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.

where V is the matrix of eigenvectors of .

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.

where g is the number of eigenvalues of greater than 0 and the denominator L’s are from the multivariate least squares means calculations.