TREND VECTOR REPRESENTATION OF MULTIPLE TRANSITION MATRICES BY PENALIZED OPTIMAL SCALING

Individuals' choices of categories observed on two occasions are described by transition frequency matrices. In this paper, a penalized optimal scaling method is presented to analyze a set of the matrices obtained from multiple sources and graphically represent a transition trend for each source as a vector. This method finds scores of individuals, those of categories, and vectors of trends, in such a way that individuals' scores become homogeneous to the scores of chosen categories and trend vectors become homogeneous to the inter-occasion changes in individuals' scores. The resulting low-dimensional configuration of trend vectors allows us easily to grasp transition trends. Further, the projection of category scores onto trend vectors gives the unidimensional scales of categories useful for scrutinizing transition trends.