Signature
YLL_aw = K * C / beta^2 * ((1 + beta * a) * exp(-beta * a) - (1 + beta * (a + L)) * exp(-beta * (a + L))) + (1 - K) * L
| Inputs | Definition | Unit |
|---|---|---|
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 |
beta | Shape parameter of the age-weight function, 0.04 in the GBD DALY | per year of age, above zero |
a | Age at death | years, zero or above |
L | Standard years of life lost at the age of death, the remaining life expectancy in the standard used | years, zero or above |
YLL_aw | Age-weighted years of life lost for one death, without discounting | years |
|---|
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
Undiscounted age-weighted years lost per death from named cells
With the constants in AgeWtModulation, AgeWtConstC and AgeWtBeta, the age at death in AgeAtDeath and the years lost in YearsLost, the cell (named YLLUndisc) returns the age-weighted years lost.
=AgeWtModulation*AgeWtConstC/AgeWtBeta^2*((1+AgeWtBeta*AgeAtDeath)*EXP(-AgeWtBeta*AgeAtDeath)-(1+AgeWtBeta*(AgeAtDeath+YearsLost))*EXP(-AgeWtBeta*(AgeAtDeath+YearsLost)))+(1-AgeWtModulation)*YearsLost
Assumptions
Undiscounted age-weighted years over a single span of standard years
The death removes the years from a to a + L in full and each is weighted by its own age, with no discounting. L comes from the standard used, the GBD standard life table in burden estimates or, in the article's worked example, the illustrative years up to age 80.
Modulation constant fixed within one set of estimates
K is at least 0 and at most 1 and is the same for every death compared. Values other than 0 and 1 give a weighted mixture of the age-weighted and uniform totals, used in sensitivity analysis.
Worked examples
Undiscounted age-weighted years lost for a death at 5
A death at 5 that removes the 75 years to an illustrative age of 80 loses about 84.07 age-weighted years, 103.625 times (0.98248 minus 0.17120), against 75 unweighted, as in the article.
K = 1; C = 0.1658; beta = 0.04; a = 5; L = 75; YLL_aw = 84.0684
Undiscounted age-weighted years lost for a death at 40
A death at 40 that removes 40 years loses about 36.66 age-weighted years against 40 unweighted, because all its lost years fall after the peak, as in the article.
K = 1; C = 0.1658; beta = 0.04; a = 40; L = 40; YLL_aw = 36.6552
Undiscounted years lost for a death at 5 without age weights
With K = 0 the result is the 75 standard years, a limiting case that returns HE-FM-YLL-001 for one death.
K = 0; C = 0.1658; beta = 0.04; a = 5; L = 75; YLL_aw = 75
Common errors
Weight at the age of death applied to every year lost
Multiplying L by the weight at the age of death, W(a) times L, ignores that the lost years are lived at later ages. For a death at 5 it gives 0.6787 times 75, about 50.90, instead of 84.07 (computed here for illustration), and turns the gain from age weighting for a child death into a loss.
Undiscounted age-weighted total treated as the standard GBD 1990 figure
The standard DALYs of the original GBD combined age weights with 3% discounting, so a death at 5 in the article's illustration counts 36.44 years (HE-FM-AGW-003), not 84.07. The undiscounted age-weighted value is a component for comparison, not the published convention.
Sources
Murray's integral for the DALYs lost by one individual
Murray CJ. Quantifying the burden of disease, the technical basis for disability-adjusted life years. Bulletin of the World Health Organization. 1994;72(3) 429-445. Page 441, which writes the DALYs lost by one individual as the integral from a to a + L of D C x e to the minus beta x times e to the minus r (x minus a), with D the disability weight (1 for a death), a the age of onset and L the duration or time lost. Setting r to zero gives the undiscounted integral here.
WHO guide on the modulation constant for age weights
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 and its DALY parameters, which define K as the age-weighting modulation constant, K = 0 for no age weights to 1 for full age weights. The guide gives K only in the discounted formula; the undiscounted (1 minus K) L term of HE-FM-AGW-002 is derived from it, not printed there.
Canonical Identity
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