Marginal Model Plot
Study guide: Exam MAS-I
A Marginal Model Plot is a regression diagnostic that overlays a smooth curve fit directly to the raw data against the smooth curve implied by the fitted Generalized Linear Model, both plotted against a predictor or the fitted values — large divergence between the two curves signals model misspecification.
\text{Marginal model plot: compare } \widehat{E}[Y \mid x] \text{ (data smooth) to } \hat{\mu}(x) \text{ (model smooth)}
- Unlike a Residual Plot, which plots residuals, a marginal model plot directly compares two fitted curves — one nonparametric smooth of the data and one implied by the model — on the original response scale
- Close agreement between the two smooths across the range of
xindicates the model adequately captures the relationship; systematic gaps indicate missing Model Structure (e.g., a missing Interaction or a needed transformation) - Marginal model plots can be drawn against any predictor, or against the fitted values to check overall calibration
- The same plot serves a GLM: the response stays on the vertical axis, the smoother is a generalized additive model with the model's own family, link and weights (the default in R's
car::mmps), and the overall plot is drawn against the linear predictor
Detecting Missing Structure with a Marginal Model Plot A Gamma GLM models claim severity as a linear function of vehicle age. The marginal model plot shows the data smooth curving upward sharply for very new and very old vehicles, while the model's smooth (fit with a single linear term) stays nearly flat in those regions. What does this suggest?
Answer The divergence between the two smooths at the extremes of vehicle age indicates the linear term is misspecified — severity is not linear in vehicle age. A quadratic or banded term for vehicle age (richer Model Structure) would likely bring the model's smooth closer to the data smooth.