ROC/AUC Calculator

ROC & AUC Calculator

True Binary Labels (Y ∈ {0, 1}) Enter true 0 (negative) and 1 (positive) values separated by commas or spaces

Please enter valid binary 0 or 1 labels.

Predicted Scores / Probabilities (p̂) Must match the number of observations in True Labels

Please enter valid numeric predicted scores matching observation length.

Significance Level (α) Default threshold is 0.05 (5%)

Significance level α must be strictly between 0 and 1 (e.g. 0.05).

Binary Classifier Performance

What Is ROC and AUC?

A Receiver Operating Characteristic (ROC) curve is a graphical plot that illustrates the diagnostic ability of a binary classifier system across all possible classification discrimination thresholds.

The Area Under the Curve (AUC) quantifies the overall performance of the model by measuring the entire two-dimensional area underneath the ROC curve from $(0,0)$ to $(1,1)$. An $AUC = 1.0$ represents a perfect model, while $AUC = 0.5$ represents random guessing.

When to Use It

Evaluating Diagnostic Tests: Assessing medical screening tests where sensitivity and specificity trade off across thresholds.
Machine Learning Validation: Comparing predictive models (e.g., Logistic Regression vs. Random Forest) on imbalanced binary data.
Finding Optimal Thresholds: Using Youden's J Statistic ($J = \text{Sensitivity} + \text{Specificity} - 1$) to establish the best decision cutoff.
Ranking Performance: Measuring the probability that a model ranks a random positive instance higher than a random negative instance.

The Mathematical Formulas

$$\text{True Positive Rate (TPR)} = \frac{TP}{TP + FN}, \quad \text{False Positive Rate (FPR)} = \frac{FP}{FP + TN}$$ $$\text{AUC (Trapezoidal Rule)} = \sum_{i=1}^{m} (FPR_i - FPR_{i-1}) \cdot \frac{TPR_i + TPR_{i-1}}{2}$$ $$\text{Youden's J Statistic} = TPR - FPR = \text{Sensitivity} + \text{Specificity} - 1$$

Where $TP, FP, TN, FN$ are true positives, false positives, true negatives, and false negatives evaluated at each unique probability score threshold.

Worked Example

Scenario: 8 subjects with true labels $[0, 0, 1, 0, 1, 1, 0, 1]$ ($P=4, N=4$) and predicted probabilities $[0.1, 0.4, 0.35, 0.8, 0.6, 0.7, 0.2, 0.9]$.

Step 1: Sort by Score Descending:
Scores: $0.9 (1), 0.8 (0), 0.7 (1), 0.6 (1), 0.4 (0), 0.35 (1), 0.2 (0), 0.1 (0)$.

Step 2: Compute TPR & FPR at Each Threshold:
Trapezoidal integration across coordinate points yields $\text{AUC} = 0.7500$.

Step 3: Optimal Threshold (Youden's J):
Maximum $J = 0.50$ occurs at threshold score $= 0.60$ ($\text{Sensitivity} = 75\%, \text{Specificity} = 75\%$).

Conclusion: $\text{AUC} = 0.7500$ indicates fair-to-acceptable discriminative classification performance.

ROC Analysis Complete

ROC & AUC Results

Area Under the Curve (AUC)

AUC = 0.8125

Good Discriminative Classification Performance (0.80 ≤ AUC < 0.90)

Area Under Curve (AUC) 0.8125

Optimal Threshold 0.6000

Max Youden's J 0.5000

Positives (P) 4

Negatives (N) 4

Total Sample (N) 8

ROC Curve Threshold Coordinates

Threshold (p̂)	True Pos (TP)	False Pos (FP)	True Neg (TN)	False Neg (FN)	Sensitivity (TPR)	1 - Specificity (FPR)	Youden's J (TPR - FPR)

ROC/AUC Calculator