2019 Volume 8 Issue 1 Pages 47-57
In 1986, Carroll, Green and Schaffer proposed the so-called CGS scaling, in which they tried to rectify the joint graphical display of quantification results as proposed in France and widely used throughout the world. The CGS proposal was severely criticized by Greenacre in 1989, and it was abandoned by most researchers. In 1989, Nishisato promised J.D. Carroll that he would write a paper to support the CGS scaling. As is known, the row variates and the column variates derived by quantification theory do not span the same space unless they are perfectly correlated. Yet, the traditional joint graphical display is based on the condition that row variates and column variates occupy the same space. 30 years since then, Nishisato finally succeeded in his paper accepted for publication. This is the current paper which demonstrates that we must double the space for graphical display of rows and columns of the contingency table. He employed the response-pattern table format for the contingency table, and identified contingency space, dual space and residual space, all of which constitute quantification space, and demonstrated that dual space is for the joint graphical display with interesting pair-wise subspace of dual space which shows how discrepant the traditional symmetric graph is from the exact coordinates. It is unfortunate that the CGS scaling did not fully discuss the quantification space but initiated the search for mathematically correct joint graphical display.
