Concentration Index

Concentration Index Formula, Sign, Bounds and Corrections in Health Equity Analysis

The concentration index equals twice the area between a concentration curve and the line of equality, and in practice it is computed as twice the covariance between a health variable and each person's fractional rank in the living standards distribution, divided by the variable's mean. A negative value shows that the variable sits disproportionately with poorer people. For ill health or mortality that is inequality to the disadvantage of the poor, while for good health, coverage or health care use it means the poor hold more than their population share. Health economists use it to compare health inequality across countries, years and services, to test horizontal equity in care and to describe who gains from health spending. Bounded and binary variables call for the Wagstaff and Erreygers corrections, and a need-standardised version measures horizontal inequity.

From the concentration curve to a single number

The concentration curve shows the share of a health variable held by the poorest fraction of the population, and the index condenses it into one figure that can be compared across settings. O'Donnell, van Doorslaer, Wagstaff and Lindelow, in the World Bank guide to analysing health equity with household survey data, define the index from the curve as follows.

$$C = 1 - 2\int_{0}^{1} L_h(p)\,dp$$

where $C$ is the concentration index, $p$ is the cumulative proportion of the population ranked from poorest to richest and $L_h(p)$ is the cumulative proportion of the health variable held by that group. The index is zero when there is no socioeconomic-related inequality, negative when the curve lies above the line of equality and positive when it lies below. It can also be zero when the curve crosses the line and the areas on each side cancel, so the guide advises reading the index together with the curve. The index depends only on the relationship between the health variable and people's rank in the living standards distribution, so a change in income inequality need not change it.

The covariance formula and how it is estimated

With individual survey data the guide recommends the convenient covariance form. Any statistical package can compute it from the variable, its mean and each person's rank.

$$C = \frac{2}{\mu}\,\mathrm{cov}(h_i, r_i), \qquad r_i = \frac{i - 0.5}{N}$$

where $h_i$ is the health variable for person $i$, $\mu$ is its mean, $N$ is the number of people and $r_i$ is the fractional rank, with $i = 1$ for the poorest and $i = N$ for the richest. With survey weights scaled to sum to one, the rank becomes the sum of the weights of everyone poorer plus half the person's own weight, and the guide states that the weights should also enter the covariance and the mean. Because the fractional rank has mean 0.5, the same index can be written as $C = \frac{2}{N\mu}\sum_{i} h_i r_i - 1$, and with grouped data each group's mean and mid-rank replace the individual values, weighted by each group's population share.

An equivalent estimate comes from a convenient regression of $(2\sigma_r^2/\mu)\,h_i$ on the fractional rank, where $\sigma_r^2$ is the variance of the rank; the slope is the concentration index. This form gives standard errors that can allow for heteroskedasticity and cluster sampling. In the guide's example of the public subsidy to hospital outpatient care in Vietnam in 1998, the index was 0.167, showing that the better-off received more of the subsidy, and allowing for within-cluster correlation raised the standard error from 0.034016 to 0.041988.

Reading the sign for ill health, good health and health services

The sign gives the direction of the link between the variable and living standards, and its meaning depends on whether the variable is good or bad. The guide's convention is that a negative value indicates disproportionate concentration among the poor, and that for a bad such as ill health a negative value means ill health is higher among the poor.

An example from the guide shows the reading. Under-five deaths in India in 1982 to 1992, grouped by wealth quintile, gave an index of -0.1694, reflecting higher mortality among poorer children. For health services a negative index is not on its own evidence of fairness, because poorer people often have greater need; that question belongs to the horizontal inequity index below. The magnitude reflects both the strength of the relationship and the variability of the health variable. The guide reports Koolman and van Doorslaer's result that 75 times the index gives the percentage of the variable that would need to be linearly redistributed from the richer half to the poorer half, where inequality favours the rich, to bring the index to zero.

Range, bounded variables and the Wagstaff and Erreygers corrections

The index lies between -1 and 1 when the variable is measured on a ratio scale with non-negative values. It is unchanged when the variable is multiplied by a constant, such as converting monthly use to yearly use, but it changes when a constant is added. If the variable can take negative values the index is no longer bounded by -1 and 1, and with a mean of zero it is undefined.

Binary variables raise a further problem. Wagstaff showed that, in large samples, the minimum and maximum of the index for a binary variable are $\mu - 1$ and $(1 - \mu)$, so the feasible range shrinks as the mean rises, which complicates comparing immunisation or mortality inequality across countries with different means. His normalisation divides the index by $(1 - \mu)$. Erreygers proposed a corrected index meeting four requirements, namely transfer, level independence, cardinal invariance and mirror. For a variable bounded between $a$ and $b$, the formulas set out by Erreygers and Van Ourti give it as follows.

$$E = \frac{4\mu}{b - a}\,C = \frac{8}{b - a}\,\mathrm{cov}(h_i, r_i)$$

where $E$ is the Erreygers corrected index, $a$ and $b$ are the lower and upper bounds of the variable and the other terms are as above. The mirror property requires an index of health and the same index of the matching ill health variable to be equal in size and opposite in sign; the standard index fails it. Erreygers and Van Ourti show that the corrected index and the generalised form of Wagstaff's normalisation 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 Erreygers index ends up measuring absolute inequality.

Worked example of limiting illness across five income quintiles

The figures in this example are illustrative. A survey reports the share of adults with a limiting long-term illness in each income quintile, from poorest to richest, of 0.40, 0.30, 0.25, 0.15 and 0.10. Each quintile holds 20 per cent of the population, and every member of a quintile is given the quintile's mid-rank.

1. Ranks and mean. The fractional ranks are 0.1, 0.3, 0.5, 0.7 and 0.9. The mean prevalence is $\mu = 1.20 / 5 = 0.24$.

2. Covariance. The products of prevalence and rank are 0.04, 0.09, 0.125, 0.105 and 0.09, which sum to 0.45 and average 0.09. The covariance is the mean product minus the product of the means.

$$\mathrm{cov}(h, r) = 0.09 - 0.24 \times 0.5 = -0.03$$

where $h$ is illness prevalence and $r$ is the fractional rank.

3. Concentration index. The index is $C = (2 \times -0.03) / 0.24 = -0.25$. Computing it from the concentration curve, with cumulative shares 0.3333, 0.5833, 0.7917, 0.9167 and 1, gives the same value.

4. Bounds and corrections. Illness is binary, so the feasible range is, in large samples, -0.76 to 0.76. The Wagstaff normalised index is $W = -0.25 / 0.76 = -0.329$. The Erreygers index, with $a = 0$ and $b = 1$, is $E = 4 \times 0.24 \times -0.25 = -0.24$, which equals 8 times the covariance.

5. The mirror check. The share without limiting illness is 0.60, 0.70, 0.75, 0.85 and 0.90, with mean 0.76. Its covariance with rank is +0.03, so its standard index is $C = (2 \times 0.03) / 0.76 = 0.079$, far smaller in size than -0.25. Its Wagstaff index is $W = 0.0789 / 0.24 = 0.329$ and its Erreygers index is $E = 8 \times 0.03 = 0.24$, exact mirrors of the illness values.

6. Interpretation. All three indices agree that illness is concentrated among poorer people. The standard index suggests the inequality is about three times larger when described as illness than as good health, an artefact of the bounded variable, which is why comparisons of binary outcomes across populations with different means should state which correction was used.

Horizontal inequity and the index of need-standardised use

An index of health care use mixes differences in need with other influences such as income. The guide measures horizontal inequity, equal treatment for equal need, as the concentration index of actual use minus that of use predicted from need indicators such as age, sex and self-assessed health, $HI = C_M - C_N$, where $C_M$ and $C_N$ are the indices of actual and need-predicted use. A positive value indicates pro-rich inequity and a negative value pro-poor inequity.

The guide's example uses the 1989 Survey of Living Conditions in Jamaica and the probability of a preventive care visit. The index was 0.0928 for actual use and -0.0452 for need-predicted use, and the index of need-standardised use ranged from 0.1318 to 0.1374 across four standardisation methods. The simple difference, 0.138, is close but not identical because need was predicted from a probit model with controls. Need was concentrated among the poor, so need-standardised use was even more pro-rich than actual use. The method assumes that variation left after controlling for observed need reflects non-need factors, which the guide notes is biased if unobserved need is correlated with income.

Uses in health economics for financing, explanation and distributional analysis

The index is a building block for other measures. In financing analysis the Kakwani index of progressivity is the concentration index of health payments minus the Gini coefficient of ability to pay, ranging from -2 to 1, with a positive value indicating progressivity. For explanation, the index of an outcome modelled as a linear function of its determinants decomposes into the sum of each determinant's elasticity times its own concentration index, plus a residual term, so analysts can show how much income-related inequality in, for example, child malnutrition is associated with household consumption, water or sanitation.

In distributional cost-effectiveness analysis (DCEA) the index plays a descriptive role. The NICE support document on health inequalities for its technology appraisal manual (PMG36) asks for net health benefit by quintile of the Index of Multiple Deprivation and for descriptive inequality metrics such as gaps, ratios or simple regressions that use the middle groups. It does not name the concentration index, but the index can be estimated as the slope of a regression on fractional rank and uses all five quintiles. Because net health benefit can be negative or close to zero on average, an absolute measure such as the generalised concentration index, $\mu C$, or the slope index of inequality is the safer fit. The same document says DCEA results should not apply health inequality aversion weights. The tutorial by Asaria, Griffin and Cookson ranks strategies by dominance and evaluates trade-offs with Atkinson and Kolm social welfare indices.

Choices that change the index and common misreadings

The ranking variable matters when the change in rank between measures is correlated with the health variable. The guide reproduces Lindelow's results for Mozambique in 1996 to 1997, where the index for health centre visits was 0.066 with households ranked by consumption and -0.136 with an asset index. Analysts should therefore report the ranking variable, the direction of the health variable, the weights, the survey design and the mean.

Common errors are reading a negative index for a bad as pro-poor, treating an index of use as inequity without standardising for need, and comparing standard indices of binary variables across populations with different means. An index near zero can hide a crossing curve, and the index describes association, not cause. The health Gini coefficient ranks people by the health variable itself and so measures pure health inequality; the concentration index ranks them by living standards and can be negative. The concentration curve page sets out how the two relate.

Sources

• Asaria M, Griffin S, Cookson R. Distributional cost-effectiveness analysis: a tutorial. Medical Decision Making. 2016;36(1):8-19.
• Erreygers G. Correcting the concentration index. Journal of Health Economics. 2009;28(2):504-515.
• 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.
• National Institute for Health and Care Excellence. Support document: health inequalities. Distributional cost-effectiveness analysis methods. In: NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). London: NICE; 2022, last updated 2026.
• 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; Chapter 13, Explaining socioeconomic-related health inequality; Chapter 15, Measuring and explaining inequity in health service delivery; Chapter 16, Who pays for health care.
• Wagstaff A. The bounds of the concentration index when the variable of interest is binary, with an application to immunization inequality. Health Economics. 2005;14(4):429-432.
• Wagstaff A. Correcting the concentration index: a comment. Journal of Health Economics. 2009;28(2):516-520.