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These WINKS statistics tutorials
explain the use and interpretation of standard statistical analysis
techniques for Medical, Pharmaceutical, Clinical Trials, Marketing or Scientific
Research. The examples include howto instructions for WINKS SDA Version 6.0
Software.
KruskalWallis Test
(Nonparametric independent group
comparisons)
Definition: A nonparametric
test (distributionfree) used to compare three or more independent groups of
sampled data.
Assumptions: Unlike the
parametric independent group ANOVA (one way ANOVA), this nonparametric test
makes no assumptions about the distribution of the data (e.g., normality).
Characteristics: This test
is an alternative to the independent group ANOVA, when the assumption of
normality or equality of variance is not met. This, like many nonparametric
tests, uses the ranks of the data rather than their raw values to calculate the
statistic. Since this test does not make a distributional assumption, it is not
as powerful as the ANOVA.
Test: The hypotheses for the
comparison of two independent groups are:
H_{o}: The samples come
from identical populations
H_{a}: They samples come
from different populations
Notice that the hypothesis makes no
assumptions about the distribution of the populations. These hypotheses are also
sometimes written as testing the equality of the central tendency of the
populations.
The test statistic for the KruskalWallis
test is H. This value is compared to a table of critical values for U based on
the sample size of each group. If H exceeds the critical value for H at some
significance level (usually 0.05) it means that there is evidence to reject the
null hypothesis in favor of the alternative hypothesis. (See the Zar reference
for details.)
Note: When sample sizes are
small in each group (< 5) and the number of groups is less than 4 a tabled value
for the KruskalWallis should be compared to the H statistic to determine the
significance level. Otherwise, a Chisquare with k1 (the number of groups1)
degrees of freedom can be used to approximate the significance level for the
test.
Graphical comparison: The
graphical comparison allows you to visually see the distribution of the two
groups. If the pvalue is low, chances are there will be little overlap between
the distributions. If the pvalue is not low, there will be a fair amount of
overlap between the groups. There are a number of options available in the
comparison graph to allow you to examine the groups. These include box plots,
means, medians, and error bars.
Location in KWIKSTAT and WINKS:
The KruskalWallis test (independent group comparison test) is located in
the Analyze/Nonparametric comparisons menu. When there are only two groups in
this comparison, the test becomes a MannWhitney test.
See Also: The independent
group (oneway) ANOVA.
Example: KruskalWallis
independent group comparisons
The KRUSKAL.DBF database on disk
contains information on the weights of four groups of animals given four
different treatments to improve weight gain.

NonParametric Independent Group Comparison
KRUSKAL.DBF

Results of NonParametric analysis:
Group variable = GROUP Observation variable = OBS
KruskalWallis H = 24.48
Pvalue for H estimated by ChiSquare with 3 degrees of freedom.
ChiSquare = 24.5 with 3 D.F. p < 0.001
Rank sum group 1 = 28. N = 7 Mean Rank = 4.
Rank sum group 2 = 77.5 N = 7 Mean Rank = 11.07
Rank sum group 3 = 171. N = 7 Mean Rank = 24.43
Rank sum group 4 = 129.5 N = 7 Mean Rank = 18.5
Critical q
Tukey Multiple Comp. Difference
Q (.05)

Rank(3)Rank(1) = 20.4286
4.647 2.639 *
(SE used = 4.3964)
Rank(3)Rank(2) = 13.3571
3.038 2.639 *
(SE used = 4.3964)
Rank(3)Rank(4) = 5.9286
1.349 2.639
(SE used = 4.3964)
Rank(4)Rank(1) = 14.5
3.298 2.639 *
(SE used = 4.3964)
Rank(4)Rank(2) = 7.4286
1.69 2.639
(SE used = 4.3964)
Rank(2)Rank(1) = 7.0714
1.608 2.639
(SE used = 4.3964)
Homogeneous Populations, groups ranked
Gp Gp Gp Gp
1 2 4 3



This is a graphical representation of the Tukey multiple comparisons test. At
the 0.05 significance level, the means of any two groups underscored by the same
line are not significantly different.
The results of this test indicate
that there is a significant difference between the weights of animals for the
four groups (p < 0.001). The Tukey multiple comparisons are used to specify
which groups differ at the 0.05 significance level. Groups are compared, and
those that are significantly different are marked with an asterisk "*". A
graphical table is also presented to show the differences between the groups.
From this information, you can conclude that weights for group 3 are higher than
for group1, weights for group 3 are higher than for group 2 and weights for
group 4 are higher than for group 1. However, there is no significant difference
in weights between groups 3 and 4, 4 and 2 and 2 and 1. Thus, although group 3
showed the most weight gain, it was not a significantly greater than the weight
gain for group 4.
You should also examine the
comparison graph to visually see the distributions of these four groups.
Exercise: KruskalWallis
Independent Group Comparison
Four different milling machines
were being considered for purchase by a manufacturer. Potentially, the company
would be purchasing hundreds of these machines, so it wanted to make sure it
made the best decision. Initially, five of each machine were borrowed, and each
was randomly assigned to one of 20 technicians (all technicians were similar in
skill). Each machine was put through a series of tasks and rated using a
standardized test. The higher the score on the test, the better the performance
of the machine. The data are:
Machine 1 
Machine 2 
Machine 3 
Machine 4 
24.5 
28.4 
26.1 
32.2 
23.5 
34.2 
28.3 
34.3 
26.4 
29.5 
24.3 
36.2 
27.1 
32.2 
26.2 
35.6 
29.9 
30.1 
27.8 
32.5 
1. Perform a KruskalWallis test on
this data. Remember that the program expects two fields, a Group field and a
Observation field. Are there differences between the groups?
2. What specific machines, if any,
are different? Which machine would you recommend for purchase?
3. Produce a graphical comparison
of these results.
4. How does this analysis compare
with a oneway ANOVA using the same data?
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