Interpretation of the coefficient of determination of an ANOVA model as a measure of polarization

Abstract

In this paper, it is demonstrated that coefficient of determination of an ANOVA linear model provides a measure of polarization. Taking as the starting point the link between polarization and dispersion, we reformulate the measure of polarization of Zhang and Kanbur using the decomposition of the variance instead of the decomposition of the Theil index. We show that the proposed measure is equivalent to the coefficient of determination of an ANOVA linear model that explains, for example, the income of the households as a function of any population characteristic such as education, gender, occupation, etc. This result provides an alternative way to analyse polarization by sub-populations characteristics and at the same time allows us to compare sub-populations via the estimated coefficients of the ANOVA model.

On Measuring Polarization for Ordinal Data: An Approach Based on the Decomposition of the Leti Index

“…Polarization is a relevant topic in studies on income distribution (Esteban and Ray, 1994;Duclos et al, 2004;Palacios-González and García-Fernández, 2012;Jenkins, 1995;Bossert and Schworm, 2008;Wang and Tsui, 2000;Yitzhaki, 2010;Gigliarano and Mosler, 2009;Foster and Wolfson, 2010) and its original notion is based on the concept of identification-alienation: individuals identify themselves with those having similar income levels, whereas they feel alienated from those with different income levels. When measuring polarization for an ordinal variable whose categories describe the status of an individual, there is polarization when groups of individuals characterized by within-group homogeneity (identification) and between-group heterogeneity (alienation) are observable.…”

On Measuring Income Polarization: An Approach Based on Regression Trees

“…As shown in Palacios-González andGarcía-Fernández (2012, p.1546), even though the model in (3) does not include the intercept, the decomposition of the total sum of squares (TSS) into the explained sum of squares (ESS) and the residual sum of squares (RSS) is valid. Then, the coefficient of determination of the model in (3)…”

“…The link between polarization and ANOVA is outlined by Palacios-González and García-Fernández (2012) in the generalized linear model framework. Palacios-González and García-Fernández follow the Zhang and Kanbur (2001) approach to polarization, which assumes that for k predetermined groups of income receivers the larger the ratio of between-group income inequality to within-group income inequality, the larger the polarization.…”

“…Recently, Palacios-González and García-Fernández (2012) have pointed out that the coefficient of determination (R 2 ) of an ANOVA linear model can be interpreted as a measure of polarization. Since R 2 increases as within-group variance decreases (i.e.…”

“…groups are internally more homogeneous), PalaciosGonzález and García-Fernández state that R 2 can be seen as a (normalised) measure of polarization. Moreover, linking the ANOVA coefficient of determination with polarization enables one to analyse polarization by the characteristics of income receivers when groups are defined by such characteristics (Palacios-González and García-Fernández, 2012).…”