The present paper proposes a method of construction of multiple uni-dimensional scales by classifying a set of qualitative variables into groups. The method consists of Hayashi's third method of quantification and fuzzy c-means clustering. Weight parameter, agi, which indicates a degree that each item belongs to each group, facilitates the classification. As numerical examples, four artificial data sets are analyzed and the characteristics of the method are examined. Two real data sets are also analyzed and it is observed that interpretable groups are obtained in both data sets.
We propose a least square procedure called ASCLUD(Attribute Scaling using CLUster-Distance)which scales categorical attributes with profiles of objects on the attributes and dissimilarities between the objects. ASCLUD assumes that in a multidimensional space an object is represented as a cluster of the points which correspond to the attributes possessed by the object. Distances between the clusters are fitted to the inter-object dissimilarities using the generalized majorization algorithm. Examples are given to illustrate the use of ASCLUD and to compare ASCLUD with a previous dissimilarity model considering the weights of attributes. Some properties of ASCLUD are discussed.
Unfolding according to the fourth quantification method is proposed.A nonmetric unfolding method may produce a degenerate configuration, but the unfolding method proposed in this paper generates a configuration that roughly shows the dominant tendency in the data. Two applications of the method to real data are shown. One analyzed semantic differential data for colors(Oyama et al., 1963), and the other analyzed rating data for harmony between colors and words(Okamoto, 1995).Both analyses resulted in configurations with meaningful interpretations. The relation of the proposed method to the third quantification method was noted. Other successful unfolding methods, which use penalty functions or the like, are recommended when monotonicity between data and distance is crucial for the user. If such monotonicity is not crucial, then the unfolding method according to the fourth quantification method is useful.
In the following four areas, (1)test of Intellectual ability, (2)the geometry of visual space, (3)the global structure of Munsell color solid, and(4)sensory evaluation in industry, two approaches with and without scaling were contrasted on the basis of the author's own studies for a period of over 35 years in Japan and U.S.A. Special emphasis was placed on testing consistencies of scaled results as quantitative representations of the underlying latent processes.