2003 Volume 30 Issue 1 Pages 87-98
In quantifying a two-way table of data, one derives weights (spacings) for rows and columns so as to maximize the correlation of data weighted by row weights and those by column weights. It is well known that one can calculate the distance between any two rows (columns), called the within-set distance, but the distance between a row and a column, the between-set distance, cannot be calculated because the row weights span the space different from that of the column weights. The present paper shows (1) that one can calculate the between-set distances, and (2) that the data as a whole require an additional dimension to accommodate the discrepancy of the row space and the column space. Since the information contained in the between-set distances is an integral part of the data structure, it is not a matter of whether the study advocates the use of the between-set distance information, but rather it is a problem that dual scaling and correspondence analysis must deal with through development of an analytic method to tap into the entire information in the data. Until then, the current formulation of these methods which ignore the information contained in the between-set distances provide at best simplified approximations to the decomposition of data structure. This further development is not easy because these methods are by the very nature of categorical data based on the chi-square metric, making an entire inter-point (within-set and between-set) distance matrix not readily amenable to any currently available analytic method.