Signature
lambda_c = max(max(max(f_1 * r_1, f_2 * r_2), f_3 * r_3), f_4 * r_4)
| Inputs | Definition | Unit |
|---|---|---|
f_1 | Equals 1 when the first programme in ascending ICER order receives a funded share above zero, and 0 otherwise | binary indicator |
r_1 | ICER of the first programme against its comparator, the lowest in the table | currency per QALY |
f_2 | Equals 1 when the second programme in ascending ICER order receives a funded share above zero, and 0 otherwise | binary indicator |
r_2 | ICER of the second programme against its comparator | currency per QALY |
f_3 | Equals 1 when the third programme in ascending ICER order receives a funded share above zero, and 0 otherwise | binary indicator |
r_3 | ICER of the third programme against its comparator | currency per QALY |
f_4 | Equals 1 when the fourth programme in ascending ICER order receives a funded share above zero, and 0 otherwise | binary indicator |
r_4 | ICER of the fourth programme against its comparator, the highest in the table | currency per QALY |
lambda_c | ICER of the last programme funded, in full or in part | currency per QALY |
|---|
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
Critical ratio from funded shares and ICERs in named ranges
With the funded shares of the fill in a column range named FundedShares and the programmes' ICERs, in ascending order, in ProgICERs, the formula returns the largest ICER among programmes with a share above zero, held in a cell named CriticalRatio. SUMPRODUCT makes older versions of Excel evaluate the array inside MAX without array entry.
=SUMPRODUCT(MAX((FundedShares>0)*ProgICERs))
Assumptions
Ratios ascending and positive with funding flags from the budget fill
The ratios r_1, r_2, r_3 and r_4 rise in that order and are above zero, and the flags come from the fill of HE-FM-CELT-001 at the stated budget, so the largest product is the ratio of the last programme funded. A programme that dominates its comparator is adopted outside the table (HE-AS-CELT-002) and takes no ratio here.
Budget that binds on the league table for the critical ratio
The critical ratio measures the opportunity cost only while the budget is exhausted. When B exceeds the cost of every programme in the table, all are funded and money is left over, so lambda_c is still the ratio of the least cost-effective programme adopted but an extra unit of budget buys no health within the table (derived). When B exactly exhausts a programme, lambda_c is that programme's ratio, while the next unit of budget would buy health at the ratio of the programme below it (HE-ER-CELT-003).
Worked examples
Critical ratio of GBP 20,000 per QALY at the article's budget of GBP 900,000
With GBP 900,000 all four of the article's illustrative programmes receive a share, D in part, so the critical ratio is D's GBP 20,000 per QALY, as in step 1 of the article. An extra GBP 100,000 would then buy 5 more QALYs of D, which has GBP 250,000 still unfunded (derived).
f_1 = 1; f_2 = 1; f_3 = 1; f_4 = 1; r_1 = 5000; r_2 = 10000; r_3 = 12500; r_4 = 20000; lambda_c = 20000
Critical ratio of GBP 12,500 per QALY with a budget of GBP 700,000
With an illustrative budget of GBP 700,000, C receives 0.8 of its cost and D nothing, so the critical ratio falls to C's GBP 12,500 per QALY. Marseille and colleagues note that the length of the list of interventions deemed cost effective varies with the budget, which is how a league table combines cost-effectiveness with affordability.
f_1 = 1; f_2 = 1; f_3 = 1; f_4 = 0; r_1 = 5000; r_2 = 10000; r_3 = 12500; r_4 = 20000; lambda_c = 12500
Critical ratio when only the first programme of the league table is funded
With an illustrative GBP 150,000 only 0.75 of A is funded, so the critical ratio is A's GBP 5,000 per QALY, a limiting case with a single flag set.
f_1 = 1; f_2 = 0; f_3 = 0; f_4 = 0; r_1 = 5000; r_2 = 10000; r_3 = 12500; r_4 = 20000; lambda_c = 5000
Common errors
Using one critical ratio for budget cuts and increases at a row boundary
With a budget of GBP 750,000 that exactly exhausts C, a cut of GBP 50,000 loses 4 QALYs at C's GBP 12,500 per QALY, while an increase of GBP 50,000 gains only 2.5 QALYs at D's GBP 20,000 (computed here for illustration). At a row boundary the health lost from a cut and the health gained from an increase are read from the two neighbouring rows.
Treating the critical ratio as fixed when the budget changes
In the article's table the critical ratio is GBP 12,500 per QALY at a budget of GBP 700,000 and GBP 20,000 at GBP 900,000. A ratio read at one budget and applied as a threshold after the budget changes adopts or rejects the wrong programmes, so it is recomputed with the fill at each budget.
Sources
Sacristán and colleagues on the fixed budget rule and the critical ratio
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 budget rule, or league table approach, interventions are ranked in increasing order of ICER and successively included until the budget is exhausted, and that the ICER of the least cost-effective intervention adopted indicates the critical ratio, a term they attribute to Weinstein and Zeckhauser 1973, or the cost-effectiveness threshold representing the opportunity cost of funding new programmes.
Dakin and Tsiachristas on shadow prices of a constraint
Dakin H, Tsiachristas A. Rationing in an era of multiple tight constraints: is cost-utility analysis still fit for purpose? Applied Health Economics and Health Policy. 2024;22(3):315-329. Section 2.1, headed Constrained optimisation, which defines shadow prices as the change in outcome from a one unit increase in the constraint, and section 2.2 on the league table, which sorts items by the reciprocal of the ICER. The equality of the shadow price of the budget and the reciprocal of the critical ratio is derived here, not stated in the paper.
Marseille and colleagues on the last intervention adopted and affordability
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 states that since the length of the list of interventions deemed cost effective varies according to the budget, league tables combine considerations of cost-effectiveness with affordability, and that the last, least cost-effective intervention in the table to be adopted is more likely to approximate society's willingness to pay for health benefits than the open-ended set of commitments implied by global thresholds.
Canonical Identity
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