Residuals vs Predicted Plot
This is a plot of the residuals versus the ascending predicted response values. It is a visual check for the assumption of constant variance. The plot should be a random scatter having a consistent top to bottom range of residuals across the predictions on the X1 axis.
An expanding variance (“megaphone pattern <”), along with smiles and frowns on this plot indicates the need for a transformation.
This plot is also used to examine outliers. Outliers are runs with residuals outside the red lines on the plot. An outlier is an observation that is not fit well by the model. This can mean the observation has a problem, the wrong model has been used, or some combination of both.
Externally Studentized Residual Limits
For externally studentized residuals the limits on the residual plots are Student’s t-quantiles,
\(\pm\,{\large t}\left(\frac{\alpha}{2n},\, n-p-1\right)\)
where n is the number of runs and p is the number of coefficients in the model including the intercept.
The n is used as both a basis for the degrees of freedom, and the Bonferroni correction to a family-wise alpha-risk, \(\alpha\), for all the runs.
See Weisberg (2014), Chapter 9, pages 217-218 for more information.
Note
Never ignore a run just because the diagnostics plots indicate it may be a problem. Verify that the data is wrong in some way before ignoring it.
References
Sanford Weisberg. Applied Linear Regression. John Wiley & Sons, Inc., fourth edition, 2014.