11.3. Test of
General Case:
Suppose there are two
variables, column variable (with m categories) and
row variable (with p categories).
We want test the hypothesis
Row variable is independent column vairalbe v.s.
Row variable is
not independent column variable.
Suppose the sample size is
n. The contingency table is
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Column Variable (m columns) |
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1 |
... |
j |
… |
m |
proportions |
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Row Variable 1 (p rows) |
1 |
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… |
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… |
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p |
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… |
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proportions |
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1 |
If 
is true, then the
expected numbers under are
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Column Variable (m columns) |
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1 |
... |
j |
… |
m |
proportions |
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Row Variable 1 (p rows) |
1 |
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… |
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… |
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p |
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proportions |
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… |
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… |
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1 |
Note:
where
and
.
Chi-Square Test:
Let
As 
for every i and j, the chi-square test with level of significance 
for
Row variable is independent column variable
vs.
Row variable is
not independent column variable.
is
to
,
where
can be obtained
by
.
In
addition,
.
Example 3:
The following data are the
number of people who are in favor of, are not in favor of, and
have no comment on, some
proposal:
|
|
Favor |
Not Favor |
No Comment |
|
Male |
252 |
145 |
203 |
|
Female |
148 |
105 |
147 |
Please test if female and
male differ in their opinions about the proposal
at 5% level of
significance.
[solution:]
The
column totals are 
while
the row totals are 
.
In
addition, the total number is 1000. The table for the expected numbers 
is
|
|
Favor |
Not Favor |
No Comment |
Row Total |
|
Male |
|
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|
600 |
|
Female |
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|
400 |
|
Column Total |
400 |
250 |
350 |
1000 |
Thus,
Since
,
we
do not reject 
.
JavaStatSoft:
Chi-square
test:
Statistics
-> Tests -> Chi-Square Tests -> Test of