Alternative weights and invariant parameters in optimal scaling

Under conditions that are commonly satisfied in optimal scaling problems, arbitrary sets of optimal weights can be obtained by choices of generalized inverse procedures. A simple relationship holds between these and the corresponding invariant item scores. The case of optimal scaling originally treated by Guttman [1941] yields a restricted form of multicategory factor analysis. It is suggested that the invariant parameters of optimal scaling should be interpreted, according to the principles of latent trait theory, rather than the arbitrary weights.This paper benefits from a number of suggestions and comments made by Professors M. J. R. Healy, H. Goldstein, and S. Nishisato, and by Mr. C. Fraser, to whom grateful acknowledgments are due. The author is solely responsible for the final form of the paper, including of course such errors as may remain in it.This research was partly supported by Grant No. A6346 from the Natural Sciences and Engineering Research Council of Canada.