The Measurement of Association in a Contingency Table: Three Extensions of the Cross-Ratios and Metrics Methods

Summary The methods of cross-ratios, equivalence classes and metrics that were introduced in a previous paper (Altham 1970) are extended to three further problems. First the interdependence of the rows, columns and layers of a three-way contingency table is discussed. There are essentially four different types of association to be considered: the complete dependence of rows, columns and layers, the dependence of rows and columns conditional on a given layer, the dependence of the two classifications “rows-and-columns” and layers, and the second-order interaction of rows, columns and layers. The relations between these four types of association are discussed in terms of the appropriate equivalence classes. In order to obtain measures of dependence of rows, columns and layers, metrics are defined on the equivalence classes. Some examples of metrics are given. The consideration of three-way tables is concluded by a discussion of the asymptotic posterior distributions of certain special metrics, which are suggested by the work of Lindley (1964), and Plackett (1962). Cross-ratios methods are also extended to the problem of measuring association in a two-way table with some cells deleted. The paper is concluded by consideration of the problem of comparing associations in two distinct contingency tables, some of whose cells may be deleted.