bms2024 kruskal wallis 2013
TRANSCRIPT
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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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Kruskal-Wallis Rank Test
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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Kruskal-Wallis Rank Test
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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Kruskal-Wallis Rank Test
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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Kruskal-Wallis Rank Test
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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Kruskal-Wallis Rank Test
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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Kruskal-Wallis Rank Test
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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Kruskal-Wallis Rank Test
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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Kruskal-Wallis Rank Test
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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Kruskal-Wallis Rank Test
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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Kruskal-Wallis Rank Test
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