Concentration Curve

Concentration Curve Ranking by Income and Testing for Dominance

In health equity analysis the concentration curve is the picture that comes before the number. Its value lies in showing where in the distribution of living standards a health variable is concentrated, for example among the poorest fifth or only among the richest, which a single summary index cannot reveal. The curve's position relative to the diagonal shows whether the variable is concentrated among the poor or the better-off, and twice the area between the two is the concentration index. Unlike the Lorenz curve, it ranks people by living standards, and it is also used to test for dominance, to compare actual with need-expected use and in distributional cost-effectiveness analysis, while the index itself belongs to its own page.

How the curve is built from ranked survey data

Two variables are needed. The health variable is the one whose distribution is of interest, and it must be measured in units that can be added up across people, such as visits, deaths or currency. The living standards variable, usually income, consumption or a wealth index, is used only to rank people from poorest to richest, so its units do not matter. With individual data sorted from poorest ($i = 1$) to richest ($i = N$), the curve is drawn through the following points.

$$p_k = \frac{k}{N}, \qquad L(p_k) = \frac{\sum_{i=1}^{k} h_i}{\sum_{i=1}^{N} h_i}$$

where $p_k$ is the cumulative share of the population up to person $k$, $N$ is the number of people, $h_i$ is the value of the health variable for person $i$ and $L(p_k)$ is the cumulative share of the health variable held by the poorest $k$ people. With survey weights, each person's weight replaces the count of one in $p_k$ and multiplies $h_i$ in both sums, so the weights change the cumulative population shares and the cumulative shares of the variable but not the order of people. The curve runs from $(0, 0)$ to $(1, 1)$. If everyone has the same value, whatever their living standards, then $L(p) = p$ at every point and the curve is the 45-degree line of equality. A curve above that line means the variable takes higher values among poorer people, and a curve below it means it is concentrated among the better-off. Because the variable can be a good, such as a subsidy, or a bad, such as illness, its direction should always be stated.

From the curve to the concentration index

The concentration index condenses the curve into one number that can be compared across many countries, years or services, which a chart of overlapping curves does not allow. It is twice the area between the curve and the line of equality, equivalently one minus twice the area under the curve, and it is negative when the curve lies above the line. Its formulae and estimation are set out on the concentration index page.

The index loses information that the curve holds. O'Donnell and colleagues point out that it can be zero either because the curve lies on the line of equality or because it crosses the line and the areas on each side cancel, so the index should be read alongside the curve.

Concentration curves and Lorenz curves

The two curves look alike but answer different questions. A Lorenz curve ranks people by the variable being studied, and the Gini coefficient is derived from it; a concentration curve ranks them by living standards. A Lorenz curve therefore never rises above the line of equality, whereas a concentration curve can lie above it, below it or on both sides.

For a non-negative health variable with a positive mean, the concentration curve can never lie below that variable's Lorenz curve, because no group of a given size can hold a smaller share of the variable than the group with the lowest values. By the same argument it can never lie above the mirror-image curve obtained by ranking from the highest values to the lowest. The concentration index is therefore bounded by the Gini coefficient $G$ of the same variable, with $-G \le C \le G$, reaching $G$ only when the two rankings coincide. The distinction matters for health inequality measurement. Wagstaff, Paci and van Doorslaer argued that only the slope index of inequality and the concentration index were likely to give an accurate picture of socioeconomic inequalities in health, and illustrated the dangers of measures such as the Lorenz curve.

The two curves also meet in incidence analysis, where a subsidy's concentration curve is compared with the line of equality and with the Lorenz curve of household consumption. The same comparison for health care payments underlies the Kakwani index of progressivity and the analysis of vertical equity in financing.

Dominance and comparing curves without choosing an index

Curves for two countries, two years or two services can be drawn on one chart. If one curve lies everywhere above the other, it is said to dominate, and the ranking by degree of inequality is unambiguous. If the curves cross, neither dominates, and a ranking requires an index, which imposes value judgements about the weight given to inequality at different points of the distribution; different indices can then rank the curves differently.

Curves estimated from survey data are subject to sampling error, so a visual impression of dominance is not enough. Testing needs standard errors of the ordinates and, for two curves from the same sample, allowance for the dependence between them. O'Donnell and colleagues describe two decision rules, given below.

• Multiple comparison approach. Dominance is accepted if at least one ordinate difference is significant in one direction and none in the other, using studentised maximum modulus critical values to allow for multiple comparisons.
• Intersection union principle. Dominance is accepted only if the difference is significant at every comparison point, a stricter rule with much lower power to detect true dominance in Monte Carlo work by Dardanoni and Forcina.

With reasonably large samples a popular choice has been to compare 19 evenly spaced points from 0.05 to 0.95, since dominance is hard to establish at the extremes. In their example from the 1995 to 1996 National Sample Survey of India, the poorest 20 per cent of individuals received 12.5 per cent of the public health subsidy; the subsidy's curve was dominated by the line of equality, but there was no dominance either way between it and the Lorenz curve of household consumption.

Need-standardised use and horizontal equity

A curve of raw health care use cannot by itself show whether care is fair, because poorer people often have greater need. Horizontal equity, equal treatment for equal need, is assessed by comparing actual use with the use that would be expected given need, which is usually predicted from demographic and health status variables.

Curves of actual use and of need-expected use are then drawn against the same income ranking. The horizontal inequity index is the concentration index of actual use minus that of need-expected use, $HI = C_M - C_N$, which equals twice the area between the two curves, counted positive where the need curve lies above the actual-use curve. A need curve that lies everywhere above the actual-use curve therefore gives a positive index, pro-rich inequity, because at every point the poor receive a smaller share of care than their share of need; the reverse gives pro-poor inequity, and curves that cross leave the sign to the net area. The comparison assumes that differences in use left after controlling for observed need reflect other factors, which is strong when unmeasured need is correlated with income.

Worked example of GP visits and inpatient days across five income quintiles

The figures in this example are illustrative. A health system has survey estimates by income quintile of general practitioner (GP) visits per person per year, need-expected visits predicted from age, sex and self-assessed health, and inpatient days per 10 people per year. Each quintile holds 20 per cent of the population, so the cumulative population shares are 0.2, 0.4, 0.6, 0.8 and 1.0, and every variable has a mean of 4.0.

1. Compute each quintile's share and the cumulative shares. Each share is the quintile value divided by the total of 20.0. For need-expected visits the shares are 0.30, 0.25, 0.20, 0.15 and 0.10, so the curve ordinates are 0.30, 0.55, 0.75, 0.90 and 1.00. For actual visits the shares are 0.225, 0.225, 0.200, 0.175 and 0.175, giving ordinates of 0.225, 0.450, 0.650, 0.825 and 1.000. For inpatient days the shares are 0.25, 0.15, 0.20, 0.15 and 0.25, giving ordinates of 0.25, 0.40, 0.60, 0.75 and 1.00.

2. Read each curve against the line of equality. The line of equality has ordinates 0.2, 0.4, 0.6 and 0.8 at the four interior points. Both GP curves lie above it at every point, so both need and use are concentrated among poorer people. The need curve lies everywhere above the actual-use curve, since the poorest 40 per cent hold 55 per cent of need but receive 45 per cent of visits. The inpatient curve is above the line at 0.2, on it at 0.4 and 0.6, and below it at 0.8, so it crosses the line of equality.

3. Compute the indices from the areas. The area under each curve is a set of trapezoids 0.2 wide, so it equals 0.1 times the sum of each pair of adjacent ordinates, starting from zero. For need-expected visits the pair sums are 0.30, 0.85, 1.30, 1.65 and 1.90, which total 6.00, so the area is 0.60 and $C_N = 1 - 2 \times 0.60 = -0.20$. For actual visits the pair sums total 5.30, so the area is 0.53 and $C_M = 1 - 2 \times 0.53 = -0.06$. For inpatient days they total 5.00, so the area is 0.50 and $C_D = 1 - 2 \times 0.50 = 0$. Here $C_N$, $C_M$ and $C_D$ are the concentration indices of need-expected visits, actual visits and inpatient days.

4. Measure horizontal inequity. The difference between the indices of actual and need-expected use is calculated as follows.

$$HI = C_M - C_N = -0.06 - (-0.20) = 0.14$$

where $HI$ is the horizontal inequity index, here equal to twice the area of 0.07 between the need curve and the actual-use curve.

5. Interpret the result. Raw GP use looks pro-poor, but set against need the picture reverses and the index of 0.14 indicates pro-rich horizontal inequity. The inpatient index is exactly zero, yet the poorest and richest quintiles both use more inpatient care than the middle; the index alone would have reported no inequality.

Concentration curves in distributional cost-effectiveness analysis

Distributional cost-effectiveness analysis estimates how the health gains and health opportunity costs of each option fall across social groups, then ranks options on both total health and its distribution. Dominance rules come first because they rank options with few restrictions on the social welfare function. The tutorial by Asaria, Griffin and Cookson based these rules on Lorenz curves of health and on generalised Lorenz curves, which multiply the Lorenz curve by mean health.

When the concern is socioeconomic inequality, the ranking can instead be by socioeconomic status, which leads to concentration curves. By analogy with the generalised Lorenz curve, a generalised concentration curve multiplies the concentration curve by the mean, as follows.

$$GC(p) = \mu \, L(p)$$

where $GC(p)$ is the generalised concentration curve ordinate at population share $p$, $\mu$ is the mean of the variable per person and $L(p)$ is the concentration curve ordinate. A generalised curve can therefore lie higher because the total is larger, because the distribution is more pro-poor, or both. Wijemunige and colleagues used generalised concentration curve dominance to rank strategies of cardiovascular disease risk screening in Sri Lanka. They simulated the distribution of quality-adjusted life years (QALYs) net of health opportunity costs by an assets index and assumed aversion to pro-rich inequality, and they found one strategy that dominated all but one of the others.

Choices that change the curve

Several analytical choices alter the curve and should be reported. Income, consumption and an assets index can order households differently, and only the ranking matters, so a change in the degree of income inequality need not change the curve or its index. Self-reported health categories have to be converted to a cardinal scale before they can be cumulated, and the result then depends on the conversion chosen. With grouped data the shape of the curve between the plotted points is assumed.

Limits and common misreadings

A common error is to treat a curve above or below the diagonal as evidence of unfairness. A concentration curve describes how a variable is distributed by living standards; whether that is inequitable depends on need and on a normative standard, and the curve does not show why the variable is concentrated where it is. Another is to infer dominance from a chart without a formal test, or to forget that the conclusion depends on the decision rule and the number of comparison points.

Sources

• Asaria M, Griffin S, Cookson R. Distributional cost-effectiveness analysis: a tutorial. Medical Decision Making. 2016;36(1):8-19.
• 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 7, Concentration curves; Chapter 8, The concentration index; Chapter 15, Measuring and explaining inequity in health service delivery.
• Wagstaff A, Paci P, van Doorslaer E. On the measurement of inequalities in health. Social Science & Medicine. 1991;33(5):545-557.
• Wijemunige N, van Baal P, Rannan-Eliya RP, O'Donnell O. Dominance-based distributional cost-effectiveness analysis of cardiovascular disease risk screening in Sri Lanka. Value in Health. 2026;29(6):1012-1020.