1. 14 - 1 © 2003 Pearson Prentice Hall Statistics for Business and Economics Nonparametric Statistics Chapter 14

2. 14 - 2 © 2003 Pearson Prentice Hall Learning Objectives 1.Distinguish Parametric & Nonparametric Test Procedures 2.Explain a Variety of Nonparametric Test Procedures 3.Solve Hypothesis Testing Problems Using Nonparametric Tests 4.Compute Spearman’s Rank Correlation

3. 14 - 3 © 2003 Pearson Prentice Hall Hypothesis Testing Procedures Many More Tests Exist!

4. 14 - 4 © 2003 Pearson Prentice Hall Parametric Test Procedures 1.Involve Population Parameters Example: Population Mean Example: Population Mean 2.Require Interval Scale or Ratio Scale Whole Numbers or Fractions Whole Numbers or Fractions Example: Height in Inches (72, 60.5, 54.7) Example: Height in Inches (72, 60.5, 54.7) 3.Have Stringent Assumptions Example: Normal Distribution Example: Normal Distribution 4.Examples: Z Test, t Test,  2 Test

5. 14 - 5 © 2003 Pearson Prentice Hall Nonparametric Test Procedures 1.Do Not Involve Population Parameters Example: Probability Distributions, Independence Example: Probability Distributions, Independence 2.Data Measured on Any Scale Ratio or Interval Ratio or Interval Ordinal Ordinal Example: Good-Better-Best Example: Good-Better-Best Nominal Nominal Example: Male-Female Example: Male-Female 3.Example: Wilcoxon Rank Sum Test

6. 14 - 6 © 2003 Pearson Prentice Hall Advantages of Nonparametric Tests 1.Used With All Scales 2.Easier to Compute Developed Originally Before Wide Computer Use Developed Originally Before Wide Computer Use 3.Make Fewer Assumptions 4.Need Not Involve Population Parameters 5.Results May Be as Exact as Parametric Procedures © 1984-1994 T/Maker Co.

7. 14 - 7 © 2003 Pearson Prentice Hall Disadvantages of Nonparametric Tests 1.May Waste Information n If Data Permit Using Parametric Procedures n Example: Converting Data From Ratio to Ordinal Scale 2.Difficult to Compute by Hand for Large Samples 3.Tables Not Widely Available © 1984-1994 T/Maker Co.

8. 14 - 8 © 2003 Pearson Prentice Hall Frequently Used Nonparametric Tests 1.Sign Test 2.Wilcoxon Rank Sum Test 3.Wilcoxon Signed Rank Test 4.Kruskal Wallis H-Test 5.Friedman’s F r -Test 6.Spearman’s Rank Correlation Coefficient

9. 14 - 9 © 2003 Pearson Prentice Hall Sign Test

10. 14 - 10 © 2003 Pearson Prentice Hall Frequently Used Nonparametric Tests 1.Sign Test 2.Wilcoxon Rank Sum Test 3.Wilcoxon Signed Rank Test 4.Kruskal Wallis H-Test 5.Friedman’s F r -Test 6.Spearman’s Rank Correlation Coefficient

11. 14 - 11 © 2003 Pearson Prentice Hall Sign Test 1.Tests One Population Median,  (eta) 2.Corresponds to t-Test for 1 Mean 3.Assumes Population Is Continuous 4.Small Sample Test Statistic: # Sample Values Above (or Below) Median Alternative Hypothesis Determines Alternative Hypothesis Determines 5.Can Use Normal Approximation If n  10

12. 14 - 12 © 2003 Pearson Prentice Hall Sign Test Uses P-Value to Make Decision Binomial: n = 8 p = 0.5 P-Value Is the Probability of Getting an Observation At Least as Extreme as We Got. If 7 of 8 Observations ‘Favor’ H a, Then P-Value = P(x  7) =.031 +.004 =.035. If  =.05, Then Reject H 0 Since P-Value  .

13. 14 - 13 © 2003 Pearson Prentice Hall Sign Test Example You’re an analyst for Chef- Boy-R-Dee. You’ve asked 7 people to rate a new ravioli on a 5-point Likert scale (1 = terrible to 5 = excellent. The ratings are: 2 5 3 4 1 4 5. At the.05 level, is there evidence that the median rating is less than 3?

14. 14 - 14 © 2003 Pearson Prentice Hall Sign Test Solution H 0 : H a :  = Test Statistic: P-Value:Decision:Conclusion:

15. 14 - 15 © 2003 Pearson Prentice Hall Sign Test Solution H 0 :  = 3 H a :  < 3  = Test Statistic: P-Value:Decision:Conclusion:

16. 14 - 16 © 2003 Pearson Prentice Hall Sign Test Solution H 0 :  = 3 H a :  < 3  =.05 Test Statistic: P-Value:Decision:Conclusion:

17. 14 - 17 © 2003 Pearson Prentice Hall Sign Test Solution H 0 :  = 3 H a :  < 3  =.05 Test Statistic: P-Value:Decision:Conclusion: S = 2 (Ratings 1 & 2 Are Less Than  = 3: 2, 5, 3, 4, 1, 4, 5)

18. 14 - 18 © 2003 Pearson Prentice Hall Sign Test Solution H 0 :  = 3 H a :  < 3  =.05 Test Statistic: P-Value:Decision:Conclusion: P(x  2) = 1 - P(x  1) =.937 (Binomial Table, n = 7, p = 0.50) S = 2 (Ratings 1 & 2 Are Less Than  = 3: 2, 5, 3, 4, 1, 4, 5)

19. 14 - 19 © 2003 Pearson Prentice Hall Sign Test Solution H 0 :  = 3 H a :  < 3  =.05 Test Statistic: P-Value:Decision:Conclusion: Do Not Reject at  =.05 P(x  2) = 1 - P(x  1) =.937 (Binomial Table, n = 7, p = 0.50) S = 2 (Ratings 1 & 2 Are Less Than  = 3: 2, 5, 3, 4, 1, 4, 5)

20. 14 - 20 © 2003 Pearson Prentice Hall Sign Test Solution H 0 :  = 3 H a :  < 3  =.05 Test Statistic: P-Value:Decision:Conclusion: Do Not Reject at  =.05 There Is No Evidence Median Is Less Than 3 P(x  2) = 1 - P(x  1) =.937 (Binomial Table, n = 7, p = 0.50) S = 2 (Ratings 1 & 2 Are Less Than  = 3: 2, 5, 3, 4, 1, 4, 5)

21. 14 - 21 © 2003 Pearson Prentice Hall Wilcoxon Rank Sum Test

22. 14 - 22 © 2003 Pearson Prentice Hall Frequently Used Nonparametric Tests 1.Sign Test 2.Wilcoxon Rank Sum Test 3.Wilcoxon Signed Rank Test 4.Kruskal Wallis H-Test 5.Friedman’s F r -Test 6.Spearman’s Rank Correlation Coefficient

23. 14 - 23 © 2003 Pearson Prentice Hall Wilcoxon Rank Sum Test 1.Tests Two Independent Population Probability Distributions 2.Corresponds to t-Test for 2 Independent Means 3.Assumptions Independent, Random Samples Independent, Random Samples Populations Are Continuous Populations Are Continuous 4.Can Use Normal Approximation If n i  10

24. 14 - 24 © 2003 Pearson Prentice Hall Wilcoxon Rank Sum Test Procedure 1.Assign Ranks, R i, to the n 1 + n 2 Sample Observations If Unequal Sample Sizes, Let n 1 Refer to Smaller-Sized Sample If Unequal Sample Sizes, Let n 1 Refer to Smaller-Sized Sample Smallest Value = 1 Smallest Value = 1 Average Ties Average Ties 2.Sum the Ranks, T i, for Each Sample 3.Test Statistic Is T A (Smallest Sample)

25. 14 - 25 © 2003 Pearson Prentice Hall Wilcoxon Rank Sum Test Example You’re a production planner. You want to see if the operating rates for 2 factories is the same. For factory 1, the rates (% of capacity) are 71, 82, 77, 92, 88. For factory 2, the rates are 85, 82, 94 & 97. Do the factory rates have the same probability distributions at the.10 level?

26. 14 - 26 © 2003 Pearson Prentice Hall Wilcoxon Rank Sum Test Solution H 0 : H a :  = n 1 =n 2 = Critical Value(s): Test Statistic: Decision:Conclusion:  Ranks

27. 14 - 27 © 2003 Pearson Prentice Hall Wilcoxon Rank Sum Test Solution H 0 : Identical Distrib. H a : Shifted Left or Right  = n 1 =n 2 = Critical Value(s): Test Statistic: Decision:Conclusion:  Ranks

28. 14 - 28 © 2003 Pearson Prentice Hall Wilcoxon Rank Sum Test Solution H 0 : Identical Distrib. H a : Shifted Left or Right  =.10 n 1 = 4 n 2 = 5 Critical Value(s): Test Statistic: Decision:Conclusion:  Ranks

29. 14 - 29 © 2003 Pearson Prentice Hall Wilcoxon Rank Sum Table (Portion)  =.05 one-tailed;  =.10 two-tailed

30. 14 - 30 © 2003 Pearson Prentice Hall Wilcoxon Rank Sum Test Solution H 0 : Identical Distrib. H a : Shifted Left or Right  =.10 n 1 = 4 n 2 = 5 Critical Value(s): Test Statistic: Decision:Conclusion: RejectReject Do Not Reject 1327  Ranks

31. 14 - 31 © 2003 Pearson Prentice Hall Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank Rank Sum

32. 14 - 32 © 2003 Pearson Prentice Hall Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 7185 82 7794 9297 88... Rank Sum

33. 14 - 33 © 2003 Pearson Prentice Hall Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 71185 82 7794 9297 88... Rank Sum

34. 14 - 34 © 2003 Pearson Prentice Hall Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 71185 82 77294 9297 88... Rank Sum

35. 14 - 35 © 2003 Pearson Prentice Hall Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 71185 823 4 77294 9297 88... Rank Sum

36. 14 - 36 © 2003 Pearson Prentice Hall Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 71185 823 3.5824 3.5 77294 9297 88... Rank Sum

37. 14 - 37 © 2003 Pearson Prentice Hall Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 711855 823 3.5824 3.5 77294 9297 88... Rank Sum

38. 14 - 38 © 2003 Pearson Prentice Hall Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 711855 823 3.5824 3.5 77294 9297 886... Rank Sum

39. 14 - 39 © 2003 Pearson Prentice Hall Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 711855 823 3.5824 3.5 77294 92797 886... Rank Sum

40. 14 - 40 © 2003 Pearson Prentice Hall Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 711855 823 3.5824 3.5 772948 92797 886... Rank Sum

41. 14 - 41 © 2003 Pearson Prentice Hall Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 711855 823 3.5824 3.5 772948 927979 886... Rank Sum

42. 14 - 42 © 2003 Pearson Prentice Hall Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 711855 823 3.5824 3.5 772948 927979 886... Rank Sum19.525.5

43. 14 - 43 © 2003 Pearson Prentice Hall Wilcoxon Rank Sum Test Solution H 0 : Identical Distrib. H a : Shifted Left or Right  =.10 n 1 = 4 n 2 = 5 Critical Value(s): Test Statistic: Decision:Conclusion: RejectReject Do Not Reject 1327  Ranks T 2 = 5 + 3.5 + 8+ 9 = 25.5 (Smallest Sample)

44. 14 - 44 © 2003 Pearson Prentice Hall Wilcoxon Rank Sum Test Solution H 0 : Identical Distrib. H a : Shifted Left or Right  =.10 n 1 = 4 n 2 = 5 Critical Value(s): Test Statistic: Decision:Conclusion: Do Not Reject at  =.10 RejectReject Do Not Reject 1327  Ranks T 2 = 5 + 3.5 + 8+ 9 = 25.5 (Smallest Sample)

45. 14 - 45 © 2003 Pearson Prentice Hall Wilcoxon Rank Sum Test Solution H 0 : Identical Distrib. H a : Shifted Left or Right  =.10 n 1 = 4 n 2 = 5 Critical Value(s): Test Statistic: Decision:Conclusion: Do Not Reject at  =.10 There Is No Evidence Distrib. Are Not Equal RejectReject Do Not Reject 1327  Ranks T 2 = 5 + 3.5 + 8+ 9 = 25.5 (Smallest Sample)

46. 14 - 46 © 2003 Pearson Prentice Hall Wilcoxon Signed Rank Test

47. 14 - 47 © 2003 Pearson Prentice Hall Frequently Used Nonparametric Tests 1.Sign Test 2.Wilcoxon Rank Sum Test 3.Wilcoxon Signed Rank Test 4.Kruskal Wallis H-Test 5.Friedman’s F r -Test 6.Spearman’s Rank Correlation Coefficient

48. 14 - 48 © 2003 Pearson Prentice Hall Wilcoxon Signed Rank Test 1.Tests Probability Distributions of 2 Related Populations 2.Corresponds to t-test for Dependent (Paired) Means 3.Assumptions Random Samples Random Samples Both Populations Are Continuous Both Populations Are Continuous 4.Can Use Normal Approximation If n  25

49. 14 - 49 © 2003 Pearson Prentice Hall Signed Rank Test Procedure 1.Obtain Difference Scores, D i = X 1i - X 2i 2.Take Absolute Value of Differences, D i 3.Delete Differences With 0 Value 4.Assign Ranks, R i, Where Smallest = 1 5.Assign Ranks Same Signs as D i 6.Sum ‘+’ Ranks (T + ) & ‘-’ Ranks (T - ) Test Statistic Is T - (One-Tailed Test) Test Statistic Is T - (One-Tailed Test) Test Statistic Is Smaller of T - or T + (2-Tail) Test Statistic Is Smaller of T - or T + (2-Tail)

50. 14 - 50 © 2003 Pearson Prentice Hall Signed Rank Test Computation Table

51. 14 - 51 © 2003 Pearson Prentice Hall Signed Rank Test Example You work in the finance department. Is the new financial package faster (.05 level)? You collect the following data entry times: UserCurrentNew Donna9.989.88 Santosha9.889.86 Sam9.909.83 Tamika9.999.80 Brian9.949.87 Jorge9.849.84 © 1984-1994 T/Maker Co.

52. 14 - 52 © 2003 Pearson Prentice Hall Signed Rank Test Solution H 0 : H a :  = n’ = Critical Value(s): Test Statistic: Decision:Conclusion: T0T0T0T0 Reject Do Not Reject

53. 14 - 53 © 2003 Pearson Prentice Hall Signed Rank Test Solution H 0 : Identical Distrib. H a : Current Shifted Right  = n’ = Critical Value(s): Test Statistic: Decision:Conclusion: T0T0T0T0 Reject Do Not Reject

54. 14 - 54 © 2003 Pearson Prentice Hall Signed Rank Test Computation Table

55. 14 - 55 © 2003 Pearson Prentice Hall Signed Rank Test Solution H 0 : Identical Distrib. H a : Current Shifted Right  =.05 n’ = 5 (not 6; 1 elim.) Critical Value(s): Test Statistic: Decision:Conclusion: Reject Do Not Reject T0T0T0T0

56. 14 - 56 © 2003 Pearson Prentice Hall Wilcoxon Signed Rank Table (Portion)

57. 14 - 57 © 2003 Pearson Prentice Hall Signed Rank Test Solution H 0 : Identical Distrib. H a : Current Shifted Right  =.05 n’ = 5 (not 6; 1 elim.) Critical Value(s): Test Statistic: Decision:Conclusion: Reject Do Not Reject 1 T0T0T0T0

58. 14 - 58 © 2003 Pearson Prentice Hall Signed Rank Test Solution H 0 : Identical Distrib. H a : Current Shifted Right  =.05 n’ = 5 (not 6; 1 elim.) Critical Value(s): Test Statistic: Decision:Conclusion: Reject Do Not Reject 1 T0T0T0T0 Since One-Tailed Test & Current Shifted Right, Use T - : T - = 0

59. 14 - 59 © 2003 Pearson Prentice Hall Signed Rank Test Solution H 0 : Identical Distrib. H a : Current Shifted Right  =.05 n’ = 5 (not 6; 1 elim.) Critical Value(s): Test Statistic: Decision:Conclusion: Reject at  =.05 Reject Do Not Reject 1 T0T0T0T0 Since One-Tailed Test & Current Shifted Right, Use T - : T - = 0

60. 14 - 60 © 2003 Pearson Prentice Hall Signed Rank Test Solution H 0 : Identical Distrib. H a : Current Shifted Right  =.05 n’ = 5 (not 6; 1 elim.) Critical Value(s): Test Statistic: Decision:Conclusion: Reject at  =.05 There Is Evidence New Package Is Faster Reject Do Not Reject 1 T0T0T0T0 Since One-Tailed Test & Current Shifted Right, Use T - : T - = 0

61. 14 - 61 © 2003 Pearson Prentice Hall Kruskal-Wallis H-Test

62. 14 - 62 © 2003 Pearson Prentice Hall Frequently Used Nonparametric Tests 1.Sign Test 2.Wilcoxon Rank Sum Test 3.Wilcoxon Signed Rank Test 4.Kruskal Wallis H-Test 5.Friedman’s F r -Test 6.Spearman’s Rank Correlation Coefficient

63. 14 - 63 © 2003 Pearson Prentice Hall Kruskal-Wallis H-Test 1.Tests the Equality of More Than 2 (p) Population Probability Distributions 2.Corresponds to ANOVA for More Than 2 Means 3.Used to Analyze Completely Randomized Experimental Designs 4.Uses  2 Distribution with p - 1 df If At Least 1 Sample Size n j > 5 If At Least 1 Sample Size n j > 5

64. 14 - 64 © 2003 Pearson Prentice Hall Kruskal-Wallis H-Test Assumptions 1.Independent, Random Samples 2.At Least 5 Observations Per Sample 3.Continuous Population Probability Distributions

65. 14 - 65 © 2003 Pearson Prentice Hall Kruskal-Wallis H-Test Procedure 1.Assign Ranks, R i, to the n Combined Observations Smallest Value = 1; Largest Value = n Smallest Value = 1; Largest Value = n Average Ties Average Ties 2.Sum Ranks for Each Group

66. 14 - 66 © 2003 Pearson Prentice Hall Kruskal-Wallis H-Test Procedure 1.Assign Ranks, R i, to the n Combined Observations Smallest Value = 1; Largest Value = n Smallest Value = 1; Largest Value = n Average Ties Average Ties 2.Sum Ranks for Each Group 3.Compute Test Statistic Squared total of each group

67. 14 - 67 © 2003 Pearson Prentice Hall Kruskal-Wallis H-Test Example As production manager, you want to see if 3 filling machines have different filling times. You assign 15 similarly trained & experienced workers, 5 per machine, to the machines. At the.05 level, is there a difference in the distribution of filling times? Mach1Mach2Mach3 25.4023.4020.00 26.3121.8022.20 24.1023.5019.75 23.7422.7520.60 25.1021.6020.40

68. 14 - 68 © 2003 Pearson Prentice Hall  2 0 Kruskal-Wallis H-Test Solution H 0 : H a :  = df = Critical Value(s): Test Statistic: Decision:Conclusion:

69. 14 - 69 © 2003 Pearson Prentice Hall  2 0 Kruskal-Wallis H-Test Solution H 0 : Identical Distrib. H a : At Least 2 Differ  = df = Critical Value(s): Test Statistic: Decision:Conclusion:

70. 14 - 70 © 2003 Pearson Prentice Hall  2 0 Kruskal-Wallis H-Test Solution H 0 : Identical Distrib. H a : At Least 2 Differ  =.05 df = p - 1 = 3 - 1 = 2 Critical Value(s): Test Statistic: Decision:Conclusion:

71. 14 - 71 © 2003 Pearson Prentice Hall  2 05.991 Kruskal-Wallis H-Test Solution H 0 : Identical Distrib. H a : At Least 2 Differ  =.05 df = p - 1 = 3 - 1 = 2 Critical Value(s): Test Statistic: Decision:Conclusion:  =.05

72. 14 - 72 © 2003 Pearson Prentice Hall Kruskal-Wallis H-Test Solution Raw Data Mach1Mach2Mach3 25.4023.4020.00 26.3121.8022.20 24.1023.5019.75 23.7422.7520.60 25.1021.6020.40 Ranks Mach1Mach2Mach3

73. 14 - 73 © 2003 Pearson Prentice Hall Kruskal-Wallis H-Test Solution Raw Data Mach1Mach2Mach3 25.4023.4020.00 26.3121.8022.20 24.1023.5019.75 23.7422.7520.60 25.1021.6020.40 Ranks Mach1Mach2Mach3 1

74. 14 - 74 © 2003 Pearson Prentice Hall Kruskal-Wallis H-Test Solution Raw Data Mach1Mach2Mach3 25.4023.4020.00 26.3121.8022.20 24.1023.5019.75 23.7422.7520.60 25.1021.6020.40 Ranks Mach1Mach2Mach3 2 1

75. 14 - 75 © 2003 Pearson Prentice Hall Kruskal-Wallis H-Test Solution Raw Data Mach1Mach2Mach3 25.4023.4020.00 26.3121.8022.20 24.1023.5019.75 23.7422.7520.60 25.1021.6020.40 Ranks Mach1Mach2Mach3 2 1 3

76. 14 - 76 © 2003 Pearson Prentice Hall Kruskal-Wallis H-Test Solution Raw Data Mach1Mach2Mach3 25.4023.4020.00 26.3121.8022.20 24.1023.5019.75 23.7422.7520.60 25.1021.6020.40 Ranks Mach1Mach2Mach3 1492 1567 12101 1184 1353

77. 14 - 77 © 2003 Pearson Prentice Hall Kruskal-Wallis H-Test Solution Raw Data Mach1Mach2Mach3 25.4023.4020.00 26.3121.8022.20 24.1023.5019.75 23.7422.7520.60 25.1021.6020.40 Ranks Mach1Mach2Mach3 1492 1567 12101 1184 1353 653817 Total

78. 14 - 78 © 2003 Pearson Prentice Hall Kruskal-Wallis H-Test Solution

79. 14 - 79 © 2003 Pearson Prentice Hall  2 05.991 Kruskal-Wallis H-Test Solution H 0 : Identical Distrib. H a : At Least 2 Differ  =.05 df = p - 1 = 3 - 1 = 2 Critical Value(s): Test Statistic: Decision:Conclusion:  =.05 H = 11.58

80. 14 - 80 © 2003 Pearson Prentice Hall  2 05.991 Kruskal-Wallis H-Test Solution H 0 : Identical Distrib. H a : At Least 2 Differ  =.05 df = p - 1 = 3 - 1 = 2 Critical Value(s): Test Statistic: Decision:Conclusion: Reject at  =.05  =.05 H = 11.58

81. 14 - 81 © 2003 Pearson Prentice Hall  2 05.991 Kruskal-Wallis H-Test Solution H 0 : Identical Distrib. H a : At Least 2 Differ  =.05 df = p - 1 = 3 - 1 = 2 Critical Value(s): Test Statistic: Decision:Conclusion: Reject at  =.05 There Is Evidence Pop. Distrib. Are Different  =.05 H = 11.58

82. 14 - 82 © 2003 Pearson Prentice Hall Friedman F r -Test for a Randomized Block Design

83. 14 - 83 © 2003 Pearson Prentice Hall Frequently Used Nonparametric Tests 1.Sign Test 2.Wilcoxon Rank Sum Test 3.Wilcoxon Signed Rank Test 4.Kruskal Wallis H-Test 5.Friedman’s F r -Test 6.Spearman’s Rank Correlation Coefficient

84. 14 - 84 © 2003 Pearson Prentice Hall Friedman F r -Test 1.Tests the Equality of More Than 2 (p) Population Probability Distributions 2.Corresponds to ANOVA for More Than 2 Means 3.Used to Analyze Randomized Block Experimental Designs 4.Uses  2 Distribution with p - 1 df If either p, the number of treatments, or b, the number of blocks, exceeds 5 If either p, the number of treatments, or b, the number of blocks, exceeds 5

85. 14 - 85 © 2003 Pearson Prentice Hall Friedman F r -Test Assumptions 1.The p treatments are randomly assigned to experimental units within the b blocks Samples 2.The measurements can be ranked within the blocks 3.Continuous population probability distributions

86. 14 - 86 © 2003 Pearson Prentice Hall Friedman F r -Test Procedure 1.Assign Ranks, R i = 1 – p, to the p treatments in each of the b blocks Smallest Value = 1; Largest Value = p Smallest Value = 1; Largest Value = p Average Ties Average Ties 2.Sum Ranks for Each Treatment

87. 14 - 87 © 2003 Pearson Prentice Hall Friedman F r -Test Procedure 1.Assign Ranks, R i = 1 – p, to the p treatments in each of the b blocks Smallest Value = 1; Largest Value = p Smallest Value = 1; Largest Value = p Average Ties Average Ties 2.Sum Ranks for Each Treatment 3.Compute Test Statistic Squared total of each treatment

88. 14 - 88 © 2003 Pearson Prentice Hall Friedman F r -Test Example Three new traps were tested to compare their ability to trap mosquitoes. Each of the traps, A, B, and C were placed side- by-side at each five different locations. The number of mosquitoes in each trap was recorded. At the.05 level, is there a difference in the distribution of number of mosquitoes caught by the three traps? TrapA TrapBTrapC 35 0 231715 1157 842 19115

89. 14 - 89 © 2003 Pearson Prentice Hall  2 0 Friedman F r -Test Solution H 0 : Identical Distrib. H a : At Least 2 Differ  = df = Critical Value(s): Test Statistic: Decision:Conclusion:

90. 14 - 90 © 2003 Pearson Prentice Hall  2 0 Friedman F r -Test Solution H 0 : Identical Distrib. H a : At Least 2 Differ  =.05 df = p - 1 = 3 - 1 = 2 Critical Value(s): Test Statistic: Decision:Conclusion:

91. 14 - 91 © 2003 Pearson Prentice Hall  2 05.991 Friedman F r -Test Solution H 0 : Identical Distrib. H a : At Least 2 Differ  =.05 df = p - 1 = 3 - 1 = 2 Critical Value(s): Test Statistic: Decision:Conclusion:  =.05

92. 14 - 92 © 2003 Pearson Prentice Hall Friedman F r -Test Solution Raw Data TrapA TrapBTrapC 35 0 231715 1157 842 19115 Ranks TrapA TrapBTrapC

93. 14 - 93 © 2003 Pearson Prentice Hall Friedman F r -Test Solution Raw Data TrapA TrapBTrapC 35 0 231715 1157 842 19115 Ranks TrapA TrapBTrapC 1

94. 14 - 94 © 2003 Pearson Prentice Hall Friedman F r -Test Solution Raw Data TrapA TrapBTrapC 35 0 231715 1157 842 19115 Ranks TrapA TrapBTrapC 231 1

95. 14 - 95 © 2003 Pearson Prentice Hall Friedman F r -Test Solution Raw Data TrapA TrapBTrapC 35 0 231715 1157 842 19115 Ranks TrapA TrapBTrapC 231 321

96. 14 - 96 © 2003 Pearson Prentice Hall Friedman F r -Test Solution Raw Data TrapA TrapBTrapC 35 0 231715 1157 842 19115 Ranks TrapA TrapBTrapC 231 321 312..

97. 14 - 97 © 2003 Pearson Prentice Hall Friedman F r -Test Solution Raw Data TrapA TrapBTrapC 35 0 231715 1157 842 19115 Ranks TrapA TrapBTrapC 231 321 312 321 321 14106 Total

98. 14 - 98 © 2003 Pearson Prentice Hall Friedman F r -Test Solution

99. 14 - 99 © 2003 Pearson Prentice Hall  2 05.991 Friedman F r -Test Solution H 0 : Identical Distrib. H a : At Least 2 Differ  =.05 df = p - 1 = 3 - 1 = 2 Critical Value(s): Test Statistic: Decision:Conclusion:  =.05 F r = 6.64

100. 14 - 100 © 2003 Pearson Prentice Hall  2 05.991 Friedman F r -Test Solution H 0 : Identical Distrib. H a : At Least 2 Differ  =.05 df = p - 1 = 3 - 1 = 2 Critical Value(s): Test Statistic: Decision:Conclusion: Reject at  =.05  =.05 F r = 6.64

101. 14 - 101 © 2003 Pearson Prentice Hall  2 05.991 Friedman F r -Test Solution H 0 : Identical Distrib. H a : At Least 2 Differ  =.05 df = p - 1 = 3 - 1 = 2 Critical Value(s): Test Statistic: Decision:Conclusion: Reject at  =.05 There Is Evidence Pop. Distrib. Are Different  =.05 F r = 6.64

102. 14 - 102 © 2003 Pearson Prentice Hall Spearman’s Rank Correlation Coefficient

103. 14 - 103 © 2003 Pearson Prentice Hall Frequently Used Nonparametric Tests 1.Sign Test 2.Wilcoxon Rank Sum Test 3.Wilcoxon Signed Rank Test 4.Kruskal Wallis H-Test 5.Friedman’s F r -Test 6.Spearman’s Rank Correlation Coefficient

104. 14 - 104 © 2003 Pearson Prentice Hall Spearman’s Rank Correlation Coefficient 1.Measures Correlation Between Ranks 2.Corresponds to Pearson Product Moment Correlation Coefficient 3.Values Range from -1 to +1

105. 14 - 105 © 2003 Pearson Prentice Hall Spearman’s Rank Correlation Coefficient 1.Measures Correlation Between Ranks 2.Corresponds to Pearson Product Moment Correlation Coefficient 3.Values Range from -1 to +1 4.Equation (Shortcut)

106. 14 - 106 © 2003 Pearson Prentice Hall Spearman’s Rank Correlation Procedure 1.Assign Ranks, R i, to the Observations of Each Variable Separately 2.Calculate Differences, d i, Between Each Pair of Ranks 3.Square Differences, d i 2, Between Ranks 4.Sum Squared Differences for Each Variable 5.Use Shortcut Approximation Formula

107. 14 - 107 © 2003 Pearson Prentice Hall Spearman’s Rank Correlation Example You’re a research assistant for the FBI. You’re investigating the relationship between a person’s attempts at deception & % changes in their pupil size. You ask subjects a series of questions, some of which they must answer dishonestly. At the.05 level, what is the correlation coefficient? Subj.DeceptionPupil 18710 2636 39511 4507 5430

108. 14 - 108 © 2003 Pearson Prentice Hall Spearman’s Rank Correlation Table

109. 14 - 109 © 2003 Pearson Prentice Hall Spearman’s Rank Correlation Table

110. 14 - 110 © 2003 Pearson Prentice Hall Spearman’s Rank Correlation Table

111. 14 - 111 © 2003 Pearson Prentice Hall Spearman’s Rank Correlation Table

112. 14 - 112 © 2003 Pearson Prentice Hall Spearman’s Rank Correlation Table

113. 14 - 113 © 2003 Pearson Prentice Hall Spearman’s Rank Correlation Table

114. 14 - 114 © 2003 Pearson Prentice Hall Spearman’s Rank Correlation Solution

115. 14 - 115 © 2003 Pearson Prentice Hall Conclusion 1.Distinguished Parametric & Nonparametric Test Procedures 2.Explained a Variety of Nonparametric Test Procedures 3.Solved Hypothesis Testing Problems Using Nonparametric Tests 4.Computed Spearman’s Rank Correlation

116. End of Chapter Any blank slides that follow are blank intentionally.