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Youden's Index

Jun 15, 20242 min read

Youden

also known as Youden’s J statistic, is a metric used to evaluate the effectiveness of a diagnostic test. It helps determine the optimal threshold value for binary classification systems by balancing sensitivity and specificity.

Formula

where:

  • Sensitivity (True Positive Rate, TPR): The proportion of actual positives correctly identified by the test.
  • Specificity (True Negative Rate, TNR): The proportion of actual negatives correctly identified by the test.

Interpretation

  • Range: Youden’s Index ranges from 0 to 1.

    • A value of 0 indicates that the test is no better than random chance.
    • A value of 1 indicates a perfect test with no false positives or false negatives.

  • Optimal Threshold: The threshold that maximizes Youden’s Index is considered the optimal threshold, achieving the best balance between sensitivity and specificity.

Application in Logistic Regression

We can use Youden’s Index to find the optimal threshold for a logistic regression model by evaluating different thresholds and selecting the one with the highest index. Here’s an example to illustrate this process.

Example: Finding the Optimal Threshold in Logistic Regression

Let’s start with a simple logistic regression model and compute the ROC curve, AUC, and optimal threshold using Youden’s Index.

Step 1: Load and Split Data

import numpy as np
import matplotlib.pyplot as plt
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LogisticRegression
from sklearn.metrics import roc_curve, roc_auc_score
 
# Load dataset
from sklearn.datasets import load_breast_cancer
data = load_breast_cancer()
X = data.data
y = data.target
 
# Split data into training and testing sets
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3, random_state=42)

Step 2: Train the Logistic Regression Model

# Train logistic regression model
model = LogisticRegression(max_iter=10000)
model.fit(X_train, y_train)

Step 3: Predict Probabilities and Compute ROC Curve

# Predict probabilities for the test set
y_prob = model.predict_proba(X_test)[:, 1]
 
# Compute ROC curve
fpr, tpr, thresholds = roc_curve(y_test, y_prob)
roc_auc = roc_auc_score(y_test, y_prob)

Step 4: Calculate Youden’s Index and Find the Optimal Threshold

# Calculate Youden's Index for each threshold
youden_index = tpr - fpr
optimal_threshold = thresholds[np.argmax(youden_index)]
 
print(f'Best Threshold: {optimal_threshold}')
print(f'ROC AUC: {roc_auc}')

Step 5: Plot the ROC Curve and Optimal Threshold

# Plot ROC curve
plt.plot(fpr, tpr, color='blue', label=f'ROC curve (area = {roc_auc:.2f})')
plt.plot([0, 1], [0, 1], color='red', linestyle='--')
plt.scatter(fpr[np.argmax(youden_index)], tpr[np.argmax(youden_index)], marker='o', color='black', label=f'Best Threshold = {optimal_threshold:.2f}')
plt.xlabel('False Positive Rate')
plt.ylabel('True Positive Rate')
plt.title('Receiver Operating Characteristic (ROC) Curve')
plt.legend(loc='lower right')
plt.show()

Graph View

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