Chapter 12 Nonparametric Methods Professor Jen-pei Liu, PhD Division of Biometry Department of Agronomy National Taiwan University 2018/12/1 Copyright by Jen-pei Liu, PhD

Nonparametric Methods 12.1 Introduction 12.2 Wilcoxon Signed-rank Test for a paired Sample 12.3 Mann-Whitney-Wilcoxon Test for Two Independent Random Samples 12.4 Kruskal-Wallis Test for t independent Random Sample 12.5 Friedman Test for a Block Design 12.6 Rank Correlation Coefficient 12.7 Summary 2018/12/1 Copyright by Jen-pei Liu, PhD

Copyright by Jen-pei Liu, PhD Introduction Assumption for t-test or correlation (regression) coefficients Normality Equal variance Independence Not all data satisfy these assumptions! 2018/12/1 Copyright by Jen-pei Liu, PhD

Copyright by Jen-pei Liu, PhD Examples: Customer’s ranking of 4 brands of laundry detergents Preference of 10 CPAS for different brands PCs Rating of tests characteristics of 5 brands of cornflakes Rating on a 5-point scale of each of 8 job applicants by 3 separate personnel managers 2018/12/1 Copyright by Jen-pei Liu, PhD

12.2 Wilcoxon Signed-rank Test Example: A frozen foods manufacturer A: own frozen product B: competitors frozen product Purchasing managers: 10-point Likert scale (High score: better product) Price, package design, variety, company promotion, et. 2018/12/1 Copyright by Jen-pei Liu, PhD

12.2 Wilcoxon Signed-rank Test Likert Score Manager A B Difference(A-B) 1 6.5 4.0 2.5 2 8.0 -4.0 3 5.5 6.0 -1.0 4 10.0 -4.5 5 7.0 6 9.0 -5.5 2018/12/1 Copyright by Jen-pei Liu, PhD

12.2 Wilcoxon Signed-rank Test Likert Scale Impression The manager is asked to mark his/her response on the scale The mark defines an assumed measure of acceptability for the product in question Vary poor Acceptable outstanding 1 2 3 4 5 6 7 8 9 10 2018/12/1 Copyright by Jen-pei Liu, PhD

12.2 Wilcoxon Signed-rank Test Exact Methods for Small Sample(n ≤ 30) Calculate the difference( xA - xB)for each of the n pairs. Differences equal to zero are eliminated, and the number of pairs, n, is reduced accordingly. Rank the absolute values of the differences, assigning 1 to the smallest, 2 to the second smallest, and so on. Tied observations are assigned the average of the rank that would have been assigned with no ties. 2018/12/1 Copyright by Jen-pei Liu, PhD

12.2 Wilcoxon Signed-rank Test Calculate the rank sum for the negative differences and label this value T-. Similarly, calculate T+, the rank sum for the positive differences. For a two-tailed test we use the smaller of these two quantities, T, as test statistic to test the null hypothesis that the two population relative frequency histograms are identical. 2018/12/1 Copyright by Jen-pei Liu, PhD

12.2 Wilcoxon Signed-rank Test Methods for Larger Samples (n>30) Null hypothesis: H0: The population relative frequency distributions for A and B are identical. Alternative hypothesis: Ha: The population relative frequency distributions differ in location (a two-tailed test). Or Ha:The population relative frequency distribution for A is shifted to the right (or left) of the relative frequency distribution for population B (a one-tailed test). 2018/12/1 Copyright by Jen-pei Liu, PhD

12.2 Wilcoxon Signed-rank Test Test statistic: and T can be either T+ or T- 2018/12/1 Copyright by Jen-pei Liu, PhD

12.2 Wilcoxon Signed-rank Test Rejection region: Reject H0 if or for a two-tailed test. For a one-tailed test, place all of  in one tail of the z distribution. To detect a shift in the distribution of A observations to the right of the distribution of B observations, let T=T+ and reject H0 when . To detect a shift in the opposite direction, let T=T- and reject H0 if . Tabulated values of z are given in Table 3 in the Appendix. 2018/12/1 Copyright by Jen-pei Liu, PhD

12.2 Wilcoxon Signed-rank Test Example(continued) Likert score Manager A B Difference (A-B) Rank of Abs. Diff Signed Rank(+) Rank(-) 1 6.5 4.0 2.5 3 2 8.0 -4.0 4 5.5 6.0 -1.0 1.5 10.0 -4.5 5 7.0 6 9.0 -5.5 Total T-=18 T+=3 2018/12/1 Copyright by Jen-pei Liu, PhD

12.2 Wilcoxon Signed-rank Test N=7<30  Exact Method Two-tailed: T=min(T-, T+)=min(18, 3)=3 =0.05, T0=2  T=3 > T0=2 Fail to reject H0: No difference 2018/12/1 Copyright by Jen-pei Liu, PhD

12.3 Mann-Whitney-Wilcoxon Test Two independent random samples Example Effectiveness of two CPA training program for CPA examinations 8 branch offices of a large accounting firm 50 junior accountants from each branch office # of junior accountants passed the CPA examination 2018/12/1 Copyright by Jen-pei Liu, PhD

12.3 Mann-Whitney-Wilcoxon Test Example Training Program 1 2 28 33 31 29 27 35 25 30 2018/12/1 Copyright by Jen-pei Liu, PhD

12.3 Mann-Whitney-Wilcoxon Test Method Rank the observations in the combined sample from the smallest (1) to the largest (n1+n2) In case of ties, use the averaged rank Compute the sum of ranks for each sample 2018/12/1 Copyright by Jen-pei Liu, PhD

12.3 Mann-Whitney-Wilcoxon Test Example Training Program 1 2 obs. Rank 28 3 33 7 31 6 29 4 27 35 8 25 30 5 sum T1=12 T2=24 2018/12/1 Copyright by Jen-pei Liu, PhD

12.3 Mann-Whitney-Wilcoxon Test Exact Method for Small Samples(n1+n2 ≤ 30) Null hypothesis: H0: The population relative frequency distributions for 1 and 2 are identical. Alternative hypothesis: Ha: The population relative frequency distributions are shifted in respect to their relative location(a two-tailed test). Or Ha:The population relative frequency distribution for population 1 is shifted to the right of the relative frequency distribution for population 2 (a one-tailed test). 2018/12/1 Copyright by Jen-pei Liu, PhD

12.3 Mann-Whitney-Wilcoxon Test Test statistics: For a two-tailed test, use U, the smaller of and Where T1 and T2 are the rank sums for samples 1 and 2, respectively. For a one-tailed test, use U1. 2018/12/1 Copyright by Jen-pei Liu, PhD

12.3 Mann-Whitney-Wilcoxon Test Rejection region: a. For the two-tailed test and a given value of α , reject H0 if U ≤ U0, where p(U≤ U0)=α/2. [Note: Observe that U0 is the value such that P(U ≤ U0) is equal to half ofα] b. For a one-tailed test and a given value of α, reject H0 if U1 ≤ U0, where P(U1 ≤ U0)= α 2018/12/1 Copyright by Jen-pei Liu, PhD

12.3 Mann-Whitney-Wilcoxon Test Method for larger Samples (n1+n2>30 ) Null hypothesis: H0: The population relative frequency distributions for 1 and 2 are identical. Alternative hypothesis: Ha: The population relative frequency distributions are not identical. Or Ha:The population relative frequency distribution for 1 is shifted to the right (or left) of the relative frequency distribution for population 2 (a one-tailed test). 2018/12/1 Copyright by Jen-pei Liu, PhD

12.3 Mann-Whitney-Wilcoxon Test Test Statistics: , and let U=U1for one-sided alternative. 2018/12/1 Copyright by Jen-pei Liu, PhD

12.3 Mann-Whitney-Wilcoxon Test Rejection region: Reject H0 if z > zα/2 or z < -zα/2 for a two- tailed test. For a one-tailed test, place all of α in one tail of the z distribution. To detect a shift in distribution 1 to the right of distribution 2, let U=U1 and reject H0 when z<-zα . To detect a shift in the opposite direction, let U=U1 and reject H0 when z>zα. Tabulated values of z are given in Table 3 in the Appendix. 2018/12/1 Copyright by Jen-pei Liu, PhD

12.3 Mann-Whitney-Wilcoxon Test Example: Fail to reject H0 of no difference between train program 2018/12/1 Copyright by Jen-pei Liu, PhD

Copyright by Jen-pei Liu, PhD 12.4 Kruskal-Wallis Test t independent samples Example: To compare difference in use of tax shelters among executives in four major industries: banking, broadcasting, electronic and retailing Random samples from 4 industries Measurements: federal tax paid as a percentage of the total income 2018/12/1 Copyright by Jen-pei Liu, PhD

Copyright by Jen-pei Liu, PhD 12.4 Kruskal-Wallis Test Example: Federal tax paid as a percentage of the total income Banking Broadcasting Electronics Retailing 16 41 32 22 37 39 30 25 21 35 38 33 29 46 28 19 40 47 20 43 27 2018/12/1 Copyright by Jen-pei Liu, PhD

Copyright by Jen-pei Liu, PhD 12.4 Kruskal-Wallis Test Methods: Rank the combined sample from the smallest (1) to the largest(n1+ n2+… + nt ) In case of ties, use the averaged rank Compute the sums of ranks for each samples 2018/12/1 Copyright by Jen-pei Liu, PhD

Copyright by Jen-pei Liu, PhD 12.4 Kruskal-Wallis Test Methods: Null hypothesis: H0: The relative frequency distributions of all of the t>2 populations are identical. Alternative hypothesis: Ha: At least two of the t relative frequency distributions differ 2018/12/1 Copyright by Jen-pei Liu, PhD

Copyright by Jen-pei Liu, PhD 12.4 Kruskal-Wallis Test Test statistics: Rejection region: Reject H0 if with (t-1) degrees of freedom 2018/12/1 Copyright by Jen-pei Liu, PhD

Copyright by Jen-pei Liu, PhD 12.4 Kruskal-Wallis Test Example: Banking Broadcasting Electronics Retailing obs. Rank 16 1 41 18 32 11 22 5 37 14 39 30 10 25 6 21 4 35 13 38 15 33 12 29 9 46 20 28 8 19 2 40 17 47 3 43 27 7 sum 103 65 2018/12/1 Copyright by Jen-pei Liu, PhD

Copyright by Jen-pei Liu, PhD 12.4 Kruskal-Wallis Test Example: Reject H0 that text shelters were used equally among the executives from banking, broadcasting, electronics and retailing. 2018/12/1 Copyright by Jen-pei Liu, PhD

12.5 Friedman Test for a Block Design Example: To compare difference in overall business climate in four locations by 5 VPs of a firm Overall business climate on a 10-point scale (1= totally unacceptable, 10= outstanding) Each VP has his/her own preference and scoring pattern(habit) VP is a blocking factor Assessment of 4 locations in a random order 2018/12/1 Copyright by Jen-pei Liu, PhD

12.5 Friedman Test for a Block Design Example: Location VP 1 2 3 4 A 8.5 8.0 3.5 6.0 B 7.5 5.5 C 9.0 4.0 7.0 D E 5.0 4.5 2018/12/1 Copyright by Jen-pei Liu, PhD

12.5 Friedman Test for a Block Design Method: Rank the observations with each block(VP) from the smallest (1) to the largest (t) Use the average rank if ties Sum the rank for each treatment(locations) 2018/12/1 Copyright by Jen-pei Liu, PhD

12.5 Friedman Test for a Block Design Compute the test statistic: Reject H0 of no difference among treatments if 2018/12/1 Copyright by Jen-pei Liu, PhD

12.5 Friedman Test for a Block Design Example Location 1 2 3 4 VP obs. Rank A 8.5 8.0 3.5 6.0 B 7.5 5.5 C 9.0 4.0 7.0 D E 5.0 4.5 5 sum 7 12 17 14 2018/12/1 Copyright by Jen-pei Liu, PhD

12.5 Friedman Test for a Block Design Example: t=4, b=5 Fail to reject H0 of no difference on overall business climate among 4 locations 2018/12/1 Copyright by Jen-pei Liu, PhD

12.6 Rank Correlation Coefficient A measure of the degree of monotonicity between the variables A coefficient of preference data 2018/12/1 Copyright by Jen-pei Liu, PhD

12.6 Rank Correlation Coefficient Example: Preference ranks and prices of TV sets Manufacturer Preference Price 1 7 $449.50 2 4 525.00 3 479.95 6 499.95 5 580.00 549.95 8 469.95 532.50 2018/12/1 Copyright by Jen-pei Liu, PhD

12.6 Rank Correlation Coefficient Method: Where Xi and yi represent the ranks of the i th pair of observations and 2018/12/1 Copyright by Jen-pei Liu, PhD

12.6 Rank Correlation Coefficient Method: 2018/12/1 Copyright by Jen-pei Liu, PhD

12.6 Rank Correlation Coefficient Method: If no ties 2018/12/1 Copyright by Jen-pei Liu, PhD

12.6 Rank Correlation Coefficient Manufacturer Preference Price obs Rank Obs di 1 7 $449.5 6 36 2 4 525.00 5 -1 3 479.95 499.95 580.00 8 -7 48 549.95 -4 16 469.95 532.50 Total 144 2018/12/1 Copyright by Jen-pei Liu, PhD

12.6 Rank Correlation Coefficient α =0.05 one-tailed γs ≤-γ0 (8)=-0.643 reject H0 of no association between preference and price of TV set. 2018/12/1 Copyright by Jen-pei Liu, PhD

12.6 Rank Correlation Coefficient Methods: Null hypothesis: H0: There is no association between the ranked pairs. Alternative hypothesis: Ha:There is an association between the ranked pairs (a two-tailed test) or the correlation between the ranked pairs is positive (or negative) (a one-tailed test) Test statistic: γs 2018/12/1 Copyright by Jen-pei Liu, PhD

12.6 Rank Correlation Coefficient Rejection region: a. For a given value of α and for a two-tailed test, reject H0 if γs > γ0 or if γs<-γ0 , where r0 is given in Table 10 in the Appendix; double the table value of α to obtain the value ofα for the two-tailed test. b. For a given value of α and for a one-tailed test, reject H0 if γs > γ0, for an upper- tailed test, or if γs<-γ0 , for a lower-tailed test, the α value for a one-tailed test is the value shown in Table 10. 2018/12/1 Copyright by Jen-pei Liu, PhD

Copyright by Jen-pei Liu, PhD 12.7 Summary Wilcoxon Signed-rank Test Mann-Whitney-Wilcoxon Test Kruskal-Wallis Test Friedman Test Rank Correlation Coefficient 2018/12/1 Copyright by Jen-pei Liu, PhD