**Assumptions:** Unlike the
parametric independent group ANOVA (one way ANOVA), this non-parametric 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 non-parametric
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 Kruskal-Wallis
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 Kruskal-Wallis should be compared to the H statistic to determine the
significance level. Otherwise, a Chi-square with k-1 (the number of groups-1)
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 p-value is low, chances are there will be little overlap between
the distributions. If the p-value 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 Kruskal-Wallis test (independent group comparison test) is located in
the Analyze/Non-parametric comparisons menu. When there are only two groups in
this comparison, the test becomes a Mann-Whitney test.

**See Also:** The independent
group (one-way) ANOVA.

**Example: Kruskal-Wallis
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.

--------------------------------------------------------------

Non-Parametric Independent Group Comparison
KRUSKAL.DBF

--------------------------------------------------------------

Results of Non-Parametric analysis:

Group variable = GROUP Observation variable = OBS

Kruskal-Wallis H = 24.48

P-value for H estimated by Chi-Square with 3 degrees of freedom.

Chi-Square = 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: Kruskal-Wallis
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 Kruskal-Wallis 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 one-way ANOVA using the same data?

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