On the Hypotheses of 'No Interaction' in Contingency Tables

The statistical analysis of data in multi-dimensional contingency tables is discussed in terms of appropriate underlying probability models. Emphasis is placed on the distinction between 'factors' (such as treatments or blocks) which have fixed marginal totals and 'responses' (such as category of performance) which have random marginal totals. Hence, four principal cases arise: (i) the 'multi-response, no factor' tables, (ii) the 'multi-response, uni-factor' tables, (iii) the 'multi-response, multifactor' tables, (iv) the 'uni-response, multi-factor' tables. For situations (i) and (ii), the concept of 'no interaction' is related to questions regarding the pattern of association among responses. However, for situation (iv), it is related to how factors combine (e.g., additively) to determine the response distribution. Finally, for situation (iii), both types of questions arise. For each of the different types of tables, the problem of formulating appropriate hypotheses of 'no interaction' is considered. The corresponding test statistics are based upon a general and computationally simple criterion of Wald 1943. The suggested methods are illustrated with several numerical examples.