Signature
YLD_awd = DW * (K * C * exp(r * a) / (r + beta)^2 * (exp(-(r + beta) * (L + a)) * (-(r + beta) * (L + a) - 1) - exp(-(r + beta) * a) * (-(r + beta) * a - 1)) + (1 - K) / r * (1 - exp(-r * L)))
| Inputs | Definition | Unit |
|---|---|---|
DW | Disability weight of the condition, 0 for full health and 1 for a state equivalent to death | none |
K | Age-weighting modulation constant, 0 for no age weights and 1 for full age weights | none, at least 0 and at most 1 |
C | Scaling constant of the age-weight function, 0.1658 in the standard GBD DALY | per year of age |
r | Continuous discount rate applied from onset, 0.03 in the standard GBD 1990 DALY | per year, above zero |
a | Age at onset of the condition | years, zero or above |
beta | Shape parameter of the age-weight function, 0.04 in the GBD DALY | per year of age, above zero |
L | Duration of the condition from onset until remission or death | years, zero or above |
YLD_awd | Age-weighted, continuously discounted years lived with disability for one incident case, valued at onset | years |
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Function
Age-weighted healthy years lost under the GBD 1990 exponential age-weight function
Maps an age, or a span of ages from the age at death or onset a over L years, together with the age-weight constants C and beta, the modulation constant K, a discount rate r and, for a case of illness, a disability weight DW, to the weight on a year lived at that age and to the age-weighted, discounted healthy years lost over the span. With K = 0 the age weights drop out and the result is the continuously discounted years of HE-FM-YLL-003 for a death and HE-FM-YLD-002 for an incident case, the per-death and per-case inputs of HE-FN-YLL-001, HE-FN-YLD-001 and the DALY records HE-FN-DALY-001 and HE-FM-DALY-004. The package adds the weight function and its peak, the undiscounted age-weighted years lost, the age-weighted discounted years lost per death of the original DALY, and the same expression for years lived with disability. Notation follows the Age Weighting article.
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Implementations
Excel
Age-weighted discounted years lived with disability per case from named cells
With DisabilityWt, AgeWtModulation, AgeWtConstC, AgeWtBeta, DiscRate, the age at onset in OnsetAge and the duration in Duration, the cell (named YLDAgeWtDisc) returns the years lived with disability for one case.
=DisabilityWt*(AgeWtModulation*AgeWtConstC*EXP(DiscRate*OnsetAge)/(DiscRate+AgeWtBeta)^2*(EXP(-(DiscRate+AgeWtBeta)*(Duration+OnsetAge))*(-(DiscRate+AgeWtBeta)*(Duration+OnsetAge)-1)-EXP(-(DiscRate+AgeWtBeta)*OnsetAge)*(-(DiscRate+AgeWtBeta)*OnsetAge-1))+(1-AgeWtModulation)/DiscRate*(1-EXP(-DiscRate*Duration)))
Assumptions
Constant disability weight over the case and discounting from onset
The weight is the same throughout the case, and each year is age-weighted at its own age and discounted from the age of onset. WHO's 2026 methods document (section 2.3, HE-SR-AGW-008) notes that in the original DALY method the start time for discounting YLDs was the year of incidence, whereas for YLLs it was the year of death.
Incidence perspective for age-weighted years lived with disability
All of a case's future health loss is assigned to its year and age of onset, the GBD 1990 incidence perspective of HE-FM-YLD-001. Prevalence-based YLDs from GBD 2010 onwards carry no age weights or discounting (HE-FM-DWT-001).
Worked examples
Age-weighted discounted years lived with disability for a case with onset at 35
An illustrative case with onset at 35, lasting 10 years at a weight of 0.6, counts about 6.955 years with full age weights and 3% discounting, 0.6 times 11.5913 (computed here for illustration).
DW = 0.6; K = 1; C = 0.1658; r = 0.03; beta = 0.04; a = 35; L = 10; YLD_awd = 6.9548
Discounted years lived with disability for the same case without age weights
With K = 0 the case counts 5.1836 years, Larson's discounted YLD for this case and the value of HE-FM-YLD-002, a limiting case that checks the second term.
DW = 0.6; K = 0; C = 0.1658; r = 0.03; beta = 0.04; a = 35; L = 10; YLD_awd = 5.1836
Common errors
Age at the end of the case entered as its duration
L is the duration, not the age at which the case ends. Entering 45 for the case with onset at 35 counts 45 years of illness and gives about 15.86 years lived with disability instead of 6.955 (computed here for illustration).
Disability weight applied to the age-weighted term only
The weight multiplies the whole expression. Leaving the (1 minus K) term unweighted at K = 0.5 gives about 7.80 years for the case with onset at 35 instead of 6.07 (computed here for illustration), and at K = 0 it counts the illness as if it were death.
Sources
WHO guide on the age-weighted discounted years lived with disability for a single event
Prüss-Üstün A, Mathers C, Corvalán C, Woodward A. Introduction and methods, assessing the environmental burden of disease at national and local levels. Environmental Burden of Disease Series No. 1. Geneva, World Health Organization, 2003. Section 3.5, page 33, which obtains the YLD for a single disabling event by replacing the standard life expectancy in the YLL formula with the duration of disease and multiplying by the disability weight, with a the age of onset and L the duration of disability.
Larson on discounted years lived with disability without age weighting
Larson BA. Calculating disability-adjusted-life-years lost (DALYs) in discrete-time. Cost Effectiveness and Resource Allocation. 2013;11 article 18. Equation 2, which gives the discounted YLD for 10 years at a weight of 0.60 and 3% as 0.6 times (1/0.03)(1 minus e to the minus 0.3), 5.1836, for a woman with onset at age 35 in the Fox-Rushby and Hanson example.
Canonical Identity
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