Business statistics, 6e, 2005, groebner CH16

Wilcoxon Signed Rank Test Used to test a hypothesis about one population median  the median is the midpoint of the distribution: 50% below, 50% above  A hypothesized median is rejec

Chapter Goals

After completing this chapter, you should be

able to:

 Recognize when and how to use the Wilcoxon

signed rank test for a population median

 Recognize the situations for which the Wilcoxon signed rank test applies and be able to use it for decision-making

 Know when and how to perform a Mann-Whitney U-test

 Perform nonparametric analysis of variance using the Kruskal-Wallis one-way ANOVA

Nonparametric Statistics

 Nonparametric Statistics

 Fewer restrictive assumptions about data levels and underlying probability distributions

 Population distributions may be skewed

 The level of data measurement may only

be ordinal or nominal

Wilcoxon Signed Rank Test

 Used to test a hypothesis about one

population median

 the median is the midpoint of the distribution: 50%

below, 50% above

 A hypothesized median is rejected if sample

results vary too much from expectations

 no highly restrictive assumptions about the shape of the population distribution are needed

The W Test Statistic

Performing the Wilcoxon Signed Rank Test

Calculate the test statistic W using these steps:

 Step 1: collect sample data

 Step 2: compute di = difference between each

value and the hypothesized median

 Step 3: convert di values to absolute differences

The W Test Statistic

Performing the Wilcoxon Signed Rank Test

 Step 4: determine the ranks for each di value

 eliminate zero di values

 Lowest di value = 1

 For ties, assign each the average rank of the

tied observations

(continued)

The W Test Statistic

Performing the Wilcoxon Signed Rank Test

 Step 5: Create R+ and R- columns

 for data values greater than the hypothesized

median, put the rank in an R+ column

 for data values less than the hypothesized

median, put the rank in an R- column

(continued)

The W Test Statistic

Performing the Wilcoxon Signed Rank Test

 Step 6: the test statistic W is the sum of the

ranks in the R+ column

 Test the hypothesis by comparing the

calculated W to the critical value from the table

in appendix P

 Note that n = the number of non-zero di values

(continued)

 The median class size is claimed to be 40

 Sample data for 8 classes is randomly obtained

 Compare each value to the hypothesized median to find difference

Class size = xi

Difference

di = xi – 40 | di |23

45 34 78 34 66 61 95

-17 5 -6 38 -6 26 21 55

17 5 6 38 6 26 21 55

 Rank the absolute differences:

| di | Rank

5 6 6 17 21 26 38 55

1 2.5 2.5 4 5 6 7 8

tied

(continued)

 Put ranks in R+ and R- columns and find sums:

Class size = xi

-17 5 -6 38 -6 26 21 55

17 5 6 38 6 26 21 55

4 1 2.5 7 2.5 6 5 8

1

7

6 5 8

4 2.5 2.5

Completing the Test

H0: Median = 40

HA: Median ≠ 40

Test at the  = 05 level:

This is a two-tailed test and n = 8, so find WL and WU in

appendix P: WL = 3 and WU = 33

The calculated test statistic is W = R+ = 27

Completing the Test

H0: Median = 40

HA: Median ≠ 40

WL = 3 and WU = 33

WL < W < WU so do not reject H0

(there is not sufficient evidence to conclude that the

median class size is different than 40)

(continued)

WL = 3 do not reject H0reject H0

W = R+ = 27

WU = 33 reject H0

If the Sample Size is Large

 The W test statistic approaches a normal distribution as n increases

 For n > 20, W can be approximated by

24

1) 1)(2n

n(n

4

1)

n(n W

where W = sum of the R+ ranks

d = number of non-zero di values

Nonparametric Tests for Two

Population Centers

Nonparametric Tests for Two Population Centers

WilcoxonMatched-PairsSigned Rank Test

Mann-Whitney

U-test

Large Samples

Small Samples

Large Samples Small

Samples

Mann-Whitney U-Test

Used to compare two samples from two populations

Assumptions:

 The two samples are independent and random

 The value measured is a continuous variable

 The measurement scale used is at least ordinal

 If they differ, the distributions of the two populations will differ only with respect to the central location

 Consider two samples

 combine into a singe list, but keep track of which sample each value came from

 rank the values in the combined list from low

to high

 For ties, assign each the average rank of the tied values

 separate back into two samples, each value keeping its assigned ranking

 sum the rankings for each sample

Mann-Whitney U-Test

(continued)

 If the sum of rankings from one sample differs enough from the sum of rankings from the other sample, we conclude there is

a difference in the population medians

Mann-Whitney U-Test

(continued)

Mann-Whitney U-TestMann-Whitney U-Statistics

2

1

R

) n

(

n n

n U

2

1

R

) n

(

n n

n U

where:

n1 and n2 are the two sample sizes

R1 and R2 = sum of ranks for samples 1 and 2

 Suppose the results are:

Class size (Math, M) Class size (English, E)

23 45 34 78 34 66 62 95 81

30 47 18 34 44 61 54 28 40

(continued)

Mann-Whitney U-Test

 Split back into the original samples:

Class size (Math, M) Rank

Class size (English, E) Rank

23 45 34 78 34 66 62 95 81

2 10 6 16 6 15 14 18 17

30 47 18 34 44 61 54 28 40

4 11 1 6 9 13 12 3 8

 = 104  = 67

(continued)

Mann-Whitney U-Test

H0: MedianM ≤ MedianE

HA: MedianM > MedianE

Claim: Median class size for

Math is larger than the

median class size for English

22

104 2

(9)(10) (9)(9)

R 2

1) (n

n n

n

2 1

59

67 2

(9)(10) (9)(9)

R 2

1) (n

n n

n

2 1

 The Mann-Whitney U tables in Appendices L

and M give the lower tail of the U-distribution

 For one-tailed tests like this one, check the

alternative hypothesis to see if U1 or U2 should

be used as the test statistic

 Since the alternative hypothesis indicates that

population 1 (Math) has a higher median, use

U1 as the test statistic

(continued)

Mann-Whitney U-Test

 Use U1 as the test statistic: U = 22

 Compare U = 22 to the critical value U from the appropriate table

 For sample sizes less than 9, use Appendix L

 For samples sizes from 9 to 20, use Appendix M

 If U < U, reject H0

(continued)

Mann-Whitney U-Test

Since U  U, do not reject H0

 Use U1 as the test statistic: U = 19

 U from Appendix M for  = 05, n1 = 9 and

Mann-Whitney U-Test for

Large Samples

 The table in Appendix M includes U values

only for sample sizes between 9 and 20

 The U statistic approaches a normal distribution

as sample sizes increase

 If samples are larger than 20, a normal

approximation can be used

Mann-Whitney U-Test for

Large Samples

 The mean and standard deviation for

Mann-Whitney U Test Statistic:

n )(

n )(

Mann-Whitney U-Test for

n )(

n )(

n (

2

n

n

U z

2 1

2 1

2 1









Large Sample Example

 U statistic is found to be U = 655

1345

1475 2

(40)(41) (40)(50)

R 2

1) (n

n n

n

2 1

655

2620 2

(50)(51) (40)(50)

R 2

1) (n

n n

n

2 1

Since the alternative hypothesis indicates that

population 2 has a higher median, use U2 as the test

statistic

 Compute the U statistics:

Large Sample Example

(continued)

Since z = -2.80 < -1.645, we reject H0

645 1

z  

Reject H0

0 Median

Median

:

H

0 Median

Median

:

H

2 1

A

2 1

) 1 50 40

)(

50 )(

40 (

1000 655

12

) 1 n

n )(

n )(

n

(

2

n n U z

2 1

2 1

2 1

Wilcoxon Matched-Pairs

Signed Rank Test

 The Mann-Whitney U-Test is used when

samples from two populations are independent

 If samples are paired, they are not independent

 Use Wilcoxon Matched-Pairs Signed Rank Test with paired samples

The Wilcoxon T Test Statistic

Performing the Small-Sample Wilcoxon

Matched Pairs Test (for n < 25)

Calculate the test statistic T using these steps:

 Step 1: collect sample data

 Step 2: compute di = difference between the

sample 1 value and its paired sample 2 value

 Step 3: rank the differences, and give each rank

the same sign as the sign of the difference value

The Wilcoxon T Test Statistic

Performing the Small-Sample Wilcoxon

Matched Pairs Test (for n < 25)

 Step 4: The test statistic is the sum of the absolute values of the ranks for the group with the smaller

Small Sample Example

 Paired samples, n = 9:

Value (before) Value (after)

38 45 34 58 30 46 42 55 41

30 47 18 34 34 31 24 38 40

b a

A

b a

0

Median Median

: H

Median Median

Small Sample Example

 Paired samples, n = 9:

Value (before)

Value (after)

30 47 18 54 38 31 24 62 40

6 -2 16 4 -8 15 18 -7 1

4 -2 8 3 -6 7 9 -5 1

 The calculated T value is T = 13

 Complete the test by comparing the calculated

T value to the critical T-value from Appendix N

 For n = 9 and  = 025 for a one-tailed test,

Small Sample Example

(continued)

Wilcoxon Matched Pairs Test

for Large Samples

 The table in Appendix N includes T values

only for sample sizes from 6 to 25

 The T statistic approaches a normal

distribution as sample size increases

 If the number of paired values is larger than 25,

a normal approximation can be used

 The mean and standard deviation for

Wilcoxon T :

(continued)

4

) 1 n

2 )(

1 n

where n is the number of paired values

Wilcoxon Matched Pairs Test

for Large Samples

Mann-Whitney U-Test for

2 )(

1 n

( n

4

) 1 n

(

n

T z

 Tests the equality of more than 2 population

medians

 Assumptions:

 variables have a continuous distribution

 the data are at least ordinal

 samples are independent

 samples come from populations whose only possible difference is that at least one may have a different central location than the others

Kruskal-Wallis One-Way

ANOVA

Kruskal-Wallis Test

Procedure

 Obtain relative rankings for each value

 In event of tie, each of the tied values gets the average rank

 Sum the rankings for data from each of the k groups

 Compute the H test statistic

Kruskal-Wallis Test

Procedure

 The Kruskal-Wallis H test statistic:

(with k – 1 degrees of freedom)

) 1 N

(

3 n

R )

1 N

( N

Ri = Sum of ranks in the i th sample

ni = Size of the i th sample

(continued)

 Complete the test by comparing the calculated H value to a critical 2 value from the chi-square distribution with k – 1

degrees of freedom(The chi-square distribution is Appendix G)

 Do different departments have different class

sizes?

Kruskal-Wallis Example

Class size (Math, M)

Class size (English, E)

Class size (History, H)

23 45 54 78 66

55 60 72 45 70

30 40 18 34 44

 Do different departments have different class

Class size (History, H) Ranking

23 41 54 78 66

2 6 9 15 12

55 60 72 45 70

10 11 14 8 13

30 40 18 34 44

3 5 1 4 7

 The H statistic is

(continued)

Kruskal-Wallis Example

72 6 )

1 15

(

3 5

20 5

56 5

44 )

1 15

( 15

12

) 1 N

(

3 n

R )

1 N

( N

12 H

2 2

Medians population

all ot N : H

Median Median

Median :

H

A

H E

M

2 05 



 Compare H = 6.72 to the critical value from the chi-square distribution for 5 – 1 = 4 degrees of freedom and  = 05:

4877

9

2 05 



There is not sufficient evidence to reject that the population medians are all equal

) t t

(

g

1 i

i

3 i

g = Number of different groups of ties

ti = Number of tied observations in the i th tied group of scores

N = Total number of observations

H Statistic Corrected for

Tied Rankings

 Corrected H statistic:

N N

) t t

( 1

) 1 N

(

3 n

R )

1 N

( N

12 H

3

g

1 i

i

3 i

k

1

2 i

Chapter Summary

 Developed and applied the Wilcoxon signed rank W-test for a population median

 Small Samples

 Large sample z approximation

 Developed and applied the Mann-Whitney U-test for two population medians

 Small Samples

 Large Sample z approximation

 Used the Wilcoxon Matched-Pairs T-test for paired

samples

 Small Samples

 Large sample z approximation

 Applied the Kruskal-Wallis H-test for multiple population medians