The paper presents a partition of the Pearson chi-squared statistic for triply ordered three-way contingency tables. The partition invokes orthogonal polynomials and identifies three-way association terms as well as each combination of two-way associations. This partition provides information about the structure of each variable by identifying important bivariate and trivariate associations in terms of location (linear), dispersion (quadratic) and higher order components. The significance of each term in the partition, and each association within each term can also be determined.
The paper compares the chi-squared partition with the log-linear models of Agresti (1994) for multi-way contingency tables with ordinal categories, by generalizing the model proposed by Haberman (1974).