Decomposition of the Gray-Williams tau in main and interaction effects by ANOVA in three-way contingency table.

The identification of meaningful relationships between two or more categorical variables is an important, and ongoing, element to the analysis of contingency tables. It involves detecting categories that are similar and/or different to other categories. Correspondence analysis can be used to detect such relationships by providing a graphical interpretation of the association between the variables, and it is especially useful when it is known that this association is of a symmetric nature. In this paper we explore the Gray-Williams index when used as the measure of association in non-symmetrical correspondence analysis (NSCA). It will be shown that, by concatenating a predictor variables of a three-way contingency table the two measure are equivalent. The will analyse the sum of squares for nominal data partitioning the sum of squares for main effects and interaction in the sense of analysis of variance giving an orthogonal decomposition of the Gray-Williams index.