Partial regression plot

In applied statistics, a partial regression plot attempts to show the effect of adding an additional variable to the model (given that one or more indpendent variables are already in the model). Partial regression plots are also referred to as added variable plots, adjusted variable plots, and individual coefficient plots.

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.

Partial regression plots are formed by:
#Compute the residuals of regressing the response variable against the independent variables but omitting X_i
#Compute the residuals from regressing "X"_i against the remaining independent variables
#Plot the residuals from (1) against the residuals from (2). Velleman and Welsch (see References below) express this mathematically as::where:"Y"_{. [i]} = residuals from regressing Y (the response variable) against all the independent variables except Xi:"X"_{i. [i]} = residuals from regressing "X"_i against the remaining independent variables. Velleman and Welsch list the following useful properties for this plot:
#The least squares linear fit to this plot has the slope Betai and intercept zero.
#The residuals from the least squares linear fit to this plot are identical to the residuals from the least squares fit of the original model (Y against all the independent variables including Xi).
#The influences of individual data values on the estimation of a coefficient are easy to see in this plot.
#It is easy to see many kinds of failures of the model or violations of the underlying assumptions (nonlinearity, heteroscedasticity, unusual patterns).

Partial regression plots are widely discussed in the regression diagnostics literature (e.g., see the References section below). Since the strengths and weaknesses of partial regression plots are widely discussed in the literature, we will not discuss that in any detail here.

Partial regression plots are related to, but distinct from, partial residual plots. Partial regression plots are most commonly used to identify data points with high leverage and influential data points that might not have high leverage. Partial residual plots are most commonly used to identify the nature of the relationship between "Y" and "X"_i (given the effect of the other independent variables in the model). Note that since the simple correlation between the two sets of residuals plotted is equal to the partial correlation between the response variable and "X"_i, partial regression plots will show the correct strength of the linear relationship between the response variable and "X"_i. This is not true for partial residual plots. On the other hand, for the partial regression plot, the x-axis is not "X"_i. This limits its usefulness in determining the need for a transformation (which is the primary purpose of the partial residual plot).

ee also

*Partial residual plot
*Partial leverage plot
*Variance inflation factor for a multi-linear fit.
*Scatter plot matrix

External links

* [http://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/partregr.htm Partial Regression Plot]

References

* cite book
title = Modern Regression Methods
author = Tom Ryan
publisher = John Wiley
year = 1997
* cite book
title = Applied Linear Statistical Models
edition = 3rd ed.
author = Neter, Wasserman, and Kunter
year = 1990
publisher = Irwin
* cite book
title = Applied Regression Analysis
edition = 3rd ed.
author = Draper and Smith
publisher = John Wiley
year = 1998
* cite book
title = Residuals and Influence in Regression
author = Cook and Weisberg
publisher = Chapman and Hall
year = 1982
* cite book
title = Regression Diagnostics
author = Belsley, Kuh, and Welsch
publisher = John Wiley
year = 1980
* cite journal
title = Efficient Computing of Regression Diagnostiocs
author = Paul Velleman
coauthor = Roy Welsch
journal = The American Statistician
month = November
year = 1981
volume = 35
number = 4
pages = 234-242