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- 1. Nonparametric Statistics
- 2. Chapter Goals <ul><li>After completing this chapter, you should be able to: </li></ul><ul><li>Recognize when and how to use the Wilcoxon signed rank test for a population median </li></ul><ul><li>Recognize the situations for which the Wilcoxon signed rank test applies and be able to use it for decision-making </li></ul><ul><li>Know when and how to perform a Mann-Whitney U-test </li></ul><ul><li>Perform nonparametric analysis of variance using the Kruskal-Wallis one-way ANOVA </li></ul>
- 3. Nonparametric Statistics <ul><li>Nonparametric Statistics </li></ul><ul><ul><li>Fewer restrictive assumptions about data levels and underlying probability distributions </li></ul></ul><ul><ul><ul><li>Population distributions may be skewed </li></ul></ul></ul><ul><ul><ul><li>The level of data measurement may only be ordinal or nominal </li></ul></ul></ul>
- 4. Wilcoxon Signed Rank Test <ul><li>Used to test a hypothesis about one population median </li></ul><ul><ul><li>the median is the midpoint of the distribution: 50% below, 50% above </li></ul></ul><ul><li>A hypothesized median is rejected if sample results vary too much from expectations </li></ul><ul><ul><li>no highly restrictive assumptions about the shape of the population distribution are needed </li></ul></ul>
- 5. The W Test Statistic <ul><li>Performing the Wilcoxon Signed Rank Test </li></ul><ul><li>Calculate the test statistic W using these steps: </li></ul><ul><li>Step 1: collect sample data </li></ul><ul><li>Step 2: compute d i = difference between each value and the hypothesized median </li></ul><ul><li>Step 3: convert d i values to absolute differences </li></ul>
- 6. The W Test Statistic <ul><li>Performing the Wilcoxon Signed Rank Test </li></ul><ul><li>Step 4: determine the ranks for each d i value </li></ul><ul><ul><li>eliminate zero d i values </li></ul></ul><ul><ul><li>Lowest d i value = 1 </li></ul></ul><ul><ul><li>For ties, assign each the average rank of the tied observations </li></ul></ul>(continued)
- 7. The W Test Statistic <ul><li>Performing the Wilcoxon Signed Rank Test </li></ul><ul><li>Step 5: Create R+ and R- columns </li></ul><ul><ul><li>for data values greater than the hypothesized median, put the rank in an R+ column </li></ul></ul><ul><ul><li>for data values less than the hypothesized median, put the rank in an R- column </li></ul></ul>(continued)
- 8. The W Test Statistic <ul><li>Performing the Wilcoxon Signed Rank Test </li></ul><ul><li>Step 6: the test statistic W is the sum of the ranks in the R+ column </li></ul><ul><li>Test the hypothesis by comparing the calculated W to the critical value from the table in appendix P </li></ul><ul><ul><li>Note that n = the number of non-zero d i values </li></ul></ul>(continued)
- 9. Example <ul><li>The median class size is claimed to be 40 </li></ul><ul><li>Sample data for 8 classes is randomly obtained </li></ul><ul><li>Compare each value to the hypothesized median to find difference </li></ul>Class size = x i Difference d i = x i – 40 | d i | 23 45 34 78 34 66 61 95 -17 5 -6 38 -6 26 21 55 17 5 6 38 6 26 21 55
- 10. Example <ul><li>Rank the absolute differences: </li></ul>tied (continued) | d i | Rank 5 6 6 17 21 26 38 55 1 2.5 2.5 4 5 6 7 8
- 11. Example <ul><li>Put ranks in R+ and R- columns </li></ul><ul><li>and find sums: </li></ul>(continued) These three are below the claimed median, the others are above Class size = x i Difference d i = x i – 40 | d i | Rank R+ R- 23 45 34 78 34 66 61 95 -17 5 -6 38 -6 26 21 55 17 5 6 38 6 26 21 55 4 1 2.5 7 2.5 6 5 8 1 7 6 5 8 4 2.5 2.5 = 27 = 9
- 12. Completing the Test <ul><li>H 0 : Median = 40 </li></ul><ul><li>H A : Median ≠ 40 </li></ul>Test at the = .05 level: This is a two-tailed test and n = 8, so find W L and W U in appendix P: W L = 3 and W U = 33 The calculated test statistic is W = R+ = 27
- 13. Completing the Test <ul><li>H 0 : Median = 40 </li></ul><ul><li>H A : Median ≠ 40 </li></ul>W L = 3 and W U = 33 W L < W < W U so do not reject H 0 (there is not sufficient evidence to conclude that the median class size is different than 40) (continued) W L = 3 do not reject H 0 reject H 0 W = R+ = 27 W U = 33 reject H 0
- 14. If the Sample Size is Large <ul><li>The W test statistic approaches a normal distribution as n increases </li></ul><ul><li>For n > 20, W can be approximated by </li></ul>where W = sum of the R+ ranks d = number of non-zero d i values
- 15. Nonparametric Tests for Two Population Centers <ul><li>Nonparametric </li></ul><ul><li>Tests for Two </li></ul><ul><li>Population Centers </li></ul>Wilcoxon Matched-Pairs Signed Rank Test Mann-Whitney U-test Large Samples Small Samples Large Samples Small Samples
- 16. Mann-Whitney U-Test Used to compare two samples from two populations Assumptions: The two samples are independent and random The value measured is a continuous variable The measurement scale used is at least ordinal If they differ, the distributions of the two populations will differ only with respect to the central location
- 17. <ul><li>Consider two samples </li></ul><ul><ul><li>combine into a singe list, but keep track of which sample each value came from </li></ul></ul><ul><ul><li>rank the values in the combined list from low to high </li></ul></ul><ul><ul><ul><li>For ties, assign each the average rank of the tied values </li></ul></ul></ul><ul><ul><li>separate back into two samples, each value keeping its assigned ranking </li></ul></ul><ul><ul><li>sum the rankings for each sample </li></ul></ul>Mann-Whitney U-Test (continued)
- 18. <ul><li>If the sum of rankings from one sample differs enough from the sum of rankings from the other sample, we conclude there is a difference in the population medians </li></ul>Mann-Whitney U-Test (continued)
- 19. Mann-Whitney U-Test (continued) Mann-Whitney U-Statistics where: n 1 and n 2 are the two sample sizes R 1 and R 2 = sum of ranks for samples 1 and 2
- 20. Mann-Whitney U-Test (continued) Claim: Median class size for Math is larger than the median class size for English A random sample of 9 Math and 9 English classes is selected (samples do not have to be of equal size) Rank the combined values and then split them back into the separate samples
- 21. <ul><li>Suppose the results are: </li></ul>(continued) Mann-Whitney U-Test Class size (Math, M) Class size (English, E) 23 45 34 78 34 66 62 95 81 30 47 18 34 44 61 54 28 40
- 22. Mann-Whitney U-Test Ranking for combined samples tied (continued) Size Rank 18 1 23 2 28 3 30 4 34 6 34 6 34 6 40 8 44 9 Size Rank 45 10 47 11 54 12 61 13 62 14 66 15 78 16 81 17 95 18
- 23. <ul><li>Split back into the original samples: </li></ul>Mann-Whitney U-Test (continued) Class size (Math, M) Rank Class size (English, E) Rank 23 45 34 78 34 66 62 95 81 2 10 6 16 6 15 14 18 17 30 47 18 34 44 61 54 28 40 4 11 1 6 9 13 12 3 8 = 104 = 67
- 24. Mann-Whitney U-Test H 0 : Median M ≤ Median E H A : Median M > Median E Claim: Median class size for Math is larger than the median class size for English Note: U 1 + U 2 = n 1 n 2 (continued) Math: English:
- 25. <ul><li>The Mann-Whitney U tables in Appendices L and M give the lower tail of the U-distribution </li></ul><ul><li>For one-tailed tests like this one, check the alternative hypothesis to see if U 1 or U 2 should be used as the test statistic </li></ul><ul><li>Since the alternative hypothesis indicates that population 1 (Math) has a higher median, use U 1 as the test statistic </li></ul>Mann-Whitney U-Test (continued)
- 26. <ul><li>Use U 1 as the test statistic: U = 22 </li></ul><ul><li>Compare U = 22 to the critical value U from the appropriate table </li></ul><ul><ul><li>For sample sizes less than 9, use Appendix L </li></ul></ul><ul><ul><li>For samples sizes from 9 to 20, use Appendix M </li></ul></ul><ul><li>If U < U , reject H 0 </li></ul>Mann-Whitney U-Test (continued)
- 27. <ul><li>Use U 1 as the test statistic: U = 19 </li></ul><ul><li>U from Appendix M for = .05, n 1 = 9 and n 2 = 9 is U = 7 </li></ul>Mann-Whitney U-Test Since U U , do not reject H 0 (continued) U = 7 U = 19 do not reject H 0 reject H 0
- 28. Mann-Whitney U-Test for Large Samples <ul><li>The table in Appendix M includes U values only for sample sizes between 9 and 20 </li></ul><ul><li>The U statistic approaches a normal distribution as sample sizes increase </li></ul><ul><li>If samples are larger than 20, a normal approximation can be used </li></ul>
- 29. Mann-Whitney U-Test for Large Samples <ul><li>The mean and standard deviation for Mann-Whitney U Test Statistic: </li></ul>(continued) Where n 1 and n 2 are sample sizes from populations 1 and 2
- 30. Mann-Whitney U-Test for Large Samples <ul><li>Normal approximation for Mann-Whitney U Test Statistic: </li></ul>(continued)
- 31. Large Sample Example <ul><li>We wish to test </li></ul><ul><li>Suppose two samples are obtained: </li></ul><ul><li>n 1 = 40 , n 2 = 50 </li></ul><ul><li>When rankings are completed, the sum of ranks for sample 1 is R 1 = 1475 </li></ul><ul><li>When rankings are completed, the sum of ranks for sample 2 is R 2 = 2620 </li></ul>H 0 : Median 1 Median 2 H A : Median 1 < Median 2
- 32. <ul><li>U statistic is found to be U = 655 </li></ul>Large Sample Example Since the alternative hypothesis indicates that population 2 has a higher median, use U 2 as the test statistic Compute the U statistics: (continued)
- 33. Large Sample Example Since z = -2.80 < -1.645, we reject H 0 Reject H 0 = .05 Do not reject H 0 0 (continued)
- 34. Wilcoxon Matched-Pairs Signed Rank Test <ul><li>The Mann-Whitney U-Test is used when samples from two populations are independent </li></ul><ul><li>If samples are paired, they are not independent </li></ul><ul><li>Use Wilcoxon Matched-Pairs Signed Rank Test with paired samples </li></ul>
- 35. The Wilcoxon T Test Statistic <ul><li>Performing the Small-Sample Wilcoxon Matched Pairs Test (for n < 25) </li></ul><ul><li>Calculate the test statistic T using these steps: </li></ul><ul><li>Step 1: collect sample data </li></ul><ul><li>Step 2: compute d i = difference between the sample 1 value and its paired sample 2 value </li></ul><ul><li>Step 3: rank the differences, and give each rank the same sign as the sign of the difference value </li></ul>
- 36. The Wilcoxon T Test Statistic <ul><li>Performing the Small-Sample Wilcoxon Matched Pairs Test (for n < 25) </li></ul><ul><li>Step 4: The test statistic is the sum of the absolute values of the ranks for the group with the smaller expected sum </li></ul><ul><ul><li>Look at the alternative hypothesis to determine the group with the smaller expected sum </li></ul></ul><ul><ul><li>For two tailed tests, just choose the smaller sum </li></ul></ul>(continued)
- 37. Small Sample Example <ul><li>Paired samples, n = 9: </li></ul>Claim: Median value is smaller after than before Value (before) Value (after) 38 45 34 58 30 46 42 55 41 30 47 18 34 34 31 24 38 40
- 38. Small Sample Example <ul><li>Paired samples, n = 9: </li></ul>(continued) Value (before) Value (after) Difference d Rank of d Ranks with smaller expected sum 36 45 34 58 30 46 42 55 41 30 47 18 54 38 31 24 62 40 6 -2 16 4 -8 15 18 -7 1 4 -2 8 3 -6 7 9 -5 1 2 6 5 = T = 13
- 39. <ul><li>The calculated T value is T = 13 </li></ul><ul><li>Complete the test by comparing the calculated T value to the critical T-value from Appendix N </li></ul><ul><li>For n = 9 and = .025 for a one-tailed test, </li></ul><ul><li>T = 6 </li></ul>Small Sample Example Since T T , do not reject H 0 T = 6 T = 13 do not reject H 0 reject H 0 (continued)
- 40. Wilcoxon Matched Pairs Test for Large Samples <ul><li>The table in Appendix N includes T values only for sample sizes from 6 to 25 </li></ul><ul><li>The T statistic approaches a normal distribution as sample size increases </li></ul><ul><li>If the number of paired values is larger than 25, a normal approximation can be used </li></ul>
- 41. <ul><li>The mean and standard deviation for Wilcoxon T : </li></ul>Wilcoxon Matched Pairs Test for Large Samples (continued) where n is the number of paired values
- 42. Mann-Whitney U-Test for Large Samples <ul><li>Normal approximation for the Wilcoxon T Test Statistic: </li></ul>(continued)
- 43. <ul><li>Tests the equality of more than 2 population medians </li></ul><ul><li>Assumptions: </li></ul><ul><ul><li>variables have a continuous distribution. </li></ul></ul><ul><ul><li>the data are at least ordinal. </li></ul></ul><ul><ul><li>samples are independent. </li></ul></ul><ul><ul><li>samples come from populations whose only possible difference is that at least one may have a different central location than the others. </li></ul></ul>Kruskal-Wallis One-Way ANOVA
- 44. Kruskal-Wallis Test Procedure <ul><li>Obtain relative rankings for each value </li></ul><ul><ul><li>In event of tie, each of the tied values gets the average rank </li></ul></ul><ul><li>Sum the rankings for data from each of the k groups </li></ul><ul><ul><li>Compute the H test statistic </li></ul></ul>
- 45. Kruskal-Wallis Test Procedure <ul><li>The Kruskal-Wallis H test statistic: </li></ul><ul><li> (with k – 1 degrees of freedom) </li></ul>where: N = Sum of sample sizes in all samples k = Number of samples R i = Sum of ranks in the i th sample n i = Size of the i th sample (continued)
- 46. <ul><li>Complete the test by comparing the calculated H value to a critical 2 value from the chi-square distribution with k – 1 degrees of freedom </li></ul><ul><li>(The chi-square distribution is Appendix G) </li></ul><ul><li>Decision rule </li></ul><ul><ul><li>Reject H 0 if test statistic H > 2 </li></ul></ul><ul><ul><li>Otherwise do not reject H 0 </li></ul></ul>Kruskal-Wallis Test Procedure (continued)
- 47. <ul><li>Do different departments have different class sizes? </li></ul>Kruskal-Wallis Example Class size (Math, M) Class size (English, E) Class size (History, H) 23 45 54 78 66 55 60 72 45 70 30 40 18 34 44
- 48. <ul><li>Do different departments have different class sizes? </li></ul>Kruskal-Wallis Example Class size (Math, M) Ranking Class size (English, E) Ranking Class size (History, H) Ranking 23 41 54 78 66 2 6 9 15 12 55 60 72 45 70 10 11 14 8 13 30 40 18 34 44 3 5 1 4 7 = 44 = 56 = 20
- 49. <ul><li>The H statistic is </li></ul>Kruskal-Wallis Example (continued)
- 50. <ul><li>Since H = 6.72 < </li></ul><ul><li>do not reject H 0 </li></ul>Kruskal-Wallis Example (continued) Compare H = 6.72 to the critical value from the chi-square distribution for 5 – 1 = 4 degrees of freedom and = .05: There is not sufficient evidence to reject that the population medians are all equal
- 51. Kruskal-Wallis Correction <ul><li>If tied rankings occur, give each observation the mean rank for which it is tied </li></ul><ul><li>The H statistic is influenced by ties, and should be corrected </li></ul><ul><li>Correction for tied rankings: </li></ul>where: g = Number of different groups of ties t i = Number of tied observations in the i th tied group of scores N = Total number of observations
- 52. H Statistic Corrected for Tied Rankings <ul><li>Corrected H statistic: </li></ul>

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