Measuring the Dependence Among Dimensions of Welfare: A Study Based on Spearman’s Footrule and Gini’s Gamma

Abstract

Welfare is multidimensional as it involves not only income, but also education, health or labour. The composite indicators of welfare are usually based on aggregating somehow the information across dimensions and individuals. However, this approach ignores the relationship between the dimensions being aggregated. To face this goal, in this paper we analyse the dependence between the dimensions included in the Human Development Index (HDI), namely income, health and schooling, through three copula-based measures of multivariate association: Spearman's footrule, Gini's gamma and Spearman's rho. We discuss their properties and prove new results on Spearman's footrule. The copula approach focuses on the positions of the individuals across dimensions, rather than the values that the variables attain for such individuals. Thus, it allows for more general types of dependence than the linear correlation. We base our study on data from 1980 till 2014 for the countries included in the 2015 Human Development Report. We find out that though the overall HDI has increased over this period, the dependence between its dimensions remains high and nearly unchanged so that the richest countries tend to be also the best ranked in both health and education.

Copula‐based analysis of multivariate dependence patterns between dimensions of poverty in Europe*

“…Pérez and Prieto (2015) extended Decancq’s results by considering other multivariate versions of Spearman’s rho to study how the dependence between the dimensions of the AROPE rate has evolved in Spain over the period 2009–2013. Also, Pérez and Prieto‐Alaiz (2016a) analysed the multivariate dependence between the dimensions of the HDI using data from 187 countries and three copula‐based measures of multivariate association: Spearman’s footrule, Gini’s gamma and Spearman’s rho.…”

On the lower bound of Spearman’s footrule

“…a value ϕ(C) in the real numbers. Due to its "simplicity, robustness and natural interpretation" (Genest et al [11]), it has already been used in several and quite di erent areas of research: This includes, for instance, genomics [12], information science [13] and economics [15]; we also refer to the impressive number of applications listed in [11] and [16].…”

Copula-based methods and their application to multidimensional poverty analysis

“…On the other hand, as it was shown by Pérez and Prieto-Alaiz (2016a), when C = M in (A2), the multivariate version of Spearman's footrule proposed by Úbeda-Flores (2005) comes up, namely:…”

“…As Pérez and Prieto-Alaiz (2016a) show, an appealing property of the AOD measures is that they have a probabilistic interpretation in terms of multivariate concordance and include, as members of this family, some of the well-known multivariate measures of association, as we will see next. In fact, recalling the definition in (2.10), the family of AOD measures of multivariate association given by (A1) can be written in terms of the probability of multivariate concordance Q d as follows:…”

“…He analysed the temporal evolution of well-being in Russia by means of the multivariate Kendall's tau and the multivariate version of Spearman's rho defined in (2.26) applied to the dimensions included in the Human Development Index (HDI). Also, Pérez and Prieto-Alaiz (2016a) analysed the multivariate dependence between the dimensions of the HDI using data from 187 countries and three copula-based measures of multivariate association: the Spearman's footrule in (A4), the Gini's gamma in (A5) and the Spearman's rho defined in (2.26). More recently, García-Gómez et al (2021) analysed the evolution of multivariate dependence between the dimensions of poverty in Europe over the period 2008-2014 using the multivariate extensions of Spearman's rho introduced in Section 2.4.…”