An Order on Cross-Tabulations and Degrees of Association

Abstract

It is often argued that an order exists in a cross-tabulation when the table’s margins have such a structure. We can free ourselves from this point of view and clearly define an order on the table itself. As Louis Guttman noted previously in the case of scalogram analysis, one must often move rows and columns about to be able to create a scale. In this case, it is the order of the table’s structure which induces an order on the margins and not the reverse. However, Goodman and Kruskal, when they proposed the gamma index that defines the strength of an association in the ordered case, only use the margins’ order, and they have since then been followed by most researchers. One should return to the original intuition of Guttman and show that at least an approximate order is almost always present in a table. The ordered cross-tabulation generated by ordered questions is only one case among many others and conversely a table with a strong order structure induces an order on question modalities. With real examples, we show that the criteria are available to define an order on a table, that there are formalized methods to reveal the associated structure, that there are also different indices to measure the degree of association, and finally that there are tests to assess the level of significance.