Logistic Regression: Binary Classification for Insurance Risk

The Logistic Model

Logistic regression models a binary outcome Y (taking values 0 or 1) by relating the probability of success to a linear combination of predictors through the logistic function: P(Y = 1 | X) = 1 / (1 + exp(-(beta_0 + beta_1*X1 + ... + beta_p*Xp))). The logistic function maps any real-valued input to the interval (0, 1), ensuring that the predicted probability is always valid.

The model is most naturally expressed in terms of the log-odds (logit): ln(p/(1-p)) = beta_0 + beta_1*X1 + ... + beta_p*Xp. The logit link transforms the probability scale to the real line, making the relationship between predictors and the transformed response linear. This is what makes logistic regression a generalized linear model with a Bernoulli response and logit link function.

Parameters are estimated by maximum likelihood rather than ordinary least squares. OLS is inappropriate because the response is binary, the errors are not normally distributed, and the relationship between the probability and predictors is inherently nonlinear.

Interpreting Coefficients as Odds Ratios

The key to interpreting logistic regression output is the odds ratio. A one-unit increase in predictor X_j multiplies the odds of the event by exp(beta_j), holding all other predictors constant. The odds are defined as p/(1-p), where p is the probability of the event.

For example, suppose you are modeling whether a policyholder files a claim (Y = 1) or not (Y = 0). If the coefficient for driver age is beta = -0.03, then each additional year of age multiplies the odds of filing a claim by exp(-0.03) = 0.970, a 3% decrease in the odds. If the coefficient for a binary indicator "prior_claims" is beta = 0.85, having prior claims multiplies the odds by exp(0.85) = 2.34, more than doubling the odds.

For a continuous predictor, a c-unit increase multiplies the odds by exp(c * beta_j). So a 10-year increase in driver age multiplies the odds by exp(10 * (-0.03)) = exp(-0.3) = 0.741, a 25.9% decrease.

Be careful to distinguish between odds and probability. If the baseline probability is 0.10, the odds are 0.10/0.90 = 0.111. Doubling the odds gives 0.222, which corresponds to a probability of 0.222/1.222 = 0.182. The probability increased by 8.2 percentage points, not doubled.

Estimation and Inference

The log-likelihood for logistic regression is l = sum of [Y_i * ln(pi_i) + (1 - Y_i) * ln(1 - pi_i)], where pi_i = P(Y_i = 1 | X_i). The MLE maximizes this function numerically using algorithms such as Newton-Raphson or iteratively reweighted least squares (IRLS). There is no closed-form solution.

The deviance is defined as D = -2 * l, which plays the role of the residual sum of squares in linear regression. The null deviance uses only an intercept; the residual deviance uses the full model. A large drop in deviance indicates that the predictors explain substantial variation in the response.

For testing individual coefficients, the Wald test computes z = beta_j_hat / SE(beta_j_hat). Under H0: beta_j = 0, this follows a standard normal distribution. The Wald confidence interval for beta_j is beta_j_hat +/- z_{alpha/2} * SE(beta_j_hat), and the corresponding interval for the odds ratio is (exp(lower), exp(upper)).

For comparing nested models, the likelihood ratio test is more reliable than the Wald test: G^2 = -2(l_reduced - l_full) follows a chi-squared distribution with degrees of freedom equal to the difference in the number of parameters. This test is preferred when sample sizes are moderate or when coefficients are large (where the Wald test can be unreliable).

Worked Example

An insurer fits a logistic regression model for claim occurrence with three predictors: age (continuous), gender (0 = female, 1 = male), and urban indicator (0 = rural, 1 = urban). The estimated model is: logit(p) = -1.20 - 0.025 * Age + 0.40 * Gender + 0.55 * Urban.

Question: What is the predicted probability of a claim for a 35-year-old male in an urban area?

Step 1: Compute the linear predictor. eta = -1.20 - 0.025(35) + 0.40(1) + 0.55(1) = -1.20 - 0.875 + 0.40 + 0.55 = -1.125.

Step 2: Apply the logistic function. p = 1 / (1 + exp(1.125)) = 1 / (1 + 3.080) = 1 / 4.080 = 0.245.

The predicted probability is approximately 24.5%.

Question: What are the odds ratios? exp(-0.025) = 0.975 for age (each year reduces odds by 2.5%), exp(0.40) = 1.492 for gender (males have 49.2% higher odds than females), and exp(0.55) = 1.733 for urban (urban policyholders have 73.3% higher odds than rural ones).

Model Assessment for Logistic Regression

Since R-squared is not directly applicable to logistic regression, several alternative measures are used. The pseudo R-squared (McFadden's) is 1 - (l_full / l_null), where l values are log-likelihoods. Values between 0.2 and 0.4 indicate a good fit. AIC = -2l + 2k is used for model comparison, with lower values preferred.

Classification performance is assessed using the confusion matrix, sensitivity, specificity, ROC curves, and AUC. The Hosmer-Lemeshow test groups observations into deciles of predicted probability and compares observed to expected counts using a chi-squared statistic. A significant result suggests poor calibration.

Residual diagnostics differ from linear regression. Deviance residuals, r_i = sign(Y_i - pi_hat_i) * sqrt(d_i), where d_i is the contribution to the deviance, should be approximately standard normal if the model fits well. Pearson residuals (Y_i - pi_hat_i) / sqrt(pi_hat_i * (1 - pi_hat_i)) serve a similar purpose.

Insurance Applications

Logistic regression is one of the most widely used techniques in insurance analytics. Common applications include modeling claim occurrence (whether a policyholder files at least one claim), lapse prediction (whether a policyholder renews), underwriting decisions (approve or decline an application), and fraud detection (flagging suspicious claims for investigation).

In pricing, logistic regression for claim frequency is often paired with a separate severity model (gamma or lognormal regression) to produce pure premiums. This frequency-severity approach is standard in property and casualty insurance.

Logistic regression is preferred over more complex methods in many regulatory environments because of its interpretability. Regulators and stakeholders can understand odds ratios and assess whether rating factors are actuarially justified.

Study Tips for Exam SRM

For Exam SRM, practice these essential skills. First, be able to compute predicted probabilities from given coefficient estimates by plugging values into the logit formula and applying the logistic function. Second, interpret odds ratios correctly, distinguishing between odds and probability. Third, understand the likelihood ratio test for comparing nested logistic models. Fourth, know the relationship between logistic regression and the broader GLM framework. Common exam question patterns include: computing the predicted probability for a new observation, interpreting an odds ratio in context, choosing between the Wald test and likelihood ratio test, and identifying the appropriate link function. Practice converting between probability, odds, and log-odds fluently, as this is the foundation of all logistic regression calculations.