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Statistics

Statistics. Chapter 14: Nonparametric Statistics. Where We’ve Been. Presented methods for making inferences about means and correlation Methods required the data or the sampling distributions to be normally distributed. Where We’re Going.

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Statistics

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  1. Statistics Chapter 14: Nonparametric Statistics

  2. Where We’ve Been • Presented methods for making inferences about means and correlation • Methods required the data or the sampling distributions to be normally distributed McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  3. Where We’re Going • Inferential techniques requiring fewer or less stringent assumptions • Nonparametric tests based on ranks McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  4. 14.1: Distribution-Free Tests • Testing non-normal data with test based on normality may lead to • P(Type I error) >  • less than maximum power of the test (1 - ). McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  5. 14.1: Distribution-Free Tests Parametric tests (z, t, F) Data or sampling distribution are normal Non-parametric tests (Rank-ordered, no assumed distribution) Data or sampling distribution are skewed, or data is ordinal McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  6. 14.1: Distribution-Free Tests Parametric tests (z, t, F) Data or sampling distribution are normal Nonparametric statistics (or tests) based on the ranks of measurements are called rank statistics (or rank tests.) Non-parametric tests (Rank-ordered, no assumed distribution) Data or sampling distribution are skewed, or data is ordinal McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  7. 14.2: Single-Population Inferences • The sign test provides inferences about population medians, or central tendencies, when skewed data or an outlier would invalidate tests based on normal distributions. McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  8. 14.2: Single-Population Inferences • One-tailed test for a Population Median,  Test Statistic: S = number of sample measurements greater than (less than) 0 • Two-tailed test for a Population Median,  Test Statistic: • S = larger of S1and S2 where S1 is the number of measurements less than 0 and S2the number greater than 0 McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  9. 14.2: Single-Population Inferences • One-tailed test for a Population Median,  Observed significance level: p-value = P(xS) • Two-tailed test for a Population Median,  • Observed significance level: • p-value = 2P(xS) where x has a binomial distribution with parameters n and p = .5. Reject H0 if p-value . McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  10. 14.2: Single-Population Inferences Median time to failure for a band of compact disc players is 5,250 hours. Twenty players from a competitor are tested, with failure times from 5 hours to 6,575 hours. Fourteen of the players exceed 5,250 hours. Do the competitor’s machines perform differently? McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  11. 14.2: Single-Population Inferences Median time to failure for a band of compact disc players is 5,250 hours. Twenty players from a competitor are tested, with failure times from 5 hours to 6,575 hours. Fourteen of the players exceed 5,250 hours. Do the competitor’s machines perform differently? McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  12. 14.2: Single-Population Inferences Median time to failure for a band of compact disc players is 5,250 hours. Twenty players from a competitor are tested, with failure times from 5 hours to 6,575 hours. Fourteen of the players exceed 5,250 hours. Do the competitor’s machines perform differently? Do not reject H0 McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  13. 14.3: Comparing Two Populations: Independent Samples • Wilcoxon Rank Sum Test • Used to test whether two independent samples have the same probability distribution • Samples must be random and independent. • Probability distributions must be continuous. McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  14. 14.3: Comparing Two Populations: Independent Samples Wilcoxon Rank Sum Test • One-tailed test H0: D1 and D2 are identical Ha: D1 is shifted right of D2orHa: D1 is shifted left of D2 Test statistic: T1, if n1 < n2 T2, if n1 > n2 Either if n1 = n2 Rejection region: T1: T1 TU or T1  TL T2: T2  TL or T2  TU • Two-tailed test H0: D1 and D2 are identical Ha: D1 is shifted either right or left of D2 Test statistic: T1, if n1 < n2 T2, if n1 > n2 Either if n1 = n2 Rejection region: T TL or T  TU McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  15. 14.3: Comparing Two Populations: Independent Samples Reaction Times of Subjects Under the Influence of Drug A or B Rank: Value: Rank: Value: H0 : DA and DBare identical Ha: DA is shifted right of DBorHa: DA is shifted left of DB  =.05 TA = 1 + 2 + 3 + 4 + 7 + 8 = 25 TB = 5 + 6 + 9 + 10 + 11 + 12 + 13 = 66 Test Statistic is TA, since nA < nB TL (=.05, nA= 6, nB= 7) = 28 > TA = 25 Reject H0 McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  16. 14.3: Comparing Two Populations: Independent Samples Wilcoxon Rank Sum Test for Large Samples • One-tailed test H0: D1 and D2 are identical Ha: D1 is shifted right of D2orHa: D1 is shifted left of D2 Rejection region: | z | > za • Two-tailed test H0: D1 and D2 are identical Ha: D1 is shifted either right or left of D2 Rejection region: | z | > za/2 Test Statistic: McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  17. 14.4: Comparing Two Populations: Paired Difference Experiment McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  18. 14.4: Comparing Two Populations: Paired Difference Experiment T-= Sum of negative ranks = 9 T+= Sum of positive ranks = 46 McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  19. 14.4: Comparing Two Populations: Paired Difference Experiment • H0: The probability distributions of the ratings for products A and B are identical • Ha: The probability distributions of the ratings differ • = .05, two-tailed test • Test statistic: T = Smaller of T+ and T- • Rejection region: T 8 (see Table XIII in Appendix A) • T- = 9 > 8 Do not reject H0 McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  20. 14.5: Comparing Three or More Populations: Completely Randomized Design • Kruskal-Wallis H – test • Compares probability distributions for k populations or treatments • No assumption about the distributions • H0 : The k probability distributions are identical • Ha: At least two of the k probability distributionsdiffer McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  21. 14.5: Comparing Three or More Populations: Completely Randomized Design • Kruskal-Wallis H – test • k samples are random and independent. • For each sample nj 5. • The k probability distributions are continuous. McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  22. 14.5: Comparing Three or More Populations: Completely Randomized Design • Kruskal-Wallis H – test • Test statistic: • n = total sample size • nj = measurements in sample j • Rj = rank sum of sample j McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  23. 14.5: Comparing Three or More Populations: Completely Randomized Design • Kruskal-Wallis H – test • H = 0 All samples have the same mean rank • Large H Larger differences between sample mean ranks • If H0 is true, H ~ 2, with df = (k-1) • Reject H0 if H >2 McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  24. 14.5: Comparing Three or More Populations: Completely Randomized Design A study of three populations yielded the following: • H0 : The k probability distributions are identical • Ha: At least two of the k probability distributionsdiffer • = .05 • df = 3-1= 2 • 2.05= 5.99147 McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  25. 14.5: Comparing Three or More Populations: Completely Randomized Design A study of three populations yielded the following: McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  26. 14.5: Comparing Three or More Populations: Completely Randomized Design A study of three populations yielded the following: • H0 : The k probability distributions are identical • Ha: At least two of the k probability distributionsdiffer • = .05 • df = 3-1= 2 • 2.05 = 5.99147 Since H = 8.19 > 2.05= 5.99147, reject H0 McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  27. 14.6: Comparing Three or More Populations: Randomized Block Design • Friedman Fr-statistic • H0 : The p probability distributions are identical • Ha: At least two of the p probability distributionsdiffer in location • Test statistic: • b = number of blocks (>5) • k = number of treatments (>5) • Rj = rank sum of treatment j McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  28. 14.6: Comparing Three or More Populations: Randomized Block Design • Friedman Fr-statistic • Treatments are randomly assigned to experimental units within the blocks. • Measurements can be ranked within blocks. • The p probability distributions from which the samples within each block are drawn are continuous. • Fr ~ 2 with k – 1 degrees of freedom. McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  29. 14.6: Comparing Three or More Populations: Randomized Block Design A study of four treatments and six blocks yielded the following: • H0 : The probability distributions for the p treatments are identical • Ha: At least two of the p probability distributionsdiffer in location • = .05 • df = 4-1= 3 • 2.05 = 7.81473 McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  30. 14.6: Comparing Three or More Populations: Randomized Block Design A study of four treatments and six blocks yielded the following: McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  31. 14.6: Comparing Three or More Populations: Randomized Block Design A study of four treatments and six blocks yielded the following: • H0 : The probability distributions for the p treatments are identical • Ha: At least two of the p probability distributionsdiffer in location • = .05 • df = 4-1= 3 • 2.05 = 7.81473 Since H = 15.2 > 2.05= 7.81473, reject H0 McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  32. 14.7: Rank Correlation Spearman’s Rank Correlation Coefficient where McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  33. 14.7: Rank Correlation Spearman’s Rank Correlation Coefficient where (cont.) ui = Rank of the ith observation in sample 1 vi = Rank of the ith observation in sample 2 n = Number of pairs of observations Shortcut Formula for rs* where * A good approximation when there are few ties relative to n McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  34. 14.7: Rank Correlation Spearman’s Rank Correlation Coefficient -1 +1 0 Perfect negative correlation Perfect positive correlation No correlation McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  35. 14.7: Rank Correlation Spearman’s Nonparametric Test for Rank Correlation Two-Tailed test Rejection region: |rs| > rs,/2 One-Tailed Test Rejection region: |rs| > rs, Test Statistics: rs Conditions 1. The sample of experimental units on which the two variables are measured must be randomly selected, and 2. The probability distributions of the two variables must be continuous. McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  36. 14.7: Rank Correlation Preseason Predictions for 2007 ACC Atlantic Division Football McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  37. 14.7: Rank Correlation Preseason Predictions for 2007 ACC Atlantic Division Football McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

  38. 14.7: Rank Correlation Preseason Predictions for 2007 ACC Atlantic Division Football From Table XIV, with n = 6, rs =.05 =.829, so H0is not rejected. McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics

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