1. 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

2. 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
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3. 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!
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4. 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
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Copyright by Jen-pei Liu, PhD

5. 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.
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6. 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
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7. 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
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8. 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

9. 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
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10. 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).
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11. 12.2 Wilcoxon Signed-rank Test
Test statistic:
and T can be either T+ or T-
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12. 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

13. 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
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Copyright by Jen-pei Liu, PhD

14. 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
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15. 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
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16. 12.3 Mann-Whitney-Wilcoxon Test
Example
Training Program 1 2 28 33 31 29 27 35 25 30
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17. 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
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18. 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
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19. 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).
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20. 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
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21. 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

22. 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).
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23. 12.3 Mann-Whitney-Wilcoxon Test
Test Statistics:
, and let U=U1for one-sided alternative.
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24. 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.
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25. 12.3 Mann-Whitney-Wilcoxon Test
Example:
Fail to reject H0 of no difference between train program
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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
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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
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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
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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
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12.4 Kruskal-Wallis Test
Test statistics:
Rejection region:
Reject H0 if with (t-1) degrees of freedom
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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
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12.4 Kruskal-Wallis Test
Example:
Reject H0 that text shelters were used equally among the executives from banking, broadcasting, electronics and retailing.
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33. 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
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34. 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
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Copyright by Jen-pei Liu, PhD

35. 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)
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36. 12.5 Friedman Test for a Block Design
Compute the test statistic:
Reject H0 of no difference among treatments if
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37. 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
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38. 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
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39. 12.6 Rank Correlation Coefficient
A measure of the degree of monotonicity between the variables
A coefficient of preference data
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40. 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
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41. 12.6 Rank Correlation Coefficient
Method:
Where Xi and yi represent the ranks of the i th pair of observations and
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42. 12.6 Rank Correlation Coefficient
Method:
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43. 12.6 Rank Correlation Coefficient
Method: If no ties
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44. 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
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45. 12.6 Rank Correlation Coefficient
α =0.05 one-tailed γs ≤-γ0 (8)= reject H0 of no association between preference and price of TV set.
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46. 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
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47. 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.
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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