Modeling and Multivariate Methods - SAS

Chapter 3 Fitting Standard Least Squares Models 103
Row Diagnostics
The points on a leverage plot for simple regression are actual data coordinates, and the horizontal line for
the constrained model is the sample mean of the response. But when the leverage plot is for one of multiple
effects, the points are no longer actual data values. The horizontal line then represents a partially constrained
model instead of a model fully constrained to one mean value. However, the intuitive interpretation of the
plot is the same whether for simple or multiple regression. The idea is to judge if the line of fit on the effect’s
leverage plot carries the points significantly better than does the horizontal line.
Figure 3.38 is a general diagram of the plots in Figure 3.39. Recall that the distance from a point to the line
of fit is the actual residual and that the distance from the point to the mean is the residual error if the
regressor is removed from the model.
Confidence Curves
The leverage plots are shown with confidence curves. These indicate whether the test is significant at the 5%
level by showing a confidence region for the line of fit. If the confidence region between the curves contains
the horizontal line, then the effect is not significant. If the curves cross the line, the effect is significant.
Compare the examples shown in Figure 3.40.
Figure 3.40 Comparison of Significance Shown in Leverage Plots
Significant Borderline Not Significant
confidence curve
crosses horizontal
line
confidence curve is
asymptotic to
horizontal line
confidence curve
does not cross
horizontal line
Interpretation of X Scales
If the modeling type of the regressor is continuous, then the x-axis is scaled like the regressor and the slope
of the line of fit in the leverage plot is the parameter estimate for the regressor. (See the left illustration in
Figure 3.41.)
If the effect is nominal or ordinal, or if a complex effect like an interaction is present instead of a simple
regressor, then the x-axis cannot represent the values of the effect directly. In this case the x-axis is scaled like
the y-axis, and the line of fit is a diagonal with a slope of 1. The whole model leverage plot is a version of
this. The x-axis turns out to be the predicted response of the whole model, as illustrated by the right-hand
plot in Figure 3.41.