Polarization measurement for ordinal data

Atkinson's Theorem (Atkinson J. Econ. Theory 2, 244-263, 1970) is a classic result in inequality measurement. It establishes Lorenz dominance as a useful criterion for comparative judgements of inequality between distributions. If distribution A Lorenz dominates distribution B, then all indices in a broad class of measures must confirm A as less unequal than B. Recent research, however, shows that standard inequality theory cannot be applied to ordinal data (Zheng Res. Econ. Inequal. 16, 177-188, 2008), such as self-reported health status or educational attainment. A new theory in development (Abul Naga and Yalcin J. Health Econ. 27(6), 1614-1625, 2008) measures disparity of ordinal data as polarization. Typically a criterion used to compare distributions is the polarization relation as proposed by Allison and Foster (J. Health Econ. 23(3), 505-524, 2004). We characterize classes of polarization measures equivalent to the AF relation analogously to Atkinson's original approach.