segregating families. All plants were individually ran-
domized at the time of sowing so that the variation
within families reflects all the non-heritable sources of
variation to which the experiment is exposed. With this
design the estimate of variance of a family mean (
V
x
)
Joint scaling test
,
valid for use in the joint scaling test, is obtained in the
usual way by dividing the variance within the family by
the number of individuals in that family. Reference to
these variances shows that the greater family size of the
segregating generations has more than compensated for
their greater expected variability in that the variances
of their family means are smaller than those of their
non-segregating families.
The procedure described above can be repeated in a
similar way to derive a test for the B-scaling test or the
C-scaling test. Indeed, sets of such scaling tests can be
devised to cover any combination of types of family that
may be available.
As an alternative, however, to testing the various
expected relationships one at a time, a procedure was
proposed by a researcher called Cavalli, which is known
