Logistic Regression: Binary Classification for Insurance Risk

The Logistic Model

Logistic regression models a binary outcome Y (0 or 1) using the logistic function: P(Y = 1 | X) = 1 / (1 + exp(-(beta_0 + beta_1*X1 + ... + beta_p*Xp))). The log-odds (logit) is a linear function of the predictors: ln(p/(1-p)) = beta_0 + beta_1*X1 + ... + beta_p*Xp. Parameters are estimated by maximum likelihood, not least squares.

Interpretation: a one-unit increase in Xj multiplies the odds by exp(beta_j). If beta_j = 0.5, the odds increase by a factor of e^0.5 = 1.65, or 65%. If beta_j < 0, the odds decrease.

Estimation and Inference

The log-likelihood is L = sum of [Yi * ln(pi_i) + (1-Yi) * ln(1 - pi_i)], where pi_i = P(Y_i = 1). The MLE is found numerically (Newton-Raphson or iteratively reweighted least squares). There is no closed-form solution. The deviance (-2 * log-likelihood) plays the role of SSE in linear regression. The Wald test (z = beta_j_hat / SE) tests individual coefficients. The likelihood ratio test compares nested models: G^2 = -2(L_reduced - L_full) follows a chi-squared distribution with degrees of freedom equal to the difference in parameters.

Insurance Applications

Logistic regression is widely used in insurance for modeling claim occurrence (claim/no claim), lapse prediction, underwriting decisions, and fraud detection. For Exam SRM, be comfortable interpreting odds ratios, computing predicted probabilities, and comparing models using deviance or AIC. A common exam question gives coefficient estimates and asks for the predicted probability for a specific set of predictor values, or asks how the odds change for a given change in a predictor.