Local Concordance and Some Applications

Abstract

One of the main issues when analyzing multidimensional phenomena such as well being is how to define a composite indicator. However a sometimes neglected collateral issue is how to take into account the joint distribution of the single components, and connected with this issue the question should be: how to compute the association among the single components of a multidimensional concept. This is precisely the aim of this paper. We suggest a counting based approach to detect positive association amongst the single components of multivariate phenomena, in particular among the best performing units and vice-versa among the worst performing units. Taking moves from Kendall’s notion of concordance/agreement and from his well known concordance coefficient W (Kendall and Babington Smith in Ann Math Stat 10:275–287, 1939), we introduce the concept of local concordance and derive a local concordance coefficient and a local concordance curve. Our intent is to have, not an overall measure of concordance, i.e. not a global indicator, but instead local concordance coefficients to detect different degrees of concordance in the head, tail or centre of the multivariate distribution of the components of a well being indicator. The local concordance curve can have many different applications. When referred to the components of a well being indicator (and thus to inequality), the local concordance coefficient obtained from the first (last) window can be seen as a measure of concentration of high-level (or low-level) attributes. We apply this approach to exploit whether different aspects of social vulnerability are equally or unequally distributed among censuary areas of a same region.