Functions & Formulae

Disability-adjusted life years for a cause or a scenario from years of life lost and years lived with disability

d(N_a, L_a, P_s, DW_s) = DALY

• DALYs for a cause as the sum of its years of life lost and years lived with disability

  DALY = YLL + YLD

  Adds the fatal and non-fatal parts of the health loss from one cause, both calculated for the same cause, sex, age group and year, as in WHO's Global Health Estimates and the Global Burden of Disease (GBD) study. One DALY is one lost year of healthy life, so a death and a period of illness are counted on one scale. Each part comes from its own formula, and the DALY is their sum.

• Undiscounted DALYs of a cohort with deaths at one age and survivors living with a condition for a fixed time

  DALY = D * L + S * DW * T

  Gives the DALYs a cohort incurs in one scenario when D people die at once at an age where the standard remaining life expectancy is L years, and S survivors live T years at disability weight DW before recovering fully. The first term is the YLLs of HE-FM-YLL-001 and the second the incidence-based YLDs of HE-FM-YLD-001, so the result is HE-FM-DALY-001 for the cohort. The article applies it to one cohort with and without a treatment programme.

• DALYs averted and cost per DALY averted by a programme against its comparator

  DALY_av = DALY_0 - DALY_1; ICER = (C_1 - C_0) / DALY_av

  Subtracts the DALYs a cohort incurs with the programme from those it incurs without it, and divides the extra cost by the difference. Because DALYs count health lost, the comparator's DALYs come first in the denominator, which makes the ratio HE-FM-ICER-001 with the effect measured as DALYs averted. Both scenarios cover the whole cohort over the same time horizon, so deaths and disability both count, unlike HE-FM-DWT-005, where only disability weights change.

• Continuously discounted DALYs of a cohort with deaths and disability beginning at time zero

  f_L = (1 - exp(-r * L)) / r; f_T = (1 - exp(-r * T)) / r; DALY_d = D * f_L + S * DW * f_T

  Replaces the L standard years lost per death and the T years lived with the condition by their present values under continuous discounting at rate r, (1 minus e to the minus rL) over r and (1 minus e to the minus rT) over r, as in HE-FM-YLL-003 and HE-FM-YLD-002. Fox-Rushby and Hanson define DALYs for economic evaluation as the present value of future years of life lost plus the present value of future years lived with disability, and the original DALY used a 3% rate. Discounting shrinks the long stream of years lost to a death more than the shorter stream of illness, so it lowers the weight of deaths prevented. The function exp is the exponential function.

• DALY rate per 100,000 population for a cause

  DALY_rate = DALY / Pop * 100000

  Divides the DALYs from a cause by the population in which they arise and scales the result to 100,000 people, the form in which GBD 2021 reports all-age and age-specific rates. A rate removes the effect of population size but not of age structure; age-standardised rates weight age-specific rates by a standard population, as in HE-FM-INEQ-001.