Signature
g = (B_j - B_i) / (K_j - K_i); lambda_g = 1 / g; K_max = K_j + (B_new - B_j) / g
| Inputs | Definition | Unit |
|---|---|---|
B_j | Benefit of intervention j, the end of the last segment | benefit units |
B_i | Benefit of intervention i, the less effective end of the last segment, on an approximately cardinal scale | benefit units |
K_j | Total net cost per patient of intervention j | currency per patient |
K_i | Total net cost per patient of intervention i, costs adjusted for cost offsets | currency per patient |
B_new | Benefit of the new intervention on the same scale, at least B_j | benefit units |
g | Incremental benefit per incremental net cost between the last two interventions on the efficiency frontier | benefit units per unit of currency |
|---|---|---|
lambda_g | Net cost per extra unit of benefit along the last segment, the sequential ICER of j against i | currency per benefit unit |
K_max | Total net cost per patient at which the new intervention lies on the linear extension of the last segment | currency per patient |
Function
Cost-effectiveness frontier from the expected costs and QALYs of mutually exclusive strategies
Maps the expected total costs C_j and expected total QALYs Q_j of a set of mutually exclusive strategies, estimated for one population, perspective and time horizon, to the strategies on the cost-effectiveness frontier, the slopes of the segments joining them and, at a value λ placed on one QALY, the strategy chosen. It builds on live records and does not restate them. The dominance screen is HE-FM-DOM-001 and HE-FM-DOM-002, the extended dominance condition and the blend of neighbours are HE-FM-EXD-001 and HE-FM-EXD-002, the ratio between adjacent frontier strategies is the sequential ICER HE-FM-CEA-001 and the maximum net monetary benefit rule is HE-FM-NMB-003. This package adds the range of λ over which each frontier strategy is chosen, the net monetary benefit a strategy gives up against the frontier choice, the gradient of the IQWiG efficiency frontier and two computational functions that build the frontier for any number of strategies. Notation follows the Cost-Effectiveness Frontier article.
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Implementations
Excel
Net cost on the extended last segment of an efficiency frontier from named cells
With the last two frontier interventions in CostPrev, BenefitPrev, CostLast and BenefitLast and the new intervention's benefit in BenefitNew, the first formula returns the gradient and the second the net cost per patient on the extension.
=(BenefitLast-BenefitPrev)/(CostLast-CostPrev); =CostLast+(BenefitNew-BenefitLast)/((BenefitLast-BenefitPrev)/(CostLast-CostPrev))
Assumptions
Benefit on an approximately cardinal scale for the efficiency frontier
IQWiG 6.1 (section 4.6.1) transfers the benefit, if required, into an approximately cardinally scaled measure and plots it in a positive value range (section 4.6.3), so that differences in benefit can be divided by differences in cost.
Last segment drawn with absolute and extended dominance applied
Intervention i and intervention j are the last two interventions on a frontier built with absolute and extended dominance. IQWiG 6.1 (section 4.6.2) notes that without extended dominance the stepped absolute frontier no longer gives a gradient that reads as the reciprocal of a willingness to pay.
Price judged by the version 6.1 extension of the last segment
The extension assumes that the gradient of the last segment represents the reciprocal of the current willingness to pay and that efficiency in the therapeutic area should not deteriorate (IQWiG 6.1, section 4.1.9). Version 8.0 keeps the frontier for evaluations outside §35b SGB V but no longer extends the last segment.
Worked examples
New intervention judged on an illustrative efficiency frontier in euros
On an illustrative frontier the last segment joins an intervention with net costs of EUR 8,000 and 40 benefit points to one with EUR 14,000 and 55 points, a gradient of 0.0025 points per euro, or EUR 400 per point. A new intervention with 61 points lies on the extended segment at a net cost of EUR 16,400 per patient.
B_i = 40; K_i = 8000; B_j = 55; K_j = 14000; B_new = 61; g = 0.0025; lambda_g = 400; K_max = 16400
New intervention with no added benefit over the last frontier intervention
With 55 points, the same benefit as the last intervention on the frontier, the net cost on the extension equals that intervention's EUR 14,000, a limiting case that checks the implementation. The figures are illustrative.
B_i = 40; K_i = 8000; B_j = 55; K_j = 14000; B_new = 55; g = 0.0025; lambda_g = 400; K_max = 14000
Common errors
Reading an efficiency frontier gradient as a cost per unit of benefit
A gradient of 0.0025 points per euro is benefit per cost, the reciprocal of EUR 400 per point. A steeper segment means greater efficiency, the opposite direction to an ICER, where a lower ratio is better.
Extending an earlier segment of the efficiency frontier
If an earlier segment joins EUR 5,000 and 25 points to EUR 8,000 and 40 points, its gradient is 0.005 points per euro. Extending it beyond EUR 8,000 and 40 points gives EUR 12,200 for the new intervention with 61 points, EUR 4,200 below the EUR 16,400 from the last segment (computed here for illustration). Only the last segment applies.
Sources
IQWiG General Methods 6.1 on the efficiency frontier gradient and its extension
Institute for Quality and Efficiency in Health Care (IQWiG). General Methods, version 6.1 of 24 January 2022. Cologne, IQWiG, 2022. Section 4.6.2, which states that the gradient of the line segment between 2 interventions gives the incremental benefit per incremental costs, section 4.6.3, which plots benefit on the vertical axis and total net costs per patient on the horizontal axis and extends the last segment for new interventions with more benefit and more costs, section 4.1.9 on the extrapolated last segment as the reciprocal of the current willingness to pay, and section 4.9.2, where the threshold for the net health benefit is the reciprocal of the gradient of the last segment and the price is the maximum cost that keeps the net health benefit at least zero.
IQWiG General Methods 8.0 on efficiency frontiers without the extended last segment
Institute for Quality and Efficiency in Health Care (IQWiG). General Methods, version 8.0 of 19 December 2025. Cologne, IQWiG, 2025. Section 4.15 on health economic evaluations outside §35b SGB V, which plots cost-effectiveness ratios for more than 2 treatments as one or more efficiency frontiers formed with absolute and extended dominance and states that it is not necessary to extend the last segment, previously intended for deriving a price recommendation.
Canonical Identity
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