Chapter 14: Nonparametric Statistics
McClave, Statistics, 11th ed. 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 McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics 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
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
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
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
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
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 S1 and S2 where S1 is the number of measurements less than 0 and S2 the number greater than 0 McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics
14.2: Single-Population Inferences One-tailed test for a Population Median, Observed significance level: p-value = P(xS) Two-tailed test for a Population Median, Observed significance level: p-value = 2P(xS) 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
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
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
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
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
14.3: Comparing Two Populations: Independent Samples Wilcoxon Rank Sum Test One-tailed test H0: D1 and D2 are identical Ha: D1 is shifted right of D2 or Ha: 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
14.3: Comparing Two Populations: Independent Samples Reaction Times of Subjects Under the Influence of Drug A or B Rank: Value: 1 2 3 4 5 6 7 1.62 1.71 1.93 1.96 2.07 2.11 2.24 8 9 10 11 12 13 2.41 2.43 2.50 2.71 2.84 2.88 H0 : DA and DB are identical Ha: DA is shifted right of DB or Ha: 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
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 D2 or Ha: 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
14.4: Comparing Two Populations: Paired Difference Experiment Judge Product A Product B A - B |A – B| Rank of 1 6 4 2 5 8 3 7.5 -1 9 7 -2 10 McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics
14.4: Comparing Two Populations: Paired Difference Experiment Judge Product A Product B A - B |A – B| Rank of 1 6 4 2 5 8 3 7.5 -1 9 7 -2 10 T- = Sum of negative ranks = 9 T+ = Sum of positive ranks = 46 McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics
14.4: Comparing Two Populations: Paired Difference Experiment Judge Product A Product B A - B |A – B| Rank of 1 6 4 2 5 8 3 7.5 -1 9 7 -2 10 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
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics 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 distributions differ McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics 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
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics 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
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics 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
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics 14.5: Comparing Three or More Populations: Completely Randomized Design A study of three populations yielded the following: Population A B C nj 15 Rj 235 439 361 Rj2 55,225 192,721 130,321 H0 : The k probability distributions are identical Ha: At least two of the k probability distributions differ = .05 df = 3-1= 2 2 .05= 5.99147 McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics 14.5: Comparing Three or More Populations: Completely Randomized Design A study of three populations yielded the following: Population A B C nj 15 Rj 235 439 361 Rj2 55,225 192,721 130,321 McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics 14.5: Comparing Three or More Populations: Completely Randomized Design A study of three populations yielded the following: Population A B C nj 15 Rj 235 439 361 Rj2 55,225 192,721 130,321 H0 : The k probability distributions are identical Ha: At least two of the k probability distributions differ = .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
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 distributions differ 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
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
14.6: Comparing Three or More Populations: Randomized Block Design A study of four treatments and six blocks yielded the following: Population A B C D Rj 11 21 7 Rj2 121 441 49 H0 : The probability distributions for the p treatments are identical Ha: At least two of the p probability distributions differ in location = .05 df = 4-1= 3 2 .05 = 7.81473 McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics
14.6: Comparing Three or More Populations: Randomized Block Design A study of four treatments and six blocks yielded the following: Population A B C D Rj 11 21 7 Rj2 121 441 49 McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics
14.6: Comparing Three or More Populations: Randomized Block Design A study of four treatments and six blocks yielded the following: Population A B C D Rj 11 21 7 Rj2 121 441 49 H0 : The probability distributions for the p treatments are identical Ha: At least two of the p probability distributions differ 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
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics 14.7: Rank Correlation Spearman’s Rank Correlation Coefficient where McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics
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
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics 14.7: Rank Correlation Spearman’s Rank Correlation Coefficient -1 +1 Perfect negative correlation Perfect positive correlation No correlation McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics
14.7: Rank Correlation Spearman’s Nonparametric Test for Rank Correlation One-Tailed Test Rejection region: |rs | > rs, Two-Tailed test Rejection region: |rs | > rs,/2 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
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics 14.7: Rank Correlation Preseason Predictions for 2007 ACC Atlantic Division Football Team Predictor A Predictor B Boston College 1 5 Florida State 2 Wake Forest 3 Clemson 4 Maryland 6 N.C. State McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics 14.7: Rank Correlation Preseason Predictions for 2007 ACC Atlantic Division Football Team Predictor A Predictor B Boston College 1 5 Florida State 2 Wake Forest 3 Clemson 4 Maryland 6 N.C. State McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics
McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics 14.7: Rank Correlation Preseason Predictions for 2007 ACC Atlantic Division Football Team Predictor A Predictor B Boston College 1 5 Florida State 2 Wake Forest 3 Clemson 4 Maryland 6 N.C. State From Table XIV, with n = 6, rs =.05 =.829, so H0 is not rejected. McClave, Statistics, 11th ed. Chapter 14: Nonparametric Statistics