This section gives formulas for the multivariate test statistics, describes the approximate F-tests, and provides details on canonical scores.

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.

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.

Approximate F-Tests

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.

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.