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Kruskal-Wallis Test
(Non-parametric independent group comparisons)

Definition: A non-parametric test (distribution-free) used to compare three or more independent groups of sampled data.

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?

For more information... we recommend:

• WINKS -- a simple to use and affordable statistical software program that will help you analyze, interpret and write-up your results. Download a free trial copy.
   

• Additional statistics tutorials.
   

• Against All Odds VIDEOS - Now in VHS or DVD formats Teaching Videos from Annenberg/PBS Click here for info
   

• BeSmartNotes Reference sheets for SAS, SAS ODS, SPSS and WINKS  - Click here for info. (www.besmartnotes.com)
   

• Statistical Analysis Quick Reference Guidebook: With SPSS Examples is a practical "cut to the chase" handbook that quickly explains the when, where, and how of statistical data analysis as it is used for real-world decision-making in a wide variety of disciplines. It contains examples using SPSS Statistics software. In this one-stop reference, the authors  provide succinct guidelines for performing an analysis, avoiding pitfalls, interpreting results, and reporting outcomes. Paperback. Sage Publishers ISBN: 1412925606 Order book from Amazon

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