Is it the same as Kruskal-Wallis?

No; it uses a specified ordering.

Can ties be included?

Yes, with appropriate scoring and inference.

AI Glossary / Statistics and probability / Advanced

Jonckheere's trend test

Definition

Jonckheere's trend test, also called the Jonckheere-Terpstra test, is a nonparametric test for an ordered alternative across independent groups. It assesses whether responses tend to increase or decrease along a group ordering specified in advance. Its statistic combines comparisons between observations from different groups, rewarding comparisons that follow the anticipated direction.

What it means

Suppose three training levels have a meaningful order: introductory, intermediate, and advanced. A general group-comparison test asks whether the groups differ. Jonckheere's test instead uses the specific expectation that responses tend to increase across that order.
The test is about ordered distributional differences. Interpreting it solely as a comparison of means or medians requires additional assumptions about how distributions differ. A significant result does not prove that every adjacent pair differs significantly or that every person in a higher group outperforms every person in a lower group.

How it works

Specify group order and test direction before examining the outcomes. For every pair of groups in that order, compare every lower-group observation with every higher-group observation. Add one when the higher-group value is larger, one half for a tie, and zero when it is smaller.
Compare the resulting statistic with its null distribution under identical group distributions and an appropriate exchangeability assumption. Exact enumeration is possible for small examples; permutation simulation or a suitable large-sample approximation can be used in other settings.
Repeated observations from the same participant cannot simply be treated as independent groups. Ties and sampling structure also affect the reference calculation, so verify how the chosen implementation handles them.

Three practical examples

An AI training study compares independently sampled participants across three pre-specified instruction levels for an anticipated increase in assessment scores.
An experiment compares independent devices under ordered processing loads when a directional response pattern is scientifically justified.
A researcher compares independent observations across ordered categories without assuming normally distributed responses, while preserving the original group order.

Think of it this way

Imagine comparing everyone in an earlier queue with everyone in a later queue and counting how often the later person has a higher score. Then combine those comparisons across all ordered queue pairs. The method uses the direction of the queues, not just the existence of several groups.

Worked example

Consider an original six-observation example with three groups of size two. The pre-specified increasing order is A, B, C. Their observed scores are A: 1 and 2; B: 3 and 5; C: 4 and 6.
There are four comparisons between A and B, four between A and C, and four between B and C, for twelve comparisons altogether. Every A value is smaller than every B value, contributing four. Every A value is also smaller than every C value, contributing another four.
Between B and C, score three is below both four and six, contributing two. Score five is below six but above four, contributing one. This group pair contributes three, so the total statistic J is four plus four plus three = eleven.
The maximum possible J is twelve. A p-value requires a reference distribution, not just a large statistic.
Under the identical-distribution null with exchangeable observations, assign the six distinct scores to the three labeled groups while keeping two observations per group. There are six choose two ways to fill A and four choose two ways to fill B; the remaining two fill C. That gives fifteen times six = ninety equally likely allocations.
Enumerating all ninety allocations produces one with J = twelve and two with J = eleven. Therefore, the exact one-sided probability of a statistic at least as large as observed is three divided by ninety = one thirtieth, approximately 0.0333.
The word at least matters: allocations tied with the observed statistic belong in this tail. Counting only J greater than eleven would produce one ninetieth and would not be the ordinary exact upper-tail calculation described here.
The null distribution is symmetric around six. Its counts for J from zero through twelve are 1, 2, 5, 7, 11, 12, 14, 12, 11, 7, 5, 2, and 1. These counts sum to ninety and verify the probability.
Reversing the anticipated group order changes the no-tie statistic from eleven to one. Choosing whichever direction looks better after seeing the data would invalidate the original one-sided interpretation. The direction is part of the planned question.
For a separate tie illustration, use A: 1 and 2; B: 2 and 4; C: 3 and 5. The A-versus-B contribution is 3.5 because one comparison is tied. A versus C contributes four, while B versus C contributes three. The total is 10.5.
Do not reuse the ninety-allocation tail counts from the distinct-score example for this tied dataset. The statistic's reference distribution has changed. An appropriate calculation must retain the observed tie structure and the intended allocation scheme.
Finally, replacing every original score by ten times that score plus seven preserves all comparisons and leaves J unchanged. The test uses ordering information, so it does not estimate the size of an increase in the original measurement units.

Illustrative case study

Fictional company Alder Training plans an ordered comparison of three AI instruction levels. Its analyst records the increasing alternative before collecting independent assessment results and reports the exact small-sample calculation alongside the group data.
The team explains that the result supports the planned trend question but does not establish the magnitude of improvement or prove every pairwise difference. This is a fictional analysis, not evidence that a particular training program works.

Common misconceptions and mistakes

Do not sort groups by their observed averages and then claim a pre-specified trend test.
Nonparametric does not mean assumption-free.
A small p-value is not the probability that the null hypothesis is true.
The pooled comparison statistic is not a regression slope.
A tied sample requires an appropriate reference calculation rather than a copied no-tie formula.

Business use

Use this test when ordered independent groups and a directional hypothesis match the decision. Present the group distributions and practical differences alongside the test result. Record ordering, tail choice, tie handling, and whether the p-value is exact, simulated, or approximate so the analysis can be reproduced.

Check your understanding

What is the original statistic?

Answer: Eleven.

How many allocations were enumerated?

Answer: Ninety.

What is the exact increasing-tail p-value?

Answer: One thirtieth.

Related terms

Rank test
Permutation test
Ordered alternative
Mann-Whitney statistic
Kruskal-Wallis test
exchangeability
statistical power

Learn next

Compare ordered and unordered alternatives, then study tie-aware permutation inference and ways to report meaningful effect sizes.

Sources

Venkatraman Seshan and Karissa Whiting, clinfun Jonckheere-Terpstra documentation: https://veseshan.github.io/clinfun/reference/jonckheere.test.html . NIST Dataplot, Jonckheere-Terpstra Test: https://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/jonck.htm . Datasets, exact enumeration, and company scenario are original illustrations.

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