Modeling and Multivariate Methods - SAS

678 Statistical Details Appendix A
Leverage Plot Details
Figure A.8 A Mosaic Plot for Categorical Data
Leverage Plot Details
Leverage plots for general linear hypotheses were introduced by Sall (1980). This section presents the details
from that paper. Leverage plots generalize the partial regression residual leverage plots of Belsley, Kuh, and
Welsch (1980) to apply to any linear hypothesis. Suppose that the estimable hypothesis of interest is
Lβ = 0
The leverage plot characterizes this test by plotting points so that the distance of each point to the sloped
regression line displays the unconstrained residual, and the distance to the x-axis displays the residual when
the fit is constrained by the hypothesis.
Of course the difference between the sums of squares of these two groups of residuals is the sum of squares
due to the hypothesis, which becomes the main ingredient of the F-test.
The parameter estimates constrained by the hypothesis can be written
b 0 = b – ( X'X) – 1 L'λ
where b is the least squares estimate
b
= ( X'X) – 1 X'y
and where lambda is the Lagrangian multiplier for the hypothesis constraint, which is calculated
λ
=
( LX'X ( ) – 1 L' ) – 1 Lb)
Compare the residuals for the unconstrained and hypothesis-constrained residuals, respectively.
r = y–
Xb
r 0
= r + X( X'X) – 1 L'λ