An Analysis for Compounded Functions of Categorical Data

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