14.3 Kruskal-Wallis Test

The Kruskal-Wallis test is a powerful tool for comparing multiple groups when data isn't normally distributed. It's like ANOVA's cool cousin who doesn't care about fancy assumptions. This test ranks all the data, then checks if the groups' ranks are significantly different.

To use it, you rank all observations, calculate sums for each group, and crunch some numbers to get the H statistic. If H is bigger than a critical value, you've found significant differences between groups. It's great for analyzing things like survey responses or treatment effects.

Kruskal-Wallis Test

Applications of Kruskal-Wallis Test

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• Compares three or more independent samples drawn from different populations or groups (age groups, treatment conditions)
• Determines the presence of significant differences between the samples without assuming the data follows a normal distribution
• Serves as a non-parametric alternative to one-way ANOVA when normality assumptions are not met or the data is ordinal
• Relies on the ranks of the data values rather than the actual measurements to assess differences between groups (test scores, survey responses)

Conducting Kruskal-Wallis Test

• Rank all observations from smallest to largest, disregarding group membership
  • Assign ranks starting with 1 for the smallest value, 2 for the second smallest, and so on
  • For tied values, assign the average rank to each tied observation (3.5 for two values tied at ranks 3 and 4)
• Calculate the sum of ranks for each group separately
• Determine the sample size for each group ($n_1$, $n_2$, ..., $n_k$) where $k$ represents the number of groups (control, treatment 1, treatment 2)
• Calculate the Kruskal-Wallis test statistic ($H$) using the formula: $H = \frac{12}{N(N+1)} \sum_{i=1}^k \frac{R_i^2}{n_i} - 3(N+1)$
  • $N$ = total sample size (sum of all $n_i$)
  • $R_i$ = sum of ranks for group $i$
  • $n_i$ = sample size for group $i$
  • $k$ = number of groups

Calculations for Kruskal-Wallis Test

• The Kruskal-Wallis test statistic ($H$) follows a chi-square distribution with $k-1$ degrees of freedom, where $k$ is the number of groups compared
• Determine the desired significance level ($\alpha$) for the test (0.05, 0.01)
• Look up the critical value from the chi-square distribution table using $k-1$ degrees of freedom and the chosen significance level
• Compare the calculated test statistic ($H$) to the critical value
  1. If $H$ exceeds the critical value, reject the null hypothesis, indicating significant differences between the groups
  2. If $H$ is less than or equal to the critical value, fail to reject the null hypothesis, suggesting insufficient evidence of significant differences

Interpretation of Kruskal-Wallis results

• When the null hypothesis is rejected:
  • The test provides evidence of significant differences between the populations from which the samples were drawn (income levels, satisfaction ratings)
  • At least one sample likely comes from a different population than the others
  • Conduct post-hoc tests such as Dunn's test to identify which specific groups differ significantly from each other
• When the null hypothesis is not rejected:
  • The test does not provide sufficient evidence to conclude that the populations differ significantly
  • The samples may have been drawn from identical or similar populations
• Interpret the results in the context of the research question or problem
  • Consider both statistical significance and practical significance of the findings (effect size, domain knowledge)
  • Integrate the Kruskal-Wallis test results with other relevant information (qualitative data, literature) to inform decision-making processes (resource allocation, treatment selection)