Partial AUC (pAUC)

1) What it is

• Partial AUC = the area under a specified portion of the ROC curve (or sometimes PR curve).
• Instead of integrating over the entire False Positive Rate (FPR) range $[0,1]$, you only compute the AUC over a restricted interval, e.g. $[0, 0.1]$.

Motivation: In many real applications, you only care about part of the curve (e.g., very low false positive rates).

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2) Formal Definition

For a ROC curve:

$pAUC = \int_{FPR=a}^{b} TPR(FPR) \, d(FPR)$

where $[a, b]$ is the FPR interval of interest (e.g., $a=0$, $b=0.1$).

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3) Intuition

• Regular AUC = overall ranking ability of the classifier.
• Partial AUC = ranking ability in a restricted operating region (e.g., high specificity).

Example:

• In cancer screening, false positives are costly → you only care about ROC performance when FPR ≤ 5%.
• A model with high overall AUC but poor pAUC in [0,0.05] is not clinically useful.

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4) Why Use Partial AUC

• Domain-specific constraints:
  • Medical tests: need high specificity (low FPR).
  • Fraud detection: small tolerance for false alarms.
• Better reflection of real-world utility than full AUC.
• Model comparison: two models may have same AUC, but different pAUC in critical regions.

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5) Normalized Partial AUC

• Since pAUC depends on interval length (b−a)(b-a)(b−a), people often normalize it:

$pAUC_{norm} = \frac{pAUC – (b-a)\cdot 0}{(b-a)\cdot 1 – (b-a)\cdot 0} = \frac{pAUC}{b-a}$

so the score is scaled to $[0,1]$, just like full AUC.

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6) Example

Suppose two classifiers:

• Model A: AUC = 0.92, pAUC (0–0.1 FPR) = 0.05
• Model B: AUC = 0.89, pAUC (0–0.1 FPR) = 0.09

Even though A has better overall AUC, Model B is preferable if your application only allows ≤10% false positives.

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7) Applications

• Medical diagnosis (screening tests).
• Fraud detection (minimize false alarms).
• Security systems (high precision zones).
• High-stakes domains where performance in a narrow region matters more than global performance.

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Summary

• Partial AUC = area under a restricted portion of the ROC (or PR) curve.
• Focuses on critical FPR (or recall) ranges.
• Often normalized for comparability.
• Useful when false positives are costly or only certain operating points matter.