# Partial residual plot

In applied statistics, a partial residual plot is a graphical technique that attempts to show the relationship between a given independent variable and the response variable given that other independent variables are also in the model.

## Background

When performing a linear regression with a single independent variable, a scatter plot of the response variable against the independent variable provides a good indication of the nature of the relationship. If there is more than one independent variable, things become more complicated. Although it can still be useful to generate scatter plots of the response variable against each of the independent variables, this does not take into account the effect of the other independent variables in the model.

## Definition

Partial residual plots are formed as:

$\mathrm{Residuals} + \hat{\beta}_iX_i \mathrm{\ versus\ } X_i$

where

Residuals = residuals from the full model
$\hat{\beta}_i$ = regression coefficient from the ith independent variable in the full model
Xi = the ith independent variable

Partial residual plots are widely discussed in the regression diagnostics literature (e.g., see the References section below). Although they can often be useful, they can also fail to indicate the proper relationship. In particular, if Xi is highly correlated with any of the other independent variables, the variance indicated by the partial residual plot can be much less than the actual variance. These issues are discussed in more detail in the references given below.

## CCPR plot

The CCPR (component and component-plus-residual) plot is a refinement of the partial residual plot, adding

$\hat{\beta}_iX_i \mathrm{\ versus\ } X_i.$

This is the "component" part of the plot and is intended to show where the "fitted line" would lie.

## References

• Tom Ryan (1997). Modern Regression Methods. John Wiley.
• Neter, Wasserman, and Kutner (1990). Applied Linear Statistical Models (3rd ed.). Irwin.
• Draper and Smith (1998). Applied Regression Analysis (3rd ed.). John Wiley.
• Cook and Weisberg (1982). Residuals and Influence in Regression. Chapman and Hall.
• Belsley, Kuh, and Welsch (1980). Regression Diagnostics. John Wiley.
• Paul Velleman; Roy Welsch (November 1981). "Efficient Computing of Regression Diagnostics". The American Statistician (American Statistical Association) 35 (4): 234–242. doi:10.2307/2683296. JSTOR 2683296.