Jonckheere's k-sample test
- Compares mean ranks across multiple (ordered) groups using ranked data rather than raw values.
- Useful when data violate parametric assumptions such as normality or homogeneity of variance.
- Requires a predefined order of samples and is generally less powerful than parametric alternatives, especially with small sample sizes.
Definition
Section titled “Definition”Jonckheere’s k-sample test is a non-parametric statistical test used to compare the mean ranks of multiple samples. It is often used when the data does not meet the assumptions of parametric tests, such as normality or homogeneity of variance.
Explanation
Section titled “Explanation”- Observations across all groups are ranked, and each sample’s mean rank is computed (the sum of ranks for that sample divided by its number of observations).
- The test statistic J is computed as the sum of the products of each sample’s mean rank and its sample size. The J statistic is then compared to a critical value from a table for Jonckheere’s test based on the number of samples and the chosen significance level. If J is greater than or equal to the critical value, the difference in mean ranks across samples is considered statistically significant.
- The test assumes the samples are in a specific, meaningful order; changing the order requires re-running the test. Jonckheere’s test is less powerful than parametric tests, so it may fail to detect differences when sample sizes are small.
Examples
Section titled “Examples”Treatment effectiveness (pain) — conceptual example
Section titled “Treatment effectiveness (pain) — conceptual example”A study comparing three treatments (A, B, and C) for chronic back pain would rank patient pain scores within each treatment group from lowest to highest, then use Jonckheere’s test to assess whether the mean ranks differ across the three groups.
School performance — conceptual example
Section titled “School performance — conceptual example”A school district comparing mean math exam scores for three schools would rank student scores within each school and apply Jonckheere’s test to determine whether the mean ranks differ among the schools.
Numerical example — mean rank
Section titled “Numerical example — mean rank”If a sample has 10 observations with ranks 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20, the mean rank is:
Numerical example — J statistic
Section titled “Numerical example — J statistic”If the mean ranks for three groups are 11, 12, and 13, each with 10 observations, the J statistic is:
Use cases
Section titled “Use cases”- When data do not meet parametric-test assumptions (for example, violations of normality or homogeneity of variance).
- When the investigator has an a priori ordering of the samples and wishes to test for ordered differences in mean ranks.
Notes or pitfalls
Section titled “Notes or pitfalls”- The samples must be in a specific order for the test to be valid; changing the order requires re-conducting the test (e.g., treatments must be in the order A, B, C).
- Jonckheere’s test is less powerful than parametric alternatives and may not detect significant differences with small sample sizes.
- The decision rule uses a table of critical values specific to Jonckheere’s test for the number of samples and chosen significance level.
Related terms
Section titled “Related terms”- Parametric tests
- Normality
- Homogeneity of variance
- Critical values (Jonckheere’s test)