Residual Plot
Study guides: Exam MAS-I, Exam PCPA
A Residual Plot is a diagnostic graph used to assess model fit by plotting residuals (observed minus fitted values) against fitted values, a predictor, or another variable. Patterns in residuals indicate model deficiencies.
e_i = y_i - \hat{y}_i \quad \text{(raw residual)}
r_i = \frac{e_i}{\hat{\sigma}\sqrt{1 - h_{ii}}} \quad \text{(standardized/studentized residual)}
How to read residual plots:
| Pattern | Implication |
|---|---|
| Random scatter around zero | Model is adequate |
| Fan shape (spread increases) | Heteroscedasticity — variance is not constant |
| Curved pattern | Non-linearity — a transform or polynomial term is needed |
| Systematic trend | A predictor is missing or improperly specified |
| Isolated points far from zero | Outliers; whether one is influential depends on its leverage h_{ii}, which a residual plot does not show — check h_{ii} or Cook's distance |
- For Generalized Linear Models, Pearson residuals
r_i^P = (y_i - \hat{\mu}_i)/\sqrt{V(\hat{\mu}_i)}and deviance residuals are used instead of raw residuals - Residuals vs. fitted: checks for non-linearity and heteroscedasticity
- Residuals vs. predictor: checks whether a variable's effect is correctly specified
- A Normal QQ Plot of residuals checks the normality assumption
Identifying Heteroscedasticity in a Residual Plot After fitting a linear regression of claim severity on vehicle value, the residual plot shows residuals spreading out in a fan shape as fitted values increase. What does this suggest?
Answer The fan shape indicates heteroscedasticity: variance of residuals increases with the fitted value. This is typical for positive, right-skewed data like claim severity. Remedies include applying a log transformation to severity or switching to a Gamma Generalized Linear Model with a log Link Function, which naturally models multiplicative variance.