1. If we can reduce our desire,
then all worries that bother us will disappear.

2. Statistical Package Usage
Topic: Basic Categorical Data Analysis
By Prof Kelly Fan, Cal. State Univ., East Bay

3. Outline Only categorical variables are discussed here.
Verify the hypothesized distribution
One-sample Chi-square test
Test the independence between two categorical variables
Chi-square test for two-way contingency table
McNemar’s test for paired data
Measure the dependence (Phil and Kappa Coefficients)
Odds ratios and relative risk
Test the trend of a binary response
Chi-square test for trend
Meta-analysis

4. Example: Hair Color Distribution
Fair
Red
Medium
Dark
Black
Frequency 76 19 83 65 3 %
30.89
7.72
33.74
26.42
1.22
Test % 30 12 25
From a random sample of 246 children

5. One-sample Chi-Square Test
Must be a random sample
The sample size must be large enough so that expected frequencies are greater than or equal to 5 for 80% or more of the categories

6. One-sample Chi-Square Test
Test statistic:
Oi = the observed frequency of i-th category
ei = the expected frequency of i-th category

7. Chi-Square Test for Specified Proportions
SAS Output
Chi-Square Test for Specified Proportions
Chi-Square
7.7602 DF 4
Pr > ChiSq
0.1008

8. Two-way Contingency Tables
Report frequencies on two variables
Such tables are also called crosstabs.

9. Contingency Tables (Crosstabs)
1991 General Social Survey
Frequency
Party Identification
Democrat
Independent
Republican
Race
White
341
105
405
Black
103 15 11

10. Crosstabs Analysis (SAS: p.88-90; SPSS: p.369-371)
Chi-square test for testing the independence between two variables:
For a fixed column, the distribution of frequencies over rows keeps the same regardless of the column
For a fixed row, the distribution of frequencies over columns keeps the same regardless of the row

11. Crosstabs Analysis
The phi coefficient measures the association between two categorical variables
-1 < phi < 1
| phi | indicates the strength of the association
If the two variables are both ordinal, then the sign of phi indicate the direction of association

12. SAS Output Statistic DF Value Prob Chi-Square 2 79.4310 <.0001
Likelihood Ratio Chi-Square <.0001
Mantel-Haenszel Chi-Square <.0001
Phi Coefficient
Contingency Coefficient
Cramer's V
Sample Size = 980

13. Fisher’s Exact Test for Independence
The Chi-squared tests are for large samples
The sample size must be large enough so that expected frequencies are greater than or equal to 5 for 80% or more of the categories

14. SAS Output Fisher's Exact Test Table Probability (P) 3.823E-22
Pr <= P E-20
Sample Size = 980

15. 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 form the other sample (eg. Twins)

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

17. Marginal Homogeneity
The probabilities of “success” for both samples are identical
Eg. The probability of approve at the first and 2nd surveys are identical

18. McNemar Test (for 2x2 Tables only)
See SAS textbook Section 3.L
Ho: marginal homogeneity
Ha: no marginal homogeneity
Exact p-value
Approximate p-value (When n12+n21>10)

19. 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

20. Comparing Proportions in 2x2 Tables
Difference of proportions: pi1-pi2
Relative risk: pi1/pi2
Odds Ratio: odds1/odds2
odds1=pi1/(1-pi1)
odds2=pi2/(1-pi2)

21. Example: Aspirin vs. Heart Attack
Prospective sampling; Row totals were fixed
Frequency
Heart attack
No Heart attack
Placebo
189
10845
Aspirin
104
10933

22. Chi-square Test for Trend
Situation: A binary response (success, failure)
+ an ordinal explanatory variable
Question: Is there a trend? Are the proportions (of success) in each of the levels of the explanatory variable increasing or decreasing in a linear fashion?

23. Example: Shoulder Harness Usage
Use?
Large Cars
Medium Cars
Small Cars No
226
165
175
Yes 83 70 71
Question: Is the proportion of shoulder harness usage increasing or decreasing linearly as the car size gets larger?

24. SAS Output Statistics for Table of response by car_size
Statistic DF Value Prob
Chi-Square
Likelihood Ratio Chi-Square
Mantel-Haenszel Chi-Square
Phi Coefficient
Contingency Coefficient
Cramer's V

25. Meta Analysis Also known as Mantel-Haenszel test; stratified analysis
Situation: When another variable (strata) may “pollute” the effect of a categorical explanatory variable on a categorical response
Goal: Study the effect of the explanatory while controlling the stratification variable

26. Example: Respiratory Improvement
Center
Treatment
Yes No
Total 1
Test 29 16 45
Placebo 14 31 43 47 90 2 37 9 24 21 61

27. SAS Output Statistics for Table 1 of trtmnt by response
Controlling for center=1
Statistic DF Value Prob
Chi-Square
Likelihood Ratio Chi-Square
Continuity Adj. Chi-Square
Mantel-Haenszel Chi-Square
Phi Coefficient
Contingency Coefficient
Cramer's V
Estimates of the Relative Risk (Row1/Row2)
Type of Study Value % Confidence Limits
Case-Control (Odds Ratio)
Cohort (Col1 Risk)
Cohort (Col2 Risk)
Sample Size = 90

28. 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
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Nonzero Correlation <.0001
Row Mean Scores Differ <.0001
General Association <.0001

29. SAS Output Estimates of the Common Relative Risk (Row1/Row2)
Type of Study Method Value % Confidence Limits
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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
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Chi-Square DF
Pr > ChiSq
Total Sample Size = 180