Concentration Index

Concept

---

Theoretically, Concentration Index is a rank-dependent measure of socioeconomic inequality in a health variable. It quantifies the extent to which health outcomes, healthcare utilisation, healthcare expenditure or other health-related quantities are concentrated among individuals with lower or higher socioeconomic status. The index is derived from the concentration curve and combines information about the health variable with the socioeconomic ranking of the population. A value of zero indicates no socioeconomic-related inequality, while the sign identifies the direction of concentration.

Mathematically, the Concentration Index is defined as twice the covariance between the health variable and the individual?s fractional socioeconomic rank, divided by the mean of the health variable. Equivalently, it is twice the area between the concentration curve and the line of equality. For a non-negative, unbounded health variable, the index conventionally ranges from ?1 to 1, although the attainable bounds depend on the distribution and measurement scale of the outcome. Corrections may therefore be required for bounded or binary variables.

In practice, individuals are ranked from the least to the most socioeconomically advantaged, fractional ranks are calculated, and the Concentration Index is estimated using covariance, weighted regression or grouped-data methods. Sampling uncertainty is assessed using analytical standard errors, bootstrap procedures or survey-design-adjusted estimation. In health economics, the index is used to evaluate inequalities in health, healthcare use, healthcare financing, treatment uptake and public expenditure.

---

Purpose

Used to quantify the magnitude and direction of socioeconomic-related inequality in health outcomes, healthcare utilisation, expenditure and financing, supporting equity assessment and comparison across populations, interventions and time periods.

---

Mathematical Formulae

Primary Formula

C = 2Cov(y, R) / ?

where:

• C = Concentration Index
• y = health variable
• R = fractional socioeconomic rank
• ? = mean of the health variable

Supporting Formulae

Fractional rank for unweighted observations:

R? = (i ? 0.5) / n

Discrete formulation:

C = (2 / (n?)) ? ?? y?R? ? 1

Concentration-curve formulation:

C = 1 ? 2 ? ??? L(p) dp

Convenient regression:

2��? ? (y? / ?) = � + ?R? + �?

C = ?

Erreygers correction for a bounded variable:

E = 4?C / (ymax ? ymin)

Related Mathematical Methods

• Concentration Curve
• Covariance
• Fractional Rank
• Lorenz Curve
• Gini Coefficient
• Wagstaff Normalisation
• Erreygers Correction
• Dominance Testing
• Decomposition of the Concentration Index

---

Example

Five patients are ranked from the lowest to the highest household income. Their annual outpatient visits are 8, 6, 4, 2 and 0. Their fractional income ranks are 0.1, 0.3, 0.5, 0.7 and 0.9.

The mean number of visits is:

? = (8 + 6 + 4 + 2 + 0) / 5

= 4

The covariance between outpatient visits and fractional income rank is:

Cov(y, R) = ?0.8

The Concentration Index is:

C = 2 ? (?0.8) / 4

= ?0.40

The negative value indicates that outpatient visits are disproportionately concentrated among patients with lower household incomes.

---

Excel Implementation

Function	Example Formula	Health Economics Application
SORTBY	=SORTBY(A2:B1001,A2:A1001,1)	Sort individuals from the lowest to the highest socioeconomic position.
RANK.AVG	=(RANK.AVG(A2,$A$2:$A$1001,1)-0.5)/COUNT($A$2:$A$1001)	Calculate the fractional socioeconomic rank for each individual.
COVARIANCE.P	=2*COVARIANCE.P(B2:B1001,C2:C1001)/AVERAGE(B2:B1001)	Calculate the Concentration Index from the health variable and fractional ranks.
SUMPRODUCT	=2*SUMPRODUCT(B2:B1001,C2:C1001)/(COUNT(B2:B1001)*AVERAGE(B2:B1001))-1	Calculate the unweighted discrete Concentration Index directly from ranked observations.
SUM	=SUM(B$2:B2)/SUM($B$2:$B$1001)	Calculate cumulative health shares for the associated concentration curve.

---

VBA (Optional)

Automate socioeconomic ranking, Concentration Index estimation, correction for bounded outcomes and production of equity-analysis summary tables.

---

Sources

• Wagstaff A, Paci P, van Doorslaer E. On the measurement of inequalities in health. Social Science & Medicine. 1991;33(5):545?557.
• Kakwani N, Wagstaff A, van Doorslaer E. Socioeconomic inequalities in health: Measurement, computation and statistical inference. Journal of Econometrics. 1997;77(1):87?103.
• Wagstaff A. The bounds of the Concentration Index when the variable of interest is binary, with an application to immunisation inequality. Health Economics. 2005;14(4):429?432.
• Erreygers G. Correcting the Concentration Index. Journal of Health Economics. 2009;28(2):504?515.
• O'Donnell O, van Doorslaer E, Wagstaff A, Lindelow M. Analyzing Health Equity Using Household Survey Data. World Bank.