1. Introduction to Categorical Data Analysis July 22, 2004

2. Categorical data
The t-test, ANOVA, and linear regression all assumed outcome variables that were continuous (normally distributed).
Even their non-parametric equivalents assumed at least many levels of the outcome (discrete quantitative or ordinal).
We haven’t discussed the case where the outcome variable is categorical.

3. Types of Variables: a taxonomy
discrete random variables
Categorical
Quantitative
binary
nominal
ordinal
discrete
continuous
2 categories +
more categories +
order matters +
numerical +
uninterrupted

4. Overview of statistical tests
Independent variable=predictor
Dependent variable=outcome
e.g., BMD= pounds age amenorrheic (1/0)
Continuous predictors
Binary predictor
Continuous outcome

5. Types of variables to be analyzed
Statistical procedure
or measure of association
Predictor (independent) variable/s
Outcome (dependent) variable
Categorical
Continuous
ANOVA
Dichotomous
Continuous
T-test
Continuous
Continuous
Simple linear regression
Multivariate
Continuous
Multiple linear regression
Categorical
Categorical
Chi-square test
Dichotomous
Dichotomous
Odds ratio, Mantel-Haenszel OR, Relative risk,
difference in proportions
Multivariate
Dichotomous
Logistic regression
Kaplan-Meier curve/ log-rank test
Categorical
Time-to-event
Time-to-event
Multivariate
Cox-proportional hazards model

6. done Today and next week Last part of course
Types of variables to be analyzed
Statistical procedure
or measure of association
Predictor (independent) variable/s
Outcome (dependent) variable
done
Categorical
Continuous
ANOVA
Dichotomous
Continuous
T-test
Continuous
Continuous
Simple linear regression
Multivariate
Continuous
Multiple linear regression
Today and next week
Categorical
Categorical
Chi-square test
Dichotomous
Dichotomous
Odds ratio, Mantel-Haenszel OR, Relative risk,
difference in proportions
Multivariate
Dichotomous
Logistic regression
Kaplan-Meier curve/ log-rank test
Last part of course
Categorical
Time-to-event
Time-to-event
Multivariate
Cox-proportional hazards model

7. Difference in proportions
Example: You poll 50 people from random districts in Florida as they exit the polls on election day You also poll 50 people from random districts in Massachusetts. 49% of pollees in Florida say that they voted for Kerry, and 53% of pollees in Massachusetts say they voted for Kerry. Is there enough evidence to reject the null hypothesis that the states voted for Kerry in equal proportions?

8. Null distribution of a difference in proportions
Standard error of a proportion=
Standard error can be estimated by=
(still normally distributed)
Standard error of the difference of two proportions=
The variance of a difference is the sum of variances (as with difference in means).
Analagous to pooled variance in the ttest

9. Null distribution of a difference in proportions
Difference of proportions
For our example, null distribution=

10. Answer to Example
We saw a difference of 4% between Florida and Massachusetts
Null distribution predicts chance variation between the two states of 10%.
P(our data/null distribution)=P(Z>.04/.10=.4)>.05
Not enough evidence to reject the null.

11. Chi-square test for comparing proportions (of a categorical variable) between groups
I. Chi-Square Test of Independence
When both your predictor and outcome variables are categorical, they may be cross-classified in a contingency table and compared using a chi-square test of independence.
A contingency table with R rows and C columns is an R x C contingency table.

12. Example
Asch, S.E. (1955). Opinions and social pressure. Scientific American, 193,

13. The Experiment
A Subject volunteers to participate in a “visual perception study.”
Everyone else in the room is actually a conspirator in the study (unbeknownst to the Subject).
The “experimenter” reveals a pair of cards…

14. The Task Cards
Standard line
Comparison lines
A, B, and C

15. The Experiment
Everyone goes around the room and says which comparison line (A, B, or C) is correct; the true Subject always answers last – after hearing all the others’ answers.
The first few times, the 7 “conspirators” give the correct answer.
Then, they start purposely giving the (obviously) wrong answer.
75% of Subjects tested went along with the group’s consensus at least once.

16. Further Results
In a further experiment, group size (number of conspirators) was altered from 2-10.
Does the group size alter the proportion of subjects who conform?

17. Number of group members?
The Chi-Square test
Conformed?
Number of group members? 2 4 6 8 10
Yes 20 50 75 60 30 No 80 25 40 70
Apparently, conformity less likely when less or more group members…

18. = 235 conformed
out of 500 experiments.
Overall likelihood of conforming = 235/500 = .47

19. Number of group members?
Expected frequencies if no association between group size and conformity…
Conformed?
Number of group members? 2 4 6 8 10
Yes 47 No 53

20. Do observed and expected differ more than expected due to chance?
Do observed and expected differ more than expected due to chance?

21. Chi-Square test
Degrees of freedom = (rows-1)*(columns-1)=(2-1)*(5-1)=4
Rule of thumb: if the chi-square statistic is much greater than it’s degrees of freedom, indicates statistical significance. Here 85>>4.

22. The Chi-Square distribution: is sum of squared normal deviates
The expected value and variance of a chi-square:
E(x)=df
Var(x)=2(df)

23. Chi-Square test
Degrees of freedom = (rows-1)*(columns-1)=(2-1)*(5-1)=4
Rule of thumb: if the chi-square statistic is much greater than it’s degrees of freedom, indicates statistical significance. Here 85>>4.

24. Caveat
**When the sample size is very small in any cell (<5), Fischer’s exact test is used as an alternative to the chi-square test.

25. Example of Fisher’s Exact Test

26. Fisher’s “Tea-tasting experiment”
Claim: Fisher’s colleague (call her “Cathy”) claimed that, when drinking tea, she could distinguish whether milk or tea was added to the cup first.
To test her claim, Fisher designed an experiment in which she tasted 8 cups of tea (4 cups had milk poured first, 4 had tea poured first).
Null hypothesis: Cathy’s guessing abilities are no better than chance.
Alternatives hypotheses:
Right-tail: She guesses right more than expected by chance.
Left-tail: She guesses wrong more than expected by chance

27. Fisher’s “Tea-tasting experiment”
Experimental Results:
Milk
Tea 3 1
Guess poured first
Poured First 4

28. Fisher’s Exact Test
Step 1: Identify tables that are as extreme or more extreme than what actually happened:
Here she identified 3 out of 4 of the milk-poured-first teas correctly. Is that good luck or real talent?
The only way she could have done better is if she identified 4 of 4 correct.
Milk
Tea 3 1
Guess poured first
Poured First 4
Milk
Tea 4
Guess poured first
Poured First

29. Fisher’s Exact Test
Step 2: Calculate the probability of the tables (assuming fixed marginals)
Milk
Tea 3 1
Guess poured first
Poured First 4
Milk
Tea 4
Guess poured first
Poured First

30. “right-hand tail probability”: p=.243
Step 3: to get the left tail and right-tail p-values, consider the probability mass function:
Probability mass function of X, where X= the number of correct identifications of the cups with milk-poured-first:
“right-hand tail probability”: p=.243
“left-hand tail probability” (testing the null hypothesis that she’s systematically wrong): p=.986

31. SAS code and output for generating Fisher’s Exact statistics for 2x2 table
Milk
Tea 3 1 4

32. data tea;
input MilkFirst GuessedMilk Freq;
datalines;
1 1 3
1 0 1
0 1 1
0 0 3
run;
data tea; *Fix quirky reversal of SAS 2x2 tables;
set tea;
MilkFirst=1-MilkFirst;
GuessedMilk=1-GuessedMilk;run;
proc freq data=tea;
tables MilkFirst*GuessedMilk /exact;
weight freq;run;

33. SAS output Statistics for Table of MilkFirst by GuessedMilk
Statistic DF Value Prob
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Chi-Square
Likelihood Ratio Chi-Square
Continuity Adj. Chi-Square
Mantel-Haenszel Chi-Square
Phi Coefficient
Contingency Coefficient
Cramer's V
WARNING: 100% of the cells have expected counts less
than 5. Chi-Square may not be a valid test.
Fisher's Exact Test
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Cell (1,1) Frequency (F)
Left-sided Pr <= F
Right-sided Pr >= F
Table Probability (P)
Two-sided Pr <= P
Sample Size = 8

34. Introduction to the 2x2 Table

35. Introduction to the 2x2 Table
Exposure (E)
No Exposure (~E)
Disease (D) a b
a+b = P(D)
No Disease (~D) c d
c+d = P(~D)
a+c = P(E)
b+d = P(~E)
Marginal probability of disease
Marginal probability of exposure

36. Cohort Studies Exposed Not Exposed Disease-free cohort Disease
Target population
Disease
Disease-free
TIME

37. The Risk Ratio, or Relative Risk (RR)
Exposure (E)
No Exposure (~E)
Disease (D) a b
No Disease (~D) c d
a+c
b+d
risk to the exposed
risk to the unexposed

38. Hypothetical Data Normal BP Congestive Heart Failure No CHF 1500 3000
Normal BP
Congestive Heart Failure
No CHF
1500
3000
High Systolic BP
400
1100
2600

39. Case-Control Studies
Sample on disease status and ask retrospectively about exposures (for rare diseases)
Marginal probabilities of exposure for cases and controls are valid.
Doesn’t require knowledge of the absolute risks of disease
For rare diseases, can approximate relative risk

40. Case-Control Studies Exposed in past Disease (Cases) Not exposed
Target population
Exposed
No Disease
(Controls)
Not Exposed

41. The Odds Ratio (OR) Exposure (E) No Exposure (~E) Disease (D)
Exposure (E)
No Exposure (~E)
Disease (D)
a = P (D& E)
b = P(D& ~E)
No Disease (~D)
c = P (~D&E)
d = P (~D&~E)

42. The Odds Ratio Via Bayes’ Rule 1 1 When disease is rare: P(~D)  1
“The Rare Disease Assumption” 1

43. Properties of the OR (simulation)

44. Properties of the lnOR
Standard deviation =
Standard deviation =

45. Hypothetical Data 30 30 Smoker Non-smoker Lung Cancer 20 10
Smoker
Non-smoker
Lung Cancer 20 10
No lung cancer 6 24 30 30
Note that the size of the smallest 2x2 cell determines the magnitude of the variance
NOTE how this means the smallest cell really determines the magnitude of the variance.

46. Example: Cell phones and brain tumors (cross-sectional data)
Brain tumor
No brain tumor
Own a cell phone 5
347
352
Don’t own a cell phone 3 88 91 8
435
453

47. Same data, but use Chi-square test or Fischer’s exact
Brain tumor
No brain tumor
Own 5
347
352
Don’t own 3 88 91 8
435
453

48. Same data, but use Odds Ratio
Brain tumor
No brain tumor
Own a cell phone 5
347
352
Don’t own a cell phone 3 88 91 8
435
453