Residual Analysis
Residual Analysis
The concepts behind residual analysis for a multiple linear regression model are similar to those for a simple linear regression model. However, they are much more important for the multiple linear regression models because of the lack of good graphical representations of the data set and the fitted model. In simple linear regression a plot of the response variable against the input variable showing the data points and the fitted regression line provides a good graphical summary of the regression analysis.
With the higher dimensions of a multiple linear regression model, similar plots cannot be obtained. Plots of the residuals against the fitted values and against the input variables may alert the experimenter to any problems with the regression model. If all is well with the regression model, then these residual plots should exhibit a random scatter of points. Any patterns in the residual plots should draw the experimenter’s attention and should be investigated. For example, a funnel shape to a plot of the residuals against the fitted values indicates a lack of homogeneity of the error variance. A series of negative, positive, and then negative values in a plot of the residuals against an input variable suggests that the model can be improved with the addition of a quadratic term.
In certain cases where the data observations are obtained sequentially over time, it is also prudent to plot the residuals against a time axis, which designates the order in which the observations are taken. Most residual plot indicates that there is a lack of independence in the error terms, with adjacent observations being positively correlated. Normal probability plots can always be used to investigate whether there is any indication that the error terms are not normally distributed.
The standardized residuals can be used to identify individual data points that do not fit the model well. Typically, computer packages alert the experimenter to points
The concepts behind residual analysis for a multiple linear regression model are similar to those for a simple linear regression model. However, they are much more important for the multiple linear regression models because of the lack of good graphical representations of the data set and the fitted model. In simple linear regression a plot of the response variable against the input variable showing the data points and the fitted regression line provides a good graphical summary of the regression analysis.
With the higher dimensions of a multiple linear regression model, similar plots cannot be obtained. Plots of the residuals against the fitted values and against the input variables may alert the experimenter to any problems with the regression model. If all is well with the regression model, then these residual plots should exhibit a random scatter of points. Any patterns in the residual plots should draw the experimenter’s attention and should be investigated. For example, a funnel shape to a plot of the residuals against the fitted values indicates a lack of homogeneity of the error variance. A series of negative, positive, and then negative values in a plot of the residuals against an input variable suggests that the model can be improved with the addition of a quadratic term.
In certain cases where the data observations are obtained sequentially over time, it is also prudent to plot the residuals against a time axis, which designates the order in which the observations are taken. Most residual plot indicates that there is a lack of independence in the error terms, with adjacent observations being positively correlated. Normal probability plots can always be used to investigate whether there is any indication that the error terms are not normally distributed.
The standardized residuals can be used to identify individual data points that do not fit the model well. Typically, computer packages alert the experimenter to points