ROC-AUC - Step by step calculation

 Let’s go through ROC-AUC just like we did for KS — with intuitive explanation, formulas, and a step-by-step example using 10 observations.

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📘 What is ROC-AUC?

🟦 ROC = Receiver Operating Characteristic Curve

It plots:

• X-axis: False Positive Rate (FPR) = FP / (FP + TN)

• Y-axis: True Positive Rate (TPR) = TP / (TP + FN)

Each point on the ROC curve represents a threshold on the predicted probability.

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🟧 AUC = Area Under the Curve

• AUC = Probability that the model ranks a random positive higher than a random negative

• AUC ranges from:

  • 1.0 → perfect model

  • 0.5 → random guessing

  • < 0.5 → worse than random

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✅ ROC-AUC Formula (Conceptually)

There are two main interpretations:

1. Integral of the ROC Curve:

$$AUC = \int_0^1 TPR(FPR) \, dFPR$$

2. Rank-Based Interpretation (Used in practice):

$$AUC = \frac{\text{Number of correct positive-negative pairs}}{\text{Total positive-negative pairs}}$$

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📊 Example: 10 Observations

We'll reuse your 10 data points:

Obs	Actual (Y)	Predicted Score
1	1	0.95
2	0	0.90
3	1	0.85
4	0	0.80
5	0	0.70
6	1	0.60
7	0	0.40
8	0	0.30
9	1	0.20
10	0	0.10

• Total Positives (P) = 4

• Total Negatives (N) = 6

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📈 Step-by-Step: Rank-Based AUC Calculation

Let’s find all (positive, negative) score pairs and count how many times:

• Positive score > Negative score → Correct

• Positive score == Negative score → 0.5 credit

• Positive score < Negative score → Wrong

Step 1: List All Positive-Negative Pairs

Positive scores: 0.95, 0.85, 0.60, 0.20
Negative scores: 0.90, 0.80, 0.70, 0.40, 0.30, 0.10

Total Pairs = 4 × 6 = 24

Step 2: Count Favorable Pairs

Pos Score	Compared to Neg Scores	Wins
0.95	> all (0.90 ... 0.10)	6
0.85	> all except 0.90	5
0.60	> 0.40, 0.30, 0.10	3
0.20	> 0.10 only	1
Total		6+5+3+1 = 15 wins

No ties, so:

$$AUC = \frac{15}{24} = \boxed{0.625}$$

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🧠 Interpretation:

• Model has 62.5% chance of ranking a random defaulter higher than a non-defaulter.

• Better than random, but not great.

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📉 ROC Curve (Optional Idea):

If we plot TPR vs FPR at various thresholds:

• Start at (0,0)

• End at (1,1)

• The area under that curve will match AUC = 0.625