Modeling and Multivariate Methods - SAS

680 Statistical Details Appendix A
Multivariate Details
Figure A.10 Cases of Significant, Borderline, and Nonsignificant Confidence Curves.
Significant Borderline Not Significant
confidence curve
crosses horizontal
line
confidence curve
asymptotic to
horizontal line
confidence curve
does not cross
horizontal line
Leverage plots make use of the same device by calculating a confidence function with respect to one variable
while holding all other variables constant at their sample mean value.
For general hypothesis tests, JMP can display a curve with the same properties: the confidence function
shows at the mean value in the middle, with adjusted curvature so that it crosses the horizon if the F-test is
significant at some α-level like 0.05.
Consider the functions
Upper(z) = z s 2 2 F α
+ t α ⁄ 2
h + ------ 2 z
F
and
Lower (z) = z s 2 2 F α
– t α ⁄ 2
h + ------ z 2
F
where F is the F-statistic for the hypothesis, F α is the reference value for significance α, and
h = xX'X ( ) – 1 x' , where x is the regressors at a suitable middle value, such as the mean.
These functions serve the same properties listed above. If the F-statistic is greater than the reference value,
the confidence functions cross the x-axis. If the F-statistic is less than the reference value, the confidence
functions do not cross. If the F-statistic is equal to the reference value, the confidence functions have the
x-axis as an asymptote. And the range between the confidence functions at the middle value is a valid
confidence region of the predicted value at the point.
Multivariate Details
The following sections show computations used for multivariate tests and related, exact and approximate
F-statistics, canonical details, and discriminant functions. In the following sections, E is the residual cross
E
product matrix and ----------- estimates the residual covariance matrix. Diagonal elements of E are the sum of
n – 1
squares for each variable. In discriminant analysis literature, this is often called W, where W stands for
within.