Modeling and Multivariate Methods - SAS

682 Statistical Details Appendix A
Multivariate Details
Table A.17 Approximate F-statistics (Continued)
Hotelling-Lawley
Trace
F
=
---------------------------------
2( sn + 1)U
s 2 ( 2m + s+
1)
s( 2m+ s + 1) 2( sn + 1)
Roy’s Max Root
F
=
Θ
--------------------------------------------------
( v – max( pq , ) + q)
max( pq , )
max( pq , ) v – max( pq , ) + q
Canonical Details
The canonical correlations are computed as
ρ i
=
λ
------------- i
1 + λ i
The canonical Y’s are calculated as
Ỹ
=
YMV
where Y is the matrix of response variables, M is the response design matrix, and V is the matrix of
eigenvectors of E – 1 H . Canonical Y’s are saved for eigenvectors corresponding to eigenvalues larger than
zero.
The total sample centroid is computed as
Grand = yMV
where V is the matrix of eigenvectors of E – 1 H .
The centroid values for effects are calculated as
E
m = ( c' 1 x j , c' 2 x j , …,
c' g x j ) wherec i = v' i
-----------
N–
r
v – 1 ⁄ 2
i
v i
and the vs are columns of V, the eigenvector matrix of E – 1 H , x j refers to the multivariate least squares
mean for the jth effect, g is the number of eigenvalues of E – 1 H greater than 0, and r is the rank of the X
matrix.
The centroid radii for effects are calculated as
d =
2
χ g ( 0.95)
-----------------------------
LX'X ( ) – 1 L'
where g is the number of eigenvalues of E – 1 H greater than 0 and the denominator L’s are from the
multivariate least squares means calculations.