In the following, E is the residual cross product matrix. Diagonal elements of E are the residual sums of squares for each variable. In the discriminant analysis literature, this is often called W, where W stands for within.
Test statistics in the multivariate results tables are functions of the eigenvalues λ of . The following list describes the computation of each test statistic.
Note: After specification of a response design, the initial E and H matrices are premultiplied by and postmultiplied by M.
 Wilks’ Lambda Pillai’s Trace Hotelling-Lawley Trace Roy’s Max Root , the maximum eigenvalue of .
The whole model L is a column of zeros (for the intercept) concatenated with an identity matrix having the number of rows and columns equal to the number of parameters in the model. L matrices for effects are subsets of rows from the whole model L matrix.
To compute F-values and degrees of freedom, let p be the rank of . Let q be the rank of , where the L matrix identifies elements of associated with the effect being tested. Let v be the error degrees of freedom and s be the minimum of p and q. Also let and .
Approximate F-statistics, gives the computation of each approximate F from the corresponding test statistic.
 Test Approximate F Numerator DF Denominator DF Wilks’ Lambda Pillai’s Trace Hotelling-Lawley Trace Roy’s Max Root
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.