Review of the methods of quantification.

In order to analyze qualitative observations, methods of quantification or optimal scaling have been proposed by Fisher, Guttman, and Hayashi. According to these methods, scores are assigned optimally in some objective and operational sense to the qualitative categories. The present paper mainly reviews Hayashi's four methods of quantification from the mathematical point of view. They are widely used, especially in Japan, in various fields such as social and marketing surveys, psychological research, medical research, etc., where information is obtained mainly in the form of qualitative categories. The first and second methods are applied to the case where an external criterion is present, and are used to predict the external criterion or to analyze the effects of factors. On the other hand, the third and fourth methods are applied to the case where no external criterion is present, and are used to construct a spatial configuration so as to grasp the mutual relationship of the data. After reviewing Hayashi's four methods, we discuss two topics which have been pointed out as the problems to be solved in applying the methods of quantification. One is quantification for ordered categories and the other is statistical consideration. With respect to these topics we review some recently developed methods including the studies due to the present author. Finally we mention briefly several computer programs available in Japan.