Logistic Regression: Odds Ratios & Customer Churn Prediction

While linear regression predicts continuous numerical outcomes, Logistic Regression is the industry standard for modeling binary categorical decisions (e.g., Churn vs Retain, Fraud vs Legitimate, Default vs Repay). By compressing linear combinations through the Sigmoid (Logit) function, logistic regression outputs calibrated probabilities bounded strictly between 0 and 1.

1. The Sigmoid Function & Log-Odds

A standard linear equation can produce predictions from $-\infty$ to $+\infty$, which is mathematically invalid for probabilities. Logistic regression applies the Sigmoid activation function:

$$P(Y=1 | X) = \sigma(z) = \frac{1}{1 + e^{-z}}$$

Where $z = \beta_0 + \beta_1 X_1 + \dots + \beta_p X_p$.

Log-Odds (Logit):

$$\ln\left(\frac{p}{1 - p}\right) = \beta_0 + \beta_1 X_1 + \dots + \beta_p X_p$$

• $\frac{p}{1-p}$ is the Odds Ratio.
• Exponentiating a coefficient $e^{\beta_j}$ reveals the multiplicative effect on customer churn odds for each unit change in $X_j$.

2. Building a Customer Churn Predictor

3. Interpreting Odds Ratios for Business Strategy

Example Insights:

• If num_support_tickets has an Odds Ratio of 1.85: Each additional customer support ticket increases churn odds by 85%.
• If contract_is_annual has an Odds Ratio of 0.32: Annual contracts reduce churn odds by 68% compared to monthly contracts!

4. Probability Threshold Tuning

By default, predict() classifies cases as positive if $P \ge 0.50$. In customer retention, proactively reaching out to high-risk customers might warrant a lower threshold (e.g., $P \ge 0.35$):