Measures of Association for Cross Classifications

Abstract

When populations are cross-classified with respect to two or more classifications or polytomies, questions often arise about the degree of association existing between the several polytomies. Most of the traditional measures or indices of association are based upon the standard chi-square statistic or on an assumption of underlying joint normality. In this paper a number of alternative measures are considered, almost all based upon a probabilistic model for activity to which the cross-classification may typically lead. Only the case in which the population is completely known is considered, so no question of sampling or measurement error appears. We hope, however, to publish before long some approximate distributions for sample estimators of the measures we propose, and approximate tests of hypotheses. Our major theme is that the measures of association used by an empirical investigator should not be blindly chosen because of tradition and convention only, although these factors may properly be given some weight, but should be constructed in a manner having operational meaning within the context of the particular problem.

This paper is partly an outgrowth of work sponsored by the Army, Navy, and Air Force through the Joint Services Advisory Committee for Research Groups in Applied Mathematics and Statistics by Contract No. N6ori-02035. We are indebted for helpful comments and criticisms to: Otis D. Duncan (University of Chicago), Churchill Eisenhart (National Bureau of Standards), Maurice G. Kendall