One area of application which has become increasingly important to statisticians and other researchers is the analysis of categorical data. Often the principal objective in such investigations is either the testing of appropriate hypotheses or the fitting of simplified models to the multi-dimensional contingency tables which arise when frequency counts are obtained for the respective cross-classifications of specific qualitative variables. Grizzle, Starmer, and Koch [1969] (subsequently abbreviated GSK) have described how linear regression models and weighted least squares can be used for this purpose. The resulting test statistics belong to the class of minimum modified chi-square due to Neyman [1949] which is equivalent to the general quadratic form criteria of Wald [1943]. As such, they have central x2-distributions when the corresponding null hypotheses are true. Two alternative approaches to this methodology are that based on maximum likelihood as formulated by Bishop [1969; 1971] and Goodman [1970; 1971a, b] and that based on minimum discrimination information as formulated by Ku et al. [1971]. In each of the previously mentioned papers, primary emphasis was given to the formulation of models and the problems of analysis under various conditions of "no interaction" (see Roy and Kastenbaum [1956] or Bhapkar