Model Structure

Study guides: Exam MAS-I, Exam MAS-II

Model Structure is the specified form of an Extended Linear Model: the distribution of the response, the Link Function, and the terms of the linear predictor — which predictors, transformations, and interactions appear. This is the CAS MAS-I usage: choosing a GLM's model structure from the behavior of the data includes choosing its link function and the distribution of the dependent variable.

Y \sim \text{chosen distribution}, \qquad g(\mu) = \eta, \qquad \eta = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 (x_1 \times x_2) + \beta_4 x_1^2 + \cdots

• \mu = E(Y) and g is the link. The distribution and link are chosen from the behavior of the response; Model Selection covers how
• Linear-predictor decisions include: which predictors to include, whether to add interaction terms, and whether to band or transform continuous variables (e.g., polynomial terms) — the choice of terms is the focus of Variable Selection
• For a Linear Mixed Model (MAS-II), the structure also covers how the model accommodates correlated observations and a variance that is not constant; the MAS-II outline names the model structure and variable selection as separate things to evaluate
• A model can have the correct distribution and link but the wrong linear predictor (e.g., a missing interaction); its adequacy is checked with a Residual Plot or a Marginal Model Plot
• Adding terms always improves in-sample fit but risks overfitting; AIC and BIC penalize unnecessary complexity when comparing structures
• Two models with the same distribution and link, one's linear predictor nested within the other's, are compared using ANOVA (analysis of deviance)

Comparing Two Candidate Model Structures An actuary fits a Poisson GLM for claim frequency two ways: Model 1 uses main effects for territory and vehicle age only (AIC = 4{,}102). Model 2 adds a territory \times vehicle age interaction (AIC = 4{,}085). Which model structure is preferred?

Answer Lower AIC indicates a better trade-off between fit and complexity. Since Model 2's AIC (4{,}085) is lower than Model 1's (4{,}102), the richer model structure — including the territory \times vehicle age Interaction — is preferred.

Territory is categorical, so the interaction is a set of coefficients, at least one per non-base territory. Judge them jointly, as this AIC comparison does or as a nested F-test of all of them at once would, not one p-value at a time in the parameter estimate table: the individual coefficients and their p-values change with the choice of base territory, and the joint test does not.