Signature
F_s = ((1 + r_H) / (1 + r_C))^s; ICER_s = ICER_0 * F_s
| Inputs | Definition | Unit |
|---|---|---|
r_H | Unit: proportion per year | — |
r_C | Unit: proportion per year | — |
s | Unit: years | — |
ICER_0 | Ratio before the delay, for example from HE-FM-DDIS-001 | pounds per QALY |
F_s | Ratio after the delay divided by the ratio before it | none |
|---|---|---|
ICER_s | Unit: pounds per QALY | — |
Function
Differential discounting of incremental costs and health effects in a cost-effectiveness ratio
Converts the yearly streams of incremental costs and incremental QALYs to present values at two different annual rates, a cost rate and a usually lower health rate, before dividing one by the other. The package covers the ratio for a programme with costs and QALYs in different years, the exact health rate implied by growth in the money value of health, the effective cost rate implied by a growing cost-effectiveness threshold, and the factor by which delay changes the ratio. Separate discounting of two streams in general is HE-FM-DR-003 on the discount rate page and the two-option ratio HE-FM-ICER-001. Notation follows the Differential Discounting article.
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Implementations
Excel
Delay factor and delayed ratio from named cells
With RateH, RateC and DelayYears named and ICERval, CostY0, CostYa, YearA, QALYb and YearB from HE-FM-DDIS-001, the formulas return the delay factor and the delayed ratio, held in DelayFactor and ICERDelayed.
=((1+RateH)/(1+RateC))^DelayYears; =ICERval*DelayFactor
Assumptions
Whole programme shifted with unchanged amounts and rates
Every cost and QALY moves by the same s years with no change in its size, and both rates stay constant; Keeler and Cretin state their delay result for the case in which the ability to produce the health effect does not diminish too quickly over time.
Worked examples
Prevention programme delayed by 10 years at 3 and 1.5 per cent
The factor is (1.015 / 1.03) to the power 10, about 0.86355, so the ratio of 23,490 pounds per QALY falls to 20,285, about 14 per cent lower for an unchanged programme, as in the article.
r_H = 0.015; r_C = 0.03; s = 10; ICER_0 = 23490; F_s = 0.86355; ICER_s = 20285
Prevention programme delayed by 10 years at 3.5 per cent for both streams
Under equal rates the factor is one and the ratio stays at 34,004 pounds per QALY (computed here for illustration).
r_H = 0.035; r_C = 0.035; s = 10; ICER_0 = 34004; F_s = 1; ICER_s = 34004
Discount base year placed five years before the start of the programme
Moving the base year earlier by five years has the same arithmetic as a five-year delay: at 3 and 1.5 per cent the factor is about 0.92927 and the ratio falls from 23,490 to about 21,829 pounds per QALY (computed here for illustration).
r_H = 0.015; r_C = 0.03; s = 5; ICER_0 = 23490; F_s = 0.92927; ICER_s = 21829
Common errors
Moving the discount base year before the start of a programme
Attema and colleagues warn that placing the base year before the actual start lowers the ratio when costs are discounted more than effects, the same arithmetic as the delay factor, so a fixed rule such as the year of treatment initiation is needed.
Concluding that a programme should be postponed because its ratio falls
The fall comes from the discounting arithmetic, not from the programme; Gravelle and Smith argue that the paradox reveals a difficulty with cost-effectiveness analysis that is irrelevant to the choice of the health rate, and Attema and colleagues note that infinite postponement was never observed in the UK before 2004 or in the Netherlands since 2004.
Sources
Keeler and Cretin on the improvement of programmes by delay
Keeler EB, Cretin S. Discounting of life-saving and other nonmonetary effects. Management Science. 1983;29(3):300-306. doi:10.1287/mnsc.29.3.300 (abstract read). Abstract: if the ability to produce the nonmonetary effect does not diminish too quickly over time, failure to discount benefits implies that programs are always improved by delay; discounting benefits and costs at different rates can lead to peculiar results.
Brouwer and colleagues on the postponing paradox
Brouwer WBF, Niessen LW, Postma MJ, Rutten FFH. Need for differential discounting of costs and health effects in cost effectiveness analyses. BMJ. 2005;331(7514):446-448. doi:10.1136/bmj.331.7514.446 (full text read). Box 2: if one uses lower discount rates for effects than for costs, postponing any given programme will improve its cost effectiveness ratio.
Attema and colleagues on postponement and the base year
Attema AE, Brouwer WBF, Claxton K. Discounting in economic evaluations. PharmacoEconomics. 2018;36(7):745-758. doi:10.1007/s40273-018-0672-z (full text read). Section Discounting of Costs and Health Effects: Brief History: the cost-effectiveness ratio of a given intervention will improve with each year it is postponed; infinite postponing was never observed in practice, including the UK before 2004 and the Netherlands since 2004. Section Normative or Positive Approach: the further the starting year is placed before the actual start of the programme, the more the ICER will be reduced.
Gravelle and Smith on the Keeler and Cretin paradox
Gravelle H, Smith D. Discounting for health effects in cost benefit and cost effectiveness analysis. CHE Technical Paper 20. York: Centre for Health Economics, University of York; 2000 (full text read). Page 1: the Keeler-Cretin paradox reveals a fundamental difficulty with cost-effectiveness analysis, though not with cost-benefit analysis, and does not arise because of the use of different discount rates for costs and health effects.
Canonical Identity
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