Kruskal-Wallis Test Calculator

The Kruskal-Wallis test ranks all observations from all groups combined, then examines whether the rank sums differ more than expected by chance. The procedure is:

1. Combine all observations and rank them from 1 to N
2. Compute the rank sum \(R_i\) for each group
3. Calculate the H-statistic

The test statistic is:

$$H = \frac{12}{N(N+1)} \sum_{i=1}^{k} \frac{R_i^2}{n_i} - 3(N+1)$$

Where \(N\) is the total sample size, \(k\) is the number of groups, \(n_i\) is the size of group \(i\), and \(R_i\) is the rank sum of group \(i\). Under the null hypothesis that all groups come from the same distribution, H approximately follows a chi-square distribution with \(k - 1\) degrees of freedom.

The expected rank sum for group \(i\) under H₀ is \(n_i(N+1)/2\). The H-statistic measures how much the observed rank sums deviate from these expected values. Larger H indicates greater between-group differences in rank distributions.

A tie correction factor can be applied: \(H_{corrected} = H / (1 - \sum(t^3 - t) / (N^3 - N))\), where \(t\) is the number of tied observations in each tied group.

Interpreting the Kruskal-Wallis test results:

• H-Statistic: Compare to the chi-square critical value with df = k − 1. For k = 3 groups and α = 0.05, the critical value is 5.991 (df = 2). If H > 5.991, reject the null hypothesis.
• Rank Sums: Higher rank sums indicate groups with generally larger values. Compare each group's rank sum to its expected value n_i(N+1)/2 to see which groups deviate.
• Post-hoc Tests: A significant H only indicates that at least one group differs. Use Dunn's test or pairwise Mann-Whitney tests with Bonferroni correction to identify which specific groups differ.
• Effect Size: Eta-squared for H: η² = (H − k + 1)/(N − k). Values of 0.01, 0.06, 0.14 correspond to small, medium, large effects.