Kruskal–Wallis Test | Nonparametric Analysis

The Kruskal–Wallis test is a rank-based method that compares three or more independent groups and is a rank-based analogue of one-way ANOVA. We combine and rank all observations and measure how far the group mean rank deviates from the common center.

Study Design and Hypotheses#

The outcome is at least ordinal
Observations between groups are independent and each observation belongs to only one group.
The test is not appropriate for repeated measurements
If interpreted as a median difference, the group distributions have approximately the same shape.

H_0:F_1=F_2=\cdots=F_k\qquad H_1:\text{At least one group's distribution is different}

Pool and Rank All Observations#

Total sample size

N=\sum_{i=1}^{k}n_i

rank sum for group i

R_i=\sum_{j=1}^{n_i}R_{ij}

mean rank

\bar R_i=\frac{R_i}{n_i}

Assign the average of the occupied ranks to tied values. The sum of all ranks is N(N+1)/2, and their common center is (N+1)/2. Under H₀, each group’s mean rank should lie near this value.

sum of all ranks

1+2+\cdots+N=\frac{N(N+1)}2

common center

\bar R_{\cdot}=\frac{N+1}{2}

Kruskal–Wallis Statistic#

mean-rank form

H=\frac{12}{N(N+1)}\sum_i n_i\left(\bar R_i-\frac{N+1}{2}\right)^2

rank-sum form

H=\frac{12}{N(N+1)}\sum_i\frac{R_i^2}{n_i}-3(N+1)

Chi-square approximation and degrees of freedom#

H\overset{\cdot}{\sim}\chi^2_{k-1}

Because the total rank sum across k groups is fixed, only k−1 rank sums can vary freely. For small samples or severe imbalance, consider exact or permutation methods. Having roughly five observations per group is only a guideline for the chi-square approximation and is unrelated to the expected-cell-count rule used for contingency tables.

Tie correction#

correction factor

C=1-\frac{\sum_j(t_j^3-t_j)}{N^3-N}

Correction H

H_{\mathrm{corrected}}=\frac HC

Post Hoc Comparisons after a Significant Test#

The significance of the overall test only indicates that at least one group is different. Use Dunn’s test or planned rank contrasts, and adjust multiplicity using Holm, Bonferroni, etc.

Perform the overall Kruskal–Wallis test
If significant, perform Dunn’s test or planned rank contrasts
Adjust for multiple comparisons
Report each group’s n, median, IQR, and mean rank, together with H, df, p value, and an effect size

Item	One-way ANOVA	Kruskal–Wallis
Input	Raw values	Pooled ranks
Comparison	Means	Mean ranks / distributions
Statistic	F	H
Reference distribution	F	χ²(k−1)
Post hoc	Tukey and related methods	Dunn and related methods