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Stats 2022n

Stats 2022n. Non-Parametric Approaches to Data Chp 15.5 & Appendix E. Outline. A note on ordinal scales. An ordinal scale :. Example – Grades. A note on ordinal scales. Ordinal scales allow ranking. Example – Grades. Why use ordinal scales?. Some data is easier collected as ordinal.

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Stats 2022n

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

  2. Outline

  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 • Ordinal data needs to be ranked before it can be tested (via non-parametric tests) • Transforming data through ranking can be a usefultool

  7. The case for ranking data Ranking data (rank transform) can be a usefultool • 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

  8. Rank Transformation

  9. Rank Transformation What if ties?....

  10. Ordinal Transformation Ranking Data, If Ties

  11. Chp 15.5Spearman 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 Example

  14. Spearman Correlation Calculation

  15. Spearman Correlation Calculation

  16. Spearman Correlation Spearman Correlation Special Formula = = =

  17. Spearman Correlation 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 • Extreme difference due to conditions • Distributions of ranks unequal • No difference due to conditions • Distributions of ranks unequal

  22. Mann - Whitney U-Test Example

  23. Mann - Whitney U-Test Computing U by hand U=27

  24. Mann - Whitney U-Test Computing U via formula RA = 73 RB= 63 U=27 = 8

  25. Mann - Whitney U-Test Evaluating Significance with U H0: H1: 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

  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=27 = 8 Note: 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 • Requirements • Two related samples (repeated measure) • Rank ordered data • Hypotheses: • H0: • H1:

  30. Wilcoxon signed-rank test

  31. Wilcoxon signed-rank test

  32. Wilcoxon signed-rank test 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 Write up 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.

  34. Wilcoxon signed-rank test A note on difference scores of zero N = 4 N = 5 N = 4

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

  36. Wilcoxon signed-rank test Evaluating Significance Using Normal Approximation T = 17 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 • For each treatment condition • n: n for each group • r: sum of ranks for each condition • Overall • k: Total groups • Uses distribution for hypothesis testing. Chi square statistic for ranks.

  44. Summary Ratio Data Ranked Data

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