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Chapter 13

Chapter 13. Chi-Square and Nonparametric Procedures. Going Forward. Your goals in this chapter are to learn: When to use nonparametric statistics The logic and use of the one-way chi square The logic and use of the two-way chi square

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Chapter 13

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  1. Chapter 13 Chi-Square and Nonparametric Procedures

  2. Going Forward Your goals in this chapter are to learn: • When to use nonparametric statistics • The logic and use of the one-way chi square • The logic and use of the two-way chi square • The names of the nonparametric procedures with ordinal scores

  3. Parametric Versus Nonparametric Statistics

  4. Nonparametric Statistics Nonparametric statisticsare inferential procedures used with either nominal or ordinal data.

  5. Chi Square Procedures

  6. Chi Square • The chi square procedure is the nonparametric procedure for testing whether the frequencies in each category in sample data represent specified frequencies in the population • The symbol for the chi square statistic is c2

  7. One-Way Chi Square:The Goodness of Fit Test

  8. One-Way Chi Square The one-way chi square test is computed when data consist of the frequencies with which participants belong to the different categories of one variable

  9. Statistical Hypotheses H0: all frequencies in the population are equal Ha: all frequencies in the population are not equal

  10. Observed Frequency • The observed frequency is the frequency with which participants fall into a category • It is symbolized by fo • The sum of the fos from all categories equals N

  11. Formula for Expected Frequencies • The expected frequency is the frequency we expect in a category if the sample data perfectly represent the distribution of frequencies in the population described by H0 • The symbol is fe

  12. Assumptions of the One-Way Chi Square • Participants are categorized along one variable having two or more categories, and we count the frequency in each category • Each participant can be in only one category • Category membership is independent • We include the responses of all participants in the study • The fe must be at least 5 per category

  13. Computing One-WayChi-Square Statistic • Where fo are the observed frequencies and fe are the expected frequencies • df = k – 1 where k is the number of categories

  14. The c2 Distribution

  15. “Goodness of Fit” Test • The one-way chi square procedure is also called the goodness of fit test • That is, how “good” is the “fit” between the data and the frequencies we expect if H0 is true

  16. The Two-Way Chi Square:The Test of Independence

  17. Two-Way Chi Square The two-way chi square procedure is used for testing whether category membership on one variable is independent of category membership on the other variable.

  18. Computing Two-WayChi Square Statistic • Where fo are the observed frequencies and fe are the expected frequencies • df = (number of rows – 1)(number of columns – 1)

  19. Two-Way Chi Square • A significant two-way chi square indicates the sample data are likely to represent variables that are dependent (correlated) in the population • When a 2 x 2 chi square test is significant, we compute the phi coefficient ( f ) to describe the strength of the relationship

  20. Nonparametric Statistics

  21. Nonparametric Tests • Spearman correlation coefficient is analogous to the Pearson correlation coefficient for ranked data • Mann-Whitney test is analogous to the independent samples t-test • Wilcoxon test is analogous to the related-samples t-test

  22. Nonparametric Tests • Kruskal-Wallis test is analogous to a one-way between-subjects ANOVA • Friedman test is analogous to a one-way within-subjects ANOVA

  23. Example A survey is conducted where respondents are asked to indicate (a) their sex and (b) their preference in pets between dogs and cats. The frequency of males and females making each pet selection is given below. Perform a two-way chi square test.

  24. Example • The expected values for each cell are: (39)(35)/104 = 13.125 (65)(39)/104 = 21.875 (39)(69)/104 = 25.875 (65)(69)/104 = 43.125

  25. Example

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