Signature
C = 2 * cov_hr / mu
| Inputs | Definition | Unit |
|---|---|---|
cov_hr | Population-weighted covariance between group mean health and the fractional rank of each group | health units, for example years of QALE |
mu | Population-weighted mean health across all groups. Above zero | health units, in the same unit as cov_hr |
C | Concentration index: twice the area between the concentration curve and the line of equality. Bounded by minus 1 and 1 for a health variable with nonnegative values | none |
|---|
Function
Health inequality measures across ordered population groups
Maps the mean health of population groups, ordered from the most to the least disadvantaged, and the share of the population in each group to summary measures of socioeconomic-related health inequality: the absolute and relative gaps between the extreme groups, the slope index of inequality, the relative index of inequality in two forms and the concentration index. The measures describe how health varies with socioeconomic position and do not say whether that variation is unfair, which is the separate judgement of health inequity. The Atkinson index, which ignores the ranking of groups, has its own records on the Atkinson index page (HE-FN-ATK-001) and is not repeated here.
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Implementations
Excel
Concentration index from grouped data in one Excel cell
With group mean health in a range named Health, population shares summing to 1 in Share and fractional ranks at the group midpoints in Rank, Excel returns the concentration index from the covariance form.
=2*SUMPRODUCT(Share,(Health-SUMPRODUCT(Share,Health))*(Rank-0.5))/SUMPRODUCT(Share,Health)
Assumptions
Ratio-scale nonnegative health for the concentration index
Health is measured on a ratio scale with nonnegative values. The index is unchanged when health is multiplied by a constant but changes when a constant is added. For a binary variable its bounds depend on the mean.
Ranking by socioeconomic position for the concentration index
Groups are ranked by socioeconomic position, such as deprivation or income, not by health. Ranking by the health variable itself would measure overall health inequality rather than socioeconomic-related health inequality.
Between-group inequality only in the grouped concentration index
Each group is given its mean health, so the index from grouped data describes inequality between the groups only.
Worked examples
Concentration index of QALE for five equal deprivation quintiles
In the article's example the covariance between QALE and rank is 1.0 and mean QALE is 68 years, so the index is about 0.0294. The positive sign shows that health is concentrated among the less deprived.
cov_hr = 1.0; mu = 68; C = 0.0294
Concentration index after a gain of 1 year in every quintile
Under Programme A in the article the covariance stays at 1.0 and mean QALE rises to 69 years, so the index falls to about 0.0290 although the absolute gap and the slope index are unchanged.
cov_hr = 1.0; mu = 69; C = 0.0290
Concentration index with unequal group sizes
In the illustrative unequal-shares example, with shares of 0.30, 0.25, 0.20, 0.15 and 0.10 from the most to the least deprived group, the covariance is 1.02375 and mean QALE 66.75 years, giving an index of about 0.0307.
cov_hr = 1.02375; mu = 66.75; C = 0.0307
Common errors
Reading a smaller concentration index as narrower absolute health inequality
The concentration index is a relative measure. Under Programme A it falls from 0.0294 to 0.0290 while the absolute gap stays at 10 years and the slope index at 12.5 years, because adding the same amount to everyone changes the index without changing any absolute difference.
Treating a zero concentration index as no socioeconomic health inequality
An index of zero can arise because the concentration curve lies on the line of equality or because it crosses the line and the areas on each side cancel out. The curve is examined alongside the index before concluding that there is no socioeconomic-related inequality.
Sources
World Bank guide to the concentration index for grouped data
O'Donnell O, van Doorslaer E, Wagstaff A, Lindelow M. Analyzing health equity using household survey data: a guide to techniques and their implementation. Washington, DC: World Bank; 2008. Chapter 8, The concentration index. Equation 8.3 (covariance form), equation 8.5 (fractional rank of a group at the midpoint of its interval), equation 8.4 (Fuller and Lury formula for grouped data), equations 8.10 to 8.12 (index from the slope of health on rank), and pages 96 to 97 on bounds, scale properties and the crossing curve.
Canonical Identity
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