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### Statistics

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

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

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

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

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

13.2: Testing Categorical Probabilities: One-Way Table

Reject the null hypothesis

McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

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

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

13.2: Testing Categorical Probabilities: One-Way Table

Example 13.2: Distribution of Opinions About Marijuana

Possession Before Television Series has Aired

Table 13.2: Distribution of Opinions About Marijuana

Possession After Television Series has Aired

McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

13.2: Testing Categorical Probabilities: One-Way Table

McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

13.2: Testing Categorical Probabilities: One-Way Table

Expected Distribution of 500 Opinions About Marijuana

Possession After Television Series has Aired

McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

13.2: Testing Categorical Probabilities: One-Way Table

Expected Distribution of 500 Opinions About Marijuana

Possession After Television Series has Aired

McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

13.2: Testing Categorical Probabilities: One-Way Table

Expected Distribution of 500 Opinions About Marijuana

Possession After Television Series has Aired

Reject the null hypothesis

McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

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

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

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.

McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

13.3: Testing Categorical Probabilities: Two-Way Table

McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

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

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

13.3: Testing Categorical Probabilities: Two-Way Table

- The expected frequencies (see Figure 13.4) are included below:

Religious Affiliation

Marital

Status

The chi-square value computed with SAS is 7.1355, with p-value = .1289.

Even at the = .10 level, we cannot reject the null hypothesis.

McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

13.3: Testing Categorical Probabilities: Two-Way Table

McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

13.4: A Word of Caution About Chi-Square Tests

McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

13.4: A Word of Caution About Chi-Square Tests

Be sure

McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

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