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    Sample size to estimate a proportional-odds odds ratio

    How many participants are needed in each of two groups to estimate the common odds ratio of an ordinal outcome, from a proportional-odds (ordinal logistic) model, within a chosen factor. Planned from the outcome’s expected distribution across its categories. Covers unequal groups, finite populations and non-response.

    Engine version 1.0.0 · Reviewed 4 October 2026

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    https://statwisely.com/methods/precision-ordinal-odds-ratio/

    When to use it

    • Two groups or interventions compared on an ordered outcome, such as a severity grade, a functional scale or a Likert response, when the aim is to estimate the effect with a stated precision.

    Not for

    • Showing that the groups differ. Use the Mann–Whitney calculation under “Detect an effect”.
    • Odds ratios that clearly differ across cut-points, where the proportional-odds assumption fails, or an odds ratio adjusted for other variables.
    • Cluster sampling, which this calculation does not cover.

    Notation

    n1, n2n_1,\ n_2
    Sample sizes of groups 1 and 2
    pˉj\bar{p}_j
    Expected share of category j, both groups together
    rar_a
    Allocation ratio n₂ / n₁
    ff
    Precision factor: the interval runs from OR / f to OR × f
    gg
    Number of groups, including the reference group
    z1−α/2z_{1-\alpha/2}
    Standard normal quantile for confidence 1 − α (1.960 for 95%)
    N1, N2N_1,\ N_2
    Population sizes of the two groups
    qq
    Non-response rate

    Formulas

    Whitehead’s variance of the log odds ratio

    n1=⌈z1−α/22⋅3(1+ra)ra(1−∑jpˉj3)(ln⁡f)2⌉,n2=⌈ra n1⌉n_1 = \left\lceil \frac{z_{1-\alpha/2}^{2}\cdot\dfrac{3(1+r_a)}{r_a\left(1-\sum_j \bar{p}_j^{3}\right)}}{(\ln f)^{2}} \right\rceil, \qquad n_2 = \lceil r_a\, n_1 \rceil

    The log odds ratio of a proportional-odds model has large-sample variance 3 / [N w(1 − w)(1 − Σ p̄ⱼ³)], with w the share in group 1 [1]. It depends on the outcome only through its category shares: the more evenly the outcome spreads over more categories, the smaller 1 − Σ p̄ⱼ³ leaves the variance, and the fewer participants are needed. With two categories it is the binary odds ratio’s variance.

    Several groups against a reference

    n1=max⁡j n1(j),ntotal=(g−1) n1+n2,α  ←  α/(g−1) (simultaneous)n_1 = \max_{j}\, n_1^{(j)}, \qquad n_{\text{total}} = (g-1)\,n_1 + n_2, \qquad \alpha \;\leftarrow\; \alpha / (g-1) \ \text{(simultaneous)}

    Optional. With g groups, each of the g − 1 others is compared with the same reference group, giving g − 1 odds ratios. Each comparison has the two-group requirement n₁⁽ʲ⁾; every comparison group is given the largest, so all intervals reach the factor, and the reference group is n₂ = r_a n₁. Choosing simultaneous intervals splits α across the g − 1 comparisons (Bonferroni). A larger reference group serves every comparison at once. Cluster sampling is not offered with several groups, and a population size applies to each group.

    Finite population correction

    ni′=⌈ni1+(ni−1)/Ni⌉,i=1,2n_i' = \left\lceil \frac{n_i}{1 + (n_i - 1)/N_i} \right\rceil, \qquad i = 1, 2

    Applied to each group against its own population when population sizes are given [3]. It matters only when a group is a sizeable share of its population, and is not offered with cluster sampling.

    Non-response

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

    The number to approach in each group so that n respond. Applied last, after any clustering.

    Parameters

    Confidence level1−α1-\alpha

    Values: 90%, 95% or 99%

    95% is the convention in health research. A higher level widens the interval, so the same margin needs a larger sample.

    Precision factorff

    Values: Above 1; 2 means the interval runs from OR / 2 to OR × 2

    A ratio is bounded multiplicatively, so its precision is a factor rather than a difference. Smaller factors need much larger samples.

    Expected share of each ordered categorypˉ1,…,pˉc\bar{p}_1, \dots, \bar{p}_c

    Values: At least two, each above 0, adding to 1

    The outcome’s expected distribution with both groups together, from a previous study or routine data. If only the control group is known, its distribution is a close substitute when the odds ratio is modest.

    Allocation ratiora=n2/n1r_a = n_2/n_1Optional

    Values: Above 0; 1 for equal groups

    The planned size of group 2 divided by group 1. Equal groups give the most precision for a given total; use another ratio when one group is easier or cheaper to recruit.

    Number of groups, including the referenceggOptional

    Values: A whole number of at least 2; 2 for one comparison

    For example, three exposure levels compared with no exposure is g = 4.

    Simultaneous intervals across comparisonsα/(g−1)\alpha / (g-1)Optional

    Values: On or off

    Turn on when the comparisons will be read together, so that every interval holds at once.

    Population sizesN1, N2N_1,\ N_2Optional

    Values: Whole numbers of at least 2; leave the second empty if both groups come from populations of the same size

    Leave empty for large populations. They matter only when a group would be more than about 5% of its population. Not available with cluster sampling.

    Expected non-response rateqqOptional

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

    The share of people approached who will not take part; use the response rate of a comparable study. Approaching more people restores the number of respondents, not their representativeness: if non-responders differ, the estimate can still be biased.

    Assumptions

    • The two groups are independent, unmatched samples, and participants respond independently.
    • Proportional odds: one odds ratio describes the groups at every cut-point between categories. Whitehead’s variance is exact as the odds ratio approaches 1 and a close approximation for moderate odds ratios.
    • Non-response is unrelated to the outcome.

    Worked examples

    Four categories, within a factor of 2

    Shares of 20%, 30%, 30% and 20%: 1 − Σ p̄ⱼ³ = 1 − 0.07 = 0.93, so n₁ = 1.96² × 6 / 0.93 / (ln 2)² = 51.6, rounded up to 52 in each group.

    Group 1: 52 · Group 2: 52 · Total: 104

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    Five categories, within a factor of 1.5

    Shares of 10%, 20%, 40%, 20% and 10% with factor 1.5: 153 in each group.

    Group 1: 153 · Group 2: 153 · Total: 306

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    Two categories: the binary odds ratio

    Shares of 50% and 50% give 64 per group, the same as the odds-ratio calculator with both proportions at 0.5.

    Group 1: 64 · Group 2: 64 · Total: 128

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    Twice as many in group 2

    Four equal categories and r_a = 2: 39 in group 1 and 78 in group 2.

    Group 1: 39 · Group 2: 78 · Total: 117

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    Three treatments against a control

    The basic example’s categories with 4 groups, each of three compared with the control and read together: α split three ways gives z = 2.394, so n = 2.394² × 6 / 0.93 / (ln 2)² = 77.0, rounded up to 77 per group and 308 in all.

    77 per group · Total: 308

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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.
    • The R package Hmisc implements Whitehead’s method in posamsize; at power 0.5 its formula is this one, and it gives totals of 103.2, 305.4 and 115.1 for the first, second and fourth examples, which StatWisely rounds up per group to 104, 306 and 117 [2]. Fitting MASS::polr to 2,000 simulated trials at each planned size gave a median interval factor of 2.02 against a target of 2, and 1.504 against 1.5.

    How to cite

    StatWisely. Sample size to estimate a proportional-odds odds ratio. Method document, engine version 1.0.0, reviewed 2026-10-04. https://statwisely.com/methods/precision-ordinal-odds-ratio/

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

    References

    1. 1.Whitehead J. Sample size calculations for ordered categorical data. Statistics in Medicine. 1993;12(24):2257–2271. doi:10.1002/sim.4780122404
    2. 2.Harrell FE Jr. Hmisc: Harrell Miscellaneous. R package version 5.3-0, function posamsize. Comprehensive R Archive Network.
    3. 3.Cochran WG. Sampling Techniques. 3rd ed. New York: John Wiley & Sons; 1977.