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

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Chapter 16 categorical data analysis

Chapter 16 – Categorical Data Analysis

Math 22

Introductory Statistics


Chi square

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 Multinomial Experiment

  • The experiment consist of n identical experiments.

  • The outcome of each trial falls into one of k categories.


The multinomial experiment1

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 experiment2

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

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 test1

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

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)


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