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14 - 1 © 2000 Prentice-Hall, Inc. Statistics Nonparametric Statistics Chapter 14.

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1 14 - 1 © 2000 Prentice-Hall, Inc. Statistics Nonparametric Statistics Chapter 14

2 14 - 2 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. Hypothesis Testing Procedures Many More Tests Exist!

4 14 - 4 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. Frequently Used Nonparametric Tests 1.Sign Test 2.Wilcoxon Rank Sum Test 3.Wilcoxon Signed Rank Test 4.Kruskal Wallis H-Test 5.Friedman F r -Test

9 14 - 9 © 2000 Prentice-Hall, Inc. Sign Test

10 14 - 10 © 2000 Prentice-Hall, Inc. Frequently Used Nonparametric Tests 1.Sign Test 2.Wilcoxon Rank Sum Test 3.Wilcoxon Signed Rank Test 4.Kruskal Wallis H-Test 5.Friedman F r -Test

11 14 - 11 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. 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 at least 3?

14 14 - 14 © 2000 Prentice-Hall, Inc. Sign Test Solution H 0 : H a :  = Test Statistic: P-Value:Decision:Conclusion:

15 14 - 15 © 2000 Prentice-Hall, Inc. Sign Test Solution H 0 :  = 3 H a :  < 3  = Test Statistic: P-Value:Decision:Conclusion:

16 14 - 16 © 2000 Prentice-Hall, Inc. Sign Test Solution H 0 :  = 3 H a :  < 3  =.05 Test Statistic: P-Value:Decision:Conclusion:

17 14 - 17 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. Wilcoxon Rank Sum Test

22 14 - 22 © 2000 Prentice-Hall, Inc. Frequently Used Nonparametric Tests 1.Sign Test 2.Wilcoxon Rank Sum Test 3.Wilcoxon Signed Rank Test 4.Kruskal Wallis H-Test 5.Friedman F r -Test

23 14 - 23 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. Wilcoxon Rank Sum Test Solution H 0 : H a :  = n 1 =n 2 = Critical Value(s): Test Statistic: Decision:Conclusion:  Ranks

27 14 - 27 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. Wilcoxon Rank Sum Table (Portion)  =.05 one-tailed;  =.10 two-tailed

30 14 - 30 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank Rank Sum

32 14 - 32 © 2000 Prentice-Hall, Inc. Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 7185 82 7794 9297 88... Rank Sum

33 14 - 33 © 2000 Prentice-Hall, Inc. Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 71185 82 7794 9297 88... Rank Sum

34 14 - 34 © 2000 Prentice-Hall, Inc. Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 71185 82 77294 9297 88... Rank Sum

35 14 - 35 © 2000 Prentice-Hall, Inc. Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 71185 823 4 77294 9297 88... Rank Sum

36 14 - 36 © 2000 Prentice-Hall, Inc. Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 71185 823 3.5824 3.5 77294 9297 88... Rank Sum

37 14 - 37 © 2000 Prentice-Hall, Inc. Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 711855 823 3.5824 3.5 77294 9297 88... Rank Sum

38 14 - 38 © 2000 Prentice-Hall, Inc. Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 711855 823 3.5824 3.5 77294 9297 886... Rank Sum

39 14 - 39 © 2000 Prentice-Hall, Inc. Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 711855 823 3.5824 3.5 77294 92797 886... Rank Sum

40 14 - 40 © 2000 Prentice-Hall, Inc. Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 711855 823 3.5824 3.5 772948 92797 886... Rank Sum

41 14 - 41 © 2000 Prentice-Hall, Inc. Wilcoxon Rank Sum Test Computation Table Factory 1Factory 2 RateRankRateRank 711855 823 3.5824 3.5 772948 927979 886... Rank Sum

42 14 - 42 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. Wilcoxon Signed Rank Test

47 14 - 47 © 2000 Prentice-Hall, Inc. Frequently Used Nonparametric Tests 1.Sign Test 2.Wilcoxon Rank Sum Test 3.Wilcoxon Signed Rank Test 4.Kruskal Wallis H-Test 5.Friedman F r -Test

48 14 - 48 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. Signed Rank Test Computation Table

51 14 - 51 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. Signed Rank Test Solution H 0 : H a :  = n’ = Critical Value(s): Test Statistic: Decision:Conclusion: T0T0T0T0 Reject Do Not Reject

53 14 - 53 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. Signed Rank Test Computation Table

55 14 - 55 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. Wilcoxon Signed Rank Table (Portion)

57 14 - 57 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. Kruskal-Wallis H-Test

62 14 - 62 © 2000 Prentice-Hall, Inc. Frequently Used Nonparametric Tests 1.Sign Test 2.Wilcoxon Rank Sum Test 3.Wilcoxon Signed Rank Test 4.Kruskal Wallis H-Test 5.Friedman F r -Test

63 14 - 63 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. Kruskal-Wallis H-Test Assumptions 1.Independent, Random Samples 2.At Least 5 Observations Per Sample 3.Continuous Population Probability Distributions

65 14 - 65 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc.  2 0 Kruskal-Wallis H-Test Solution H 0 : H a :  = df = Critical Value(s): Test Statistic: Decision:Conclusion:

69 14 - 69 © 2000 Prentice-Hall, Inc.  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 © 2000 Prentice-Hall, Inc.  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 © 2000 Prentice-Hall, Inc.  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 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. 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 © 2000 Prentice-Hall, Inc. Kruskal-Wallis H-Test Solution

79 14 - 79 © 2000 Prentice-Hall, Inc.  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 © 2000 Prentice-Hall, Inc.  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 © 2000 Prentice-Hall, Inc.  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 © 2000 Prentice-Hall, Inc. Friedman F r -Test

83 14 - 83 © 2000 Prentice-Hall, Inc. Frequently Used Nonparametric Tests 1.Sign Test 2.Wilcoxon Rank Sum Test 3.Wilcoxon Signed Rank Test 4.Kruskal Wallis H-Test 5.Friedman F r -Test

84 14 - 84 © 2000 Prentice-Hall, Inc. Friedman F r -Test 1.Tests the Equality of 2 or More (p) Population Probability Distributions When Blocking Variable Used 2.Corresponds to Randomized Block F-Test 3.Used to Analyze Randomized Block Designs 4.Uses  2 Distribution with p - 1 df If Number of Blocks or Treatments > 5 If Number of Blocks or Treatments > 5

85 14 - 85 © 2000 Prentice-Hall, Inc. Friedman F r -Test Assumptions 1.Independent, Random Samples 2.Measurements Can Be Ranked Within Blocks 3.Continuous Population Probability Distributions

86 14 - 86 © 2000 Prentice-Hall, Inc. Friedman F r -Test Procedure 1.Assign Ranks, R i, to the Observations Within Each Block Smallest Value = 1; Largest Value = n j Smallest Value = 1; Largest Value = n j Average Ties Average Ties 2.Sum Ranks Within Each Block

87 14 - 87 © 2000 Prentice-Hall, Inc. Friedman F r -Test Procedure 1.Assign Ranks, R i, to the Observations Within Each Block Smallest Value = 1; Largest Value = n j Smallest Value = 1; Largest Value = n j Average Ties Average Ties 2.Sum Ranks Within Each Block 3.Compute Test Statistic Squared total of each block

88 14 - 88 © 2000 Prentice-Hall, Inc. Friedman F r -Test Example You’re a research assistant for the NIH. You’re investigating the effects of plants on human stress. You record finger temperatures under 3 conditions: presence of a live plant, plant photo, nothing. At the.05 level, does finger temperature depend on experimental condition? Subj.LivePhotoNone 191.493.596.6 294.996.690.5 397.095.895.4 493.796.296.7 596.096.693.5

89 14 - 89 © 2000 Prentice-Hall, Inc.  2 0 Friedman F r -Test Solution H 0 : H a :  = df = Critical Value(s): Test Statistic: Decision:Conclusion:

90 14 - 90 © 2000 Prentice-Hall, Inc.  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:

91 14 - 91 © 2000 Prentice-Hall, Inc.  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:

92 14 - 92 © 2000 Prentice-Hall, Inc.  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

93 14 - 93 © 2000 Prentice-Hall, Inc. Friedman F r -Test Solution Raw Data PlantPhotoNone 91.493.596.6 94.996.690.5 97.095.895.4 93.796.296.7 96.096.693.5 Ranks PlantPhotoNone

94 14 - 94 © 2000 Prentice-Hall, Inc. Friedman F r -Test Solution Raw Data PlantPhotoNone 91.493.596.6 94.996.690.5 97.095.895.4 93.796.296.7 96.096.693.5 Ranks PlantPhotoNone 1

95 14 - 95 © 2000 Prentice-Hall, Inc. Friedman F r -Test Solution Raw Data PlantPhotoNone 91.493.596.6 94.996.690.5 97.095.895.4 93.796.296.7 96.096.693.5 Ranks PlantPhotoNone 12

96 14 - 96 © 2000 Prentice-Hall, Inc. Friedman F r -Test Solution Raw Data PlantPhotoNone 91.493.596.6 94.996.690.5 97.095.895.4 93.796.296.7 96.096.693.5 Ranks PlantPhotoNone 123

97 14 - 97 © 2000 Prentice-Hall, Inc. Friedman F r -Test Solution Raw Data PlantPhotoNone 91.493.596.6 94.996.690.5 97.095.895.4 93.796.296.7 96.096.693.5 Ranks PlantPhotoNone 123 1

98 14 - 98 © 2000 Prentice-Hall, Inc. Friedman F r -Test Solution Raw Data PlantPhotoNone 91.493.596.6 94.996.690.5 97.095.895.4 93.796.296.7 96.096.693.5 Ranks PlantPhotoNone 123 21

99 14 - 99 © 2000 Prentice-Hall, Inc. Friedman F r -Test Solution Raw Data PlantPhotoNone 91.493.596.6 94.996.690.5 97.095.895.4 93.796.296.7 96.096.693.5 Ranks PlantPhotoNone 123 231

100 14 - 100 © 2000 Prentice-Hall, Inc. Friedman F r -Test Solution Raw Data PlantPhotoNone 91.493.596.6 94.996.690.5 97.095.895.4 93.796.296.7 96.096.693.5 Ranks PlantPhotoNone 123 231 321 123 231

101 14 - 101 © 2000 Prentice-Hall, Inc. Friedman F r -Test Solution Raw Data PlantPhotoNone 91.493.596.6 94.996.690.5 97.095.895.4 93.796.296.7 96.096.693.5 Ranks PlantPhotoNone 123 231 321 123 231 9129 Total

102 14 - 102 © 2000 Prentice-Hall, Inc. Friedman F r -Test Solution

103 14 - 103 © 2000 Prentice-Hall, Inc.  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 = 1.2

104 14 - 104 © 2000 Prentice-Hall, Inc.  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: Do Not Reject at  =.05  =.05 F r = 1.2

105 14 - 105 © 2000 Prentice-Hall, Inc.  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: Do Not Reject at  =.05 There Is No Evidence Distrib. Are Different  =.05 F r = 1.2

106 14 - 106 © 2000 Prentice-Hall, Inc. Spearman’s Rank Correlation Coefficient

107 14 - 107 © 2000 Prentice-Hall, Inc. 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

108 14 - 108 © 2000 Prentice-Hall, Inc. 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)

109 14 - 109 © 2000 Prentice-Hall, Inc. 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

110 14 - 110 © 2000 Prentice-Hall, Inc. 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

111 14 - 111 © 2000 Prentice-Hall, Inc. Spearman’s Rank Correlation Table

112 14 - 112 © 2000 Prentice-Hall, Inc. Spearman’s Rank Correlation Table

113 14 - 113 © 2000 Prentice-Hall, Inc. Spearman’s Rank Correlation Table

114 14 - 114 © 2000 Prentice-Hall, Inc. Spearman’s Rank Correlation Table

115 14 - 115 © 2000 Prentice-Hall, Inc. Spearman’s Rank Correlation Table

116 14 - 116 © 2000 Prentice-Hall, Inc. Spearman’s Rank Correlation Table

117 14 - 117 © 2000 Prentice-Hall, Inc. Spearman’s Rank Correlation Solution

118 14 - 118 © 2000 Prentice-Hall, Inc. 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

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