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    Sample size for a group-sequential trial: O’Brien–Fleming

    How many participants a trial needs at most when it will be analysed at several pre-planned looks and may stop early for benefit, with O’Brien–Fleming boundaries. Starts from means, proportions or any fixed-sample size. Covers dropout.

    Engine version 1.0.0 · Reviewed 1 October 2026

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    https://statwisely.com/methods/sequential-obrien-fleming/

    When to use it

    • A trial with interim analyses by a data monitoring committee, where stopping early for clear benefit is wanted at little cost to the final analysis.

    Not for

    • Designs that may also stop for futility, or change their sample size at an interim look.

    Notation

    KK
    Number of analyses (looks), equally spaced in information
    nfixedn_{\text{fixed}}
    Per-group size of the same trial without interim analyses
    RR
    Inflation factor: maximum size divided by fixed size
    ckc_k
    Critical value at look k
    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

    The fixed-sample size

    nfixed=2σ2(z1−α/2+z1−β)2Δ2or[p1(1−p1)+p2(1−p2)](z1−α/2+z1−β)2(p1−p2)2n_{\text{fixed}} = \frac{2\sigma^{2}(z_{1-\alpha/2} + z_{1-\beta})^{2}}{\Delta^{2}} \quad\text{or}\quad \frac{\left[p_1(1-p_1) + p_2(1-p_2)\right](z_{1-\alpha/2} + z_{1-\beta})^{2}}{(p_1 - p_2)^{2}}

    For means or proportions, by the normal approximation; or entered directly from another calculation.

    O’Brien–Fleming’s boundary

    ck=cOFK/kc_k = c_{OF}\sqrt{K / k}

    Early looks demand overwhelming evidence and the last look is close to the usual 1.96, so the maximum size is barely above the fixed size. The usual choice for that reason [1].

    Maximum size

    nmax⁡=⌈R⋅nfixed⌉n_{\max} = \lceil R \cdot n_{\text{fixed}} \rceil

    The boundaries and the probability of stopping at each look are computed by the Armitage–McPherson–Rowe recursion, and R is the information that makes the overall power 1 − β [2, 3]. The expected size, reported alongside, is what the trial uses on average if the effect is real.

    Dropout

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

    Applied last, to the participants actually recruited.

    Parameters

    Number of analysesKK

    Values: A whole number of at least 1, including the final analysis

    Usually 2 to 5. Each extra look costs a little more at the maximum.

    Boundary shapeckc_k

    Values: O’Brien–Fleming or Pocock

    O’Brien–Fleming unless early stopping is the priority.

    Difference in meansΔ\DeltaOptional

    Values: Not zero

    With the SD, to base the design on means.

    Standard deviationσ\sigmaOptional

    Values: Above 0

    Common to both groups.

    Two proportionsp1, p2p_1,\ p_2Optional

    Values: Between 0 and 1, different

    To base the design on proportions instead.

    Fixed-sample size per groupnfixedn_{\text{fixed}}Optional

    Values: Above 0

    From any other calculation, such as a survival or cluster design, to inflate it for interim looks.

    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

    • The looks are pre-specified and equally spaced in information, and the stopping rule is followed. Any change needs its own statistical assessment.
    • Stopping is only for benefit: there is no futility boundary.
    • Dropout is unrelated to the outcome; recruiting more participants restores the numbers but cannot remove bias.

    Worked examples

    Four looks, from means

    Difference 5, SD 10: the fixed trial needs 63 per group; with 4 looks R = 1.024, so at most 65 per group, and 53 on average if the effect is real.

    Group 1: 65 · Group 2: 65 · Total: 130

    Protocol statement

    To detect a difference in means of 5 with 80% power at a two-sided significance level of 0.05, using a group-sequential design with 4 analyses and O’Brien–Fleming boundaries, and assuming a standard deviation of 10, the required sample size is at most 65 per group (130 in total), against 63 without interim analyses and 53 on average if the effect is real. Calculated with StatWisely (statwisely.com) using: maximum n per group = R × fixed-sample n; boundaries spend α across the looks (Armitage-McPherson-Rowe recursion). These are planning estimates; confirm the final design with a qualified statistician.

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    From a fixed size

    A fixed design of 100 per group becomes at most 103.

    Group 1: 103 · Group 2: 103 · Total: 206

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    From proportions

    0.3 against 0.45: fixed 160, maximum 164, expected 134 per group.

    Group 1: 164 · Group 2: 164 · Total: 328

    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.
    • The inflation factors agree with Jennison and Turnbull’s tables for α = 0.05 two-sided and 80% power: 1.024 for O’Brien–Fleming and 1.202 for Pocock with four looks, and 1.110 for Pocock with two; the O’Brien–Fleming critical values for four looks are 4.049, 2.863, 2.337 and 2.024 [3].

    How to cite

    StatWisely. Sample size for a group-sequential trial: O’Brien–Fleming. Method document, engine version 1.0.0, reviewed 2026-10-01. https://statwisely.com/methods/sequential-obrien-fleming/

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

    References

    1. 1.O’Brien PC, Fleming TR. A multiple testing procedure for clinical trials. Biometrics. 1979;35(3):549–556.
    2. 2.Armitage P, McPherson CK, Rowe BC. Repeated significance tests on accumulating data. Journal of the Royal Statistical Society, Series A. 1969;132(2):235–244.
    3. 3.Jennison C, Turnbull BW. Group Sequential Methods with Applications to Clinical Trials. Boca Raton: Chapman & Hall/CRC; 2000.