where Y is the matrix of response variables, M is the response design matrix, and V is the matrix of eigenvectors of 
for the given test. Canonical Y’s are saved for eigenvectors corresponding to eigenvalues larger than zero.
for the given test. Canonical Y’s are saved for eigenvectors corresponding to eigenvalues larger than zero.
.
where
and the vs are columns of V, the eigenvector matrix of 
, 
refers to the multivariate least squares mean for the jth effect, g is the number of eigenvalues of 
greater than 0, and r is the rank of the X matrix.
, refers to the multivariate least squares mean for the jth effect, g is the number of eigenvalues of greater than 0, and r is the rank of the X matrix.
where g is the number of eigenvalues of 
greater than 0 and the denominator L’s are from the multivariate least squares means calculations.
greater than 0 and the denominator L’s are from the multivariate least squares means calculations.