Explain the steps in computation of Kruskal-Wallis ANOVA with the help of a suitable example.

1. Introduction to Kruskal-Wallis ANOVA

Kruskal-Wallis ANOVA is a non-parametric test used to compare the medians of three or more independent groups when the assumptions of parametric ANOVA cannot be met. It is suitable for ordinal or interval-ratio data and does not require the data to be normally distributed. The test ranks the observations from all groups and compares the mean ranks to determine if there are significant differences between the groups.

2. Example Scenario

Consider a study investigating the effect of different teaching methods on students' test scores. Three teaching methods (A, B, and C) are tested on separate groups of students, and their test scores are recorded. The null hypothesis is that there is no significant difference in test scores between the three teaching methods.

3. Steps in Computation

3.1 Rank the Data:
Combine the data from all groups and rank the observations from lowest to highest, assigning a rank to each observation. Ties are assigned the average of the ranks they would occupy if not tied.

3.2 Calculate the Sum of Ranks for Each Group:
Sum the ranks for each group separately. This gives the sum of ranks for groups A, B, and C.

3.3 Calculate the Test Statistic:
Use the formula for the Kruskal-Wallis test statistic:

[ H = \frac{{12}}{{N(N+1)}} \left[ \sum_{j=1}^{k} \frac{{T_j^2}}{{n_j}} \right] – 3(N+1) ]

where:

• ( H ) is the Kruskal-Wallis test statistic,
• ( N ) is the total number of observations,
• ( k ) is the number of groups,
• ( T_j ) is the sum of ranks for group ( j ),
• ( n_j ) is the sample size of group ( j ).

3.4 Determine the Critical Value:
Consult the Kruskal-Wallis table or use statistical software to find the critical value of ( H ) at a specified significance level (e.g., ( \alpha = 0.05 )) with degrees of freedom equal to ( k – 1 ).

3.5 Compare the Test Statistic to the Critical Value:
If the calculated test statistic is greater than the critical value, reject the null hypothesis and conclude that there are significant differences between the groups. If the test statistic is less than the critical value, fail to reject the null hypothesis.

4. Application to the Example

In our example, suppose we have the following test scores for each teaching method:

Group A: 55, 60, 65, 70
Group B: 50, 55, 60, 65
Group C: 45, 50, 55, 60

After ranking all the scores and summing the ranks for each group, we compute the test statistic using the formula. We then compare the calculated test statistic to the critical value from the Kruskal-Wallis table at the chosen significance level.

5. Interpretation of Results

If the calculated test statistic exceeds the critical value, we reject the null hypothesis and conclude that there are significant differences in test scores between the teaching methods. If the test statistic does not exceed the critical value, we fail to reject the null hypothesis, indicating no significant differences between the groups.

Conclusion

The Kruskal-Wallis ANOVA is a valuable statistical test for comparing the medians of multiple independent groups when parametric assumptions cannot be met. By following the steps outlined above and applying the appropriate formula, researchers can effectively analyze ordinal or interval-ratio data and draw meaningful conclusions about group differences.