Chapter 16 – Categorical Data Analysis

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

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Chapter 16 – Categorical Data Analysis

Math 22

Introductory Statistics

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