Regression & Correlation

ROC Curve & AUC

Computes the ROC curve (Receiver Operating Characteristic) and the Area Under the Curve (AUC) for binary classification.

Statusvalidated

Version1.0.0

Minitab equivalentStat > Regression > Binary Logistic Regression > ROC Curve

Description

The ROC curve plots True Positive Rate (sensitivity) against False Positive Rate (1 − specificity) across all possible decision thresholds. The AUC summarises classifier discrimination in a single metric: AUC = 1.0 indicates a perfect classifier, AUC = 0.5 a random one. Computation uses the trapezoidal rule over sorted thresholds.

Formulas

TPR (Sensitivity)

TPR = \frac{TP}{TP + FN}

True Positive Rate (Sensitivity): proportion of correctly identified positive cases

FPR (1 − Specificity)

FPR = \frac{FP}{FP + TN}

False Positive Rate (1 − Specificity): proportion of negative cases incorrectly classified as positive

AUC (trapezoidal)

AUC = i = 1 \sum n (FPR_{i} - FPR_{i - 1}) \cdot \frac{TPR _{i} + TPR _{i - 1}}{2}

Area Under the ROC Curve, computed using the trapezoidal rule

Perfect classifier

AUC = 1.0

A perfect classifier achieves AUC = 1.0 (complete class separation)

Random classifier

AUC = 0.5

A random classifier achieves AUC = 0.5 (ROC curve on the diagonal)

Assumptions

Binary classification with labels 0 and 1
Probabilities represent posterior P(Y=1|X)
Higher probability indicates the positive class
Observations are independent

Limitations

Tied probabilities can affect curve shape
AUC is insensitive to calibration (only rank-based)
A single AUC value may mislead with class imbalance — inspect the full curve
All-positive or all-negative Y defaults to AUC = 0.5

References

Fawcett, T., An Introduction to ROC Analysis, Pattern Recognition Letters, 2006
Hosmer, D.W., Lemeshow, S., Applied Logistic Regression, 3rd Ed., Wiley, Ch. 5
Hanley, J.A., McNeil, B.J., The Meaning and Use of the Area under a ROC Curve, Radiology, 1982