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    Method document

    Sample size to estimate the area under the ROC curve

    How many people must be tested to estimate the area under the ROC curve (AUC) of a continuous diagnostic marker within a chosen margin, from the expected AUC and the prevalence in a consecutive sample. Reports the expected reference-standard positives and negatives. Covers non-response.

    Engine version 1.0.0 · Reviewed 6 October 2026

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

    When to use it

    • Validating a continuous marker, such as a biomarker or a risk score, against a reference standard, when its overall discrimination is the aim.

    Not for

    • Comparing the AUCs of two markers measured on the same people; that needs their correlation.
    • A marker with a strongly skewed or very unequal spread between the groups; the binormal variance then needs its second parameter.

    Notation

    AA
    Expected area under the ROC curve
    n1, n2n_1,\ n_2
    Reference-standard positives and negatives, N Prev and N(1 − Prev)
    QQ
    Probability that one diseased value exceeds two non-diseased values, or two diseased exceed one non-diseased
    EE
    Margin of error, in AUC units
    z1−α/2z_{1-\alpha/2}
    Standard normal quantile for confidence 1 − α (1.960 for 95%)
    qq
    Non-response rate

    Formulas

    Variance of the empirical AUC

    V(A^)=A(1−A)+(n1−1)(Q−A2)+(n2−1)(Q−A2)n1n2,Q=Φ2 ⁣(δ,δ;12), δ=Φ−1(A)V(\hat A) = \frac{A(1-A) + (n_1 - 1)(Q - A^{2}) + (n_2 - 1)(Q - A^{2})}{n_1 n_2}, \qquad Q = \Phi_2\!\left(\delta, \delta; \tfrac{1}{2}\right),\ \delta = \Phi^{-1}(A)

    The structure is Hanley and McNeil’s [1]. Their own Q₁ = A/(2 − A) and Q₂ = 2A²/(1 + A) come from an exponential model and overstate the variance of a continuous binormal marker, so both are computed exactly under the equal-variance binormal model. N is the smallest total with z_(1−α/2)√V ≤ E.

    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

    In AUC units: 0.05 means the interval runs from AUC − 0.05 to AUC + 0.05.

    Expected AUCAA

    Values: Above 0.5 and below 1

    From a previous study of the marker. An AUC near 0.5 is the hardest to estimate precisely.

    Expected prevalencePrevPrev

    Values: Greater than 0 and less than 1

    In the people who will actually be tested. It splits the sample into reference-standard positives and negatives.

    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; the marker is roughly normal with similar spread in diseased and non-diseased people, possibly after a transformation; the reference standard is accurate.
    • Non-response is unrelated to the outcome.

    Worked examples

    Balanced disease status

    AUC 0.8, prevalence 0.5, margin ±0.05: N = 292, half diseased.

    Sample size: 292

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    A stronger marker at 20% prevalence

    AUC 0.9 and prevalence 0.2: N = 218, about 44 of them diseased.

    Sample size: 218

    Open this example in the calculator

    A wider margin

    AUC 0.75, prevalence 0.3, margin ±0.1: N = 107.

    Sample size: 107

    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.
    • Simulating 4,000 binormal samples in R at each planned size gave empirical-AUC interval half-widths of 0.0497, 0.0495, 0.0993 and 0.0296 against targets of 0.05, 0.05, 0.10 and 0.03. With Hanley and McNeil’s exponential Q terms, the 20%-prevalence example would have needed 345 people for a half-width of 0.039. An independent R integration of Q reproduces the engine’s sizes.

    How to cite

    StatWisely. Sample size to estimate the area under the ROC curve. Method document, engine version 1.0.0, reviewed 2026-10-06. https://statwisely.com/methods/precision-auc/

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

    References

    1. 1.Hanley JA, McNeil BJ. The meaning and use of the area under a receiver operating characteristic (ROC) curve. Radiology. 1982;143(1):29–36. doi:10.1148/radiology.143.1.7063747