1. Stats 2022n Non-Parametric Approaches to Data Chp 15.5 & Appendix E

2. Outline Chp 15.5alternative to Spearman Correlation ExamplePearson correlation Appendix E Mann - Whitney U-Test Exampleindependent measures t test Wilcoxon signed-rank test Examplerepeated-measures t test Kruskal – Wallace Test independent measures ANOVA) Friedman Test (repeated measures ANOVA)

3. A note on ordinal scales An ordinal scale : Example – Grades

4. A note on ordinal scales Ordinal scales allow ranking Example – Grades

5. Why use ordinal scales? Some data is easier collected as ordinal –

6. The case for ranking data 1.Ordinal data needs to be ranked before it can be tested (via non-parametric tests) 2.Transforming data through ranking can be a useful tool

7. Ranking data (rank transform) can be a useful tool – If assumptions of a test are not (or cannot be) met… – Common if data has: Non linear relationship … Unequal variance… High variance … – Data sometimes requires rank transformation for analysis The case for ranking data

8. Rank Transformation Group AGroup B 854 9882 5892 7823 A RanksB Ranks 13 86 47 52

9. Rank Transformation Group AGroup B 86 86 82 71 What if ties?....

10. Ordinal Transformation Ranking Data, If Ties Group scores (ordered)rank rank (tie adjusted) B111 B222 B633.5 B64 A755 A867 A877 A887 Group AGroup B 86 86 82 71 A RanksB Ranks 15 27 3.57 7

11. Chp 15.5 Spearman Correlation

12. Spearman Correlation Only requirement – ability to rank order data Data already ranked Rank transformed data Rank transform useful if relationship non-linear…

13. Spearman Correlation Participantxy A49 B26 C22 D10 E38 F7 Participantxyx ranky rank A4944 B261.52 C22 1 D10 65.5 E3833 F71055.5 Example

14. Spearman Correlation x ranky rankxyx2x2 y2y2 4416 1.5232.254 1.51 2.251 65.5333630.25 33999 55.527.52530.25 21 9090.5 21 90.5 90 Calculation

15. Spearman Correlation 21 90.5 90 Calculation

16. Spearman Correlation x ranky rankDD2D2 4400 1.520.50.25 1.51-0.50.25 65.5-0.50.25 3300 55.50.50.25 01 Spearman Correlation Special Formula

17. Spearman Correlation x ranky rankDD2D2 4400 1.520.50.25 1.51-0.50.25 65.5-0.50.25 3300 55.50.50.25 01 Spearman Correlation Special Formula v.s.

18. Hypothesis testing with spearman Same process as Pearson – (still using table B.7)

19. Appendix E Mann - Whitney U-Test Wilcoxon signed-rank test Kruskal – Wallace Test Friedman Test

20. Mann - Whitney U-Test – Requirements – Hypotheses:

21. Mann - Whitney U-Test Illustration Sample A Ranks Sample B Ranks 16 27 38 49 510 Sample A Ranks Sample B Ranks 12 34 56 78 910 Extreme difference due to conditions Distributions of ranks unequal No difference due to conditions Distributions of ranks unequal

22. Mann - Whitney U-Test Example GroupScore A8 A98 A58 A78 A42 A14 A63 A84 B54 B82 B92 B23 B53 B41 B28 B25 Group AGroup B 854 9882 5892 7823 4253 1441 6328 8425 ranked (sorted) according to values GroupScoreRank A81 A142 B233 B254 B285 B416 A427 B538 B549 A5810 A6311 A7812 B8213 A8414 B9215 A9816

23. Mann - Whitney U-Test GroupRank A1 A2 B3 B4 B5 B6 A7 B8 B9 A10 A11 A12 B13 A14 B15 A16 A RanksB Ranks 13 24 75 106 118 129 1413 1615 A RanksB Ranks 1032 2042 7452 10662 11683 12693 147136 168157 UAUA 37UBUB 27 verify:8*8= 6437+27=64 U=27 Computing U by hand

24. Mann - Whitney U-Test Computing U via formula A RanksB Ranks 13 24 75 106 118 129 1413 1615 7363 U=27

25. Mann - Whitney U-Test Evaluating Significance with U U=27 alpha = 0.05, 2 tails, df(8,8) Critical value = 13 U > critical value, we fail to reject the null The ranks are equally distributed between samples H0:H1:H0:H1:

26. Mann - Whitney U-Test Write-Up The original scores were ranked ordered and a Mann-Whitney U-test was used to compare the ranks for the n = 8 participants in treatment A and the n = 8 participants in treatment B. The results indicate no significant difference between treatments, U = 27, p >.05, with the sum of the ranks equal to 27 for treatment A and 37 for treatment B.

27. Mann - Whitney U-Test Evaluating Significance Using Normal Approximation With n>20, the MW-U distribution tends to approximate a normal shape, and so, can be evaluated using a z-score statistic as an alternative to the MW-U table. U=27Note: n not > 20!

28. Mann - Whitney U-Test Evaluating Significance Using Normal Approximation alpha = 0.05 2 tails Critical value: z = ± 1.96 -0.5251 is not in the critical region Fail to reject the null.

29. Wilcoxon signed-rank test Hypotheses: H 0 : H 1 : participantCondition 1Ciondition 2difference A13-2 B624 C910 D710-3 E945 F39-6 G220 H918 I918 J35-2 K14-3 Requirements Two related samples (repeated measure) Rank ordered data

30. Wilcoxon signed-rank test ParticipantDifference A-2 B4 C D-3 E5 F-6 H8 I8 J-2 K-3 Sorted and ranked by magnitude ParticipantDifferenceRank C1 A-22.5 J-22.5 D-34 B45 E56 F-67 H88.5 I8

31. Wilcoxon signed-rank test Sorted and ranked by magnitude ParticipantDifferenceRank C1 A-22.5 J-22.5 D-34 B45 E56 F-67 H88.5 I8 Positive rank scores Negative rank scores 51 62.5 8.52.5 8.54 7 2817 T=17

32. Wilcoxon signed-rank test T=17 n=10 alpha =.05 two tales critical value = 8 T obtained > critical value, fail to reject the null The difference scores are not systematically positive or systematically negative.

33. Wilcoxon signed-rank test The 11 participants were rank ordered by the magnitude of their difference scores and a Wilcoxon T was used to evaluate the significance of the difference between treatments. One sample was removed due to having a zero difference score. The results indicate no significant difference, n = 10, T = 17, p <.05, with the positive ranks totaling 28 and the negative ranks totaling 17. Write up

34. Wilcoxon signed-rank test ParticipantDifferenceRank C01.5 A0 J-23 D-34 B45 Positive rank scores Negative rank scores 1.5 43 5.54.5 A note on difference scores of zero ParticipantDifferenceRank C01 A01.5 J0 D-33 B44 ParticipantDifferenceRank C0 A21.5 J-21.5 D-33 B44 N = 4 N = 5 N = 4 Positive rank scores Negative rank scores 1.5 53 4 6.58.5 Positive rank scores Negative rank scores 1.5 43 5.54.5

35. Wilcoxon signed-rank test Evaluating Significance Using Normal Approximation T=17n=10 Note: n not > 20!

36. Wilcoxon signed-rank test Evaluating Significance Using Normal Approximation alpha = 0.05 2 tails Critical value: z = ± 1.96 -0.21847 is not in the critical region Fail to reject the null.

37. Interim Summary Calculation of Mann-Whitney or Wilcoxon is fair game on test. When to use Mann-Whitney or Wilcoxon If data is already ordinal or ranked If assumptions of parametric test are not met

38. Kruskal – Wallace Test Alternative to independent measures ANOVA Expands Mann – Whitney Requirements Null –

39. Kruskal – Wallace Test Rank ordered data (all conditions)

40. Kruskal – Wallace Test For each treatment condition n: n for each group T: sum of ranks for each group Overall N: Total participants Statistic identified with H Distribution approximates same distribution as chi-squared (i.e. use the chi squared table)

41. Friedman Test Alternative to repeated measures ANOVA Expands Wilcoxon test Requirements Null

42. Friedman Test Rank ordered data (within each participant)

43. Friedman Test

44. Summary Groups23+ Independent measure Repeated measure Groups23+ Independen t measure Repeated measure Ratio Data Ranked Data