The mode and the measurement of inequality with single and multiple ordinal variables

The extensive use of multivariate ordered categorical data in the social sciences presents challenges for the measurement of socioeconomic inequities. The ambiguities inherently associated with artificial attribution of scale to ordinal categories, preclude the use of standard distance-based inequality and polarization measures. These issues have been surmounted in the univariate world by employing notions of likelihood distance (the increasing likelihood that an outcome between two categories will occur the bigger is their categorical gap) and aggregating outcome distances from the median category as a reference point, adapting the transfer principle using Hammond transfers, or with probability-based notions of status. Unfortunately, the median category is not always the ideal reference point of complete commonality and is not uniquely defined in multivariate environments. However, as the most frequently observed outcome, the modal category provides a natural measure of the extent of commonality or equity in the population, thus providing a readymade reference point from which to measure likelihood outcome distance. We provide axiomatic foundations and characterize classes of modally focused inequality measures for univariate and multivariate ordered categorical environments together with their asymptotic distributions for inference purposes. We also identify the partial ordering induced by our proposed mode-clustering transfers which provides a useful robustness test for inequality indices in the spirit of stochastic dominance conditions. In an empirical illustration we study the evolution of inequality in educational attainment and experience among men and women in Canada.