Signature
YLL_death = (1 - exp(-r * L)) / r; YLL_d = N * YLL_death
| Inputs | Definition | Unit |
|---|---|---|
r | 0.03 in the original DALY | proportion per year, above zero |
L | Standard remaining life expectancy for the age at death, as in HE-FM-YLL-001 | years |
N | Number of deaths at the same age | deaths |
YLL_death | Present value of the standard years lost by one death | years |
|---|---|---|
YLL_d | Discounted years of life lost from N deaths at the same age | years |
Function
Years of life lost from deaths weighted by a loss function of age at death
Converts deaths from a cause into years of life lost by multiplying the deaths at each age by the years that a loss function assigns to a death at that age, then summing over ages. In the Global Burden of Disease (GBD) study from 2010 onwards and in WHO's Global Health Estimates (GHE) the loss function is the remaining life expectancy at the age of death in a normative standard life table (HE-FM-YLL-001). Potential years of life lost replace it with the years from the age at death to a fixed cut-off age (HE-FM-YLL-002), and the original DALY discounted the standard years (HE-FM-YLL-003). YLLs added to years lived with disability (HE-FN-YLD-001, HE-FM-DWT-001) give DALYs. Notation follows the Years of Life Lost article.
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Implementations
Excel
Discounted years of life lost per death and per group from named cells
With DiscRate, StdLE and Deaths named, the first formula returns the discounted years per death, held in YLLDeath, and the second the discounted total for the group.
=(1-EXP(-DiscRate*StdLE))/DiscRate; =Deaths*YLLDeath
Assumptions
Continuous discounting from the time of death without age weighting
Each standard year is discounted continuously at a constant rate from the time of death, and the age weights of the original DALY are left out, as in the article and in Larson's equation 1.
Discounted years of life lost for evaluation and historical comparison only
GBD from 2010 onwards and WHO since 2012 do not discount YLLs, so the formula reproduces the original DALY convention or an economic evaluation that discounts health; current burden estimates use HE-FM-YLL-001.
Worked examples
Larson's death with 10 years of remaining life expectancy at 3 per cent
One death with 10 remaining years discounted continuously at 3 per cent counts 8.6394 years, Larson's continuous-time value, against 8.786 with discrete annual discounting.
r = 0.03; L = 10; N = 1; YLL_death = 8.6394; YLL_d = 8.6394
Death removing 80 standard years discounted at 3 per cent
A death that removes 80 standard years counts about 30.3 years at 3 per cent, as in the article.
r = 0.03; L = 80; N = 1; YLL_death = 30.3094; YLL_d = 30.3094
Twenty child deaths at ages 1 to 4 discounted at 3 per cent
Each death counts about 30.62 years in place of 83.63, and the 20 deaths 612.4 in place of 1,672.6, a cut of 63 per cent, as in the article.
r = 0.03; L = 83.63; N = 20; YLL_death = 30.6214; YLL_d = 612.43
One hundred deaths at ages 75 to 79 discounted at 3 per cent
Each death counts about 10.66 years in place of 12.85, a cut of 17 per cent, so a child death counts 2.9 times an older death against 6.5 times undiscounted. With 1,224.5 from the 50 deaths at ages 40 to 44, the discounted total is 2,903.2, or 2,902.9 in the article, which rounds each death to two decimals first.
r = 0.03; L = 12.85; N = 100; YLL_death = 10.6630; YLL_d = 1066.3
Common errors
Mixing continuous and discrete discounting of years of life lost
Counting each of 10 lost years at the start of the year and discounting at 3 per cent gives 8.786 against the continuous 8.6394, a difference of 1.7 per cent. Larson regards the difference as minor, but an analysis should state which convention it uses.
Comparing discounted years of life lost with current GBD or WHO estimates
GBD 2010 calculated YLLs without discounting and WHO adopted a 0 per cent rate after 2012, so a discounted figure, such as the article's 2,902.9 against 5,169.1, is not comparable with current burden tables.
Treating discounting of years of life lost as neutral between ages
Discounting compresses the difference between young and old deaths: in the article's example the older deaths rise from 25 to 37 per cent of the total. The rate therefore changes cause rankings as well as totals.
Sources
Larson on continuously discounted years of life lost without age weighting
Larson BA. Calculating disability-adjusted-life-years lost (DALYs) in discrete-time. Cost Effectiveness and Resource Allocation. 2013;11:18 (full text read). Equation 1: in the absence of age weighting, YLL(r, 0) equals (1/r)(1 minus e to the minus rL), where L is the age-specific life expectancy of the person who died; with a 3% rate and 10 years the answer is 8.6394. Table 1: discrete annual discounting gives 8.786, a difference of 1.7% described as minor.
WHO on the 3 per cent discount rate and its replacement by 0 per cent
World Health Organization. WHO methods and data sources for global burden of disease estimates 2000-2021. Global Health Estimates Technical Paper WHO/DDI/DNA/GHE/2024.3. Geneva: WHO; 2024 (full text read). Section 2.3: the GBD 1990 study and subsequent WHO updates published DALYs computed with a 3% discount rate and an alternative set with a 0% rate; following informal consultations in 2012 WHO adopted the GBD 2010 approach, with a time discount rate of 0% and no age weighting.
Canonical Identity
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