1. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter Fifteen Chi-Square and Other Nonparametric Procedures

2. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 15 - 2 Nonparametric Statistics Use nonparametric statistics when –dependent scores form skewed or otherwise nonnormal distributions, –the population variance is not homogeneous, or –scores are measured using ordinal or nominal scales

3. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 15 - 3 One-Way Chi Square: The Goodness of Fit Test

4. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 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. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 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. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 15 - 6 Assumptions of the One-Way Chi Square 1.Participants are categorized along one variable having two or more categories, and we count the frequency in each category 2.Each participant can be in only one category 3.Category membership is independent 4.We include the responses of all participants in the study 5.The f e in any category must be at least 5

7. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 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. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 15 - 8 The Two-Way Chi Square: The Test of Independence

9. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 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. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 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. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. A Significant Two-Way Chi-Square When a two-way chi square test is significant, you must compute either the phi coefficient or the contingency coefficient Chapter 15 - 11 Phi Coefficient Contingency Coefficient

12. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 15 - 12 Nonparametric Statistics

13. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 15 - 13 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

14. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 15 - 14 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, which is the smaller of U 1 and U 2 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 7.If U obt is significant, ignore sample size and compute eta squared by using the following Rank Sums Test

15. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 15 - 15 Rank Sums Test Used to test two independent samples of ranks and either n is greater than 20

16. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 15 - 16 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

17. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 15 - 17 The Wilcoxon T test is used to test two related samples of ranked data 1.Determine the difference score for each pair of scores 2.Determine the N of the nonzero difference scores 3.Assign ranks to the nonzero difference scores 4.Separate the ranks, using the sign of the difference scores 5.Compute the sums of ranks for the positive and negative difference scores 6.Determine the Wilcoxon T obt. In the two-tailed test, the T obt equals the smallest Wilcoxon T Test

18. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Wilcoxon T Test 7.Find the critical value of T in the “Critical Values of the Wilcoxon T” table 8.Compare T obt to T crit [Careful! T obt is significant if it is equal to or less than T crit ] Chapter 15 - 18

19. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Kruskal-Walls H Test The 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 1.Assign ranks, using all scores in the experiment 2.Compute the sum of the ranks in each condition 3.Compute the sum of squares between groups SS bn Chapter 15 - 19

20. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Kruskal-Walls H Test 4.Compute H obt where N is the total N of the study 5.Find the critical value of H in the tables using df = k – 1 6.Compare the obtained value of H to the critical value of. The H obt is significant if it is larger than the critical value. 7.Perform post hoc comparisons using the rank sums test 8.If H obt is significant, compute Chapter 15 - 20

21. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Friedman 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) 1.Assign ranks within the scores of each participant 2.Compute the sum of the ranks in each condition 3.Compute the sum of squares between groups SS bn 4.Compute the Friedman Chapter 15 - 21

22. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 5.Find the critical value of in the tables using df = k – 1 6.Compare to the critical value of 7.When the is significant, use Nemenyi’s Procedure 8.If is significant, compute eta squared Chapter 15 - 22 Friedman

23. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 15 - 23 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.

24. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 15 - 24 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

25. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 15 - 25 Example 1

26. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 15 - 26 Example 1  2 crit for df = (2 -1)(2 - 1) = 1 is 3.84 Since  2 obt >  2 crit, we reject the null hypothesis

27. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 15 - 27 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

28. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 15 - 28 Group 1Group 2 Ranks Group 1 Ranks Group 2 51149 7967 420312 617511 21018 123102 Example 2

29. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 15 - 29 Ranks Group 1 Ranks Group 2 49 67 312 511 18 102 Example 2

30. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 15 - 30 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.

31. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Key Terms,  2 -distribution chi-square procedure contingency coefficient expected frequency Friedman  2 test goodness-of-fit test Kruskal-Wallis H test Chapter 15 - 31 Mann-Whitney U test Nemenyi’s procedure nonparametric statistics observed frequency one-way chi-square phi coefficient

32. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Key Terms (Cont’d) Chapter 15 - 32 rank sums test test of independence two-way chi square Wilcoxon T test