Signature
f_L = (1 - exp(-r * L)) / r; f_T = (1 - exp(-r * T)) / r; DALY_d = D * f_L + S * DW * f_T
| Inputs | Definition | Unit |
|---|---|---|
r | Continuous discount rate as a decimal, 0.03 in the article's example and in the original DALY | proportion per year, above zero |
L | Standard remaining life expectancy at the age of death | years |
T | Years each survivor lives with the condition before full recovery | years |
D | Number of cohort members who die at the start, all at the same age | deaths |
S | Number of cohort members who survive and live with the condition from time zero | people |
DW | Disability weight of the condition in this scenario | none |
f_L | Discounted value of the L years lost to one death | years |
|---|---|---|
f_T | Discounted value of T years lived with the condition, before weighting | years |
DALY_d | Present value of the DALYs the cohort incurs in one scenario | discounted years of healthy life lost |
Function
Disability-adjusted life years for a cause or a scenario from years of life lost and years lived with disability
Maps the deaths and the non-fatal health loss from a cause, or in one scenario of an economic evaluation, to disability-adjusted life years (DALYs) by adding the years of life lost (YLL) to the years lived with disability (YLD), each calculated by its own formula. YLLs are deaths times a standard loss function of the age at death (HE-FN-YLL-001, HE-FM-YLL-001, with the discounted form HE-FM-YLL-003 and the age-group total HE-CF-YLL-001), and YLDs are prevalent cases times a disability weight (HE-FM-DWT-001) or incident cases times the weight and the duration (HE-FN-YLD-001, HE-FM-YLD-001, discounted in HE-FM-YLD-002). This package adds the sum itself, the DALYs of a cohort in one scenario, DALYs averted and cost per DALY averted between two scenarios, the continuously discounted cohort form and the DALY rate per 100,000 population. DALYs averted by a change in disability weights alone, with mortality unchanged, are HE-FM-DWT-005, and the general incremental ratio is HE-FM-ICER-001. Notation follows the Disability-Adjusted Life Year article.
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Implementations
Excel
Continuously discounted cohort DALYs from named cells
With Deaths, StdLE, Survivors, DisWeight, DurationYrs and DiscRate named, the cell returns the discounted DALYs of one scenario, held in DALYDisc.
=Deaths*(1-EXP(-DiscRate*StdLE))/DiscRate+Survivors*DisWeight*(1-EXP(-DiscRate*DurationYrs))/DiscRate
Assumptions
Deaths and onset of illness both at time zero in a discounted cohort DALY
Each death and the start of each survivor's illness fall at time 0, so both streams are discounted from the same point. When death follows T years of illness the years lost begin at T, and their present value is multiplied by e to the minus rT. Larson's recalculation of the Fox-Rushby and Hanson case gives 5.1836 discounted YLDs for 10 years at a weight of 0.6 and 15.9823 discounted YLLs for the 34.73 years lost after them, 21.166 DALYs in all.
Continuous discounting at one rate with no age weighting
One constant rate applies throughout, in continuous time as in Larson's equation 1, and the age weights of the original DALY are left out. Discounted DALYs serve economic evaluation or historical comparison, since GBD from 2010 onwards and WHO after its 2012 decision use a 0% rate.
Worked examples
Article's untreated cohort discounted continuously at 3 per cent
Discounting the 45 standard years per death to 24.692 and the 10 years of illness to 8.639 gives 960.37 DALYs in place of 1,440. The article shows 960.35 because it multiplies the rounded factors.
D = 20; L = 45; S = 180; DW = 0.3; T = 10; r = 0.03; f_L = 24.6920; f_T = 8.6394; DALY_d = 960.37
Article's treated cohort discounted continuously at 3 per cent
With 8 deaths and 192 survivors at a weight of 0.1 the discounted total is 363.41 DALYs, as in the article.
D = 8; L = 45; S = 192; DW = 0.1; T = 10; r = 0.03; f_L = 24.6920; f_T = 8.6394; DALY_d = 363.41
Discounted cohort DALYs at a rate close to zero as a limiting case
At a rate of 0.0000001 the factors are close to 45 and 10 years and the total returns the undiscounted 1,440 DALYs of HE-FM-DALY-002 to two decimals, which checks the implementation.
D = 20; L = 45; S = 180; DW = 0.3; T = 10; r = 0.0000001; f_L = 44.9999; f_T = 10.0000; DALY_d = 1440.00
Common errors
Discrete annual factors used in place of continuous discounting for cohort DALYs
Counting each year at its start and discounting by 1/(1.03)^t gives factors of 25.254 and 8.786 and 979.54 DALYs for the untreated cohort, about 2% more than the continuous 960.37 (computed here for illustration). Larson regards differences of this size as minor, but the convention is stated and used in both scenarios.
Discounted cost per DALY averted compared with undiscounted results
Discounting raises the article's cost per DALY averted from USD 500 to about USD 744, so a discounted ratio set against an undiscounted one, or against burden estimates that use a 0% rate, misstates the difference between programmes. WHO-CHOICE presents health benefits both undiscounted and discounted at 3%.
Sources
Larson on continuously discounted DALYs without age weighting
Larson BA. Calculating disability-adjusted-life-years lost (DALYs) in discrete-time. Cost Effectiveness and Resource Allocation. 2013;11 article 18 (full text read). Equation 1, YLL(r,0) equal to (1/r)(1 minus e to the minus rL) without age weighting, 8.6394 for 10 years at 3%. Equations 2 and 3, the Fox-Rushby and Hanson case with 5.1836 discounted YLDs and 15.9823 discounted YLLs starting after 10 years, 21.166 DALYs. Table 1, the discrete value 8.786, a difference of 1.7% described as minor.
Fox-Rushby and Hanson definition of DALYs as present values
Fox-Rushby JA, Hanson K. Calculating and presenting disability adjusted life years (DALYs) in cost-effectiveness analysis. Health Policy and Planning. 2001;16(3) 326-331 (abstract read). DALYs are the sum of the present value of future years of lifetime lost through premature mortality and the present value of years of future lifetime adjusted for the average severity of any disability.
Murray on the 3 per cent discount rate of the original DALY
Murray CJL. Quantifying the burden of disease, the technical basis for disability-adjusted life years. Bulletin of the World Health Organization. 1994;72(3) 429-445 (abstract read). The four social choices of the original DALY, a standard based on model life table West Level 26, exponential age weights, six severity classes each assigned a severity weight between 0 and 1, and a 3% discount rate.
Canonical Identity
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