Multidimensional polarization for ordinal data

Abstract

The dominant approach to evaluating distributional features of ordinal variables (e.g. selfreported health status) has been the Allison-Foster bipolarization ordering (henceforth AF). It has not yet been extended to a multidimensional setting. Here we fill this gap. A multidimensional extension of the AF relation is characterized by a sequence of median-preserving spreads on each dimension and association-changing switches. This extension does not pay attention to the dimensions' association. We then offer one that does and characterize it in terms of classes of polarization measures and welfare functions. Based on these two orderings we construct polarization indices and develop statistical inference for them. We measure bidimensional polarization in educational attainment and life satisfaction across OECD members. Dependence does not affect whether or not countries dominate each other bidimensionally.

A coefficient of variation for ordered categorical data: Analyzing relative health inequality and ageing in the UK and relative human resource inequality and gender in Canada

“…Without artificial scaling, 𝐶𝑂𝑉 and 𝐶𝑂𝑉 -1 have no natural analogues in ordered categorical data environments since that paradigm is bereft of cardinal measure. In its absence, inequality and polarization researchers have used notions of probabilistic distance (Mendelson 1987) and the construct of a median preserving spread in order to quantify variation for the purpose of measuring inequality and polarization (Blair and Lacy 2000, Allison and Foster 2004, Kobus and Kurek 2019. The probabilistic distance of a given category from the median focus category is measured in terms of the likelihood of an outcome occurring in the given or any other category between it and the median category, the higher that probability is, the further apart are the categories deemed to be.…”