1. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-1 Business Statistics, 4e by Ken Black Chapter 17 Nonparametric Statistics

2. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-2 Learning Objectives Recognize the advantages and disadvantages of nonparametric statistics. Understand how to use the runs test to test for randomness. Know when and how to use the Mann-Whitney U test, the Wilcoxon matched-pairs signed rank test, the Kruskal-Wallis test, and the Friedman test. Learn when and how to measure correlation using Spearman’s rank correlation measurement.

3. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-3 Parametric vs Nonparametric Statistics Parametric Statistics are statistical techniques based on assumptions about the population from which the sample data are collected. –Assumption that data being analyzed are randomly selected from a normally distributed population. –Requires quantitative measurement that yield interval or ratio level data. Nonparametric Statistics are based on fewer assumptions about the population and the parameters. –Sometimes called “distribution-free” statistics. –A variety of nonparametric statistics are available for use with nominal or ordinal data.

4. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-4 Advantages of Nonparametric Techniques Sometimes there is no parametric alternative to the use of nonparametric statistics. Certain nonparametric test can be used to analyze nominal data. Certain nonparametric test can be used to analyze ordinal data. The computations on nonparametric statistics are usually less complicated than those for parametric statistics, particularly for small samples. Probability statements obtained from most nonparametric tests are exact probabilities.

5. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-5 Disadvantages of Nonparametric Statistics Nonparametric tests can be wasteful of data if parametric tests are available for use with the data. Nonparametric tests are usually not as widely available and well know as parametric tests. For large samples, the calculations for many nonparametric statistics can be tedious.

6. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-6 Runs Test Test for randomness - is the order or sequence of observations in a sample random or not Each sample item possesses one of two possible characteristics Run - a succession of observations which possess the same characteristic Example with two runs: F, F, F, F, F, F, F, F, M, M, M, M, M, M, M Example with fifteen runs: F, M, F, M, F, M, F, M, F, M, F, M, F, M, F

7. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-7 Runs Test: Sample Size Consideration Sample size: n Number of sample member possessing the first characteristic: n 1 Number of sample members possessing the second characteristic: n 2 n = n 1 + n 2 If both n 1 and n 2 are  20, the small sample runs test is appropriate.

8. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-8 Runs Test: Small Sample Example H 0 : The observations in the sample are randomly generated. H a : The observations in the sample are not randomly generated.  =.05 n 1 = 18 n 2 = 8 If 7  R  17, do not reject H 0 Otherwise, reject H 0. 1 2 3 4 5 6 7 8 9 10 11 12 D CCCCC D CC D CCCC D C D CCC DDD CCC R = 12 Since 7  R = 12  17, do not reject H 0

9. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-9 Runs Test: Large Sample If either n 1 or n 2 is > 20, the sampling distribution of R is approximately normal.

10. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-10 Runs Test: Large Sample Example H 0 : The observations in the sample are randomly generated. H a : The observations in the sample are not randomly generated.  =.05 n 1 = 40 n 2 = 10 If -1.96  Z  1.96, do not reject H 0 Otherwise, reject H 0. 1 1 2 3 4 5 6 7 8 9 0 11 NNN F NNNNNNN F NN FF NNNNNN F NNNN F NNNNN 12 13 FFFF NNNNNNNNNNNN R = 13 H 0 : The observations in the sample are randomly generated. H a : The observations in the sample are not randomly generated.  =.05 n 1 = 40 n 2 = 10 If -1.96  Z  1.96, do not reject H 0 Otherwise, reject H 0. 1 1 2 3 4 5 6 7 8 9 0 11 NNN F NNNNNNN F NN FF NNNNNN F NNNN F NNNNN 12 13 FFFF NNNNNNNNNNNN R = 13

11. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-11 Runs Test: Large Sample Example -1.96  Z = -1.81  1.96, do not reject H 0

12. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-12 Mann-Whitney U Test Nonparametric counterpart of the t test for independent samples Does not require normally distributed populations May be applied to ordinal data Assumptions –Independent Samples –At Least Ordinal Data

13. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-13 Mann-Whitney U Test: Sample Size Consideration Size of sample one: n 1 Size of sample two: n 2 If both n 1 and n 2 are  10, the small sample procedure is appropriate. If either n 1 or n 2 is greater than 10, the large sample procedure is appropriate.

14. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-14 Mann-Whitney U Test: Small Sample Example Service HealthEducational Service 20.1026.19 19.8023.88 22.3625.50 18.7521.64 21.9024.85 22.9625.30 20.7524.12 23.45 H 0 : The health service population is identical to the educational service population on employee compensation H a : The health service population is not identical to the educational service population on employee compensation

15. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-15 Mann-Whitney U Test: Small Sample Example  =.05 If the final p-value <.05, reject H 0. W 1 = 1 + 2 + 3 + 4 + 6 + 7 + 8 = 31 W 2 = 5 + 9 + 10 + 11 + 12 + 13 + 14 + 15 = 89 CompensationRankGroup 18.751H 19.802H 20.103H 20.754H 21.645E 21.906H 22.367H 22.968H 23.459E 23.8810E 24.1211E 24.8512E 25.3013E 25.5014E 26.1915E

16. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-16 Mann-Whitney U Test: Small Sample Example Since U 2 < U 1, U = 3. p-value =.0011 <.05, reject H 0.

17. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-17 Mann-Whitney U Test: Formulas for Large Sample Case

18. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-18 Incomes of PBS and Non-PBS Viewers PBSNon-PBS 24,50041,000 39,40032,500 36,80033,000 44,30021,000 57,96040,500 32,00032,400 61,00016,000 34,00021,500 43,50039,500 55,00027,600 39,00043,500 62,50051,900 61,40027,800 53,000 n 1 = 14 n 2 = 13 H o : The incomes for PBS viewers and non-PBS viewers are identical H a : The incomes for PBS viewers and non-PBS viewers are not identical

19. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-19 Ranks of Income from Combined Groups of PBS and Non-PBS Viewers IncomeRankGroupIncomeRankGroup 16,0001Non-PBS39,50015Non-PBS 21,0002Non-PBS40,50016Non-PBS 21,5003Non-PBS41,00017Non-PBS 24,5004PBS43,00018PBS 27,6005Non-PBS43,50019.5PBS 27,8006Non-PBS43,50019.5Non-PBS 32,0007PBS51,90021Non-PBS 32,4008Non-PBS53,00022PBS 32,5009Non-PBS55,00023PBS 33,00010Non-PBS57,96024PBS 34,00011PBS61,00025PBS 36,80012PBS61,40026PBS 39,00013PBS62,50027PBS 39,40014PBS

20. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-20 PBS and Non-PBS Viewers: Calculation of U

21. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-21 PBS and Non-PBS Viewers: Conclusion

22. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-22 Wilcoxon Matched-Pairs Signed Rank Test A nonparametric alternative to the t test for related samples Before and After studies Studies in which measures are taken on the same person or object under different conditions Studies or twins or other relatives

23. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-23 Wilcoxon Matched-Pairs Signed Rank Test Differences of the scores of the two matched samples Differences are ranked, ignoring the sign Ranks are given the sign of the difference Positive ranks are summed Negative ranks are summed T is the smaller sum of ranks

24. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-24 Wilcoxon Matched-Pairs Signed Rank Test: Sample Size Consideration n is the number of matched pairs If n > 15, T is approximately normally distributed, and a Z test is used. If n  15, a special “small sample” procedure is followed. –The paired data are randomly selected. –The underlying distributions are symmetrical.

25. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-25 Wilcoxon Matched-Pairs Signed Rank Test: Small Sample Example Family PairPittsburghOakland 11,950 1,760 21,840 1,870 32,015 1,810 41,580 1,660 51,790 1,340 61,925 1,765 H 0 : M d = 0 H a : M d  0 n = 6  =0.05 If T observed  1, reject H 0.

26. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-26 Wilcoxon Matched-Pairs Signed Rank Test: Small Sample Example Family PairPittsburghOaklanddRank 11,950 1,760 190 21,840 1,870 -30 32,015 1,810 205 41,580 1,660 -80 51,790 1,340 450 61,925 1,765 160 +4 +5 -2 +6 +3 T = minimum( T +, T - ) T + = 4 + 5 + 6 + 3= 18 T - = 1 + 2 = 3 T = 3 T = 3 > T crit = 1, do not reject H 0.

27. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-27 Wilcoxon Matched-Pairs Signed Rank Test: Large Sample Formulas

28. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-28 Airline Cost Data for 17 Cities, 1997 and 1999 City19791999dRankCity19791999dRank 120.322.8-2.5-81020.320.9-0.6 219.512.76.8171119.222.6-3.4-11.5 318.614.14.5131219.516.92.69 420.916.14.8151318.720.6-1.9-6.5 519.925.2-5.3-161417.718.5-0.8-2 618.620.2-1.6-41521.623.4-1.8-5 719.614.94.7141622.421.31.13 823.221.31.96.51720.817.43.411.5 921.818.73.110 H 0 : M d = 0 H a : M d  0

29. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-29 Airline Cost: T Calculation

30. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-30 Airline Cost: Conclusion

31. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-31 Kruskal-Wallis Test A nonparametric alternative to one-way analysis of variance May used to analyze ordinal data No assumed population shape Assumes that the C groups are independent Assumes random selection of individual items

32. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-32 Kruskal-Wallis K Statistic

33. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-33 Number of Patients per Day per Physician in Three Organizational Categories Two Partners Three or More PartnersHMO 132426 151622 201931 182227 232528 1433 17 H o : The three populations are identical H a : At least one of the three populations is different

34. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-34 Patients per Day Data: Kruskal-Wallis Preliminary Calculations n = n 1 + n 2 + n 3 = 5 + 7 + 6 = 18 Two Partners Three or More PartnersHMO PatientsRankPatientsRankPatientsRank 13124122614 153164229.5 2081973117 186229.52715 231125132816 1423318 175 T 1 = 29T 2 = 52.5T 3 = 89.5 n 1 = 5n 2 = 7n 3 = 6

35. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-35 Patients per Day Data: Kruskal-Wallis Calculations and Conclusion

36. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-36 Friedman Test A nonparametric alternative to the randomized block design Assumptions –The blocks are independent. –There is no interaction between blocks and treatments. –Observations within each block can be ranked. Hypotheses – H o : The treatment populations are equal – H a :At least one treatment population yields larger values than at least one other treatment population

37. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-37 Friedman Test

38. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-38 Friedman Test: Tensile Strength of Plastic Housings Supplier 1Supplier 2Supplier 3Supplier 4 Monday62635761 Tuesday63615965 Wednesday61625663 Thursday62605764 Friday64635866 H o :The supplier populations are equal H a :At least one supplier population yields larger values than at least one other supplier population

39. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-39 Friedman Test: Tensile Strength of Plastic Housings

40. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-40 Friedman Test: Tensile Strength of Plastic Housings Supplier 1Supplier 2Supplier 3Supplier 4 Monday3412 Tuesday3214 Wednesday2314 Thursday3214 Friday3214 1413518 19616925324 j R 2 j R

41. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-41 Friedman Test: Tensile Strength of Plastic Housings

42. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-42 Spearman’s Rank Correlation Analyze the degree of association of two variables Applicable to ordinal level data (ranks)

43. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-43 Spearman’s Rank Correlation for Cattle and Lamb Prices Year Cattle Prices ($/100 lb) Lamb Prices ($/100 lb) Rank Cattle Rank: Lambdd2 198866.6069.10671 198969.5066.109639 199074.6055.5013211121 199172.7052.2012111121 199271.3059.50103749 199372.6064.40114749 199466.7065.607524 199561.8078.20310-749 199658.7082.80112-11121 199763.1090.30413-981 199859.6072.3028-636 199963.4074.5059-416 200068.6079.40811-39 666

44. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 17-44 Spearman’s Rank Correlation for Cattle and Lamb Prices