Chapter 13: Categorical Data Analysis

1
Statistics

• Chapter 13 Categorical Data Analysis

2
Where Weve 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

3
Where Were 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

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.

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?

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

7
13.2 Testing Categorical Probabilities One-Way
Table
Reject the null hypothesis
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.

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.

10
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
11
13.2 Testing Categorical Probabilities One-Way
Table
12
13.2 Testing Categorical Probabilities One-Way
Table
Expected Distribution of 500 Opinions About
Marijuana Possession After Television Series has
Aired
13
13.2 Testing Categorical Probabilities One-Way
Table
Expected Distribution of 500 Opinions About
Marijuana Possession After Television Series has
Aired
14
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
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

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?

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.
18
13.3 Testing Categorical Probabilities Two-Way
Table
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
Marital Status
H0 Marital status and religious affiliation are
independent Ha Marital status and religious
affiliation are dependent
20
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.
21
13.3 Testing Categorical Probabilities Two-Way
Table
22
13.4 A Word of Caution About Chi-Square Tests
23
13.4 A Word of Caution About Chi-Square Tests
Be sure