bms2024 kruskal wallis 2013

8/10/2019 BMS2024 Kruskal Wallis 2013

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Advanced Managerial

Statistics

Nonparametric Analysis:

(Comparing More Than TwoPopulations)

Kruskal-Wallis Test


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Kruskal-Wallis Rank Test

Tests the equality of more than 2 population

medians

Use when the normality assumption for one-wayANOVA is violated

Assumptions:

The samples are random and independentThe data are either ordinal or quantitative but

not normal


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Obtain overall 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 c

groups

Compute the H test statistic


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The Kruskal-Wallis H test statistic:(with c1degrees of freedom)

)1(3)1(

12

1

2

nn

T

nnH

c

j j

j

where:

n = total number of values over the combined samples

c = Number of groups

Tj= Sum of ranks in the jthsample

nj= Size of the jthsample


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Decision rule:

Reject H0if test statistic H

> 2

U

Otherwise Do Not Reject H0

Complete the test by comparing the calculated H value to a

critical 2value from the chi-square distribution with (c 1)

degrees of freedom (refer to Table 8 of Cambridge)

U = 2, (c1)

0

Reject H0Do Not Reject

H0


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Example 1 Do different branch offices have a different

median number of employees?

Office size

(Chicago, C)

Office size

(Denver, D)

Office size

(Houston, H)

23

41

54

78

66

55

60

72

45

70

30

40

18

34

44


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Example 1 Do different branch offices have a different

median number of employees?

Office size(Chicago, C)

Ranking Office size(Denver, D)

Ranking Class size(Houston, H)

Ranking

23

41

5478

66

2

6

915

12

55

60

7245

70

10

11

148

13

30

40

1834

44

3

5

14

7

= 44 = 56 = 20


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Example 1

The H statistic is

72.6)115(35

20

5

56

5

44

)115(15

12

)1(3

)1(

12

222

1

2

n

n

T

nn

Hc

j j

j

equalareMedianspopulationallNot:H

MedianMedianMedian:H

A

HDC0


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Example 1

Since H = 6.72 > ,So, Reject H0

Compare H= 6.72 to the critical value from the chi-square

distribution for (31) = 2 degrees of freedom and = .05:

There is sufficient evidence of a

difference in the population

median number of employees in

the branch offices.

5.9912U

2U = 5.991

0

= 0.05

Reject H0Do Not Reject

H0


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Example 2Saturday morning cartoons have been

subjected to much criticism by educators,

psychologists and politicians because of the

excessive amount of violence. In order to

determine if certain networks have more

violent cartoons than others, severalparents of children who regularly watch

cartoons were asked to judge on a five

point scale the level of violence (1 = no

violence and 5 = extreme violence) shown

in the cartoon. Six parents were asked to

judge TV3 cartoons, another six judgedNTV7 cartoons and another six rated Astro

cartoon network programs. Can we

conclude at the 5% level of significance that

the networks differ in the degree of

violence shown in their cartoons?

Network Rating

TV3 NTV7 Astro

4 2 5

5 3 52 3 4

3 2 5

5 4 5

4 2 4


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Example 2What type of data are these?

Are they dependent orindependent samples?

Do the data qualify for a

parametric test?

Network Rating

TV3 NTV7 Astro

4 2 5

5 3 5

2 3 4

3 2 5

5 4 5

4 2 4


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Example 2Can we conclude at the 5% level of

significance that the networks

differ in the degree of violenceshown in their cartoons?

Network Rating

TV3 NTV7 Astro

4 2 5

5 3 5

2 3 4

3 2 5

5 4 5

4 2 4


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Summary

Determined when to use nonparametric tests

Assigned rank

Applied Kruskal-Wallis Test Used both Critical Value approach to conclude

Interpreted the result from using the Kruskal-Wallis Test

bms2024 kruskal wallis 2013

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