CD-ROM Chap 16-1 A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc. A Course In Business Statistics 4 th Edition CD-ROM Chapter 16 Introduction

CD-ROM Chap 16-1A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc.

A Course In Business Statistics

4th Edition

CD-ROM Chapter 16Introduction to

Nonparametric Statistics


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




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)




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)




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)


Example

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


Example

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)


Example

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

size = xi

Difference

di = xi – 40| di | Rank R+ R-

23

45

34

78

34

66

61

95

-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

= 27 = 9

(continued)

These three are below the claimed median, the others are above


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 = 3do not reject H0reject H0

W = R+ = 27

WU = 33reject 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

241)1)(2nn(n

41)n(n

Wz

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-Pairs

Signed 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)


(continued)

Mann-Whitney U-Test

Mann-Whitney U-Statistics

111

211 2

1R

)n(nnnU

222

212 2

1R

)n(nnnU

where:

n1 and n2 are the two sample sizes

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


(continued)

Mann-Whitney U-Test

Claim: Median class size for Math is larger than the median class size for English

A random sample of 9 Math and 9 English classes is selected (samples do not have to be of equal size)

Rank the combined values and then split them back into the separate samples


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


Size Rank

18 1

23 2

28 3

30 4

34 6

34 6

34 6

40 8

44 9

Size Rank

45 10

47 11

54 12

61 13

62 14

66 15

78 16

81 17

95 18

Ranking for combined samples

tied

(continued)

Mann-Whitney U-Test


Split back into the original samples:

Class size (Math, M)

RankClass 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

221042

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

2

1)(nnnnU 1

11211

59672

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

2

1)(nnnnU 2

22212

Note: U1 + U2 = n1n2

(continued)

Mann-Whitney U-Test

Math:

English:


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

n2 = 9 is U = 7

(continued)

Mann-Whitney U-Test

U = 7

U = 19

do not reject H0reject H0


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



The mean and standard deviation for Mann-Whitney U Test Statistic:

(continued)

2

nn 21

12

)1nn)(n)(n( 2121

Where n1 and n2 are sample sizes from populations 1 and 2



Normal approximation for Mann-Whitney U Test Statistic:

(continued)

12)1nn)(n)(n(

2nn

Uz

2121

21


Large Sample Example

We wish to test

Suppose two samples are obtained: n1 = 40 , n2 = 50 When rankings are completed, the sum of ranks

for sample 1 is R1 = 1475 When rankings are completed, the sum of ranks

for sample 2 is R2 = 2620

H0: Median1 Median2

HA: Median1 < Median2


U statistic is found to be U = 655

134514752

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

2

1)(nnnnU 1

11211

65526202

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

2

1)(nnnnU 2

22212

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

645.1z

Reject H0

0MedianMedian :H

0MedianMedian :H

21A

210

80.2

12)15040)(50)(40(

1000655

12)1nn)(n)(n(

2nn

Uz

2121

21

= .05

Do not reject H0

0

Large Sample Example(continued)


Wilcoxon Matched-PairsSigned 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 expected sum Look at the alternative hypothesis to determine

the group with the smaller expected sum For two tailed tests, just choose the smaller

sum

(continued)


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

baA

ba0

MedianMedian :H

MedianMedian :H

Claim: Median value is smaller after than before


Small Sample Example

Paired samples, n = 9:Value

(before)Value (after)

Difference

d

Rank

of dRanks with smaller

expected sum

36

45

34

58

30

46

42

55

41

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

2

6

5

= T = 13

(continued)


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, T = 6

Since T T, do not reject H0

T = 6

T = 13

do not reject H0reject H0

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

)1n(n

24

)1n2)(1n)(n(

where n is the number of paired values

Wilcoxon Matched Pairs Test for Large Samples



Normal approximation for the Wilcoxon T Test Statistic:

(continued)

24)1n2)(1n(n

4)1n(n

Tz


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



The Kruskal-Wallis H test statistic: (with k – 1 degrees of freedom)

)1N(3n

R

)1N(N

12H

k

1i i

2i

where:N = Sum of sample sizes in all samplesk = Number of samplesRi = Sum of ranks in the ith sampleni = Size of the ith 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) Decision rule

Reject H0 if test statistic H > 2

Otherwise do not reject H0

(continued)



Do different departments have different class sizes?

Kruskal-Wallis Example


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 sizes?



RankingClass size

(English, E)Ranking

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

= 44 = 56 = 20


The H statistic is

(continued)


72.6)115(35

20

5

56

5

44

)115(15

12

)1N(3n

R

)1N(N

12H

222

k

1i i

2i

equal are Medians population all otN :H

MedianMedianMedian :H

A

HEM0


Since H = 6.72 <

do not reject H0

(continued)


4877.9205.

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

4877.9205.

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


Kruskal-Wallis Correction

If tied rankings occur, give each observation the mean rank for which it is tied

The H statistic is influenced by ties, and should be corrected

Correction for tied rankings:

NN

)tt(1

3

g

1ii

3i

where:g = Number of different groups of tiesti = Number of tied observations in the ith tied group of scoresN = Total number of observations


H Statistic Corrected for Tied Rankings

Corrected H statistic:

NN

)tt(1

)1N(3nR

)1N(N12

H

3

g

1ii

3i

k

1i i

2i


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

Documents

CD-ROM Chap 16-1 A Course In Business Statistics, 4th © 2006 Prentice-Hall, Inc. A Course In Business Statistics 4 th Edition CD-ROM Chapter 16 Introduction