In analyses of contingency tables made up of categorical variables, the study of relationship between the variables is usually the major objective. So far, many association measures and association models have been used to measure  the association structure present in the table. Although the association measures merely determine the degree of strength of association between the study variables, the association models illustrate the details and components of association structure. These measure and models have found vast application in many disciplines. For more details see Goodman and Kruskal (1954, 1979), Leibetrau (1983), Goodman (1972, 1979, 1985, 1991) and Ghoreishi and Meshkani (2006, 2008).

When one is interested in demonstrating the association components of two ordinal categorical variables while there is at least one explanatory categorical variable, it is natural to think of partial association between the two ordinal variables while the effect of the explanatory variable(s) are averaged out in some sense. We believe that this approach is the best way to incorporate both the extra information available in explanatory variable(s) into analysis and interpret the role and share of various polynomial trends.

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