How does logistic regression work and what's the intuition behind the logit link?

Q: How does logistic regression work and what's the intuition behind the logit link?

Logistic regression predicts binary outcomes (default/no-default, buy/no-buy) by modeling the probability of the positive class. The logit link transforms bounded probabilities into an unbounded linear predictor, solving fundamental problems with naive OLS on binary data.\n\nIf you ran OLS with Y = 0 or 1 as the dependent variable:\n- Predictions can exceed [0, 1], which is nonsensical for probabilities.\n- Errors are heteroskedastic (variance depends on X).\n- Normality of errors is violated.\n\nLogit solves this. The log-odds (logit) of probability p is: logit(p) = log(p / (1−p)). This maps p∈(0,1) to the real line (−∞, +∞). We model logit(p) = β₀ + β₁X₁ + … + βₖXₖ, then invert using the sigmoid function: p = 1 / (1 + e^(−Xβ)).\n\nWorked example: suppose Ironcrest Credit Union predicts loan default using debt-to-income ratio (DTI) and credit score. Fitted model: logit(p) = −5 + 0.08 × DTI − 0.01 × CreditScore. A borrower with DTI=40 and score=650 gives: logit = −5 + 3.2 − 6.5 = −8.3, so p = 1/(1+e^8.3) ≈ 0.025 or 2.5% default probability.\n\nCoefficient interpretation: exp(βⱼ) is the odds ratio — a one-unit increase in Xⱼ multiplies the odds by exp(βⱼ). Here exp(0.08) ≈ 1.083, meaning each additional DTI point increases default odds by 8.3%.\n\nEstimation is via maximum likelihood, not OLS. Key diagnostics: AUC-ROC (discriminative power), pseudo R² (McFadden's), confusion matrices, calibration plots. Logistic regression is foundational in credit scoring, marketing response models, and bankruptcy prediction — expect it on the CFA quant exam.

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Logistic regression predicts binary outcomes using a logit link that transforms bounded probabilities into unbounded log-odds, allowing linear modeling...

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If you ran OLS with Y = 0 or 1 as the dependent variable:
- Predictions can exceed [0, 1], which is nonsensical for probabilities.
- Errors are heteroskedastic (variance depends on X).
- Normality of errors is violated.

Logit solves this. The log-odds (logit) of probability p is: logit(p) = log(p / (1−p)). This maps p∈(0,1) to the real line (−∞, +∞). We model logit(p) = β₀ + β₁X₁ + … + βₖXₖ, then invert using the sigmoid function: p = 1 / (1 + e^(−Xβ)).

Worked example: suppose Ironcrest Credit Union predicts loan default using debt-to-income ratio (DTI) and credit score. Fitted model: logit(p) = −5 + 0.08 × DTI − 0.01 × CreditScore. A borrower with DTI=40 and score=650 gives: logit = −5 + 3.2 − 6.5 = −8.3, so p = 1/(1+e^8.3) ≈ 0.025 or 2.5% default probability.

Coefficient interpretation: exp(βⱼ) is the odds ratio — a one-unit increase in Xⱼ multiplies the odds by exp(βⱼ). Here exp(0.08) ≈ 1.083, meaning each additional DTI point increases default odds by 8.3%.

Estimation is via maximum likelihood, not OLS. Key diagnostics: AUC-ROC (discriminative power), pseudo R² (McFadden's), confusion matrices, calibration plots. Logistic regression is foundational in credit scoring, marketing response models, and bankruptcy prediction — expect it on the CFA quant exam.

How does logistic regression work and what's the intuition behind the logit link?

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