Chapter 16 – Categorical Data Analysis

1. Chapter 16 – Categorical Data Analysis Math 22 Introductory Statistics

2. Chi-Square • Categorical data are statistically analyzed by means of a chi-square statistic. • A single variable is analyzed with the chi-square goodness-of-fit test. • The goodness-of-fit test consists of determining whether the frequency counts in the categories of the variable agree with a specific distribution.

3. The Multinomial Experiment • The experiment consist of n identical experiments. • The outcome of each trial falls into one of k categories.

4. The Multinomial Experiment • The probabilities associated with the k outcomes denoted by p1, p2, p3,…,pk remain the same from trial to trial. Since there are k possible outcome we have:

5. The Multinomial Experiment • The experimenter records the values o1, o2,....,ok where oj (j = 1, 2, .....,k) is equal to the number of trials in which the outcome is in category j. • Note: o1+o2+......+ok = n

6. Chi-Square Goodness-of-Fit Test • Application: Multinomial experiments. • Assumptions: • The experiment satisfies the properties of a multinomial experiment. • No expected cell counts, ej, is less than 1, and no more than 20% of the ej‘s are less than 5. (This is so the chi-square approximation will be good)

7. Chi-Square Goodness-of-Fit Test • The test is a right-tailed test, where the p-value is found in the chi-square table with k-1 degrees of freedom. Usually the exact value cannot be found, but bounds for it can be found from the closest to the observed value of the chi-square statistic. • Chi-Square Statistic:

8. Chi-Square Test of Independence • Application: Test the independence of the classifying variables Assumptions: • The experiment satisfies the properties of a multinomial experiment. • No expected cell counts, ej, is less than 1, and no more than 20% of the ej‘s are less than 5. (This is so the chi-square approximation will be good)