Signature
W_X = QA_X + w * QB_X; W_Y = QA_Y + w * QB_Y; w_s = (QA_X - QA_Y) / (QB_Y - QB_X)
| Inputs | Definition | Unit |
|---|---|---|
QA_X | Total QALYs of group A under X | QALYs |
w | Relative value of a QALY accruing to group B, 1 for the utilitarian sum | dimensionless |
QB_X | Total QALYs of group B under X | QALYs |
QA_Y | Total QALYs of group A under Y | QALYs |
QB_Y | Total QALYs of group B under Y, different from QB_X | QALYs |
W_X | Weighted total QALYs under option X | weighted QALYs |
|---|---|---|
W_Y | Weighted total QALYs under option Y | weighted QALYs |
w_s | Weight on group B at which X and Y are ranked equally | dimensionless |
Function
Welfare change measurement and aggregation function
Converts each person's gain or loss from a change into money, with the compensating or equivalent variation or the consumer surplus approximation, and then ranks the change for society: by adding the money measures, as the Kaldor-Hicks potential compensation test does, or by combining individual outcomes with an explicit social welfare function that states how gains to one person are set against losses to another.
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Implementations
Excel
Weighted welfare scores and switching weight
With the QALY totals in QAlyAX, QAlyBX, QAlyAY and QAlyBY and the weight in WeightB, the formulas return the two weighted scores and the switching weight.
=QAlyAX+WeightB*QAlyBX; =QAlyAY+WeightB*QAlyBY; =(QAlyAX-QAlyAY)/(QAlyBY-QAlyBX)
Assumptions
Linear weighting within each group
Every QALY in a group receives the same weight, so the function is linear. An inequality-averse function that is concave in each person's health gives weights that change with the level of health.
Weights are value judgements
The weight expresses a distributional judgement, even when elicited from public preference surveys. Reporting the switching weight lets decision-makers see how strong that judgement must be to change the ranking.
Worked examples
Reconfiguration against outreach with a weight of 2
In the article's example, option X gives groups A and B 20,030 and 11,990 QALYs and option Y 20,000 and 12,015. With a weight of 2 on group B, X scores 44,010 and Y 44,030, so Y is preferred; the switching weight is 30 divided by 25, or 1.2.
QA_X = 20030; QB_X = 11990; QA_Y = 20000; QB_Y = 12015; w = 2; W_X = 44010; W_Y = 44030; w_s = 1.2
Same options under the utilitarian weight
With a weight of 1 the function is the utilitarian sum: X scores 32,020 and Y 32,015, so X is preferred, as in the cost-benefit result.
QA_X = 20030; QB_X = 11990; QA_Y = 20000; QB_Y = 12015; w = 1; W_X = 32020; W_Y = 32015; w_s = 1.2
Common errors
Reading a switching weight below 1 as favouring the better-off group
The switching weight only says where the ranking changes. Its direction depends on which option does more for group B; here Y does, so every weight above 1.2 favours Y and every weight below it favours X.
Applying group totals when group sizes differ
Comparing group totals ranks options in the same way as QALYs per person only when the groups are the same size and everyone within a group is affected alike. Otherwise a Rawlsian or per-person comparison needs per-person values.
Sources
Social welfare functions over the distribution of QALYs
Wagstaff A. QALYs and the equity-efficiency trade-off. Journal of Health Economics. 1991;10(1):21-41. A social welfare function defined over individuals' health that weights QALYs by whom they accrue to, making the trade-off between total QALYs and their distribution explicit.
Canonical Identity
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