Defining and testing hypotheses in multidimensional contingency tables.
Juliet Popper Shaffer · Psychological Bulletin · 1973
The log-linear model is described as a framework for analyzing effects in multi-dimensional contingency tables, that is, tables of frequencies formed by two or more variables of classification. Variables are considered to have nominal categories. A general purpose analysis is proposed for such tables, similar to analysis of variance. Two test procedures are considered: (a) maximum likelihood estimation of expected cell frequencies and associated chi-square tests and (b) chi-square tests based on logarithms of adjusted cell frequencies. In addition, two multiple-comparison methods, related to the latter approach, are considered as supplementary or alter-native procedures. Since 1935, when the first article dealing with the estimation of interaction in multidi-mensional contingency tables was published (Bartlett, 1935), a large literature has accumu-lated concerning the problems of defining and testing complex relationships in such tables. A previous article in this journal (Sutcliile, 1957) presented the methods developed by Lancaster (1951) based on the partitioning of a total chi-square statistic in a way analogous to the partitioning of a total sum of squares in the analysis of variance. However, the majority of statisticians working in this area have re-jected these methods on the grounds that when relationships among variables are defined in an intuitively acceptable manner, the partitioned chi-square values do not correspond to tests of these relationships (see, e.g., Goodman, 1964b). The purposes of this article are (a) to discuss definitions of relationships among variables in contingency tables and (b) to describe prac-tical methods of testing hypotheses about the existence of effects of different degrees of com-plexity. The article is limited to the considera-tion of variables with nominal categories (i.e., unordered categories), or where order of categories is to be ignored;2 and it is con-cerned primarily with significance testing rather than estimation.