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ROC Curve & AUC Calculator with Hanley-McNeil Confidence Intervals

ROC curve, AUC (two methods cross-checked), Hanley-McNeil confidence intervals, Youden optimal threshold, and Mann-Whitney test from raw scores or a known AUC.

Trapezoidal:  compute (1−specificity, sensitivity) at every possible threshold, connect the points, integrate trapezoidally
Concordance:  AUC = [ #(positive > negative) + 0.5 × #(positive = negative) ] ÷ (n₁ × n₂)

SE(AUC) = √{ [AUC(1−AUC) + (N₁−1)(Q₁−AUC²) + (N₂−1)(Q₂−AUC²)] ÷ (N₁N₂) }
Q₁ = AUC ÷ (2−AUC)        Q₂ = 2·AUC² ÷ (1+AUC)
CI  = AUC ± z(α/2) × SE(AUC)

FAQ

What does AUC actually mean?

**The probability that a randomly chosen positive individual scores higher than a randomly chosen negative individual.** 0.5 is a coin flip; 1.0 is complete separation of the two groups. This is not an analogy — it is exact: Hanley & McNeil (Radiology 1982;143:29-36) proved that the trapezoidal area under the empirical ROC curve equals the normalized Mann-Whitney statistic, which is an unbiased estimate of precisely this probability. This tool computes both paths and displays them side by side so you can see this directly.

My AUC is below 0.5. Does that mean the marker is worse than chance?

Almost always, **the direction is wrong**. An AUC of 0.3 means lower scores are associated with the positive group — switch the direction option to "lower score → more likely positive" and you will get 0.7. A marker that is genuinely worse than random is extremely rare in practice. When you see AUC < 0.5, check the direction first rather than concluding the marker is useless.

What do I enter for N₁ and N₂ in the SE formula?

**N₁ is the number of positive (diseased) cases; N₂ is the number of negative (control) cases — not the total sample size.** They appear separately in the denominator and in the two bracket terms. Entering the total sample size makes the standard error too small and the interval too narrow, giving a false sense of precision. This is the most common mistake with this formula, which is why the tool asks for them separately.

The upper confidence limit exceeds 1. What should I do?

That is a limitation of the normal approximation near the boundary. With AUC 0.965 and 12 cases per group, the upper limit reaches 1.043. This tool clips the displayed value to 1.0 **and explicitly warns you** — silently clipping would make it look as though the interval truly ends at 1.0, which is a different claim. For this situation, consider bootstrap confidence intervals or apply a logit transform first and then back-transform.

Can I use the Youden-optimal threshold directly in practice?

**Not without checking the cost assumptions.** J = sensitivity + specificity − 1 implicitly assumes that false negatives and false positives carry equal cost. In screening, a missed diagnosis is usually much more serious; in that case, first fix an acceptable minimum sensitivity (e.g., ≥ 95%) and then select the threshold with the highest specificity that meets it. Also, with small samples J is a step function and multiple thresholds can tie for the maximum — the tool reports the tie count. A tie means the data cannot distinguish between those thresholds; you still need to choose based on clinical costs.

Can I compare the AUC values from two models?

This tool does not do that. Comparing two ROC curves depends on whether they come from the **same individuals**: if so, you need a paired method (DeLong, or the Hanley-McNeil 1983 paired formula), which uses the correlation between the two scores on the same sample; otherwise the samples can be treated as independent. Visually checking whether two confidence intervals overlap is incorrect — it double-counts the shared variation that is cancelled out in a paired analysis.

Does a high AUC mean the test is clinically useful?

Not necessarily. AUC is a **threshold-independent** summary of discriminative ability, but clinical use depends on a specific threshold. Two curves with identical AUC can differ substantially in the range of specificity you actually care about (say, specificity > 90%). To pick a threshold, read the ROC table row by row rather than relying on a single AUC. Also note that AUC is unaffected by prevalence, whereas positive predictive value is — in a low-prevalence population a marker with AUC 0.9 can still yield mostly false positives. The [Diagnostic 2×2 table calculator](../diagnostic-2x2/) on this site handles that side of the analysis.

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