The Multivariate Analysis of Qualitative Data: Interactions among Multiple Classifications

Abstract

For the m-way contingency table, we discuss both the direct estimation of the multiplicative interactions among the m variables, and the indirect testing of hypotheses pertaining to these interactions. We consider, among other things, hierarchical hypotheses pertaining to the interactions among the m variables, including hypotheses that can be expressed in terms of one or more of the following kinds of concepts: (a) the usual concepts of independence and equiprobability; (b) concepts describing conditional properties (e.g., conditional independence) pertaining to a subset of the m variables, given the level of some of the remaining variables; (c) concepts related to the usual logit-analysis or to a generalized form of logit-analysis; and (d) concepts related to a more general log-linear model. Methods of partitioning these hypotheses are introduced which provide, among other things, insight into the relationship between tests applied to the m-way table and tests applied to marginal tables formed from the m-way table. We also show, by example, how the combined use of direct estimation and indirect testing can lead to the discovery of hypotheses (models) that, fit the data in the m-way table better than the hypotheses that have been fitted in the earlier literature.