Agriculture Reference
In-Depth Information
It can be seen that all the terms on the left-hand side
of the last equation cancel out. This test is called the
A-scaling test
.
Another relationship is:
2 B
2
−
F
1
−
P
2
=
be attributed to chance, then the model can be provi-
sionally accepted as an adequate description of reality.
If this explanation is too unlikely, then the hypothe-
ses (that A, B and C all equal zero within the bounds
of sampling error) must be rejected along with the
additive-dominant model on which they are based. The
statistical tests are called the A-scaling test, B-scaling test
and C-scaling test (and there are many others). How-
ever, the A-scaling test will be used here to represent the
principles involved in them all.
0
and:
2
m
1
2
[
1
2
[
−
a
]+
d
]
−
(
m
+[
d
]
)
−
(
m
−[
a
]
)
=
0
(
2
m
−[
a
]+[
d
]
)
−
(
m
+[
d
]
)
−
(
m
−[
a
]
)
=
0
The A-scaling test
The basis of the A-scaling test is that the value of A is
compared to the value predicted on the assumption that
an additive-dominant model is adequate (i.e. A
2
m
−[
a
]+[
d
]−
m
−[
d
]−
m
+[
a
]=
0
Once again, all the terms on the left-hand side cancel
out. This test is called the
B-scaling test
.
A final relationship, known as the
C-scaling test
, is:
4 F
2
−
=
0).
The question is then asked:
2 F
1
−
P
1
−
P
2
=
0
If the hypothesis is true (i.e. A
0), what is the proba-
bility that any difference between observation (A) and
prediction (zero) could be due to chance?
=
These relationships are based on the predicted means
of the various generations. Would you expect this rela-
tionship to hold if we substituted the means that we had
actually measured? In other words would you expect the
above equations to equal zero exactly? In fact, it would
be quite surprising if, for example, the sum of the means
of the P
2
and F
1
generations exactly equalled twice the
mean of the B
2
generation. While P
2
+
F
1
might be
approximately equal to 2 B
2
, error variation would give
rise to random variation in all three means resulting in
some overall discrepancy.
Thus, thinking now in terms of measured generation
means:
Conventionally, if the probability that the difference
was due to chance is less than 0.05 (i.e. 5%, or 1 in 20)
then the null hypothesis (in this case, that A
is equal to
0)
is rejected and the alternative hypothesis (that A
=
0)
is accepted. In accepting the alternative hypothesis, it is
also accepted that 2 B
1
−
F
1
−
P
1
is not equal to zero,
and that an additive-dominant model is inadequate in
this particular instance.
In comparing the actual value of A with its predicted
value (zero), what factors must be taken into account?
Clearly the magnitude of the discrepancy (i.e. the actual
value of A itself, 2 B
1
−
P
1
−
F
1
)
2 B
1
−
F
1
−
P
1
=
A
must be considered.
Another important factor is the
variability in A
from
one experiment to another. If A varied enormously from
one experiment to the next, then the mean value of A
would have to be relatively large for it to be significantly
different from zero. On the other hand, if the value of A
were relatively constant from experiment to experiment,
then even quite a small value of A could be accepted as
significantly different from zero.
Finally, values based on relatively few plants are likely
to be less convincing than values based on the measure-
ment of many plants. Thus sample size is also highly
relevant.
All of this perhaps sounds like a pretty tall order. In
fact, statisticians have provided us with a method of
relating the difference between the actual and predicted
2 B
2
−
F
1
−
P
2
=
B
4 F
2
−
2 F
1
−
P
1
−
P
2
=
C
where, A, B and C are all expected to equal zero. If they
do equal zero, or at least are not too far from it, then
there is no reason to suspect that the additive-dominant
model is inadequate as an explanation of the inheritance
of the continuously varying character. On the other
hand, if they do deviate markedly from zero, then there
is reason to doubt the adequacy of the model as an expla-
nation of the inheritance of the character in question.
This is a classic instance of the need for objective statis-
tical tests to decide whether A, B or C differ significantly
from zero, or whether any discrepancies observed could
be due to chance. If the discrepancies could reasonably
