Generalized Linear Models in Depth for MAS-I

GLM Components

A generalized linear model (GLM) extends ordinary linear regression to handle response variables that are not normally distributed. Every GLM has three components. The random component specifies the distribution of the response variable, which must belong to the exponential family: Normal, Poisson, Binomial, Gamma, or Inverse Gaussian. The systematic component is the linear predictor eta = X * beta, a linear combination of the predictor variables. The link function g connects the mean response to the linear predictor: g(mu) = eta.

The exponential family form is f(y; theta, phi) = exp((y*theta - b(theta))/a(phi) + c(y, phi)), where theta is the natural parameter, phi is the dispersion parameter, and b(theta) determines the mean and variance through E[Y] = b'(theta) and Var(Y) = b''(theta) * a(phi). The variance function V(mu) = b''(theta) characterizes the mean-variance relationship, which is a defining feature of each distribution family.

Link Functions and Canonical Links

The link function g(mu) = eta maps the mean of the response to the linear predictor. Any monotonic, differentiable function can serve as a link, but the canonical link (where g(mu) = theta, the natural parameter) simplifies estimation and has desirable statistical properties. The canonical links are: identity (g(mu) = mu) for Normal, log (g(mu) = ln(mu)) for Poisson, logit (g(mu) = ln(mu/(1-mu))) for Binomial, and reciprocal (g(mu) = 1/mu) for Gamma.

The choice of link function determines how predictors affect the mean. With a log link, the model predicts ln(mu) = X*beta, so mu = exp(X*beta) and effects are multiplicative. With an identity link, mu = X*beta and effects are additive. For insurance applications, the log link is natural for claim frequency (ensuring positive predicted counts) and claim severity (ensuring positive predicted amounts). The logit link maps probabilities to the entire real line, making it appropriate for binary outcomes.

Insurance Applications of GLMs

GLMs are the standard tool for insurance pricing and classification. Claim frequency is modeled using a Poisson GLM with a log link and an offset term for exposure (the natural log of the number of policy-years). The model predicts the expected number of claims per unit of exposure. Claim severity is modeled using a Gamma GLM with a log link. The Gamma distribution naturally handles the right-skewed, positive-valued nature of claim amounts.

Rating factors enter the model as predictor variables. Categorical factors (territory, vehicle type) are represented by indicator variables. The exponentiated coefficient exp(beta_j) gives the multiplicative relativity for factor level j compared to the reference level. For example, if the territory coefficient for "urban" is 0.25, then urban territory has a claim frequency relativity of exp(0.25) = 1.284, meaning 28.4% more claims per exposure than the reference territory.

Estimation via IRLS

GLM parameters are estimated by maximum likelihood. Unlike ordinary linear regression, there is generally no closed-form solution, so an iterative algorithm is used. Iteratively Reweighted Least Squares (IRLS) is the standard method. At each iteration, the algorithm constructs a working response variable and a weight matrix based on the current parameter estimates, then performs a weighted least squares regression. The process repeats until convergence.

The convergence of IRLS is guaranteed for canonical links and typically occurs within 5 to 15 iterations. The resulting parameter estimates are maximum likelihood estimates with all the associated properties: consistency, asymptotic normality, and asymptotic efficiency.

Deviance and Model Comparison

The deviance D = 2 * (ln L_saturated - ln L_model) measures the discrepancy between the fitted model and a saturated model (which has one parameter per observation and fits the data perfectly). The deviance plays a role analogous to the residual sum of squares in linear regression. A large deviance relative to its degrees of freedom suggests poor fit.

For comparing nested models, the deviance difference D_reduced - D_full follows approximately a chi-squared distribution with degrees of freedom equal to the difference in the number of parameters. This provides a likelihood ratio test for whether the additional parameters in the full model significantly improve the fit. AIC = D + 2k and BIC = D + k * ln(n) balance fit against complexity for non-nested model comparisons.

Residual Analysis

GLMs have two main types of residuals. Pearson residuals are r_P = (y_i - mu-hat_i) / sqrt(V(mu-hat_i)), which standardize the raw residual by the estimated standard deviation. Deviance residuals are defined so that the sum of their squares equals the total deviance. Deviance residuals tend to be more normally distributed than Pearson residuals, making them better for diagnostic plots.

Residual plots for GLMs follow the same logic as for linear regression: plot residuals against fitted values and against each predictor to check for patterns. A well-fitting model produces residuals that scatter randomly around zero with no discernible pattern. Systematic patterns suggest missing predictors, incorrect link function, or wrong distributional assumption.

Common Pitfalls

A frequent mistake is using an identity link for count data, which can produce negative predicted counts. Always use a log link for Poisson and Gamma models in insurance applications. Another common error is forgetting the exposure offset in frequency models. Without the offset, the model treats each policy identically regardless of how long it was in force, biasing the results.

Students sometimes confuse deviance with the deviance statistic divided by its degrees of freedom. The deviance itself is a total measure; dividing by degrees of freedom gives an estimate of the dispersion parameter (for distributions with a free dispersion parameter like Gamma, but not for Poisson where phi = 1 by definition).

Study Tips

Exam MAS-I tests GLM setup for insurance applications, parameter interpretation, and model comparison. Practice specifying the appropriate distribution and link function for different insurance applications. Know how to interpret exponentiated coefficients as multiplicative relativities. Be comfortable with deviance-based model comparisons and understand the role of the dispersion parameter. Work through problems that require reading GLM output tables and making pricing decisions based on the results.