Correcting the Concentration Index

https://doi.org/10.1016/j.jhealeco.2008.02.003Get rights and content

Abstract

In recent years attention has been drawn to several shortcomings of the Concentration Index, a frequently used indicator of the socioeconomic inequality of health. Some modifications have been suggested, but these are only partial remedies. This paper proposes a corrected version of the Concentration Index which is superior to the original Concentration Index and its variants, in the sense that it is a rank-dependent indicator which satisfies four key requirements (transfer, level independence, cardinal invariance, and mirror). The paper also shows how the corrected Concentration Index can be decomposed and generalized.

Introduction

The Concentration Index enjoys an increasing popularity as an indicator of the inequality of health in relation to the socioeconomic position of individuals. Among the factors which have contributed to its success as a measure of the socioeconomic inequality of health we can mention its affinity with the Gini coefficient, its visual representation by means of the Concentration Curve, and the ease with which it can be decomposed. But there is also a growing awareness that it is far from being a perfect measure. In recent years attention has been drawn to the fact that the bounds of the Concentration Index may depend upon the mean of the health variable and hence make a comparison of populations with different mean health levels problematic (Wagstaff, 2005), that different rankings are obtained if inequalities in ill health rather than inequalities in health are considered (Clarke et al., 2002), and that the value of the index is to a large extent arbitrary if the health variable is of a qualitative nature Erreygers, 2006, Zheng, 2006.

Various suggestions have been made to remedy some of these defects. What is lacking, though, is an attempt to come to grips with all of these problems. In this paper, I propose a corrected version of the Concentration Index which goes a long way in that direction. After a discussion of the three issues, I analyze the modifications which have been proposed in the literature. In the core of the paper I follow an axiomatic approach which allows me to derive a new correction formula. I define a family of rank-dependent indicators and show which members of this family satisfy four key requirements. I also indicate how the approach can be generalized. I end with a brief empirical illustration.

Section snippets

Assumptions

Let us consider a given population of n individuals represented by the set N={1,2,,n}. To each individual we assign a rank λi based upon this person’s socioeconomic position, with the best well-off individual ranked first and the least well-off ranked last. In the case of ties, we assign to each member of the tied group the average rank of the group. The variable in whose distribution we are interested is health. Suppose that we have information about the health situation of the population,

The Wagstaff normalization

Wagstaff (2005) has suggested a normalization formula specifically aimed at remedying the bounds issue. For a binary health variable with ah=0 and bh=1, he proposes to divide the health Concentration Index by its (upper) bound, which leads to a normalized health Concentration Index equal to C(h)/(1μh). Generalizing this procedure to all possible cases with lower bound ah and finite upper bound bh, we can define the Wagstaff-normalized health Concentration Index W(h) as follows:W(h)μh(bhah)(bh

A family of rank-dependent indicators

So far we have encountered three closely related measures of socioeconomic inequality of health: the Concentration Index C(h), the Wagstaff index W(h), and the Generalized Concentration Index V(h). These can all be seen as members of a family of rank-dependent socioeconomic indicators. The general form of the indicators of this family is the following:I(h)=f(H)i=1nzihiwhere:zi=n+12λiThe general form consists of two parts. The part which is common to all indices, i=1nzihi, expresses the

Concluding remarks

I started from a rather negative position, in the sense that I drew up a list of a number of shortcomings of the “traditional” health Concentration Index C(h). I also showed that modified versions of it, such as Wagstaff’s index W(h) and the Generalized Concentration Index V(h), manage to remedy some of these shortcomings, but not of all of them. But gradually I adopted a more constructive attitude. Using an axiomatic approach I formulated four key requirements – transfer, level independence,

Acknowledgements

The comments of two referees of this journal have led to substantial improvements in the paper. I am also very grateful to Tadele Ferede, Peter Lambert, Ellen Van de Poel, Tom Van Ourti and Buhong Zheng for discussions on the subject matter of this note, and for comments on previous drafts. All errors are my responsibility.

References (16)

There are more references available in the full text version of this article.

Cited by (472)

View all citing articles on Scopus
View full text