Partial Area Under the ROC Curve

The standard AUC integrates test performance across the entire range of false positive rates from 0 to 1. In many clinical situations this is too broad a summary. A mass screening programme may require that the false positive rate stays below 10% — flooding the follow-up pipeline with false positives beyond that threshold is operationally unacceptable. A confirmatory test may need to operate at very high specificity: if a positive result triggers an irreversible intervention, even a few false positives at high specificity may be tolerable, but false positives at low specificity are not. In these settings, performance outside the clinically relevant range of false positive rates is not merely uninteresting — it is noise that can obscure meaningful differences between tests.

The partial AUC (pAUC) addresses this by restricting the integral to a pre-specified interval of interest, providing a summary of diagnostic accuracy that is directly calibrated to the clinical context.

Definition

The partial AUC over a specificity interval is defined as the area under the ROC curve for false positive rates (FPR) between a lower bound $t_1$ and an upper bound $t_2$, where $0 \le t_1 < t_2 \le 1$:

$$ \text{pAUC}(t_1, t_2) = \int_{t_1}^{t_2} \text{ROC}(t)\, dt $$

The interval can be specified either on the FPR axis (equivalent to a specificity interval) or on the sensitivity axis. A specificity interval of 80–100% corresponds to an FPR interval of 0–0.20. A sensitivity interval of 80–100% restricts performance to the portion of the curve where at least 80% of cases are detected, regardless of the false positive cost.

ROC curve with the partial AUC shaded in blue for false positive rates from 0 to 0.20, corresponding to a specificity interval of 80–100%. The boundary is marked by a dashed vertical line at FPR = 0.20. The unshaded area to the right represents the portion of the curve outside the clinically relevant region.

MedCalc calculates the pAUC non-parametrically using the trapezoidal method: the area is approximated by summing the areas of trapezoids formed between adjacent sensitivity–FPR pairs along the observed ROC curve within the specified interval. If the boundary of the interval falls between two observed thresholds, linear interpolation is used to estimate the sensitivity at the exact boundary point.

The Standardized Partial AUC

A raw pAUC has an awkward property: its maximum possible value depends on the width of the chosen interval. Over the full FPR range [0, 1] the maximum is 1.0; over the interval [0, 0.2] the maximum is only 0.2 (a perfect test achieves sensitivity = 1 throughout, giving an area of 0.2 × 1 = 0.2). This makes raw pAUC values hard to interpret and impossible to compare across studies that used different interval widths.

McClish (1989) proposed a standardization that maps the pAUC back onto the familiar [0.5, 1.0] scale — the same scale as the full AUC:

$$ \text{pAUC}_s = \frac{1}{2}\left(1 + \frac{\text{pAUC} - \text{pAUC}_{\min}}{\text{pAUC}_{\max} - \text{pAUC}_{\min}}\right) $$

where $\text{pAUC}_{\min}$ is the area of a useless test (whose ROC curve lies on the diagonal) over the same interval, and $\text{pAUC}_{\max}$ is the area of a perfect test over the same interval. The standardized value $\text{pAUC}_s = 0.5$ corresponds to no discrimination (same as AUC = 0.5) and $\text{pAUC}_s = 1.0$ to perfect discrimination, making it directly interpretable alongside the standard AUC guidelines.

Example: Over the FPR interval [0, 0.20], a useless test has pAUC_min = 0.5 × 0.20 × 0.20 = 0.020 (the diagonal triangle) and a perfect test has pAUC_max = 0.20 × 1.0 = 0.200 (full rectangle). If a test achieves pAUC = 0.148, then pAUC_s = ½ × (1 + (0.148 − 0.020)/(0.200 − 0.020)) = ½ × (1 + 0.711) = 0.856 — indicating excellent performance in the high-specificity region.

Confidence Intervals and Hypothesis Testing

Unlike the full AUC, there is no clean closed-form expression for the variance of the pAUC. MedCalc therefore estimates confidence intervals for both pAUC and pAUC_s using bootstrapping (Efron & Tibshirani, 1993): the original dataset is resampled with replacement many times, the pAUC is recalculated for each bootstrap replicate, and the 2.5th and 97.5th percentiles of the resulting distribution form the 95% confidence interval.

The null hypothesis of no discriminative ability is tested by examining whether the 95% bootstrap CI for pAUC_s excludes 0.5. If it does ($p < 0.05$), the test performs significantly better than chance in the specified interval.

Comparing Partial AUCs

The same paired/independent sample distinction that applies to comparing full AUCs applies here. In a paired design both tests are measured in the same patients; in an independent design they are measured in different groups.

Two ROC curves compared over the specificity interval 80–100% (FPR 0–0.20). Test A (blue) has a substantially larger partial AUC in the restricted region. Test B (orange) has a higher overall AUC — the curves cross outside the restricted interval, illustrating a case where the full AUC would favour Test B but the clinically relevant partial AUC favours Test A.

The comparison is based on the difference in pAUC (or pAUC_s) between the two tests. Because no closed-form variance estimator is available for the difference either, MedCalc again uses bootstrapping: at each replicate, both pAUC values are recalculated from the same resampled dataset (for paired designs) or from independently resampled datasets (for independent designs), and the difference is recorded. The 95% CI for the difference is read from the 2.5th and 97.5th percentiles of the bootstrap distribution. If the CI excludes zero, the difference is statistically significant at $p < 0.05$.

When this matters: Two cancer biomarkers are each measured in the same 200 patients. The full AUC is 0.84 for both — statistically indistinguishable. However, the clinical protocol requires that the false positive rate be kept below 10% (only patients who test positive proceed to an invasive biopsy). Over the FPR interval [0, 0.10], Biomarker A achieves pAUC_s = 0.81 and Biomarker B achieves pAUC_s = 0.71. The 95% bootstrap CI for the difference (0.10; 95% CI 0.03–0.17) excludes zero. The full AUC comparison would have concluded the tests are equivalent; the partial AUC comparison correctly identifies Biomarker A as the superior choice for this clinical use case.

Choosing the Interval

The interval must be chosen $before$ the analysis, based on clinical requirements — not after inspecting the data to find the region where the preferred test looks best. Post-hoc interval selection is a form of data dredging that inflates the false positive rate of the comparison. The interval should be justified in the study protocol by reference to:

The maximum acceptable false positive rate in the downstream clinical pathway (e.g., determined by the capacity of follow-up services or the harms of unnecessary procedures).
The minimum clinically required sensitivity (e.g., a sensitivity below 80% is unacceptable for a serious and treatable condition).
Established clinical guidelines or regulatory precedent for the specific application.

MedCalc implements partial AUC analysis in two procedures:

ROC curve – partial area — computes the pAUC and standardized pAUC with 95% bootstrap confidence intervals for a single test over a specified FPR or sensitivity interval.
Comparison of partial areas under ROC curves — compares the pAUC of two tests (paired or independent design) and reports the difference with a 95% bootstrap confidence interval.