A System of Paired Asymmetric Measures of Association for Use with Ordinal Dependent Variables

Many variables of concern to sociologists are ordinally measured. Evaluating causal theoretical statements involving such variables requires refined asymmetric measures of association. We develop a typology of asymmetric, proportionate reduction in error (PRE) measures for describing the relationship between a nominal or an ordinal independent variable (X) and an ordinal dependent variable (Y). Measures are classified according to prediction rule and method for handling ties in estimating predictive error. This scheme is used to compare several existing measures of association (Y, dyx, d′yx, and ϴ) and to derive two new measures. The typology yields three sets of paired measures useful for assessing relationships in terms of departure from monotonicity. This usage is analogous to the use of r2 and ϵ2 to evaluate linearity and nonlinearity.