2. Categorical Data Analysis
Chapter 3: Inference for Contingency Tables

3. Estimation of Association Parameters
Proportion difference
point and interval estimators
Relative risk
Odds ratio
Example

4. IxJ Contingency Tables
Inference on difference, RR, odds ratio
Are X and Y independent?
Ho : independence

5. Measure the Lack of Independence
Pearson chi-square statistic:
Likelihood ratio (LR) test statistic:
If the statistic is too large, then we have a strong evidence against independence.

6. 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)

7. 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)

8. Example: Oral Contraceptive vs. Heart Attack
Case-Control study: Retrospective sampling; Column totals were fixed
Oral
Contra-ceptives
Heart attack
Yes No
Used 23 34
Never used 35
132
Total 58
166

9. Follow-up Chi-squared Tests
Pearson and standardized residuals
Partitioning Chi-squared

10. Residuals
Pearson residual:
Standardized Pearson residual:

11. 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

12. Rules for Independent Partitioning
S df for the subtables = (I-1)(J-1)
Each cell count nij must appear in one and only one subtable
Each marginal total (ni+ or n+j) must be a marginal total for one and only one subtable

13. Example: Aspirin vs. Heart Attack
Prospective sampling; Row totals were fixed
Fatal H.A.
Non-fatal H.A.
No H.A.
Placebo 18
171
10845
Aspirin 5 99
10933

14. Ordinary X: Trend Tests
Test for Linear Trend alternative: M^2
Choice of scores
Example: Table General Social Survey
Job Satisfaction
Income
Very dissatisfied
Little dissatisfied
Moderately satisfied
Very satisfied
<15K 1 3 10 6
15K-25K 2 7
25K-40K 14 12
>40K 9 11

15. 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

16. 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