Functions & Formulae

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

Sullivan active life expectancy from a period life table and dependency prevalence

a(l_x, L_i, pi_i) = (ALE_x, e_x, DLE_x, s_A)

Maps the survivors of a period life table at a starting age, the person-years it assigns to each later age band and the proportion of each band who are dependent in activities of daily living to active life expectancy, life expectancy and the expected years lived dependent at that age. Sullivan's method splits the person-years of each band by the dependency prevalence (HE-FM-ALE-001), and the Computational Function HE-CF-ALE-001 adds the standard binomial variance and a 95% interval for any number of bands. The sum takes the form of HE-FM-ASF-002 with r of 0 and of HE-FM-QALY-001, but both records are defined for utility values, while the weight here is the share of survivors free of dependency, with dependent years counted as zero. HALE weights non-healthy time by severity and draws on years lived with disability (HE-FM-DWT-001), and the years of life lost and DALY records (HE-FM-YLL-001, HE-FN-DALY-001) measure health gaps, not health expectancies. Expected care costs per person are HE-FM-COST-001 with the expected dependent years as the quantity and a cost per dependent year as the unit cost. A multistate life table follows transitions between independent, dependent and dead states; with constant rates and no recovery the expected time in each living state is HE-FM-CONT-002, and HE-FM-MKCH-002 splits expected time in a state that can be re-entered into episodes, so those records are cited, not restated. Notation follows the Active Life Expectancy article.

  • Sullivan active life expectancy over three age bands with life expectancy and dependent years

    e_x = (L_1 + L_2 + L_3) / l_x; ALE_x = ((1 - pi_1) * L_1 + (1 - pi_2) * L_2 + (1 - pi_3) * L_3) / l_x; DLE_x = e_x - ALE_x; s_A = ALE_x / e_x

    Weights the person-years that a period life table assigns to each age band after the starting age x by the share of the band free of dependency, adds them and divides by the survivors at x, as Sullivan's method does. The article's three bands (65 to 74, 75 to 84, and 85 and over) are written term by term; more bands add terms in the same way, and the last band is open-ended so that all remaining life is counted. Setting every pi_i to 0 returns life expectancy e_x, person-years over survivors (Imai and Soneji equation 7). That first part is HE-FM-ASF-002 with U_t of 1 and r of 0, its single-year L_t summed within each band to give L_i over l_x (derived), and is cited, not restated. DLE_x, the expected years lived dependent, is the difference and s_A is the share of remaining life lived active. The sum has the form of HE-FM-ASF-002 with r of 0 and U_t replaced by 1 minus pi_i, a use that record does not cover because its U_t is a utility norm (HE-VR-ASF-007); here the weight is a prevalence share with dependent time counting zero.