The Kruskal-Wallis Test was developed by Kruskal and Wallis jointly and is named after them. The Kruskal-Wallis test is a nonparametric (distribution free) test, which is used to compare three or more groups of sample data. Kruskal-Wallis Test is used when assumptions of ANOVA are not met. ANOVA is a statistical data analysis technique that is used when the independent variable groups are more than two. In ANOVA, we assume that distribution of each group should be normally distributed. In Kruskal-Wallis Test, we do not assume any assumption about the distribution. So Kruskal-Wallis Test is a distribution free test. If normality assumptions are met, then the Kruskal-Wallis Test is not as powerful as ANOVA. Kruskal-Wallis Test is also an improvement over the Sign test and Wilcoxon’s sign rank test, which ignores the actual magnitude of the paired magnitude.
Hypothesis in Kruskal-Wallis Test:
Null hypothesis: In Kruskal-Wallis Test, null hypothesis assumes that the samples are from identical populations.
Alternative hypothesis: In Kruskal-Wallis Test, alternative hypothesis assumes that the sample comes from different populations.
Hypothesis in Kruskal-Wallis Test:
1. In Kruskal-Wallis Test, we assume that the samples drawn from the population are random.
2. In Kruskal-Wallis Test, we also assume that the cases of each group are independent.
3. The measurement scale for Kruskal-Wallis Test should be at least ordinal.
Procedure for Kruskal-Wallis Test:
1. Arrange the data of both samples in a single series in ascending order.
2. Assign rank to them in ascending order. In the case of a repeated value, assign ranks to them by averaging their rank position.
3. Once this is complete, ranks or the different samples are separated and summed up as R1 R2 R3, etc.
4. To calculate the value of Kruskal-Wallis Test, apply the following formula:
Where,
H = Kruskal-Wallis Test
n = total number of observations in all samples
Ri = Rank of the sample
Kruskal-Wallis Test statistics is approximately a chi-square distrubution, with k-1 degree of freedom where ni should be greater than 5. If the calculated value of Kruskal-Wallis Test is less than the chi-square table value, then the null hypothesis will be accepted. If the calculated value of Kruskal-Wallis Test H is greater than the chi-square table value, then we will reject the null hypothesis and say that the sample comes from a different population.
Kruskal-Wallis Test and SPSS:
In most statisitcal software, they have an option for Kruskal-Wallis Test. To perform the Kruskal-Wallis Test in SPSS, we have to perform the following procedures:
1. Click on the “SPSS 18″ icon from the start menu.
2. Click on the “open data” icon and select the data.
3. Click on the “analysis” menu and select “nonperametric test” from the analysis menu.
4. Select “K independent sample test” from the nonparametric option. The following window will appear as we will click on the K independent sample test.
Select the grouping variable, insert them into the right side grouping variable box and define the group. Select the dependent variable and insert them into the test variable list box. Click on the “option” button and select “descriptive statistics” from there. Select “Kruskal-Wallis Test” as a test type and click on the “ok” button. The result of the Kruskal-Wallis Test will appear. The table below will be used from the SPSS output for Kruskal-Wallis Test:
The descriptive statistics table for the Kruskal-Wallis Test will show the N, the Mean, the SD, the minimum and the maximum value for the test variable.
The Kruskal-Wallis Test rank table will show the number of groups in the independent variable, the total number in each group and the mean ranks of each group.
The Kruskal-Wallis Test statistics table will show the chi-square value, the degree of freedom and the associated significance value. By using this significance value, we can
accept or reject the null hypothesis for the Kruskal-Wallis Test.
