Skip to main content
    Sample Size CalculatorBeta

    Method document

    Sample size to estimate multinomial odds ratios per standard deviation

    How many participants are needed to estimate, for a nominal outcome of three or more categories, the odds ratio per standard deviation of a numerical predictor for each category against a reference category, from multinomial logistic regression, within a chosen factor. Allows for other covariates. Covers non-response.

    Engine version 1.0.0 · Reviewed 4 October 2026

    Open in calculator

    https://statwisely.com/methods/precision-multinomial-or-per-sd/

    When to use it

    • A nominal outcome, such as the choice among three treatments or a diagnosis among several, related to a measured predictor.

    Not for

    • Ordered categories. Use the odds ratio per SD, which uses the ordering.
    • Showing that any association exists. Use the multinomial logistic test under “Detect an effect”.

    Notation

    nn
    Number of participants
    p0, pjp_0,\ p_j
    Shares of the reference category and of category j at the predictor’s mean
    ff
    Precision factor: each interval runs from OR / f to OR × f
    RX∣Z2R^{2}_{X|Z}
    Share of the predictor’s variance explained by the other covariates
    V(βj)V(\beta_j)
    Per-participant variance of category j’s log odds ratio per SD
    z1−α/2z_{1-\alpha/2}
    Standard normal quantile for confidence 1 − α (1.960 for 95%)
    qq
    Non-response rate

    Formulas

    Small effects: a closed form

    n=⌈z1−α/22(ln⁡f)2⋅max⁡j ⁣(1pj+1p0)⋅11−RX∣Z2⌉n = \left\lceil \frac{z_{1-\alpha/2}^{2}}{(\ln f)^{2}} \cdot \max_j\!\left(\frac{1}{p_j} + \frac{1}{p_0}\right) \cdot \frac{1}{1-R^{2}_{X|Z}} \right\rceil

    With no effect the slopes of a multinomial logit have covariance W⁻¹, W = diag(p) − pp′ over the non-reference categories, whose diagonal is 1/pⱼ + 1/p₀ [1]. The category needing most participants sets the size. Simultaneous intervals split α across the C − 1 categories.

    At expected odds ratios

    V(βj)=[I−1]βjβj,I=E ⁣[zz⊤⊗W(x)]V(\beta_j) = \left[\mathcal{I}^{-1}\right]_{\beta_j\beta_j}, \qquad \mathcal{I} = E\!\left[z z^{\top} \otimes W(x)\right]

    Given the expected odds ratios, the multinomial-logit information is integrated over a standard-normal predictor by Gauss–Hermite quadrature, with z = (1, x).

    Non-response

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

    The number to approach 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

    The same factor for every category’s odds ratio.

    Category shares at the predictor’s mean, reference firstp0,…,pC−1p_0, \dots, p_{C-1}

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

    A small reference category makes every comparison imprecise; choose a common one.

    Expected odds ratios per SDORj\mathrm{OR}_jOptional

    Values: One per non-reference category, each above 0

    Leave empty to plan as if the effects were small.

    R² of the predictor on the other covariatesRX∣Z2R^{2}_{X|Z}Optional

    Values: At least 0 and below 1

    0 with no other covariates.

    Simultaneous intervals across categoriesα/(C−1)\alpha/(C-1)Optional

    Values: On or off

    Turn on when the categories will be read together.

    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

    • Participants are independent, the predictor is approximately normal, and each category’s log odds against the reference change linearly with it (independence of irrelevant alternatives).
    • Non-response is unrelated to the outcome.

    Worked examples

    Three categories, small effects

    Shares of 50%, 30% and 20%: the third category needs most, 1/0.2 + 1/0.5 = 7, so n = 3.8415 × 7 / (ln 1.5)² = 163.6, rounded up to 164.

    Sample size: 164

    Open this example in the calculator

    Planning at the expected effects

    Expecting odds ratios of 1.5 and 1.3 per SD: n = 174.

    Sample size: 174

    Open this example in the calculator

    Both intervals together

    Splitting α across the two comparisons: n = 214.

    Sample size: 214

    Open this example in the calculator

    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.
    • At no effect the quadrature reproduces 1/pⱼ + 1/p₀ exactly. In R, nnet::multinom fitted to simulated samples at the planned sizes gave a median factor of 1.515 for the widest interval against a target of 1.5, at no effect and at odds ratios of 1.5 and 1.3. An independent R integration of the information reproduces the engine’s sizes.

    How to cite

    StatWisely. Sample size to estimate multinomial odds ratios per standard deviation. Method document, engine version 1.0.0, reviewed 2026-10-04. https://statwisely.com/methods/precision-multinomial-or-per-sd/

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

    References

    1. 1.Agresti A. Categorical Data Analysis. 3rd ed. Hoboken, NJ: John Wiley & Sons; 2013.