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    Method document

    Sample size to estimate an odds ratio per standard deviation

    How many participants are needed to estimate how much the odds of a binary outcome, or of a higher category of an ordinal one, change per standard deviation of a numerical predictor, within a chosen factor. Uses logistic regression for two categories and the proportional-odds model for more. Allows for other covariates. Covers non-response.

    Engine version 1.0.0 · Reviewed 4 October 2026

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

    When to use it

    • Estimating the association of a measured predictor, such as age or a biomarker, with a yes/no or ordered outcome, when the effect with its interval rather than a test is the aim.
    • A case-control study of a numerical exposure: enter the share of cases in the sample as the outcome share.

    Not for

    • Showing that the association exists. Use logistic regression with a numerical predictor under “Detect an effect”.
    • A strongly skewed predictor, or one with few distinct values; the plan assumes a normal predictor.

    Notation

    nn
    Number of participants
    pjp_j
    Share of outcome category j for a participant at the predictor’s mean
    ff
    Precision factor: the 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(β)V(\beta)
    Per-participant variance of the 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⋅31−∑jpj3⋅11−RX∣Z2⌉n = \left\lceil \frac{z_{1-\alpha/2}^{2}}{(\ln f)^{2}} \cdot \frac{3}{1-\sum_j p_j^{3}} \cdot \frac{1}{1-R^{2}_{X|Z}} \right\rceil

    The variance of the log odds ratio per SD when the effect is small [1]. With two categories 1 − Σ pⱼ³ = 3p(1 − p), so this is Hsieh’s logistic form 1 / [p(1 − p)] [2]. The last factor inflates the sample for other covariates.

    At an expected odds ratio

    n=⌈z1−α/22 V(β)(ln⁡f)2 (1−RX∣Z2)⌉,V(β)=[I−1]ββn = \left\lceil \frac{z_{1-\alpha/2}^{2}\, V(\beta)}{(\ln f)^{2}\,(1-R^{2}_{X|Z})} \right\rceil, \qquad V(\beta) = \left[\mathcal{I}^{-1}\right]_{\beta\beta}

    A strong effect leaves less information, so the closed form understates the sample. Given the expected odds ratio, V(β) is taken from the model’s expected Fisher information, integrated over a standard-normal predictor by Gauss–Hermite quadrature; at no effect it equals the closed form.

    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

    1.5 means the interval runs from OR / 1.5 to OR × 1.5.

    Outcome category shares at the predictor’s meanp1,…,pcp_1, \dots, p_c

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

    For a binary outcome, the non-event and event shares, e.g. 0.8 and 0.2.

    Expected odds ratio per SDOR\mathrm{OR}Optional

    Values: Above 0; empty for a small effect

    An odds ratio and its reciprocal give the same plan.

    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.

    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 the log odds change linearly with it, by the same amount at every cut-point for an ordinal outcome.
    • Non-response is unrelated to the outcome.

    Worked examples

    A common outcome, small effect

    Half the participants with the outcome at the predictor’s mean, factor 1.5: 3.8415 / (0.25 × 0.1644) = 93.5, so n = 94.

    Sample size: 94

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    Planning at the expected effect

    Expecting an odds ratio of 1.5 per SD: n = 105, about 12% more than the closed form.

    Sample size: 105

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    An ordinal outcome

    Four categories shared 20/30/30/20%: 1 − Σ pⱼ³ = 0.93, so n = 3.8415 × 3.226 / 0.1644 = 75.4, rounded up to 76.

    Sample size: 76

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    Adjusting for related covariates

    If the other covariates explain 0.3 of the predictor’s variance: 93.5 / 0.7 = 133.5, so n = 134.

    Sample size: 134

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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.
    • At no effect the quadrature reproduces the closed form exactly. In R, glm and MASS::polr fitted to simulated samples at the planned sizes gave median interval factors of 1.51 against a target of 1.5, both at no effect and at an expected odds ratio of 1.5 per SD; without planning at the expected odds ratio, the factor was 1.55. An independent R integration of the logistic information reproduces the engine’s sizes.

    How to cite

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

    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.Hsieh FY, Bloch DA, Larsen MD. A simple method of sample size calculation for linear and logistic regression. Statistics in Medicine. 1998;17(14):1623–1634.