Partial Leverage Plots

Fit Analyses

Partial Leverage Plots

For linear models, the partial leverage plot for a selected explanatory variable can be obtained by plotting the residuals for the response variable against the residuals for the selected explanatory variable. The residuals for the response variable are calculated from a model having the selected explanatory variable omitted, and the residuals for the selected explanatory variable are calculated from a model where the selected explanatory variable is regressed on the remaining explanatory variables.

Let X_[j] be the n×(p-1) matrix formed from the design matrix X by removing the jth column, X_j. Let r_y[j] be the partial leverage Y variable containing the residuals that result from regressing y on X_[j] and let r_x[j] be the partial leverage X variable containing the residuals that result from regressing X_j on X_[j]. Then a partial leverage plot is a scatter plot of r_y[j] against r_x[j]. Partial leverage plots for two explanatory variables are illustrated by Figure 39.26.

Figure 39.26: Partial Leverage Plots

In a partial leverage plot, the partial leverage Y variable r_y[j] can also be computed as

$r_{y[j]i} = r_{x[j]i} b_{j} + ( y_{i}-\hat{ \mu_{i}})$

For generalized linear models, the partial leverage Y is also computed as

$r_{y[j]i} = r_{x[j]i} b_{j} + ( y_{i}-\hat{ \mu_{i}}) {g'}(\hat{ \mu_{i}})$

Two reference lines are also displayed in the plots. One is the horizontal line of Y = 0, and the other is the fitted regression of r_y[j] against r_x[j]. The latter has an intercept of 0 and a slope equal to the parameter estimate associated with the explanatory variable in the model. The leverage plot shows the changes in the residuals for the model with and without the explanatory variable. For a given data point in the plot, its residual without the explanatory variable is the vertical distance between the point and the horizontal line; its residual with the explanatory variable is the vertical distance between the point and the fitted line.

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