Proportional Reduction in Variation Measure for Normal-Ordinal Contingency Tables

Abstract

For a two-way contingency table with a nominal explanatory and a nominal response variables, Goodman and Kruskal (1954), and Theil (1970) proposed the measures which describe the proportional reduction in variation (PRV) from the marginal distribution to the conditional distributions of the response. Tomizawa, Seo and Ebi {1997) proposed a generalization of those measures. This paper proposes a PRV measure for a two-way contingency table with a nominal explanatory and an ordinal response variables (instead of those with a nominal response variable). The proposed measure is expressed using the Patil and Taillie's (1982) diversity index, defined for the cumulative probabilities that an observation will fall in the response category j or below (or, j + 1 or above).

The measure depends on the order of listing the response categories. It is useful for comparing the degrees of PRV in several tables with ordered response categories.