Signature
E = 4 * mu * C / (b - a)
| Inputs | Definition | Unit |
|---|---|---|
mu | Mean of the health variable, on the same scale as a and b and strictly above zero | units of the health variable |
C | Standard concentration index of the bounded variable by fractional rank in the living standards distribution, as in HE-FM-HINQ-005 | none |
b | Largest value the measurement scale allows, above a; 1 for a binary variable | units of the health variable |
a | Smallest value the measurement scale allows, 0 for a binary variable | units of the health variable |
E | Corrected concentration index; negative when the variable is concentrated among the poor and zero when there is no socioeconomic-related inequality | none |
|---|
Function
Bound corrections of the concentration index for binary and bounded health variables
Maps the standard concentration index of a health variable, its mean and the bounds of its measurement scale to an index whose feasible range does not depend on the mean. For a binary variable the standard index can only lie between mu minus 1 and 1 minus mu in large samples, so Wagstaff divides it by 1 minus the mean, while Erreygers multiplies it by 4 times the mean over the range of the scale, which equals 8 times the covariance of health and fractional rank over that range. The standard index itself, C = 2 cov_hr / mu, is HE-FM-HINQ-005, computed from grouped data in HE-CF-HINQ-001; the curve form by trapezoids is HE-FM-VEQ-002, the horizontal inequity index C_M minus C_N is HE-FM-HEQ-003 and the rule of 75 is HE-FM-HEQ-004. Notation follows the Concentration Index article.
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Implementations
Excel
Erreygers corrected concentration index in Excel
With the standard index in ConcIndex, the mean in MeanH and the scale bounds in LowerBound and UpperBound, the first formula returns the corrected index; the second returns the same value from the covariance of health and fractional rank held in CovHR. Both return #N/A when the upper bound is not above the lower bound.
=IF(UpperBound<=LowerBound,NA(),4*MeanH*ConcIndex/(UpperBound-LowerBound)); =IF(UpperBound<=LowerBound,NA(),8*CovHR/(UpperBound-LowerBound))
Assumptions
Fixed bounds of the measurement scale for the Erreygers correction
The variable has a known lower bound a and upper bound b set by its measurement scale, such as 0 and 1 for a binary indicator or 0 and 100 for a score, and not by the smallest and largest values observed in the sample. C and mu are computed with the same sample, weights and living standards ranking.
Value judgement behind choosing the Erreygers correction
Erreygers and Van Ourti show that the corrected index and the generalised Wagstaff index both pass the mirror and cardinal invariance tests and that further value judgements are needed to choose between them. Wagstaff replied that the standard index reflects a commitment to relative inequality, whereas the corrected index ends up as a measure of absolute inequality, so the choice of index is reported with the results.
Worked examples
Erreygers corrected index of limiting illness across five income quintiles
In the article's example the mean prevalence is 0.24 and the standard index minus 0.25, so with a of 0 and b of 1 the corrected index is minus 0.24, which equals 8 times the covariance of minus 0.03, as in the article.
C = -0.25; mu = 0.24; a = 0; b = 1; E = -0.24
Erreygers corrected index of being free of limiting illness
For the share without limiting illness the mean is 0.76, the standard index about 0.0789 and the covariance 0.03, so the corrected index is 0.24, the exact mirror of the illness value, as in the article.
C = 0.078947; mu = 0.76; a = 0; b = 1; E = 0.24
Erreygers corrected immunisation index at 50 per cent coverage
For the illustrative country with half of children immunised and a standard index of 0.10, the corrected index is 0.2, the same as the Wagstaff index in HE-EX-CIX-003, because the two corrections coincide when the mean of a binary variable is 0.5 (computed here for illustration).
C = 0.10; mu = 0.5; a = 0; b = 1; E = 0.2
Erreygers corrected immunisation index at 90 per cent coverage
At 90 per cent coverage the same standard index of 0.10 gives a corrected index of 0.36, against a Wagstaff index of 1 in HE-EX-CIX-004. Both corrections place the second country above the first, but they disagree on how far apart the two are (computed here for illustration).
C = 0.10; mu = 0.9; a = 0; b = 1; E = 0.36
Common errors
Reading the Erreygers corrected index as a measure of relative inequality
If illness prevalence doubles in every quintile of the article's example, to 0.80, 0.60, 0.50, 0.30 and 0.20, the standard index stays at minus 0.25 but the corrected index doubles from minus 0.24 to minus 0.48 (computed here for illustration). A larger corrected index can therefore come from a proportional rise in illness with no change in relative inequality, in line with Wagstaff's point that the index measures absolute inequality.
Using the observed range in place of the scale bounds in the Erreygers correction
Taking a and b as the lowest and highest quintile prevalences in the article's example, 0.10 and 0.40, divides by 0.30 in place of 1 and gives minus 0.8 in place of minus 0.24 (computed here for illustration). The bounds belong to the measurement scale and stay fixed across the populations being compared.
Sources
Erreygers paper proposing the corrected concentration index
Erreygers G. Correcting the concentration index. Journal of Health Economics. 2009;28(2):504-515. doi:10.1016/j.jhealeco.2008.02.003. Abstract: a corrected, rank-dependent version of the concentration index that satisfies four requirements, transfer, level independence, cardinal invariance and mirror.
Erreygers and Van Ourti on corrected and Wagstaff indices for bounded variables
Erreygers G, Van Ourti T. Measuring socioeconomic inequality in health, health care and health financing by means of rank-dependent indices: a recipe for good practice. Journal of Health Economics. 2011;30(4):685-694. doi:10.1016/j.jhealeco.2011.04.004. Equations 2a, 4 and 5 give the standard, Wagstaff and Erreygers indices as weighted sums of health with weights equal to rank minus (n + 1)/2; section 4.3 defines the mirror property, and the introduction states that the corrected index and the generalised Wagstaff index both pass the mirror and cardinal invariance tests.
Wagstaff comment on the Erreygers corrected concentration index
Wagstaff A. Correcting the concentration index: a comment. Journal of Health Economics. 2009;28(2):516-520. doi:10.1016/j.jhealeco.2008.12.003. Abstract: the standard index reflects a commitment to relative inequality, whereas the Erreygers index ends up as a measure of absolute inequality.
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