Geometric Mean Type of Proportional Reduction in Variation Measure for Two-Way Contingency Tables

Abstract

Traditional analysis of two-way contingency tables with explanatory and response variables focuses on the independence of two variables. However, if the variables do not show independence or a clear relationship, the analysis shifts to the degree of association. Various measures have been proposed to calculate the degree of association. One is the proportional reduction in variation (PRV) measure. This measure describes the PRV from the marginal distribution to the conditional distribution of the response variable. Although conventional PRV measures can assess the association of the entire contingency table, they cannot accurately assess the association for each explanatory variable. In this paper, we propose a geometric mean type of PRV (geoPRV) measure, which aims to sensitively capture the association of each explanatory variable to the response variable. Our approach uses a geometric mean, and enabling analysis without underestimating the values when the cells in the contingency table are partially biased. The geoPRV measure can be constructed using any function that satisfies specific conditions. This approach has practical advantages, and in special cases, conventional PRV measures can be expressed as geometric mean types.