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    Method document

    Sample size for multinomial logistic regression with a numerical predictor

    How many participants are needed to show that a numerical predictor is associated with a nominal outcome of three or more categories, from the category shares at the predictor’s mean and the odds ratio per standard deviation expected for each category against a reference. Allows for other covariates. Covers dropout.

    Engine version 1.0.0 · Reviewed 4 October 2026

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    https://statwisely.com/methods/regression-multinomial/

    When to use it

    • Testing whether a measured predictor shifts the distribution of an unordered outcome, such as treatment choice or diagnosis.

    Not for

    • Ordered categories. Use logistic regression with a numerical predictor, or a proportional-odds model.
    • A grouping predictor. Use the chi-square test of independence.

    Notation

    NN
    Number of participants
    CC
    Number of outcome categories
    β\beta
    Log odds ratios per SD of the C − 1 categories against the reference
    VV
    Per-participant covariance of the slopes at the expected odds ratios
    RX∣Z2R^{2}_{X|Z}
    Share of the predictor’s variance explained by the other covariates
    α\alpha
    Significance level: the chance of a false positive
    1−β1-\beta
    Power: the chance of detecting the effect if it is real
    qq
    Expected dropout

    Formulas

    Noncentral chi-square power

    Pr⁡ ⁣(χC−1,λ′2>χC−1, 1−α2)≥1−β,λ=N β⊤V−1β (1−RX∣Z2)\Pr\!\left(\chi'^{2}_{C-1,\lambda} > \chi^{2}_{C-1,\,1-\alpha}\right) \ge 1-\beta, \qquad \lambda = N\,\beta^{\top} V^{-1} \beta\,(1 - R^{2}_{X|Z})

    The likelihood-ratio test of all C − 1 slopes being zero has, in large samples, the noncentrality of the Wald statistic [1]. V comes from the multinomial-logit information integrated over a standard-normal predictor by Gauss–Hermite quadrature.

    Dropout

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

    Applied last, to the participants actually recruited.

    Parameters

    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

    Routine data or a previous study.

    Expected odds ratios per SDORj\mathrm{OR}_j

    Values: One per non-reference category, above 0, not all 1

    The smallest pattern of effects worth detecting.

    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.

    Predictor tested11

    Values: One numerical predictor

    Other covariates enter only through R²_X|Z.

    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.

    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, the predictor is approximately normal, and each category’s log odds against the reference change linearly with it.
    • The Wald noncentrality is slightly conservative for the likelihood-ratio test.
    • Dropout is unrelated to the outcome; recruiting more participants restores the numbers but cannot remove bias.

    Worked examples

    Three categories

    Shares of 50%, 30% and 20% with odds ratios of 1.5 and 1.3 per SD, α = 0.05, 80% power: N = 322.

    Sample size: 322

    Protocol statement

    To detect odds ratios of 1.5, 1.3 per standard deviation with 80% power at a significance level of 0.05, using a likelihood-ratio test in a multinomial logistic regression of a 3-category outcome on a numerical predictor, the required sample size is 322 participants. Calculated with StatWisely (statwisely.com) using: N: noncentral χ² power on C-1 df, λ=N·β'V⁻¹β·(1-R²_X|Z). These are planning estimates; confirm the final design with a qualified statistician.

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    Four categories, one unaffected

    Shares of 40/30/20/10% with odds ratios of 1.3, 1.6 and 1.0: N = 341 on 3 degrees of freedom.

    Sample size: 341

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    Adjusting for related covariates

    If the other covariates explain 0.2 of the predictor’s variance: N = 402.

    Sample size: 402

    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.
    • 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 nnet::multinom likelihood-ratio tests in R at the planned sizes gave power of 0.83 for both the three- and four-category examples, against a target of 0.80: the Wald noncentrality is slightly conservative. An independent R integration of the information reproduces the engine’s sizes.

    How to cite

    StatWisely. Sample size for multinomial logistic regression with a numerical predictor. Method document, engine version 1.0.0, reviewed 2026-10-04. https://statwisely.com/methods/regression-multinomial/

    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.