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    Method document

    Sample size to estimate a rate ratio per standard deviation

    How many participants, each followed for a stated time, are needed to estimate how much an event rate changes per standard deviation of a numerical predictor, from Poisson or negative binomial regression, within a chosen factor. Allows for other covariates. Covers non-response.

    Engine version 1.0.0 · Reviewed 9 October 2026

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    https://statwisely.com/methods/precision-rate-ratio-per-sd/

    When to use it

    • A cohort reporting the incidence rate ratio for a measured predictor, such as a biomarker or age, when the ratio with its interval is the aim.

    Not for

    • A yes/no outcome. Use the risk ratio per SD.

    Notation

    nn
    Number of participants
    λ\lambda
    Event rate per participant per unit of time at the predictor’s mean
    tt
    Follow-up per participant, in the rate’s time unit
    kk
    Negative binomial dispersion: variance μ + kμ²; 0 is Poisson
    ff
    Precision factor: the interval runs from RR / f to RR × 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 rate 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⋅(1λt+k)⋅11−RX∣Z2⌉n = \left\lceil \frac{z_{1-\alpha/2}^{2}}{(\ln f)^{2}} \cdot \left(\frac{1}{\lambda t} + k\right) \cdot \frac{1}{1-R^{2}_{X|Z}} \right\rceil

    With no effect, the slope of a standard-normal predictor in a count regression has variance 1/(λt) + k per participant: one over the expected events, plus the dispersion [1, 2].

    At an expected rate ratio

    V(β)=[A−1]ββ,A=E ⁣[μ1+kμ zz⊤],μ=λt RRxV(\beta) = \left[A^{-1}\right]_{\beta\beta}, \quad A = E\!\left[\frac{\mu}{1 + k\mu}\, z z^{\top}\right], \quad \mu = \lambda t\,\mathrm{RR}^{x}

    Given the expected rate ratio, the model information is integrated over the predictor by Gauss–Hermite quadrature, with z = (1, x). A strong effect raises the events above the mean, so the plan can be smaller than 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

    1.25 means the interval runs from RR / 1.25 to RR × 1.25.

    Event rate at the predictor’s meanλ\lambda

    Values: Above 0

    Rare events need many more participants, or longer follow-up: the variance grows as 1/(λt).

    Follow-up per participantttOptional

    Values: Above 0; 1 by default

    In the rate’s time unit.

    Dispersion (negative binomial k)kkOptional

    Values: 0 or more; 0 for Poisson counts

    Take k from earlier data and try plausible values; the plan grows with k.

    Expected rate ratio per SDRR\mathrm{RR}Optional

    Values: Above 0; empty for a small effect

    Sent to the engine under the shared name odds_ratio.

    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, each followed for the same time t; the predictor is approximately normal, and the log rate changes linearly with it.
    • Non-response is unrelated to the outcome.

    Worked examples

    Poisson counts, small effect

    A rate of 2 at the mean over one unit of follow-up, factor 1.25: 3.8415 × 0.5 / (ln 1.25)² = 38.6, so n = 39.

    Sample size: 39

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    Over-dispersed counts

    With dispersion k = 0.5 the variance doubles to 0.5 + 0.5 = 1: n = 78.

    Sample size: 78

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    Planning at the expected effect

    Expecting a rate ratio of 1.3 per SD with k = 0.5, the information is taken at that effect: n = 79.

    Sample size: 79

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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 1/(λt) + k exactly. Poisson and negative binomial regressions fitted to simulated samples at the planned sizes gave median interval factors within 0.6% of the target and coverage of 95% to 96%. An independent implementation in R recomputes these sizes and is compared with the engine’s.

    How to cite

    StatWisely. Sample size to estimate a rate ratio per standard deviation. Method document, engine version 1.0.0, reviewed 2026-10-09. https://statwisely.com/methods/precision-rate-ratio-per-sd/

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

    References

    1. 1.Signorini DF. Sample size for Poisson regression. Biometrika. 1991;78(2):446–450. doi:10.1093/biomet/78.2.446
    2. 2.Zhu H, Lakkis H. Sample size calculation for comparing two negative binomial rates. Statistics in Medicine. 2014;33(3):376–387.