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    Sample size to estimate an event rate

    How many participants, each followed for a stated time, are needed to estimate an event rate (events per unit of follow-up) within a chosen margin, for Poisson or over-dispersed negative binomial counts. Covers non-response.

    Engine version 1.0.0 · Reviewed 9 October 2026

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    https://statwisely.com/methods/precision-rate/

    When to use it

    • A cohort or registry estimating how often an event occurs, such as asthma attacks or falls per person-year.

    Not for

    • A yes/no outcome over the follow-up, such as whether anyone had an attack. Use a proportion.
    • Follow-up that differs between participants because of censoring or drop-out tied to the outcome; that needs a time-to-event or person-time model.

    Notation

    nn
    Number of participants
    λ\lambda
    Expected event rate per participant per unit of time
    tt
    Follow-up per participant, in the rate’s time unit
    kk
    Negative binomial dispersion: a count with mean μ has variance μ + kμ²; 0 is Poisson
    EE
    Margin of error, in events per participant per unit of time
    z1−α/2z_{1-\alpha/2}
    Standard normal quantile for confidence 1 − α (1.960 for 95%)
    qq
    Non-response rate

    Formulas

    Poisson or negative binomial counts

    n=⌈z1−α/22 λ (1+kλt)t E2⌉n = \left\lceil \frac{z_{1-\alpha/2}^{2}\, \lambda\,(1 + k\lambda t)}{t\,E^{2}} \right\rceil

    A participant followed for t has a count with mean μ = λt and variance μ(1 + kμ). The rate, total events over total person-time, then has variance λ(1 + kλt)/(n t) [1]. At k = 0 this is the Poisson variance λ/(n t), so following each participant twice as long halves n; with over-dispersion longer follow-up helps less, because the kλ part does not shrink with t.

    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.

    Margin of errorEE

    Values: Above 0, in the rate’s units

    0.2 with a rate of 2 attacks per person-year means the interval runs from about 1.8 to 2.2.

    Expected event rateλ\lambda

    Values: Above 0

    From earlier studies or routine data, in events per participant per unit of time.

    Follow-up per participantttOptional

    Values: Above 0; 1 by default

    In the same time unit as the rates: with rates per person-year, two years of follow-up is 2.

    Dispersion (negative binomial k)kkOptional

    Values: 0 or more; 0 for Poisson counts

    Clinical events, such as exacerbations or falls, usually vary between people more than a Poisson process allows. Take k from a negative binomial fit to earlier data, and try plausible values: the plan grows with k.

    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, and the event rate is constant over that time.
    • Counts are Poisson, or negative binomial with the stated dispersion; the interval is the large-sample Wald interval, which needs enough events to apply.
    • Non-response is unrelated to the outcome.

    Worked examples

    Two attacks per person-year

    A rate of 2 per person-year over one year, margin ±0.2, Poisson counts: 1.96² × 2 / 0.2² = 192.1, so n = 193.

    Sample size: 193

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    Counts that vary more between people

    With dispersion k = 0.5, each count varies 1 + 0.5 × 2 = 2 times as much as a Poisson count: 192.1 × 2 = 384.2, so n = 385.

    Sample size: 385

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    A rarer event followed for longer

    A rate of 0.5 a year, each participant followed 2 years, margin ±0.1: 1.96² × 0.5 / (2 × 0.1²) = 96.0, so n = 97.

    Sample size: 97

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    Allowing for loss to follow-up

    Expecting 10% to provide no follow-up: 193 / 0.9 = 214.4, so 215 are recruited.

    Needed for analysis: 193 · To approach: 215

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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 basic result agrees with the closed form: 1.96² × 2 / 0.2² = 192.1, rounded up to 193. In simulation, Poisson and negative binomial counts drawn at the planned sizes and analysed with Wald intervals on the empirical variance of the counts gave median half-widths and factors within 1% of the targets, and coverage of 94.4% to 95.2% for a 95% interval. An independent implementation in R recomputes these sizes and is compared with the engine’s.

    How to cite

    StatWisely. Sample size to estimate an event rate. Method document, engine version 1.0.0, reviewed 2026-10-09. https://statwisely.com/methods/precision-rate/

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

    References

    1. 1.Zhu H, Lakkis H. Sample size calculation for comparing two negative binomial rates. Statistics in Medicine. 2014;33(3):376–387.