1. Chapter Fifteen Chi-Square and Other Nonparametric Procedures

2. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 15 - 2 Nonparametric Statistics Nonparametric statistics are used when dependent scores form skewed or otherwise nonnormal distributions, when the population variance is not homogeneous, or when scores are measured using ordinal or nominal scales.

3. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 15 - 3 One-Way Chi Square: The Goodness of Fit Test

4. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 15 - 4 One-Way Chi Square The one-way chi square test is used when data consist of the frequencies with which participants belong to the different categories of one variable

5. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 15 - 5 Statistical Hypotheses H 0 : all frequencies in the population are equal H a : all frequencies in the population are not equal

6. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 15 - 6 Assumptions of the One-Way Chi Square 1.Participants are categorized along one variable having two or more categories counting the frequency in each category 2.Each participant can be in only one category 3.Category membership is independent 4.The computations include the responses of all participants in the study 5.The f e in any category must be at least 5

7. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 15 - 7 Where f o are the observed frequencies and f e are the expected frequencies df = k - 1 where k is the number of categories Computing One-Way

8. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 15 - 8 The Two-Way Chi Square: The Test of Independence

9. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 15 - 9 Two-Way Chi Square: the Test of Independence The two-way chi square procedure is used when you count the frequency of category membership along two variables

10. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 15 - 10 Computing Two-Way Where f o are the observed frequencies and f e are the expected frequencies df = (number of rows - 1)(number of columns - 1)

11. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 15 - 11 Nonparametric Statistics

12. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 15 - 12 The Mann-Whitney U test Used to test two independent samples of ranks when the n in each condition is equal to or less than 20

13. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 15 - 13 The Mann-Whitney U test 1.Assign ranks to all scores in the experiment 2.Compute the sum of the ranks for each group 3.Compute U 1 and U 2 4.Determine the U obt 5.Find the critical value of U 6.Compare U obt to U crit. U obt is significant if it is equal to or less than U crit.

14. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 15 - 14 Rank Sums Test Used to test two independent samples of ranks and either n is greater than 20

15. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 15 - 15 Rank Sums Test 1.Assign ranks to the scores in the experiment 2.Choose one group and compute the sum of the ranks 3.Compute the expected sum of ranks for the chosen group (  R exp ) 4.Compute the rank sums statistic z obt 5.Find the critical value of z 6.Compare z obt to z crit 7.Describe a significant relationship using eta squared

16. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 15 - 16 Wilcoxon T test is used to test two related samples of ranked data Kruskal-Wallis H test is used to study one factor involving at least three conditions where each is tested using independent samples and at least five participants in each sample Friedman test is used to study one factor involving at least three conditions where the samples in each are related (either matching or repeated measures) Other Nonparametric Tests

17. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 15 - 17 MalesFemales Dogs2411 Cats1554 Example 1 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.

18. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 15 - 18 MalesFemales Dogs13.12521.875 Cats25.87543.125 Example 1 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

19. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 15 - 19 Example 1

20. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 15 - 20 Example 1  2 crit for df = (2 -1)(2 - 1) = 1 is 3.84 Since  2 obt >  2 crit, we reject the null hypothesis

21. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 15 - 21 Group 1 Group 2 511 79 420 617 210 123 Example 2 Using the following data set, conduct a two-tailed Mann- Whitney U test with  = 0.05

22. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 15 - 22 Group 1Group 2 Ranks Group 1 Ranks Group 2 51149 7967 420312 617511 21018 123102 Example 2

23. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 15 - 23 Ranks Group 1 Ranks Group 2 49 67 312 511 18 102 Example 2

24. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 15 - 24 Example 2 Since this is a two-tailed test, U obt is the smaller of U 1 and U 2. Then U obt = 8 For a two-tailed test with n 1 = 6 and n 2 = 6, U crit = 5 In the Mann-Whitney U test, to be significant U obt must be equal to or less than the critical value. Here, the test is not significant.