Geoscience Reference
In-Depth Information
The chi-square test is
k
2
∑
(
Xm
m
−
)
χ
2
i
i
=
,with (
k
−
1
) degreessof freedom
i
i
=
1
where
k
= Number of groups recognized.
X
i
= Observed count for the
i
th.
M
i
= Count expected in the
i
ith group if the hypothesis is true.
For the above data,
2
2
)
2
2
(
542 588
588
−
)
(
401 392
392
−
)
(
164
−
147
(
118
98
98
−
)
χ
2
dt
=
+
+
+
=
985
.
147
This value exceeds the tabular χ
2
with 3 degrees of freedom at the 0.05 level (i.e., it is greater than
7.81). Hence, the hypothesis would be rejected (if the geneticist believed in testing at the 0.05 level).
7.14.3 b
artlett
'
s
t
est
oF
h
omogeneity
oF
v
arianCe
Many of the statistical methods described later are valid only if the variance is homogeneous (i.e.,
variance within each of the populations is equal). The
t
test of the following section assumes that
the variance is the same for each group, and so does the analysis of variance. The fitting of an
unweighted regression as described in the last section also assumes that the dependent variable has
the same degree of variability (variance) for all levels of the independent variables.
Bartlett's test offers a means of evaluating this assumption. Suppose that we have taken random
samples in each of four groups and obtained variances (
s
2
) of 84.2, 63.8, 88.6, and 72.1 based on
samples of 9, 21, 5, and 11 units, respectively. We would like to know if these variances could have
come from populations all having the same variance. The quantities needed for Bartlett's test are
tabulated here:
Corrected Sum
of Squares (
SS
)
Group
Variance (
s
2
)
(
n
- 1)
1/(
n
- 1)
log
s
2
(
n
- 1)(log
s
2
)
1
84.2
8
673.6
0.125
1.92531
15.40248
2
63.8
20
1276.0
0.050
1.80482
36.09640
3
88.6
5
443.0
0.200
1.94743
9.73715
4
72.1
10
721.0
0.100
1.85794
18.57940
Sums
—
43
3113.6
0.475
—
79.81543
where the number of groups (
k
) = 4, and the corrected sum of squares (
SS
) is
−
(
)
2
∑
X
n
∑
2
2
X
=−
(
ns
1
)
From this we compute the pooled within-group variance:
∑
∑
(
SS
n
3113 6
43
.
i
2
s
=
=
=
72 4093
.
−
1
)
i
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