Partial residual plot
Encyclopedia
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.
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.
where
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.
This is the "component" part of the plot and is intended to show where the "fitted line" would lie.
Background
When performing a linear regressionLinear regression
In statistics, linear regression is an approach to modeling the relationship between a scalar variable y and one or more explanatory variables denoted X. The case of one explanatory variable is called simple regression...
with a single independent variable
Independent variable
The terms "dependent variable" and "independent variable" are used in similar but subtly different ways in mathematics and statistics as part of the standard terminology in those subjects...
, 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:where
- Residuals = residualsErrors and residuals in statisticsIn statistics and optimization, statistical errors and residuals are two closely related and easily confused measures of the deviation of a sample from its "theoretical value"...
from the full model
= 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, addingThis is the "component" part of the plot and is intended to show where the "fitted line" would lie.
See also
- partial regression plotPartial regression plotIn applied statistics, a partial regression plot attempts to show the effect of adding an additional variable to the model...
- partial leverage plotPartial leverage plotIn statistics, high-leverage points are those that are outliers with respect to the independent variables. Leverage points are those that cause large changes in the parameter estimates when they are deleted. Although a leverage point will typically have high leverage, a high leverage point is not...
- variance inflation factorVariance inflation factorIn statistics, the variance inflation factor quantifies the severity of multicollinearity in an ordinary least squares regression analysis...
s for a multi-linear fit. - scatter plot matrix