Functions & Formulae

Each applied formula has its own function page, with a signature, implementations, and tests.

Age-weighted healthy years lost under the GBD 1990 exponential age-weight function

g(x, a, L, r, K, C, beta, DW) = W, Y

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.

  • GBD age-weight function with its age and value of peak weight

    W = C * x * exp(-beta * x); x_peak = 1 / beta; W_peak = C / beta * exp(-1)

    Gives the weight on a year of life lived at age x as C times x times e to the minus beta x, the exponential form Murray used for the original DALY. The weight is zero at birth, rises through childhood and declines slowly after its peak, which falls where the slope C e to the minus beta x times (1 minus beta x) is zero, at the age 1/beta. The weight at the peak is C/(beta e). With the standard constants of 0.1658 and 0.04 the peak is at age 25 with a weight of about 1.525, as in the article.

  • Undiscounted age-weighted years of life lost per death with a modulation constant

    YLL_aw = K * C / beta^2 * ((1 + beta * a) * exp(-beta * a) - (1 + beta * (a + L)) * exp(-beta * (a + L))) + (1 - K) * L

    Integrates the weight of HE-FM-AGW-001 over the L years lost from the age at death a, which gives C/beta squared times the difference of (1 + beta a) e to the minus beta a and the same term at a + L. The modulation constant K mixes this with the unweighted L years, so K = 1 is the article's undiscounted age-weighted YLL and K = 0 returns the L standard years of HE-FM-YLL-001 for one death. The (1 minus K) L term is not printed in the sources cited, which give the K = 1 integral and the discounted form with K. It is derived here as the limit of the second term of HE-FM-AGW-003 as r falls to zero, and the numerical integral of HE-CF-AGW-002 confirms it. The whole expression is the r = 0 case of HE-FM-AGW-003, which cannot be evaluated directly at r = 0 because it divides by r.

  • Age-weighted continuously discounted years of life lost per death of the original DALY

    YLL_awd = 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))

    Gives the years of life lost for one death at age a with L standard years remaining, each year weighted by HE-FM-AGW-001 and discounted continuously at rate r from the time of death, in the form the WHO environmental burden of disease guide gives for a single death. The first term is the age-weighted integral, scaled by K, and the second term, with weight 1 minus K, is the continuously discounted years of HE-FM-YLL-003, so K = 0 returns that record and HE-FM-DISC-003 with an amount of 1 and its continuous rate rho set equal to r. With r = 0.03 and K = 1 it gives the YLL per death of the standard GBD 1990 DALY, which GBD 2010 and WHO later dropped. The undiscounted case is HE-FM-AGW-002.

  • Age-weighted continuously discounted years lived with disability per incident case

    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)))

    Applies the expression of HE-FM-AGW-003 to a case of a non-fatal condition, with a the age at onset and L the duration, and multiplies it by the disability weight, as the WHO environmental burden of disease guide gives for a single disabling event. With K = 0 it is the continuously discounted YLD per case of HE-FM-YLD-002, and multiplying by the incident cases gives the incidence-based total of HE-FN-YLD-001 under the original DALY conventions. Disability weights are on the Disability Weight page (HE-FN-DWT-001).