Agriculture Reference
In-Depth Information
than the recombinant types. The question therefore,
is this linkage, or simple random sampling variation.
To determine this we would use the
accept chance alone as being responsible for this partic-
ular deviation; it represents an event that would occur,
on a chance basis, much less often than the one-time-in-
twenty that we have agreed on as our point of rejection;
this event would occur less often than even the one-
time-in-a-hundred that we decided to regard as highly
significant.
In the case of the example we noted for barley, the
expected frequency of each of the 4 phenotypes if no
linkage is present would be 100 (total of 400, with
four equal expectations), which would lead to devia-
tions of 12, 11, 7 and 6, these squared and divided by
the expected value (144
2
χ
test.
Tall,
Tall,
Short, Short,
6-row 2-row
6-row
2-row
Observed (obs)
112
89
93
106
Expected (exp)
100
100
100
100
obs
−
exp(
d
)
12
11
7
6
d
2
144
121
49
36
/
100
+
121
/
100
+
49
/
100
+
d
2
/exp
1.44
1.21
0.49
0.36
2
value of 3.5. This value is com-
pared in probability tables for
36
/
100) gives a
χ
3df
=
(
d
2
2
χ
/
exp
)
=
3
.
50 n.s.
2
values and it falls just
below the 50% probability table value with 3 degrees
of freedom, clearly not close to the accepted 5% prob-
ability we accept as showing significance. We therefore
say that there is no evidence that linkage exists and that
the observed deviations are likely to have happened by
random chance (sampling error).
χ
The number of degrees of freedom in tests of genetic
ratios is almost always
one less than the number of classes.
To be more precise, it is the number of observable data
that are independent. For example, in a two gene test
cross there are four possible phenotypes, expected in
equal frequency. If 400 plants are observed and there
are 50 in the first group, 120 in the second group and
140 in the third group; then by definition there must
be 90 is the last group (400
Improper use of chi-square
The two most important reservations regarding the
straightforward
use
of
the
chi-square
method
in
genetics are:
=
90). A test of 1:2:1 ratio would have two degrees of
freedom. In just the same way, if we were testing two
groups in tests of 1 : 1 or 3 : 1 ratios there is one degree of
freedom.
Do not confuse assigning degrees of freedom to
genetic frequencies with degrees of freedom in
−
−
−
50
120
140
•
Chi-square can usually be applied only to numer-
ical frequencies themselves, not to percentages or
ratios derived from the frequencies. For example, if
in an experiment one expects equal numbers in each
of two classes, but observes 8 in one class and 12
in the other, we might express the observed num-
bers as 40% and 60%, and the expected as 50% in
each class. A chi-square value computed from these
percentages can not be used directly for the determi-
nation of
p
. When the classes are large, a chi-square
value computed from percentages can be used, if it is
first multiplied by
n
/100, where
n
is the total number
of individuals observed.
2
con-
tingency tables. In a two-way contingency table, with
pre-assigned row and column total, one value can be
filled arbitrarily, but the others are then fixed by the
fact that the total must add up to the precise number
of observations involved in that row or column. When
there are four classes, any three are usually free, but the
fourth is fixed. Thus, when there are four classes, there
are usually three degrees of freedom.
Remembering the example given earlier, calculated
chi-square
χ
•
Chi-square cannot properly be applied to distribu-
tions in which the expected frequency of any class
is less than 5. In fact, some statisticians suggest that
a particular correction
be
applied if the frequency
of any class is less than 50. However, the approxi-
mations involved in chi-squares are close enough for
most practical purposes when there are more than 5
expected in each class.
=
10 for one degree of freedom for the
sample of 40 individuals. We can look this value up
in the probability tables for chi-square and in this case
chance alone would be expected in considerably less
than one in a hundred independent trials to produce as
large a deviation as that obtained. We cannot reasonably
