Signature
r_C_star = (1 + r_H) * (1 + g_k) - 1; H_PV = C_t / (k_0 * (1 + g_k)^t * (1 + r_H)^t)
| Inputs | Definition | Unit |
|---|---|---|
r_H | Decision maker's rate of time preference for health | proportion per year |
g_k | Growth in the cost per QALY of the care displaced by extra spending | proportion per year |
C_t | Extra spending in year t | pounds |
k_0 | Cost per QALY of the care displaced at the margin today | pounds per QALY |
t | Years after the discount base year | years |
r_C_star | Rate that converts a future cost into present health forgone in money terms | proportion per year |
|---|---|---|
H_PV | QALYs forgone, discounted to year 0 | QALYs |
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
Effective cost rate and discounted health forgone from named cells
With RateH, GrowthK, CostT, ThreshK0 and YearT named, the formulas return the effective cost rate and the present health forgone, held in EffCostRate and HealthForgone.
=(1+RateH)*(1+GrowthK)-1; =CostT/(ThreshK0*(1+GrowthK)^YearT*(1+RateH)^YearT)
Assumptions
Fixed budget with the threshold set by displaced care
Extra spending displaces other care at the margin, so health forgone in year t is the cost divided by the threshold k_t, and k_t grows at a constant rate g_k, k_(t+1) = k_t (1 + g_k), as Attema and colleagues set out.
Health forgone discounted at the same rate as health gained
Displaced and gained health are discounted at one health rate r_H, so the comparison of a programme's gains with the health its costs displace is made in present QALYs.
Worked examples
Threshold growing at 1.5 per cent with a health rate of 1.5 per cent
An effective cost rate of about 3.02 per cent follows, and 12,000 pounds spent in year 10 against a threshold of 20,000 pounds per QALY today displaces about 0.4455 present QALYs (computed here for illustration).
r_H = 0.015; g_k = 0.015; C_t = 12000; k_0 = 20000; t = 10; r_C_star = 0.030225; H_PV = 0.4455
Constant threshold with a health rate of 1.5 per cent
With no threshold growth the effective cost rate equals the health rate and the same cost displaces about 0.5170 present QALYs, the case in which Claxton and colleagues find equal rates correct (computed here for illustration).
r_H = 0.015; g_k = 0; C_t = 12000; k_0 = 20000; t = 10; r_C_star = 0.015; H_PV = 0.517
Common errors
Discounting at equal rates while expecting the threshold to grow
Claxton and colleagues show that, with a fixed budget and decisions based on incremental ratios, discounting costs and health at the same rate is correct only if the threshold stays constant.
Taking the effective cost rate to be the consumption rate
Attema and colleagues give the health rate as the consumption rate minus g_v and the cost rate as the health rate plus g_k, so the effective cost rate equals the consumption rate only if the threshold and the consumption value of health grow at the same rate.
Sources
Attema and colleagues on health forgone and a growing threshold
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 Recent Developments: health forgone is the cost divided by the marginal cost effectiveness of current spending, k_t; k changes at a growth rate g_k, giving k_(t+1) = k_t (1 + g_k); differential discounting is necessary if r_c is approximately equal to r_h + g_k. Full framework: the discount rate for health effects is d_h = r_c minus g_v and for costs d_c = d_h + g_k.
Claxton and colleagues on equal rates and a constant threshold
Claxton K, Paulden M, Gravelle H, Brouwer W, Culyer AJ. Discounting and decision making in the economic evaluation of health-care technologies. Health Economics. 2011;20(1):2-15. doi:10.1002/hec.1612 (abstract read). Abstract: if the budget for health care is fixed and decisions are based on incremental cost effectiveness ratios, discounting costs and health gains at the same rate is correct only if the threshold remains constant.
Canonical Identity
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