1. Categorical Data Analysis
Chapter 13
Categorical Data Analysis
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2. Categorical Data and the Multinomial Distribution
Properties of the Multinomial Experiment
Experiment has n identical trials
There are k possible outcomes to each trial, called classes, categories or cells
Probabilities of the k outcomes remain constant from trial to trial
Trials are independent
Variables of interest are the cell counts, n1, n2…nk, the number of observations that fall into each of the k classes

3. Testing Category Probabilities: One-Way Table
In a multinomial experiment with categorical data from a single qualitative variable, we summarize data in a one-way table.

4. Testing Category Probabilities: One-Way Table
Hypothesis Testing for a One-Way Table
Based on the 2 statistic, which allows comparison between the observed distribution of counts and an expected distribution of counts across the k classes
Expected distribution = E(nk)=npk, where n is the total number of trials, and pk is the hypothesized probability of being in class k according to H0
The test statistic, 2, is calculated as and the rejection region is determined by the 2 distribution using k-1 df and the desired 

5. Testing Category Probabilities: One-Way Table
Hypothesis Testing for a One-Way Table
The null hypothesis is often formulated as a no difference, where H0: p1=p2=p3=…=pk=1/k, but can be formulated with non-equivalent probabilities
Alternate hypothesis states that Ha: at least one of the multinomial probabilities does not equal its hypothesized value

6. Testing Category Probabilities: One-Way Table
Hypothesis Testing for a One-Way Table
The null hypothesis is often formulated as a no difference, where H0: p1=p2=p3=…=pk=1/k, but can be formulated with non-equivalent probabilities
Alternate hypothesis states that Ha: at least one of the multinomial probabilities does not equal its hypothesized value

7. Testing Category Probabilities: One-Way Table
One-Way Tables: an example
H0: pLegal=.07, pdecrim=.18, pexistlaw=.65, pnone=.10
Ha: At least 2 proportions differ from proposed plan
Rejection region with =.01, df = k-1 = 3 is
Since the test statistic falls in the rejection region, we reject H0

8. Testing Category Probabilities: One-Way Table
Conditions Required for a valid 2 Test
Multinomial experiment has been conducted
Sample size is large, with E(ni) at least 5 for every cell

9. Testing Category Probabilities: Two-Way (Contingency) Table
Used when classifying with two qualitative variables
H0: The two classifications are independent
Ha: The two classifications are dependent
Test Statistic:
Rejection region:2>2, where 2 has (r-1)(c-1) df

10. Testing Category Probabilities: Two-Way (Contingency) Table
Conditions Required for a valid 2 Test
N observed counts are a random sample from the population of interest
Sample size is large, with E(ni) at least 5 for every cell

11. Testing Category Probabilities: Two-Way (Contingency) Table
Sample Statistical package output

12. A Word of Caution about Chi-Square Tests
When an expected cell count is less than 5, 2 probability distribution should not be used
If H0 is not rejected, do not accept H0 that the classifications are independent, due to the implications of a Type II error.
Do not infer causality when H0 is rejected. Contingency table analysis determines statistical dependence only.