Kruskal–Wallis Exact Ties
Inspect tied ranks and compare a small exact allocation test with its asymptotic reference.
By Riano · Published · Updated · Report a correction
Kruskal–Wallis Exact Ties
The Kruskal–Wallis test is a non-parametric method used for comparing two or more independent samples of equal or different sample sizes. It extends the Mann-Whitney U test, which is used for comparing only two groups. When observations have identical values, they receive averaged ranks. This tie-correction scales the final test statistic H, which affects the exact permutation test probability distribution.
Worked example: Tied Data with 3 Groups of Size 2
Let us consider the "Tied" dataset where six values are sorted and partitioned into three groups: [0, 1], [1, 2], [2, 3]. First, we pool the data and compute the average ranks:
| Value | Raw Ranks | Avg Rank |
|---|---|---|
| 0 | 1 | 1.0 |
| 1 | 2, 3 | 2.5 |
| 2 | 4, 5 | 4.5 |
| 3 | 6 | 6.0 |
Using the pooled ranks, the group sum of ranks are:
- Group 1 [0, 1] ranks: 1.0 + 2.5 = 3.5
- Group 2 [1, 2] ranks: 2.5 + 4.5 = 7.0
- Group 3 [2, 3] ranks: 4.5 + 6.0 = 10.5
The uncorrected Kruskal-Wallis statistic Hraw is calculated as: Hraw = [12 / (6 × 7)] × [ (3.5²)/2 + (7.0²)/2 + (10.5²)/2 ] - 3 × 7 = 3.5. Since there are two ties (values 1 and 2, each with multiplicity t=2), the tie correction C is: C = 1 - [ (2³ - 2) + (2³ - 2) ] / (6³ - 6) = 1 - 12 / 210 = 0.9428571428571428. The corrected statistic H is Hraw / C = 3.712121212121212.
Interactive Experiment
You can interact with the Kruskal–Wallis Exact Ties module to see how ties and group configurations affect the test statistic and exact permutation probabilities.
- Select the Data control and set it to Separated to observe the behavior when there are no ties (C = 1.0).
- Change the Data control to Tied to observe the tie correction take effect, increasing the final H and altering the permutation reference.
- Change the Group probe control to view the specific Selected mean rank for each individual group without changing the global H statistic.
Limitations and Assumptions
The asymptotic Chi-Square reference p-value (which uses df = k-1 = 2) is often highly inaccurate for very small sample sizes (such as groups of size 2 used here). In this small data regime, the exact permutation test over the 90 possible group allocations provides the mathematically correct p-value.
References
More information about the Kruskal-Wallis test and its implementation details, including tie corrections, can be found in the SciPy documentation: scipy.stats.kruskal.
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Kruskal–Wallis Exact Ties module.
Try Kruskal–Wallis Exact Ties on Riano →