Interaction

Study guide: Exam MAS-I

An Interaction between two predictors in a Generalized Linear Model exists when the effect of one predictor on the response depends on the level or value of the other — the combined effect is not simply the sum of each predictor's separate (main) effect.

$\eta = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 (x_1 \times x_2)$

$\beta_3$ is the interaction coefficient: it measures how much the effect of $x_1$ on $\eta$ changes per unit increase in $x_2$ (and symmetrically for $x_2$ given $x_1$ )
If $\beta_3 = 0$ , the two predictors act additively on the link scale — no interaction
Interactions can occur between two categorical predictors (the effect differs by combination of levels), a categorical and a continuous predictor (the slope differs by group), or two continuous predictors
Interaction terms add to the Model Structure and are tested for significance via the parameter estimate table or a nested-model ANOVA (likelihood-ratio) test; unnecessary interactions increase the risk of overfitting. An interaction with a categorical predictor of more than two levels has several coefficients; test them jointly with the nested-model test, since the individual coefficients and their p-values depend on which level is the base
Hierarchical principle: if an interaction is included, its main effects are included too, even if their p-values are not significant. $x_1 \times x_2$ is typically correlated with $x_1$ and $x_2$ , so leaving the main effects out tends to alter the meaning of the interaction

Interpreting an Interaction Coefficient A Poisson GLM for claim frequency includes vehicle age ( $x_1$ ), an indicator for sports car ( $x_2$ ), and their interaction, with $\hat{\beta}_1 = -0.02$ , $\hat{\beta}_2 = 0.10$ , $\hat{\beta}_3 = 0.03$ . How does the effect of one additional year of vehicle age on log-frequency differ for sports cars versus non-sports cars?

Answer For non-sports cars ( $x_2 = 0$ ), one more year of age changes $\eta$ by $\hat{\beta}_1 = -0.02$ . For sports cars ( $x_2 = 1$ ), one more year of age changes $\eta$ by $\hat{\beta}_1 + \hat{\beta}_3 = -0.02 + 0.03 = 0.01$ . The interaction reverses the sign of the age effect for sports cars: frequency decreases with age for non-sports cars but slightly increases with age for sports cars.