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Statistics for Business and Economics

Statistics for Business and Economics. Nonparametric Statistics Chapter 14. Learning Objectives. 1. Distinguish Parametric & Nonparametric Test Procedures 2. Explain a Variety of Nonparametric Test Procedures 3. Solve Hypothesis Testing Problems Using Nonparametric Tests

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Statistics for Business and Economics

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  1. Statistics for Business and Economics Nonparametric StatisticsChapter 14

  2. 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. Hypothesis Testing Procedures Many More Tests Exist!

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

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

  6. Advantages of Nonparametric Tests • 1. Used With All Scales • 2. Easier to Compute • 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. Disadvantages of Nonparametric Tests • 1. May Waste Information • If Data Permit Using Parametric Procedures • 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. Frequently Used Nonparametric Tests • 1. Sign Test • 2. Wilcoxon Rank Sum Test • 3. Wilcoxon Signed Rank Test • 4. Kruskal Wallis H-Test • Friedman’s Fr-Test • Spearman’s Rank Correlation Coefficient

  9. Sign Test

  10. Frequently Used Nonparametric Tests • 1. Sign Test • 2. Wilcoxon Rank Sum Test • 3. Wilcoxon Signed Rank Test • 4. Kruskal Wallis H-Test • Friedman’s Fr-Test • Spearman’s Rank Correlation Coefficient

  11. 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 • 5. Can Use Normal Approximation If n 10

  12. 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’ Ha, Then P-Value = P(x 7) = .031 + .004 = .035. If  = .05, Then Reject H0 Since P-Value .

  13. 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. Sign Test Solution P-Value: Decision: Conclusion: • H0: • Ha: •  = • Test Statistic:

  15. Sign Test Solution P-Value: Decision: Conclusion: • H0:  =3 • Ha:  < 3 •  = • Test Statistic:

  16. Sign Test Solution P-Value: Decision: Conclusion: • H0:  =3 • Ha:  < 3 •  = .05 • Test Statistic:

  17. Sign Test Solution P-Value: Decision: Conclusion: • H0:  =3 • Ha:  < 3 •  = .05 • Test Statistic: S = 2 (Ratings 1 & 2 Are Less Than  =3:2, 5, 3, 4, 1, 4, 5)

  18. Sign Test Solution P-Value: Decision: Conclusion: • H0:  =3 • Ha:  < 3 •  = .05 • Test Statistic: 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. Sign Test Solution P-Value: Decision: Conclusion: • H0:  =3 • Ha:  < 3 •  = .05 • Test Statistic: 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) Do Not Reject at  = .05

  20. Sign Test Solution P-Value: Decision: Conclusion: • H0:  =3 • Ha:  < 3 •  = .05 • Test Statistic: 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) Do Not Reject at  = .05 There Is No Evidence Median Is Less Than 3

  21. Wilcoxon Rank Sum Test

  22. Frequently Used Nonparametric Tests • 1. Sign Test • 2. Wilcoxon Rank Sum Test • 3. Wilcoxon Signed Rank Test • 4. Kruskal Wallis H-Test • Friedman’s Fr-Test • Spearman’s Rank Correlation Coefficient

  23. 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 • Populations Are Continuous • 4. Can Use Normal Approximation If ni 10

  24. Wilcoxon Rank Sum Test Procedure • 1. Assign Ranks, Ri, to the n1 + n2 Sample Observations • If Unequal Sample Sizes, Let n1 Refer to Smaller-Sized Sample • Smallest Value = 1 • Average Ties • 2. Sum the Ranks, Ti, for Each Sample • 3. Test Statistic Is TA (Smallest Sample)

  25. 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. Wilcoxon Rank Sum Test Solution • H0: • Ha: •  = • n1 = n2 = • Critical Value(s): Test Statistic: Decision: Conclusion:  Ranks

  27. Wilcoxon Rank Sum Test Solution • H0: Identical Distrib. • Ha: Shifted Left or Right •  = • n1 = n2 = • Critical Value(s): Test Statistic: Decision: Conclusion:  Ranks

  28. Wilcoxon Rank Sum Test Solution • H0: Identical Distrib. • Ha: Shifted Left or Right •  = .10 • n1 = 4 n2 = 5 • Critical Value(s): Test Statistic: Decision: Conclusion:  Ranks

  29. Wilcoxon Rank Sum Table (Portion)  = .05 one-tailed;  = .10 two-tailed

  30. Wilcoxon Rank Sum Test Solution • H0: Identical Distrib. • Ha: Shifted Left or Right •  = .10 • n1 = 4 n2 = 5 • Critical Value(s): Test Statistic: Decision: Conclusion: Do Not Reject Reject Reject 13 27  Ranks

  31. Wilcoxon Rank Sum Test Computation Table Factory 1 Factory 2 Rate Rank Rate Rank Rank Sum

  32. Wilcoxon Rank Sum Test Computation Table Factory 1 Factory 2 Rate Rank Rate Rank 71 85 82 82 77 94 92 97 88 ... ... Rank Sum

  33. Wilcoxon Rank Sum Test Computation Table Factory 1 Factory 2 Rate Rank Rate Rank 71 1 85 82 82 77 94 92 97 88 ... ... Rank Sum

  34. Wilcoxon Rank Sum Test Computation Table Factory 1 Factory 2 Rate Rank Rate Rank 71 1 85 82 82 77 2 94 92 97 88 ... ... Rank Sum

  35. Wilcoxon Rank Sum Test Computation Table Factory 1 Factory 2 Rate Rank Rate Rank 71 1 85 82 3 82 4 77 2 94 92 97 88 ... ... Rank Sum

  36. Wilcoxon Rank Sum Test Computation Table Factory 1 Factory 2 Rate Rank Rate Rank 71 1 85 82 3 3.5 82 4 3.5 77 2 94 92 97 88 ... ... Rank Sum

  37. Wilcoxon Rank Sum Test Computation Table Factory 1 Factory 2 Rate Rank Rate Rank 71 1 85 5 82 3 3.5 82 4 3.5 77 2 94 92 97 88 ... ... Rank Sum

  38. Wilcoxon Rank Sum Test Computation Table Factory 1 Factory 2 Rate Rank Rate Rank 71 1 85 5 82 3 3.5 82 4 3.5 77 2 94 92 97 88 6 ... ... Rank Sum

  39. Wilcoxon Rank Sum Test Computation Table Factory 1 Factory 2 Rate Rank Rate Rank 71 1 85 5 82 3 3.5 82 4 3.5 77 2 94 92 7 97 88 6 ... ... Rank Sum

  40. Wilcoxon Rank Sum Test Computation Table Factory 1 Factory 2 Rate Rank Rate Rank 71 1 85 5 82 3 3.5 82 4 3.5 77 2 94 8 92 7 97 88 6 ... ... Rank Sum

  41. Wilcoxon Rank Sum Test Computation Table Factory 1 Factory 2 Rate Rank Rate Rank 71 1 85 5 82 3 3.5 82 4 3.5 77 2 94 8 92 7 97 9 88 6 ... ... Rank Sum

  42. Wilcoxon Rank Sum Test Computation Table Factory 1 Factory 2 Rate Rank Rate Rank 71 1 85 5 82 3 3.5 82 4 3.5 77 2 94 8 92 7 97 9 88 6 ... ... Rank Sum 19.5 25.5

  43. Wilcoxon Rank Sum Test Solution • H0: Identical Distrib. • Ha: Shifted Left or Right •  = .10 • n1 = 4 n2 = 5 • Critical Value(s): Test Statistic: Decision: Conclusion: T2 = 5 + 3.5 + 8+ 9 = 25.5 (Smallest Sample) Do Not Reject Reject Reject 13 27  Ranks

  44. Wilcoxon Rank Sum Test Solution • H0: Identical Distrib. • Ha: Shifted Left or Right •  = .10 • n1 = 4 n2 = 5 • Critical Value(s): Test Statistic: Decision: Conclusion: T2 = 5 + 3.5 + 8+ 9 = 25.5 (Smallest Sample) Do Not Reject at  = .10 Do Not Reject Reject Reject 13 27  Ranks

  45. Wilcoxon Rank Sum Test Solution • H0: Identical Distrib. • Ha: Shifted Left or Right •  = .10 • n1 = 4 n2 = 5 • Critical Value(s): Test Statistic: Decision: Conclusion: T2 = 5 + 3.5 + 8+ 9 = 25.5 (Smallest Sample) Do Not Reject at  = .10 Do Not Reject Reject Reject There Is No Evidence Distrib. Are Not Equal 13 27  Ranks

  46. Wilcoxon Signed Rank Test

  47. Frequently Used Nonparametric Tests • 1. Sign Test • 2. Wilcoxon Rank Sum Test • 3. Wilcoxon Signed Rank Test • 4. Kruskal Wallis H-Test • Friedman’s Fr-Test • Spearman’s Rank Correlation Coefficient

  48. 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 • Both Populations Are Continuous • 4. Can Use Normal Approximation If n 25

  49. Signed Rank Test Procedure • 1. Obtain Difference Scores, Di= X1i- X2i • 2. Take Absolute Value of Differences, Di • 3. Delete Differences With 0 Value • 4. Assign Ranks, Ri, Where Smallest = 1 • 5. Assign Ranks Same Signs as Di • 6. Sum ‘+’ Ranks (T+) & ‘-’ Ranks (T-) • Test Statistic Is T- (One-Tailed Test) • Test Statistic Is Smaller of T- or T+ (2-Tail)

  50. Signed Rank Test Computation Table

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