1. If we live with a deep sense of gratitude, our life will be greatly embellished.

2. Categorical Data Analysis
Chapter 10: Tests for Matched Pairs

3. Meta Analysis Also known as stratified analysis
Section 6.3.2: Cochran-Mantel-Haenszel test; test for conditional independence
Situation: When another variable (strata Z) may “pollute” the effect of a categorical explanatory variable X on a categorical response Y
Goal: Study the effect of X on Y while controlling the stratification variable Z without assuming a model

4. Example: Respiratory Improvement (SAS textbook, P. 46)
Center
Treatment
Yes No
Total 1
Test 29 16 45
Placebo 14 31 43 47 90 2 37 9 24 21 61

5. SAS Output Summary Statistics for trtmnt by response
Controlling for center
Cochran-Mantel-Haenszel Statistics (Based on Table Scores)
Statistic Alternative Hypothesis DF Value Prob
ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
Nonzero Correlation <.0001
Row Mean Scores Differ <.0001
General Association <.0001

6. What to Do if Dependent
(Section 6.3.5) When X and Y are NOT conditionally independent given Z, we would like to test for homogeneous association
(Section 6.3.6) If X, Y, Z have homogeneous association, we would like to estimate the common conditional odds ratio for X, Y given Z

7. SAS Output Estimates of the Common Relative Risk (Row1/Row2)
Type of Study Method Value % Confidence Limits
ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
Case-Control Mantel-Haenszel
(Odds Ratio) Logit
Cohort Mantel-Haenszel
(Col1 Risk) Logit
Cohort Mantel-Haenszel
(Col2 Risk) Logit
Breslow-Day Test for
Homogeneity of the Odds Ratios
ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
Chi-Square DF
Pr > ChiSq
Total Sample Size = 180

8. Matched-pair Data
Comparing categorical responses for two “paired” samples
When either
Each sample has the same subjects (or say subjects are measured twice) Or
A natural pairing exists between each subject in one sample and a subject from the other sample (eg. Twins)

9. Example: Rating for Prime Minister
Second Survey
First Survey
Approve
Disapprove
794
150 86
570

10. Marginal Homogeneity
The probabilities of “success” for both samples are identical (The data table shows “symmetry” across the main diagonal)
Eg. The probability of approve at the first and 2nd surveys are identical

11. Estimating Differences of Proportions
Sample estimate: P+1-P1+
Standard error of P+1-P1+ (based on the multinomial distribution of data):
Asymptotical (1-a) confidence interval:

12. McNemar Test (for 2x2 Tables only)
See SAS textbook Sec 3.7 (p. 40)
Ho: marginal homogeneity
Ha: no marginal homogeneity
A special case of C-M-H test; an approximate test (when n*=n12+n21>10)
Exact test (when n*=n12+n21<10)

13. Level of Agreement: Kappa Coefficient
The larger the Kappa coefficient is; the stronger the agreement is
The difference between observed agreement and that expected under independence compared to the maximum possible difference is called Kappa coefficient

14. SAS Output McNemar's Test Statistic (S) 17.3559 DF 1
Asymptotic Pr > S <.0001
Exact Pr >= S E-05
Simple Kappa Coefficient
Kappa
ASE
95% Lower Conf Limit
95% Upper Conf Limit
Sample Size = 1600
Level of agreement

15. Chi-square Test for Square Tables
Consider a IxI table
Marginal homogeneity:
Symmetry: for all pairs of cells,
Symmetry => marginal homogeneity
<=

16. Chi-square Test for Square Tables
Ho: symmetry vs. Ha: not symmetry
Fitted values:
Standardized Pearson residuals:
Pearson Chi-square Test statistic:
X^2 follows approximately Chi-square with df = I(I-1)/2

17. Example: Coffee Purchase
2nd purchase
1st purchase
High point
Taster’s
Sanka
Nescafe
Brim 93 17 44 7 10 9 46 11
155 12 6 4 15 2 27

18. Example: Coffee Purchase
X^2 = 20.4 and df is 5(5-1)/2=10
 lack of fit (reject Ho: symmetry)
 which pairs of cells cause the lack of fit? Examine their standardized Pearson residuals
 The pair (1,3) and (3,1) contribute the most; other pairs are fine (rij^2 is around 1 or less)