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    Sample size to show that two methods agree within a tolerance

    How many participants, each measured by two methods, are needed to show that both 95% Bland–Altman limits of agreement, with their confidence limits, lie within a clinical tolerance ±Δ. Covers dropout.

    Engine version 1.0.0 · Reviewed 9 October 2026

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    See also: Sample size to estimate Bland–Altman limits of agreement

    https://statwisely.com/methods/measurement-limits-of-agreement/

    When to use it

    • A method-comparison study that must show a new device can replace a reference one.

    Not for

    • Reporting the limits with confidence intervals, with no tolerance to meet. Use the limits of agreement under “Estimate a value”.

    Notation

    nn
    Number of participants
    μ, s\mu,\ s
    Expected mean and standard deviation of the differences
    Δ\Delta
    Largest acceptable difference
    SE\mathrm{SE}
    Standard error of each limit, s√(1/n + 1.96²/(2(n − 1)))
    Φ\Phi
    Standard normal distribution function
    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

    Both limits’ confidence limits inside ±Δ

    1−β≈Φ ⁣(Δ−μ−1.96sSE−z1−α/2)+Φ ⁣(Δ+μ−1.96sSE−z1−α/2)−11-\beta \approx \Phi\!\left(\frac{\Delta - \mu - 1.96s}{\mathrm{SE}} - z_{1-\alpha/2}\right) + \Phi\!\left(\frac{\Delta + \mu - 1.96s}{\mathrm{SE}} - z_{1-\alpha/2}\right) - 1

    The upper limit’s upper confidence limit must fall below Δ and the lower limit’s lower one above −Δ; the power of both is approximated from normal limits, after Lu et al. [2, 1].

    Dropout

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

    Applied last, to the participants actually recruited.

    Parameters

    Expected mean differenceμ\muOptional

    Values: Any value; 0 by default

    A systematic bias moves one limit towards the tolerance and needs many more participants.

    SD of the differencesss

    Values: Above 0

    From a pilot comparison of the same methods.

    Largest acceptable differenceΔ\Delta

    Values: Beyond the expected limits, |μ| + 1.96s

    Set clinically, before the study: the largest disagreement that would not change a decision.

    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 differences are normal, with a mean and spread that do not change across the range measured.
    • Dropout is unrelated to the outcome; recruiting more participants restores the numbers but cannot remove bias.

    Worked examples

    Limits of ±1.96 against ±2.5

    No bias, s = 1 and Δ = 2.5: n = 106, with power 0.800591872273523.

    Sample size: 106

    Protocol statement

    To show that the two methods differ by less than 2.5 for 95% of participants with 80% power at a significance level of 0.05, using a Bland–Altman test that both 95% limits of agreement, with their two-sided confidence limits, lie within the tolerance, and assuming a mean difference of 0 between methods and a standard deviation of the differences of 1, the required sample size is 106 participants. Calculated with StatWisely (statwisely.com) using: N: P(upper limit's CI < Δ and lower limit's CI > -Δ), SE = s√(1/n + z_0.975²/(2(n-1))) (Lu et al.). These are planning estimates; confirm the final design with a qualified statistician.

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    A small bias

    A bias of 0.2: n = 199.

    Sample size: 199

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    A wider tolerance

    Δ = 3: n = 30.

    Sample size: 30

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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.
    • An independent implementation in R recomputes this test every night and before each production release, and the results are compared with the engine’s.
    • Simulated method comparisons at the planned size gave power of 0.80 against a target of 0.80.

    How to cite

    StatWisely. Sample size to show that two methods agree within a tolerance. Method document, engine version 1.0.0, reviewed 2026-10-09. https://statwisely.com/methods/measurement-limits-of-agreement/

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

    References

    1. 1.Bland JM, Altman DG. Measuring agreement in method comparison studies. Statistical Methods in Medical Research. 1999;8(2):135–160.
    2. 2.Lu MJ, Zhong WH, Liu YX, Miao HZ, Li YC, Ji MH. Sample size for assessing agreement between two methods of measurement by Bland-Altman method. International Journal of Biostatistics. 2016;12(2).