Signature
a_1 = (Delta_C_1 <= lambda * Delta_E_1); a_2 = (Delta_C_2 <= lambda * Delta_E_2); a_3 = (Delta_C_3 <= lambda * Delta_E_3); a_4 = (Delta_C_4 <= lambda * Delta_E_4); B_lambda = a_1 * Delta_C_1 + a_2 * Delta_C_2 + a_3 * Delta_C_3 + a_4 * Delta_C_4
| Inputs | Definition | Unit |
|---|---|---|
Delta_C_1 | Incremental cost of the first programme against its comparator, above zero | currency |
lambda | Cost-effectiveness threshold at or below which a programme's ICER leads to adoption | currency per QALY |
Delta_E_1 | Incremental health gain of the first programme against its comparator, above zero | QALYs |
Delta_C_2 | Incremental cost of the second programme against its comparator, above zero | currency |
Delta_E_2 | Incremental health gain of the second programme against its comparator, above zero | QALYs |
Delta_C_3 | Incremental cost of the third programme against its comparator, above zero | currency |
Delta_E_3 | Incremental health gain of the third programme against its comparator, above zero | QALYs |
Delta_C_4 | Incremental cost of the fourth programme against its comparator, above zero | currency |
Delta_E_4 | Incremental health gain of the fourth programme against its comparator, above zero | QALYs |
a_1 | Equals 1 when the first programme's incremental cost is at most lambda times its incremental health gain, and 0 otherwise | binary indicator |
|---|---|---|
a_2 | Equals 1 when the second programme's incremental cost is at most lambda times its incremental health gain, and 0 otherwise | binary indicator |
a_3 | Equals 1 when the third programme's incremental cost is at most lambda times its incremental health gain, and 0 otherwise | binary indicator |
a_4 | Equals 1 when the fourth programme's incremental cost is at most lambda times its incremental health gain, and 0 otherwise | binary indicator |
B_lambda | Total incremental cost of the programmes adopted at the threshold lambda | currency |
Function
League-table allocation of a fixed budget across independent programmes ranked by ICER
Takes the incremental cost Delta_C_i and incremental health gain Delta_E_i of each of a set of independent programmes, each measured against its own comparator, and a fixed budget B. It returns the share of each programme funded when programmes are funded in ascending order of ICER until the budget is exhausted, the health the budget buys, the critical ratio of the last programme funded and, under the alternative fixed ratio rule, the budget that a stated threshold implies. Each programme's ratio is the two-option ICER of HE-FM-ICER-001, or the ratio against doing nothing of HE-FM-ACER-002 when doing nothing is the comparator, and is not restated here. A league table ranks independent programmes competing for one budget, while mutually exclusive options for the same patients are ranked on the cost-effectiveness frontier (HE-FN-CEF-001, HE-CF-CEF-001), and the table takes such a set at its sequential ratios along the frontier (HE-FM-CEA-001). Ties and cost-saving programmes follow the dominance rule HE-FM-DOM-001, the threshold as marginal spending over marginal health is HE-FM-CET-001, and the health a new programme displaces at a threshold, Delta_C divided by k, and its incremental net health benefit are HE-FM-CET-002. This package adds the funded share of each programme, the critical ratio, the fill for programmes that must be funded whole and the budget implied by a fixed ratio, with a computational function that fills the table for any number of programmes. Notation follows the Cost-Effectiveness League Table article.
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Implementations
Excel
Budget implied by a threshold from ranges of incremental costs and QALYs
With the incremental costs in a column range named IncCosts, the incremental QALYs in IncQALYs and the threshold in ThresholdValue, the formula returns the implied budget, held in a cell named ImpliedBudget.
=SUMPRODUCT(--(IncCosts<=ThresholdValue*IncQALYs),IncCosts)
Assumptions
Ties at the threshold count as adopted under the fixed ratio rule
A programme whose ICER equals lambda has an incremental net monetary benefit of zero and is adopted, following the wording of Sacristán and colleagues that its ICER does not exceed the threshold. The article phrases the rule as adopting what lies below the threshold, and the two readings differ only for such ties. Both increments are above zero, so the test agrees with the ICER rule of HE-AS-NMB-004.
Critical ratio as the threshold at which the two rules agree for divisible programmes
For divisible programmes the two rules agree at lambda equal to the critical ratio of HE-FM-CELT-002, where the marginal programme's incremental net monetary benefit is zero, so any share of it is consistent with the threshold. The budget then lies at or above the implied budget with ties left out and at or below the implied budget with ties counted in, while the budget binds (derived), which HE-VT-CELT-007 checks.
Worked examples
Implied budget of GBP 750,000 at a threshold of GBP 15,000 per QALY
With the article's illustrative programmes A, B, C and D, a threshold of GBP 15,000 per QALY adopts A, B and C and implies a budget of GBP 750,000, GBP 150,000 below the GBP 900,000 available.
lambda = 15000; Delta_C_1 = 200000; Delta_E_1 = 40; Delta_C_2 = 300000; Delta_E_2 = 30; Delta_C_3 = 250000; Delta_E_3 = 20; Delta_C_4 = 400000; Delta_E_4 = 20; a_1 = 1; a_2 = 1; a_3 = 1; a_4 = 0; B_lambda = 750000
Implied budget at a threshold equal to the critical ratio of GBP 20,000 per QALY
At GBP 20,000 per QALY D's ICER equals the threshold, so D is adopted as a tie and the implied budget is GBP 1,150,000, GBP 250,000 above the budget. The two rules coincide here only with D part-funded.
lambda = 20000; Delta_C_1 = 200000; Delta_E_1 = 40; Delta_C_2 = 300000; Delta_E_2 = 30; Delta_C_3 = 250000; Delta_E_3 = 20; Delta_C_4 = 400000; Delta_E_4 = 20; a_1 = 1; a_2 = 1; a_3 = 1; a_4 = 1; B_lambda = 1150000
Zero implied budget at a threshold below every ratio in the table
At an illustrative GBP 4,000 per QALY, below A's GBP 5,000, no programme is adopted and the implied budget is zero, a limiting case.
lambda = 4000; Delta_C_1 = 200000; Delta_E_1 = 40; Delta_C_2 = 300000; Delta_E_2 = 30; Delta_C_3 = 250000; Delta_E_3 = 20; Delta_C_4 = 400000; Delta_E_4 = 20; a_1 = 0; a_2 = 0; a_3 = 0; a_4 = 0; B_lambda = 0
Common errors
Adopting the marginal programme in full at the critical ratio
Setting the threshold at the critical ratio of GBP 20,000 per QALY and adopting every programme at or below it funds all of D and spends GBP 1,150,000, GBP 250,000 more than the budget. Under a fixed budget the tie at the critical ratio is settled by the share of HE-FM-CELT-001, not by the threshold.
Setting the threshold below the critical ratio under a fixed budget
A threshold of GBP 15,000 per QALY adopts A, B and C and leaves GBP 150,000 of the GBP 900,000 unspent, which at D's GBP 20,000 per QALY would have bought 7.5 QALYs (computed here for illustration).
Sources
Sacristán and colleagues on the fixed ratio rule and when the two rules coincide
Sacristán JA, Abellán-Perpiñán JM, Dilla T, Soto J, Oliva J. Some reflections on the use of inappropriate comparators in CEA. Cost Effectiveness and Resource Allocation. 2020;18:29. Introduction, which states that under the fixed ratio rule, or threshold approach, a new intervention is adopted if its ICER does not exceed a cost per QALY threshold, and that both decision rules coincide if the budget implicitly determined by the fixed ratio rule is the same as the budget constraint assumed in the fixed budget rule.
Marseille and colleagues on league tables and the threshold question
Marseille E, Larson B, Kazi DS, Kahn JG, Rosen S. Thresholds for the cost-effectiveness of interventions: alternative approaches. Bulletin of the World Health Organization. 2015;93(2):118-124. Section League tables, which describes the league table as an approach that side-steps the threshold question and focuses instead on getting the largest health impact for the budget.
Canonical Identity
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