Horizontal inequity index as a difference of concentration indices

Signature

HI = C_M - C_N

Function

Horizontal equity measurement function for health care delivery and finance

Computational function

• Computational function: horizontal inequity index from group shares, actual use and need-expected use

  Takes the inputs a horizontal equity table usually holds, the population share, mean actual use and mean need-expected use of each income group from poorest to richest, and returns the concentration indices of actual and need-expected use and the horizontal inequity index. It builds each group's fractional rank at the midpoint of its interval, applies the grouped covariance form of the concentration index (HE-FM-HINQ-005) to each column, written as in the article for a mean rank of one half, and then applies HE-FM-HEQ-003. The inputs therefore differ from the formula's variables: HE-FM-HEQ-003 needs the two indices already computed, and the function builds them from group means, for groups of any size.

  Inputs and outputs: f_t: Share of the population in income group t, ordered from poorest to richest; required, each above zero. The R and Python versions divide by the total, so counts can be entered. Unit: proportion.; mu_t: Mean actual use of care in group t; required, zero or above, with an overall mean above zero. Unit: units of use, for example visits per person per year.; muX_t: Mean need-expected use in group t from HE-FM-HEQ-001; required, above zero. Unit: units of use.; R_t: Fractional rank of group t, the shares before it plus half its own, returned as an intermediate output. Unit: none.; mu: Population-weighted mean actual use, an intermediate output. Unit: units of use.; muX: Population-weighted mean need-expected use, an intermediate output. Unit: units of use.; C_M: Concentration index of actual use. Unit: index without unit.; C_N: Concentration index of need-expected use. Unit: index without unit.; HI: Horizontal inequity index. Unit: index without unit.

  Assumption: Groups are ranked by income or another living standards measure, and each group is represented by its mean, so only inequality between groups enters. When mu and muX are equal, as with linear standardisation, HI also equals the concentration index of the need-standardised group means.

  Worked example (Doctor visits across five income quintiles): The article's quintiles give concentration indices of minus 0.04 for actual visits and minus 0.10 for need-expected visits, and a horizontal inequity index of 0.06. f_t = [0.2,0.2,0.2,0.2,0.2]; mu_t = [4.4,4.2,4.0,3.8,3.6]; muX_t = [5.0,4.5,4.0,3.5,3.0]; mu = 4.0; muX = 4.0; C_M = -0.04; C_N = -0.10; HI = 0.06

  Worked example (Jamaica quintile means for preventive care): Entering the quintile means of actual and need-predicted use from Box 15.1 of the World Bank guide gives indices of 0.0802 and minus 0.0440 and a horizontal inequity index of 0.1242. These are smaller in absolute value than the guide's 0.0928, minus 0.0452 and 0.1374 from individual data, mainly because quintile means leave out inequality within quintiles. f_t = [0.2,0.2,0.2,0.2,0.2]; mu_t = [0.1717,0.2003,0.2052,0.2157,0.2706]; muX_t = [0.2363,0.2158,0.2119,0.1954,0.1888]; mu = 0.2127; muX = 0.2096; C_M = 0.0802; C_N = -0.0440; HI = 0.1242

  Worked example (Use in line with need gives a zero index): When actual and need-expected use are equal in every group, the two indices coincide and the horizontal inequity index is zero, a limiting case that checks the implementation. f_t = [0.2,0.2,0.2,0.2,0.2]; mu_t = [5.0,4.5,4.0,3.5,3.0]; muX_t = [5.0,4.5,4.0,3.5,3.0]; mu = 4.0; muX = 4.0; C_M = -0.10; C_N = -0.10; HI = 0

  Excel: =LET(ranks,SCAN(0,Pop,LAMBDA(a,b,a+b))-Pop/2,2*SUMPRODUCT(Pop,Actual,ranks)/SUMPRODUCT(Pop,Actual)-2*SUMPRODUCT(Pop,Expected,ranks)/SUMPRODUCT(Pop,Expected)) With population shares summing to 1 in a range named Pop and the group means of actual and need-expected use in Actual and Expected, poorest group first, SCAN builds the midpoint ranks and the formula returns the horizontal inequity index; the minus 1 in each concentration index cancels. SCAN and LAMBDA need Excel for Microsoft 365 or Excel 2024.

  R: hi_grouped <- function(f, m, mx) { f <- f/sum(f); r <- cumsum(f)-f/2; conc <- function(y) 2*sum(f*y*r)/sum(f*y)-1; c(C_M = conc(m), C_N = conc(mx), HI = conc(m)-conc(mx)) } Returns the two concentration indices and the horizontal inequity index; the shares are rescaled to sum to 1 inside the function.

  Python: def hi_grouped(f, m, mx): w = [x/sum(f) for x in f]; r = [sum(w[:t])+w[t]/2 for t in range(len(w))]; conc = lambda y: 2*sum(a*b*c for a, b, c in zip(w, y, r))/sum(a*b for a, b in zip(w, y))-1; return {'C_M': conc(m), 'C_N': conc(mx), 'HI': conc(m)-conc(mx)} Plain Python with no imports; the shares are rescaled to sum to 1 inside the function.

  Test (Population-weighted mean rank is one half): With midpoint ranks built from shares that sum to 1, the weighted mean rank is 0.5, which the grouped formula relies on. Expected result: TRUE. Excel check: =ABS(SUMPRODUCT(Pop,Ranks)-0.5)<1E-9

  Test (Index matches the concentration index of standardised group means): When mu and muX are equal, the function's index equals the concentration index of the need-standardised group means, actual minus need-expected use plus the overall mean, computed separately and held in ConcStandardised. In the article's example both are 0.06. Expected result: TRUE. Excel check: =ABS(HIGrouped-ConcStandardised)<1E-9

  Common error (Groups ordered from richest to poorest): Entering the richest group first reverses the ranks and flips the sign of every index: the article's data then give concentration indices of 0.04 and 0.10 and a horizontal inequity index of minus 0.06, so a pro-rich distribution appears pro-poor.

  Source: 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, equations 8.3 and 8.5, on the covariance form of the concentration index and the fractional rank of a group at the midpoint of its interval, and chapter 15, pages 178 to 180, on the horizontal inequity index as the difference between the concentration indices of actual and need-predicted use, with the Jamaica figures in Box 15.1.

  R_t = sum_(k=1)^(t-1) [f_k] + f_t / 2; mu = sum_(t=1)^T [f_t * mu_t]; muX = sum_(t=1)^T [f_t * muX_t]; C_M = 2 / mu * sum_(t=1)^T [f_t * mu_t * R_t] - 1; C_N = 2 / muX * sum_(t=1)^T [f_t * muX_t * R_t] - 1; HI = C_M - C_N

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Implementations

• Excel

  Horizontal inequity index in one Excel cell

  With the concentration index of actual use in a cell named ConcActual and that of need-expected use in ConcNeed, both computed with the same income ranking, Excel returns the index.

  =ConcActual-ConcNeed

Assumptions

• Same people and income ranking for both concentration indices

  Both indices use the same sample, weights and fractional ranks in the income distribution. Ranking by use or by need instead of income would measure a different inequality.

• Equal means of actual and need-expected use in the difference form

  The difference equals the concentration index of need-standardised use exactly when mean need-expected use equals mean actual use, as with linear least squares. When the means differ, as with the probit predictions in the World Bank guide's Jamaica example, the two calculations diverge slightly.

• Remaining inequality in use read as horizontal inequity

  After observed need is allowed for, any remaining inequality in use by income is read as horizontal inequity. If poorer people have need that the survey does not record, their need-expected use is set too low, and the index understates pro-rich inequity or can show pro-poor inequity where none exists.

Worked examples

• Horizontal inequity in doctor visits across five income quintiles

  In the article's illustrative quintile data, actual visits have a concentration index of minus 0.04 and need-expected visits minus 0.10, so the horizontal inequity index is 0.06. Raw use looks slightly pro-poor, but at equal need the distribution is pro-rich.

  C_M = -0.04; C_N = -0.10; HI = 0.06

• Horizontal inequity in preventive care in Jamaica in 1989

  In the World Bank guide's Jamaica example the concentration index of actual preventive care use is 0.0928 and that of need-predicted use from a probit with controls is minus 0.0452, so the difference is 0.1380. The guide reports 0.1374 for the concentration index of need-standardised use, because mean predicted use, 0.2097, is below mean actual use, 0.2127; scaling C_N by the ratio of the two means before subtracting reproduces 0.1374.

  C_M = 0.0928; C_N = -0.0452; HI = 0.1380

Common errors

• Reading the concentration index of actual use as horizontal inequity

  Unstandardised inequality in use mixes differences in need with inequity. In the article's example actual visits have an index of minus 0.04, which suggests that the poor are favoured, while the horizontal inequity index of 0.06 shows pro-rich inequity at equal need.

• Subtracting the index of actual use from the index of need-expected use

  C_N minus C_M reverses the sign of the result. The article's data would give minus 0.06 and suggest pro-poor inequity where the distribution at equal need is pro-rich.

• Treating a horizontal inequity index near zero as equal treatment

  An index near zero can come from a concentration curve of standardised use that crosses the line of equality, with the areas on each side cancelling out. The curve is read alongside the index before concluding that there is no inequity.

Sources

• World Bank guide on the horizontal inequity index

  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 15, pages 178 to 180: inequity measured by the concentration index of need-standardised utilisation, referred to as HI_WV, or equivalently as the difference between the concentration indices of actual and need-predicted utilisation; the assumption that residual variation after observed need is due to non-need factors; and Box 15.1 with the Jamaica 1989 figures.

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• World Bank guide on the concentration index behind the horizontal inequity index

  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, pages 95 to 96: the concentration index as twice the area between the concentration curve and the line of equality, its covariance form (equation 8.3), the sign convention, and the warning that an index of zero can hide a curve that crosses the line of equality.

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Canonical Identity