Added Variable Plot

Study guide: Exam MAS-I

An Added Variable Plot (partial regression plot) isolates the relationship between a single predictor $x_j$ and the response after removing (partialling out) the linear effects of all other predictors in the model — used to see $x_j$ 's adjusted effect and to spot influential observations.

$e_Y = Y - \hat{Y}^{(-j)}, \qquad e_{x_j} = x_j - \hat{x}_j^{(-j)}$

$\text{Plot } e_Y \text{ vs. } e_{x_j} \text{; the slope equals } \hat{\beta}_j \text{ from the full model}$

$\hat{Y}^{(-j)}$ and $\hat{x}_j^{(-j)}$ are the fitted values from regressing $Y$ , and $x_j$ respectively, on all other predictors — the residuals $e_Y$ and $e_{x_j}$ represent what's left after removing those other predictors' effects
The least squares line through the plot has intercept $0$ and slope $\hat{\beta}_j$ , and its residuals are the full model's residuals. Curvature can flag non-linearity in $x_j$ , but the plot is a weak guide to the functional form: its horizontal axis is not $x_j$ , and its construction forces the relationship to be linear. A partial residual (component + residual) plot is the tool for choosing a transformation
Unlike a simple scatter plot of $Y$ against $x_j$ , the added variable plot isolates $x_j$ 's partial (adjusted) contribution after controlling for the other predictors, including any Control Variables
Points far out on the $e_{x_j}$ axis are high-leverage observations for $x_j$ ; points far from the fitted line are outliers (large residuals). A point that is both — at the left or right of the plot and off the general trend — is likely influential and worth investigating, complementing what a Residual Plot shows for overall fit

Reading an Added Variable Plot The added variable plot for vehicle age (after controlling for territory and driver age) shows a clear U-shape rather than a straight line. What does this indicate, and what model change would address it?

Answer The U-shape suggests that vehicle age's effect on the response, net of the other predictors, is not linear, so a single linear term may be misspecified. An added variable plot is only a rough guide to the form, so confirm the curve with a partial residual plot of vehicle age; if it is there, a quadratic term (or banding vehicle age) in the Model Structure would capture the relationship better.