Signature
WW = (d_H >= 0) and (d_I <= 0) and (d_H > 0 or d_I < 0); LL = (d_H <= 0) and (d_I >= 0) and (d_H < 0 or d_I > 0); WL = (d_H > 0) and (d_I > 0); LW = (d_H < 0) and (d_I < 0)
| Inputs | Definition | Unit |
|---|---|---|
d_H | Change in total health against the comparator net of the health opportunity cost, for example NHB_tot of HE-FM-DCEA-001 or d_mu of HE-FM-DCEA-002 | QALYs, for the population or per person |
d_I | Change in a chosen inequality index against the comparator, oriented so that a positive value means more inequality, for example d_A of HE-FM-DCEA-002 or the change in the absolute gap of HE-FM-HINQ-001 | unit of the chosen index |
WW | Equals 1 when the programme gains total health and reduces inequality, or improves on one axis with no change on the other, and 0 otherwise | binary indicator |
|---|---|---|
LL | Equals 1 when the programme loses total health and widens inequality, or worsens on one axis with no change on the other, and 0 otherwise | binary indicator |
WL | Equals 1 when the programme gains total health and widens inequality, and 0 otherwise | binary indicator |
LW | Equals 1 when the programme loses total health and reduces inequality, and 0 otherwise | binary indicator |
Function
Distributional cost-effectiveness comparison of options by social group function
Maps each social group's baseline health, its QALY gain from a programme, its share of the health opportunity cost and its size, together with a chosen inequality aversion, to the group's health under the programme, the change in equally distributed equivalent (EDE) health and in the Atkinson index against the comparator, the programme's quadrant on the equity-efficiency impact plane, and the inequality aversion at which two options give equal EDE health. The health opportunity cost of the incremental cost is the live record HE-FM-CET-002, and the EDE and Atkinson index of each distribution are the live records HE-FM-ATK-002, HE-FM-ATK-003, HE-CF-ATK-001 and HE-FM-ATK-001, which are cited here and not redrafted. Other inequality indices for grouped data, such as the absolute gap and the concentration index, are in HE-FN-HINQ-001, and direct equity weights on QALY gains are in HE-FN-EQWA-001. The two-group closed approximation to the switching aversion belongs to the equity-efficiency trade-off package, while the computational function here solves the EDE comparison itself for any number of groups. Notation follows the Distributional Cost-Effectiveness article.
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Implementations
Excel
Impact plane quadrant label from named cells
With the change in total health in a cell named HealthChange and the change in the inequality index in InequalityChange, the formula returns the quadrant label, or no change at the origin, held in a cell named ImpactQuadrant. Rounding both inputs to the model's precision first keeps the tie branches meaningful.
=IF(AND(HealthChange>=0,InequalityChange<=0,OR(HealthChange>0,InequalityChange<0)),"win-win",IF(AND(HealthChange<=0,InequalityChange>=0,OR(HealthChange<0,InequalityChange>0)),"lose-lose",IF(AND(HealthChange>0,InequalityChange>0),"win-lose",IF(AND(HealthChange<0,InequalityChange<0),"lose-win","no change"))))
Assumptions
Comparator at the origin of the equity-efficiency impact plane
Both changes are the programme less the same comparator, which sits at the origin of the plane, as in Figure 1 of Cookson and colleagues 2021. A different comparator moves every programme on the plane.
Inequality index oriented so that a higher value means more inequality
The change d_I is positive when inequality widens. The Atkinson index meets this, and so does the absolute gap of HE-FM-HINQ-001 while the less deprived group remains the healthier. An index that falls as inequality widens is negated before it is entered.
Total health on the impact plane counted net of the opportunity cost
The health axis counts the health displaced by the programme's cost, as in Cookson and colleagues 2017, whose vertical axis is the total health benefit less the health forgone by spending the same money on other programmes, and whose equity axis also allows for opportunity costs. The inequality change therefore uses the distributions of HE-FM-DCEA-001 after the shares of H_d.
Points on an axis of the impact plane follow the dominance tie rule
A zero change on one axis with an improvement on the other is classified with the improving quadrant, as an option of equal cost and greater health dominates under HE-FM-DOM-001, and identical health and inequality give no quadrant. Both changes are rounded to the precision of the model before the signs are read, so that rounding noise does not decide the quadrant.
Worked examples
Article's programme on the gap between groups with equal sharing
With equal sharing the article's programme raises total health by 600 QALYs and widens the gap between the groups by 0.020 years, to 8.020, so it lies in the win-lose quadrant, as the article states.
d_H = 600; d_I = 0.020; WW = 0; LL = 0; WL = 1; LW = 0
Article's programme on the Atkinson index at an aversion of 2
With the Atkinson index at epsilon of 2 as the inequality measure, the change is about 0.0000167 from HE-FM-DCEA-002, so the same programme stays in the win-lose quadrant.
d_H = 600; d_I = 0.0000167; WW = 0; LL = 0; WL = 1; LW = 0
Programme with no change in total health that narrows inequality
An illustrative programme that leaves total health unchanged and narrows the gap by 0.01 years is classified win-win under the tie rule of HE-FM-DOM-001.
d_H = 0; d_I = -0.01; WW = 1; LL = 0; WL = 0; LW = 0
Programme with no change on either axis of the impact plane
With both changes at zero no quadrant applies and every indicator is 0, the limiting case of the comparator itself.
d_H = 0; d_I = 0; WW = 0; LL = 0; WL = 0; LW = 0
Common errors
Reversing the orientation of the inequality index on the impact plane
Measuring the gap as the more deprived group's health less the other group's turns the article's widening of 0.020 years into minus 0.020 and moves the programme out of the win-lose quadrant into the win-win one, reporting a fall in inequality that did not happen.
Treating a win-lose programme as one to reject
The quadrant shows that a trade-off exists and leaves open which objective prevails. The article's programme is win-lose, yet at the cited public aversion of 10.95 it still raises EDE health by 31.8 EDE QALYs under equal sharing, so the decision rests on HE-FM-DCEA-002 or on an explicit judgement about the trade-off.
Sources
Cookson and colleagues 2021 on the equity-efficiency impact plane
Cookson R, Griffin S, Norheim OF, Culyer AJ, Chalkidou K. Distributional cost-effectiveness analysis comes of age. Value in Health. 2021;24(1):118-120. Section The Equity-Efficiency Impact Plane and Figure 1, which place the comparator at the origin, set out the four logical possibilities as the win-win, lose-lose, win-lose and lose-win quadrants, and allow equity impact in units of interest to the decision maker, including the gap between the best and worst off. Section Equity-Efficiency Trade-Offs, which requires trade-off analysis in the win-lose and lose-win cases.
Cookson and colleagues 2017 on the health equity impact plane
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 Health Equity Trade-offs and Figure 1, the health equity impact plane, whose vertical axis is the net total health impact, total health benefits less the forgone health benefits of spending the same money on other programmes, and whose horizontal axis is the net health equity impact after allowing for programme opportunity costs, with quadrants I to IV named win-win, win-lose, lose-lose and lose-win.
Canonical Identity
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