## 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
*j*th 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

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

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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