Nonlinear programming approach to optimal scaling of partially ordered categories

A modified technique of separable programming was used to maximize the squared correlation ratio of weighted responses to partially ordered categories. The technique employs a polygonal approximation to each single-variable function by choosing mesh points around the initial approximation supplied by Nishisato's method. The major characteristics of this approach are: (i) it does not require any grid refinement; (ii) the entire process of computation quickly converges to the acceptable level of accuracy, and (iii) the method employs specific sets of mesh points for specific variables, whereby it reduces the number of variables for the separable programming technique. Numerical examples were provided to illustrate the procedure.     

Piecewise method of reciprocal averages for dual scaling of multiple-choice data

The proposed method handles the classical method of reciprocal averages (MRA) in a piecewise (item-by-item) mode, whereby one can deal with smaller matrices and attain faster convergence to a solution than the MRA. A new concept the principle of constant proportionality is introduced to provide an interesting interpretation for scaling multiple-choice data a la Guttman. A small example is presented for discussion of the technique.This study was supported by the Natural Sciences and Engineering Research Council Canada Grant (No. A4581) to S. Nishisato. The authors are indebted to reviewers for valuable comments.     

Some results on proper eigenvalues and eigenvectors with applications to scaling

Conditions are given under which the stationary points and values of a ratio of quadratic forms in two singular matrices can be obtained by a series of simple matrix operations. It is shown that two classes of optimal weighting problems, based respectively on the grouping of variables and on the grouping of observations, satisfy these conditions. The classical treatment of optimal scaling of forced-choice multicategory data is extended for these cases. It is shown that previously suggested methods based on reparameterization will work only under very special conditions.