Some generalized methods of optimal scaling and their asymptotic theories: The case of multiple responses-multiple factors

Seven generalized criteria are proposed with the corresponding formulations for optimal scaling of multiple responses. Then, on the basis of a natural probabilistic model, the asymptotic theories are derived concerning the test statistics on factor-response relationships as well as the distributions of sample criteria (eigenvalues of the determinantal equation) and optimal scores (eigenvectors) by means of socalled δ-method. A numerical example is provided for illustrations.