1. Chapter 13: Categorical Data Analysis
Statistics
Chapter 13: Categorical Data Analysis

2. McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis
Where We’ve Been
Presented methods for making inferences about the population proportion associated with a two-level qualitative variable (i.e., a binomial variable)
Presented methods for making inferences about the difference between two binomial proportions
McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

3. McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis
Where We’re Going
Discuss qualitative (categorical) data with more than two outcomes
Present a chi-square hypothesis test for comparing the category proportions associated with a single qualitative variable – called a one-way analysis
Present a chi-square hypothesis test relating two qualitative variables – called a two-way analysis
McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

4. 13.1: Categorical Data and the Multinomial Experiment
Properties of the Multinomial Experiment
The experiment consists of n identical trials.
There are k possible outcomes (called classes, categories or cells) to each trial.
The probabilities of the k outcomes, denoted by p1, p2, …, pk, where p1+ p2+ … + pk = 1, remain the same from trial to trial.
The trials are independent.
The random variables of interest are the cell counts n1, n2, …, nk of the number of observations that fall into each of the k categories.
McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

5. 13.2: Testing Categorical Probabilities: One-Way Table
Suppose three candidates are running for office, and 150 voters are asked their preferences.
Candidate 1 is the choice of 61 voters.
Candidate 2 is the choice of 53 voters.
Candidate 3 is the choice of 36 voters.
Do these data suggest the population may prefer one candidate over the others?
McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

6. 13.2: Testing Categorical Probabilities: One-Way Table
Candidate 1 is the
choice of 61 voters.
Candidate 2 is the
choice of 53 voters.
Candidate 3 is the
choice of 36 voters.
n =150
McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

7. 13.2: Testing Categorical Probabilities: One-Way Table
Reject the null hypothesis
McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

8. 13.2: Testing Categorical Probabilities: One-Way Table
Test of a Hypothesis about Multinomial Probabilities: One-Way Table H0: p1 = p1,0, p2 = p2,0, … , pk = pk,0 where p1,0, p2,0, …, pk,0 represent the hypothesized values of the multinomial probabilities Ha: At least one of the multinomial probabilities does not equal its hypothesized value where Ei = np1,0, is the expected cell count given the null hypothesis.
McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

9. 13.2: Testing Categorical Probabilities: One-Way Table
Conditions Required for a Valid 2 Test:
One-Way Table
A multinomial experiment has been conducted.
The sample size n will be large enough so that, for every cell, the expected cell count E(ni) will be equal to 5 or more.
McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

10. 13.2: Testing Categorical Probabilities: One-Way Table
Example 13.2: Distribution of Opinions About Marijuana
Possession Before Television Series has Aired
Legalization
Decriminalization
Existing Law
No Opinion 7%
18%
65%
10%
Table 13.2: Distribution of Opinions About Marijuana
Possession After Television Series has Aired
Legalization
Decriminalization
Existing Law
No Opinion 39 99
336 26
McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

11. 13.2: Testing Categorical Probabilities: One-Way Table
McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

12. 13.2: Testing Categorical Probabilities: One-Way Table
Expected Distribution of 500 Opinions About Marijuana
Possession After Television Series has Aired
Legalization
Decriminalization
Existing Law
No Opinion
500(.07)=35
500(.18)=90
500(.65)=325
500(.10)=50
McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

13. 13.2: Testing Categorical Probabilities: One-Way Table
Expected Distribution of 500 Opinions About Marijuana
Possession After Television Series has Aired
Legalization
Decriminalization
Existing Law
No Opinion
500(.07)=35
500(.18)=90
500(.65)=325
500(.10)=50
McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

14. 13.2: Testing Categorical Probabilities: One-Way Table
Expected Distribution of 500 Opinions About Marijuana
Possession After Television Series has Aired
Legalization
Decriminalization
Existing Law
No Opinion
500(.07)=35
500(.18)=90
500(.65)=325
500(.10)=50
Reject the null hypothesis
McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

15. 13.2: Testing Categorical Probabilities: One-Way Table
Inferences can be made on any single proportion as well:
95% confidence interval on the proportion of citizens in the viewing area with no opinion is
McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

16. 13.3: Testing Categorical Probabilities: Two-Way Table
Chi-square analysis can also be used to investigate studies based on qualitative factors.
Does having one characteristic make it more/less likely to exhibit another characteristic?
McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

17. 13.3: Testing Categorical Probabilities: Two-Way Table
The columns are divided according to the subcategories for one
qualitative variable and the rows for the other qualitative variable.
Column 1 2  c
Row Totals
n11
n12
n1c R1
Row
n21
n22
n2c R2  r
nr1
nr2
nrc Rr
Column Totals C1 n
McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

18. 13.3: Testing Categorical Probabilities: Two-Way Table
McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

19. 13.3: Testing Categorical Probabilities: Two-Way Table
The results of a survey regarding marital status and religious affiliation are reported below (Example 13.3 in the text).
Religious Affiliation A B C D
None
Totals
Divorced 39 19 12 28 18
116
Married, never divorced
172 61 44 70 37
384
211 80 56 98 55
500
Marital
Status
H0: Marital status and religious affiliation are independent
Ha: Marital status and religious affiliation are dependent
McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

20. 13.3: Testing Categorical Probabilities: Two-Way Table
The expected frequencies (see Figure 13.4) are included below:
Religious Affiliation A B C D
None
Totals
Divorced 39
(48.95) 19
(18.56) 12
(12.99) 28
(27.74) 18
(12.76)
116
Married,
never
divorced
172
(162.05) 61
(61.44) 44
(43.01) 70
(75.26) 37
(42.24)
384
211 80 56 98 55
500
Marital
Status
The chi-square value computed with SAS is , with p-value =
Even at the  = .10 level, we cannot reject the null hypothesis.
McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

21. 13.3: Testing Categorical Probabilities: Two-Way Table
McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

22. 13.4: A Word of Caution About Chi-Square Tests
Relative ease of use
Widespread applications
Misuse and misinterpretation
McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

23. 13.4: A Word of Caution About Chi-Square Tests
Sample is from the correct population
Expected counts are ≥ 5
Avoid Type II errors by
not accepting non-rejected
null hypotheses
Avoid mistaking
dependence with causation
To produce (possibly) valid 2 results
Be sure
McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis