1
Categorical
Data Analysis
Chapter 3: Inference for
Contingency Tables
2
Estimation of Association
Parameters
• Proportion difference
– point and interval estimators
• Relative risk
– point and interval estimators
• Odds ratio
– point and interval estimators
– Example
3
IxJ Contingency Tables
• Inference on difference, RR, odds
ratio
• Are X and Y independent?
• Ho : independence
4
Measure the Lack of
Independence
• Pearson chi-square statistic:
2
ˆ
(
n

u
)
ij
ij
2
 
uˆij
i, j
• Likelihood ratio (LR) test statistic:
nij
G  2 nij log
uˆij
i, j
2
If the statistic is too large, then we have a
strong evidence against independence.
5
Tests for Independence
• For Pearson or LR test, the df of the
chi-square test is the dimension of
the whole parameter space (Q) – the
dimension of the hypothesized
parameter space (Q0), i.e.
df = dim(Q) – dim(Q0)
6
• Poisson sampling:
df = IJ-(I+J-1)
=(I-1)(J-1)
• Single multinomial sampling:
df = (IJ-1)-(I+J-2)
= (I-1)(J-1)
• Independent multinomial sampling:
df = I(J-1)-(J-1)
= (I-1)(J-1)
7
Example: Oral Contraceptive
vs. Heart Attack
• Case-Control study: Retrospective
sampling; Column totals were fixed
Heart attack
Oral
Contra- Used
ceptives
Never used
Total
Yes
No
23
34
35
132
58
166
8
Follow-up Chi-squared Tests
• Pearson and standardized residuals
• Partitioning Chi-squared
9
Residuals
• Pearson residual:
eij 
nij  uˆij
uˆij
1/ 2
• Standardized Pearson residual:
e ij 
s
nij  uˆij
[uˆij (1  pi  )(1  p  j )]
1/2
10
Partitioning Chi-squared
• Describing association in IxJ table
• Partition a IxJ table to (I-1)(J-1) sub
2x2 tables
• Chi-squared= the sum of independent
(I-1)(J-1) chi-squareds
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Rules for Independent
Partitioning
1. S df for the subtables = (I-1)(J-1)
2. Each cell count nij must appear in
one and only one subtable
3. Each marginal total (ni+ or n+j) must
be a marginal total for one and only
one subtable
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Example: Aspirin vs. Heart
Attack
• Prospective sampling; Row totals were
fixed
Fatal H.A. Non-fatal No H.A.
H.A.
Placebo
18
171
10845
Aspirin
5
99
10933
13
Ordinary X: Trend Tests
• Test for Linear Trend alternative: M^2
• Choice of scores
• Example: Table 2.8. 1996 General Social Survey
Job Satisfaction
Income
Very
dissatisfied
Little
dissatisfied
Moderately
satisfied
Very
satisfied
<15K
1
3
10
6
15K-25K
2
3
10
7
25K-40K
1
6
14
12
>40K
0
1
9
11
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Exact Test for Independence
• The Chi-squared tests are for large
samples
• The Chi-squared tests are valid only
when The sample size is large
enough so that expected frequencies
are greater than or equal to 5 for
80% or more of the categories
15
Fisher’s Exact Test
• Consider a 2x2 table
• Under the three sampling methods,
what is the distribution of n11
conditional on n1+, n2+, n+1, n+2?
• Example: Table 3.8
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