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    Method document

    Sample size to estimate survival at a fixed time

    How many participants are needed to estimate the share event-free at a prespecified time, such as two-year survival, with a Kaplan–Meier interval of a chosen margin, allowing for losses to follow-up. Covers non-response.

    Engine version 1.0.0 · Reviewed 10 October 2026

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    https://statwisely.com/methods/precision-survival-at-time/

    When to use it

    • A cohort or single-arm study reporting survival at a landmark time, such as one-year survival after surgery.

    Not for

    • Comparing two groups. Use a hazard ratio or the log-rank test.

    Notation

    nn
    Number of participants
    SS
    Expected survival at time t
    tt
    Time at which survival is estimated; everyone is followed to it unless lost
    λ\lambda
    Event hazard, −ln S / t
    η\eta
    Loss-to-follow-up hazard
    EE
    Margin of error, in proportion units
    z1−α/2z_{1-\alpha/2}
    Standard normal quantile for confidence 1 − α (1.960 for 95%)
    qq
    Non-response rate

    Formulas

    Greenwood’s variance with exponential survival

    n=⌈z1−α/22 S2λ(e(λ+η)t−1)(λ+η) E2⌉n = \left\lceil \frac{z_{1-\alpha/2}^{2}\, S^{2} \lambda \left(e^{(\lambda + \eta)t} - 1\right)}{(\lambda + \eta)\, E^{2}} \right\rceil

    Greenwood’s variance of the Kaplan–Meier estimate, S² ∫ λ / (S(u) G(u)) du / n, with exponential survival and losses G(u) = e^(−ηu) [1, 2]. With no losses it is S(1 − S)/n, a binomial proportion, since everyone is followed to 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 and below 0.5

    0.05 means the interval runs 5 percentage points either side of the estimate.

    Expected survival at the timeSS

    Values: Between 0 and 1

    Survival nearest 0.5 needs the most participants.

    Time at which survival is estimatedttOptional

    Values: Above 0; 1 by default

    Use one time unit for every field, such as years.

    Loss-to-follow-up rateη\etaOptional

    Values: 0 or more, per unit of time

    Participants lost before the time are censored, which leaves fewer events and widens the interval.

    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

    • Survival is exponential up to t, and losses are unrelated to the outcome.
    • Non-response is unrelated to the outcome.

    Worked examples

    One-year survival of 70%

    0.7 at one year, margin ±0.05, no losses: 1.96² × 0.7 × 0.3 / 0.05² = 322.7, rounded up to 323.

    Sample size: 323

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    Two-year survival with losses

    0.7 at 2 years with losses at 0.1 a year: the variance grows from 0.210 to 0.234, so n = 360.

    Sample size: 360

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    Allowing for non-response

    Expecting 10% not to take part: 323 / 0.9 = 358.9, so 359 are approached.

    Needed for analysis: 323 · To approach: 359

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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.
    • With no losses the result is the binomial proportion’s. Kaplan–Meier estimates with Greenwood intervals, from simulated samples at the planned sizes, gave median half-widths of 0.0499 to 0.0500 against 0.05 and coverage of 94.5% to 95.0%. An independent implementation in R recomputes these sizes and is compared with the engine’s.

    How to cite

    StatWisely. Sample size to estimate survival at a fixed time. Method document, engine version 1.0.0, reviewed 2026-10-10. https://statwisely.com/methods/precision-survival-at-time/

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

    References

    1. 1.Greenwood M. The natural duration of cancer. Reports on Public Health and Medical Subjects. London: HMSO; 1926;33:1–26.
    2. 2.Collett D. Modelling Survival Data in Medical Research. 3rd ed. Boca Raton: CRC Press; 2015.