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    Sample size for a Jonckheere–Terpstra trend test of an ordinal outcome

    How many participants are needed to show that an ordinal outcome rises or falls across three or more ordered groups, such as doses, from the expected probability of each category in each group. Allows for ties and unequal groups. Covers dropout.

    Engine version 1.0.0 · Reviewed 4 October 2026

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    https://statwisely.com/methods/nonparametric-jonckheere-terpstra/

    When to use it

    • An ordered outcome, such as a severity grade, compared across ordered groups, when the question is a trend.

    Not for

    • Unordered groups, or any difference rather than a trend. Use the Kruskal–Wallis test.
    • A binary outcome. Use the Cochran–Armitage test for trend.

    Notation

    NN
    Total number of participants
    nin_i
    Participants in ordered group i
    θij\theta_{ij}
    Probability that an observation in later group j exceeds one in earlier group i, ties counting half
    tct_c
    Expected number of participants in outcome category c
    JJ
    The Jonckheere–Terpstra statistic
    α\alpha
    Significance level: the chance of a false positive
    1−β1-\beta
    Power: the chance of detecting the effect if it is real
    τ\tau
    Number of tails: 2 for a two-sided test, 1 for one-sided
    qq
    Expected dropout

    Formulas

    How far the trend moves the statistic

    E(J)−E0(J)=∑i<jninj(θij−12)E(J) - E_0(J) = \sum_{i<j} n_i n_j \left(\theta_{ij} - \tfrac{1}{2}\right)

    J counts, over every pair of groups, how often the later group’s observation is the higher [1].

    Power from the tie-corrected null variance

    1−β=Φ ⁣(∣E(J)−E0(J)∣Var0(J)−z1−α/s)1-\beta = \Phi\!\left(\frac{|E(J) - E_0(J)|}{\sqrt{\mathrm{Var}_0(J)}} - z_{1-\alpha/s}\right)

    Var₀(J) is Hollander and Wolfe’s null variance corrected for ties, evaluated at the expected category counts t_c [2]. With two-sided tests the opposite tail is added. The first group gets the smallest whole number reaching the power, and every other group that number times its relative size.

    Dropout

    nrecruit=⌈n1−q⌉n_{\text{recruit}} = \left\lceil \frac{n}{1 - q} \right\rceil

    Applied last, to the participants actually recruited.

    Parameters

    Expected category probabilities in each ordered groupπic\pi_{ic}

    Values: At least three groups in trend order, with the same ordered categories; each adds to 1

    The smallest trend worth detecting, as the share of each group in each category.

    Relative group sizesni/n1n_i / n_1Optional

    Values: One per group, each above 0; equal if omitted

    How many participants each group gets relative to the others.

    Significance levelα\alpha

    Values: Between 0 and 1; usually 0.05

    The false-positive rate you accept. A smaller α, for example to allow for several primary outcomes, needs a larger sample.

    Power1−β1-\beta

    Values: Between 0 and 1; usually 0.80 or 0.90

    The chance of a significant result if the assumed effect is real. Funders and ethics committees commonly expect at least 0.80.

    One- or two-sidedτ\tau

    Values: 2 (two-sided) or 1 (one-sided)

    Two-sided is the default. A one-sided test is justified only when an effect in the other direction would be treated exactly as no effect, and should be stated in advance.

    Expected dropoutqqOptional

    Values: 0% up to, but not including, 100%

    The share of participants expected to provide no outcome, from comparable studies. Recruiting more restores the numbers, not the comparability of the groups: if dropout is related to the outcome, the estimate can still be biased.

    Assumptions

    • Participants are independent, each in exactly one group, and both the groups and the outcome categories are ordered.
    • The null variance is used throughout, as in Noether’s formulas [3].
    • Dropout is unrelated to the outcome; recruiting more participants restores the numbers but cannot remove bias.

    Worked examples

    Three ordered groups, four categories

    The same groups as the Kruskal–Wallis example, tested for a trend: 135 per group, N = 405, against 164 per group for Kruskal–Wallis.

    135 per group · Total: 405

    Protocol statement

    To detect the expected distributions of the outcome categories in each group with 80% power at a two-sided significance level of 0.05, using a Jonckheere–Terpstra trend test of an ordinal outcome across 3 ordered groups, the required sample size is 135 per group (405 in total, across 3 groups). Calculated with StatWisely (statwisely.com) using: N: normal power of J=Σᵢ<ⱼUᵢⱼ, E(J)-E₀(J)=Σnᵢnⱼ(θᵢⱼ-½), tie-corrected Var₀(J). These are planning estimates; confirm the final design with a qualified statistician.

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    Only an increasing trend matters

    One-sided: 107 per group, N = 321.

    107 per group · Total: 321

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    Twice as many in the first group

    Allocating 2 : 1 : 1: 208 in the first group and 104 in each of the others.

    group1: 208 · group2: 104 · group3: 104 · Total: 416

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    Validation

    • Every worked example on this page is recomputed by the StatWisely engine whenever the software is built. If a number or formula here stopped matching the calculator, the release would be stopped until this document was corrected.
    • An independent implementation in R recomputes this test every night and before each production release, and the results are compared with the engine’s.
    • Simulating the tie-corrected Jonckheere–Terpstra test in R, 4,000 trials at each planned size gave power of 0.816 (two-sided), 0.807 (one-sided), 0.808 (2 : 1 : 1) and 0.817 (two categories), against a target of 0.80, and a type I error of 0.049 when the groups did not differ.

    How to cite

    StatWisely. Sample size for a Jonckheere–Terpstra trend test of an ordinal outcome. Method document, engine version 1.0.0, reviewed 2026-10-04. https://statwisely.com/methods/nonparametric-jonckheere-terpstra/

    Cite the original references for the formulas your design uses as well.

    References

    1. 1.Jonckheere AR. A distribution-free k-sample test against ordered alternatives. Biometrika. 1954;41(1–2):133–145. doi:10.1093/biomet/41.1-2.133
    2. 2.Hollander M, Wolfe DA. Nonparametric Statistical Methods. 2nd ed. New York: John Wiley & Sons; 1999.
    3. 3.Noether GE. Sample size determination for some common nonparametric tests. Journal of the American Statistical Association. 1987;82(398):645–647.