Total health forgone for equity between a health-maximising and a fairer option

Signature

H_max = s_worse * g_worse_max + s_better * g_better_max; H_fair = s_worse * g_worse_fair + s_better * g_better_fair; H_forgone = H_max - H_fair; f_forgone = H_forgone / H_max

Function

Health forgone for equity, switching weight and implied inequality aversion function

Computational function

• Computational function for health forgone by each option against the health-maximising option

  Takes the population share of each group and the net health gain per person of each group under each of several options of equal cost, and returns each option's mean net gain, the highest mean gain and the health each option forgoes against it. This is the equity constraint analysis of HE-FM-EET-001 run over a set of options, so the health-maximising option is found by the procedure rather than named beforehand. The Math / Code line shows two groups and three options, and the R and Python versions take any number of groups and options.

  Inputs and outputs: s_worse and s_better are the population shares, required, zero or above and adding to 1; g_worse_1, g_worse_2 and g_worse_3 are the worse-off group's net gains per person under options 1, 2 and 3; g_better_1, g_better_2 and g_better_3 are the better-off group's; all gains in years of quality-adjusted life expectancy. H_1, H_2 and H_3 are the mean net gains per person, H_top is the highest of them, and F_1, F_2 and F_3 are the health forgone by each option against H_top, zero for the health-maximising option.

  Assumption: All options draw on the same budget, and each gain is net of the health displaced in that group. Two options with the same mean gain both forgo zero, and since neither dominates the other on cost and health under HE-FM-DOM-001 the choice between them rests on equity alone.

  Worked example (Programmes E and F and doing nothing): With the article's illustrative shares of 0.4 and 0.6 and net gains of 0.10 and 0.60 under E, 0.50 and 0.25 under F and zero under doing nothing, the mean gains are 0.40, 0.35 and 0, E is health-maximising, F forgoes 0.05 years per person and doing nothing 0.40. s_worse = 0.4; s_better = 0.6; g_worse_1 = 0.10; g_better_1 = 0.60; g_worse_2 = 0.50; g_better_2 = 0.25; g_worse_3 = 0; g_better_3 = 0; H_1 = 0.4; H_2 = 0.35; H_3 = 0; H_top = 0.4; F_1 = 0; F_2 = 0.05; F_3 = 0.4

  Worked example (A tie at the top): Replacing doing nothing with an illustrative option giving 0.40 years to both groups ties it with E at a mean gain of 0.40, so both forgo zero health. s_worse = 0.4; s_better = 0.6; g_worse_1 = 0.10; g_better_1 = 0.60; g_worse_2 = 0.50; g_better_2 = 0.25; g_worse_3 = 0.40; g_better_3 = 0.40; H_1 = 0.4; H_2 = 0.35; H_3 = 0.4; H_top = 0.4; F_1 = 0; F_2 = 0.05; F_3 = 0

  Excel: =SUMPRODUCT(ShareGroup,INDEX(GainMatrix,0,COLUMN()-MIN(COLUMN(GainMatrix))+1)) With one row per group, shares in a column range named ShareGroup and net gains in a range named GainMatrix with one column per option, the formula entered in each cell of a helper row named MeanGainRow, set in the same columns as GainMatrix, returns that option's mean gain, the COLUMN terms giving the option number. =MAX(MeanGainRow)-INDEX(MeanGainRow,COLUMN()-MIN(COLUMN(GainMatrix))+1) entered in each cell of a helper row named ForgoneRow, in the same columns, then returns the health that option forgoes.

  R: eq_forgone <- function(s, G) { s <- s/sum(s); H <- colSums(s*G); list(H = H, H_top = max(H), forgone = max(H)-H) } Takes a vector of shares and a matrix of gains with one row per group and one column per option, and normalises the shares so counts can be entered.

  Python: def eq_forgone(s, G): t = sum(s); H = [sum(si*row[k] for si, row in zip(s, G))/t for k in range(len(G[0]))]; top = max(H); return H, top, [top-h for h in H] Takes a list of shares and a list of rows of gains, one row per group and one entry per option.

  Test (Article's health forgone by F reproduced): The health forgone by F against the best of the three options is 0.05 years per person. Expected result TRUE. Excel check =ABS(MAX(SUMPRODUCT({0.4;0.6},{0.1;0.6}),SUMPRODUCT({0.4;0.6},{0.5;0.25}),0)-SUMPRODUCT({0.4;0.6},{0.5;0.25})-0.05)<0.000001

  Test (No option forgoes negative health): With the forgone values held in a helper row named ForgoneRow, every value is zero or above because each is measured against the highest mean gain. Expected result TRUE. FALSE shows forgone health measured against a comparator such as doing nothing, which gives minus 0.40 for E and minus 0.35 for F. Excel check =MIN(ForgoneRow)>=0

  Common error (Measuring health forgone against the comparator): Taking F against doing nothing gives minus 0.35 years, a gain, and hides the 0.05 years F gives up against E. Health forgone for equity is measured against the health-maximising option.

  Source: Cookson R, Mirelman AJ, Griffin S, Asaria M, Dawkins B, Norheim OF, Verguet S, Culyer AJ. Using cost-effectiveness analysis to address health equity concerns. Value in Health. 2017;20(2):206-212. Section on equity constraint analysis, which compares two or more options given a fixed budget and treats the health loss of the more equitable option as the cost of the equity constraint. Asaria M, Griffin S, Cookson R. Distributional cost-effectiveness analysis: a tutorial. Medical Decision Making. 2016;36(1):8-19. Section on comparing interventions in terms of total health and unfair health inequality, on the health opportunity cost as the difference in total health.

  H_1 = s_worse * g_worse_1 + s_better * g_better_1; H_2 = s_worse * g_worse_2 + s_better * g_better_2; H_3 = s_worse * g_worse_3 + s_better * g_better_3; H_top = max(H_1, max(H_2, H_3)); F_1 = H_top - H_1; F_2 = H_top - H_2; F_3 = H_top - H_3

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Implementations

• Excel

  Health forgone for equity from named share and gain cells

  With the shares in ShareWorse and ShareBetter and the net gains in GainWorseMax, GainBetterMax, GainWorseFair and GainBetterFair, the first formula returns the health forgone, held in a cell named HealthForgone, and the second its share of the health-maximising gain.

  =(ShareWorse*GainWorseMax+ShareBetter*GainBetterMax)-(ShareWorse*GainWorseFair+ShareBetter*GainBetterFair); =HealthForgone/(ShareWorse*GainWorseMax+ShareBetter*GainBetterMax)

Assumptions

• Net gains after the displaced health in each group

  Each g value is the gross gain less the health displaced elsewhere in the system in that group. The article's example spreads 0.10 years per person evenly across both groups, the base case that the NICE support document on health inequalities sets for the distribution of health opportunity costs, with gradients towards more deprived groups explored as scenarios.

• Two options of equal cost from one budget in the health forgone for equity

  The two options draw on the same budget, so the comparison is between health outcomes alone. Where costs differ, the displaced health of each option is computed at its own cost before the g values are formed, and the difference then includes the extra opportunity cost.

• Population shares sum to one and gains are per person

  The shares s_worse and s_better add to 1 and each g is a gain per person within its group, so H_max and H_fair are per person of the whole population. Multiplying by the population size gives total health forgone.

Worked examples

• Health forgone choosing programme F over programme E

  In the article's illustrative example the more deprived group is 40% of the population. Programme E gives net gains of 0.10 and 0.60 years and programme F 0.50 and 0.25 years. The mean net gains are 0.40 years under E and 0.35 under F, so choosing F forgoes 0.05 years per person, one-eighth of E's net gain.

  s_worse = 0.4; s_better = 0.6; g_worse_max = 0.10; g_better_max = 0.60; g_worse_fair = 0.50; g_better_fair = 0.25; H_max = 0.40; H_fair = 0.35; H_forgone = 0.05; f_forgone = 0.125

• No health forgone when the fairer option matches total health

  An illustrative fairer option that gives 0.40 years per person to both groups has the same mean gain as programme E, so no health is forgone and choosing it involves no equity-efficiency trade-off. This is the limiting case of the win-win cell of the health equity impact plane.

  s_worse = 0.4; s_better = 0.6; g_worse_max = 0.10; g_better_max = 0.60; g_worse_fair = 0.40; g_better_fair = 0.40; H_max = 0.4; H_fair = 0.4; H_forgone = 0; f_forgone = 0

Common errors

• Averaging group gains without population shares

  Taking the simple average of the two groups' gains gives 0.35 years for E and 0.375 for F, so F appears to gain 0.025 years more and the trade-off disappears. With the 40% and 60% shares E leads by 0.05 years per person.

• Stating the share forgone on gross gains

  Using gains before the 0.10 years of displaced health gives mean gains of 0.50 and 0.45 years. The difference is still 0.05 because both options displace the same health, but the share forgone falls to one-tenth instead of one-eighth, understating the cost of equity relative to what E delivers.

• Reading the switching weight as removing the health forgone

  A weight on the worse-off group above the switching weight of 1.3125 from HE-FM-WE-004 changes the ranking but not the health outcome. Choosing F still gives up 0.05 years per person of total health, which is why the article recommends reporting weighted results beside the unweighted ones.

Sources

• Asaria and colleagues on the health opportunity cost of reducing inequality

  Asaria M, Griffin S, Cookson R. Distributional cost-effectiveness analysis: a tutorial. Medical Decision Making. 2016;36(1):8-19. Section on comparing interventions in terms of total health and unfair health inequality, which states that the health opportunity cost of choosing an intervention that reduces health inequality is the difference in total health between the intervention and a comparator.

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• Cookson and colleagues on equity constraint analysis

  Cookson R, Mirelman AJ, Griffin S, Asaria M, Dawkins B, Norheim OF, Verguet S, Culyer AJ. Using cost-effectiveness analysis to address health equity concerns. Value in Health. 2017;20(2):206-212. Section Equity Constraint Analysis (Counting the Cost of Equity), which counts the cost of choosing fairer but less cost-effective options and states that the health loss of the more equitable option indicates the value the decision maker places on the equity constraint, compared within a fixed budget.

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• NICE support document on the distribution of health opportunity costs

  National Institute for Health and Care Excellence. Support document, health inequalities. Technology evaluation methods support document for NICE technology appraisal and highly specialised technologies guidance, the manual (PMG36). London, NICE, 2022, last updated 13 October 2026; page last updated 13 May 2025. Section Distributional cost-effectiveness analysis methods, subsection Health opportunity cost and Table 2, which state that health opportunity costs should be equally distributed between social groups in the base-case analysis, accompanied by scenario analyses of light and moderate gradients, labelled slight and moderate in Table 2, with more of the displacement in more disadvantaged groups.

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• Williams and Cookson on the opportunity cost of equity concerns

  Williams AH, Cookson RA. Equity-efficiency trade-offs in health technology assessment. International Journal of Technology Assessment in Health Care. 2006;22(1):1-9. Abstract, which states that every departure from a strict cost-effectiveness approach has an opportunity cost and that its size tests how much weight an equity concern is deemed to merit.

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