Life Expectancy Is the Death-Weighted Average of the Reciprocal of the Survival-Specific Force of Mortality

Abstract The hazard of mortality is usually presented as a function of age, but can be defined as a function of the fraction of survivors. This definition enables us to derive new relationships for life expectancy. Specifically, in a life-table population with a positive age-specific force of mortality at all ages, the expectation of life at age x is the average of the reciprocal of the survival-specific force of mortality at ages after x, weighted by life-table deaths at each age after x, as shown in (6). Equivalently, the expectation of life when the surviving fraction in the life table is s is the average of the reciprocal of the survival-specific force of mortality over surviving proportions less than s, weighted by life-table deaths at surviving proportions less than s, as shown in (8). Application of these concepts to the 2004 life tables of the United States population and eight subpopulations shows that usually the younger the age at which survival falls to half (the median life length), the longer the life expectancy at that age, contrary to what would be expected from a negative exponential life table. (ProQuest: ... denotes formulae omitted.) 1. Background and relationships 1.1 Background The life table l(x), constant in time, with continuous age x, is the proportion of a cohort (whether a birth cohort or a synthetic period cohort) that survives to age x or longer. In probabilistic terms, l(x) is one minus the cumulative distribution function of length of life x. The maximum possible age w may be finite or infinite. If w = x, then some individuals may live longer than any finite bound. By definition, l(0) = 1 and l(w) = 0. Assume l(x) is a continuous, differentiable function of x, 0 ≤ x ≤ w, and assume life expectancy at age 0 is finite. The age-specific force of mortality at age x is, by definition, (1) ... Assume u(x) > 0 for all 0 ≤ x ≤ w. The life table l(x) is strictly decreasing from l(0) = 1 to l(w) = 0 so there is a one-to-one correspondence between age x in [0,w] and the proportion s in [0, 1] of the cohort that survives to age x or longer. One direction of this correspondence is given by the life table function s = l(x) (illustrated schematically in Figure 1 and for the United States population in 2004 in Figure 3A). There appears to be no standard demographic term for the inverse function that maps the proportion surviving s, 0 ≤ s ≤ 1, to the corresponding age x, so I propose to call it the age function a (illustrated schematically in Figure 2 and for the United States population in 2004 in Figure 3D). In words, the age a(s) at which the fraction s of the birth cohort survives is the age x at which the life table function l(x) is s. By definition, under the assumption u(x) > 0 for all 0 ≤ x ≤ 1, a(s) = x if and only if l(x) = s. Equivalently, by definition, for every s in 0 ≤ s ≤ 1 and every x in 0 ≤ x ≤ w, a(l(x)) = x and l(a(s)) = s. We define a(1/2) as the median life length, that is, the age by which half the cohort has died. For every s in 0 ≤ s ≤ 1, we define the survival-specific force of mortality λ(s) in terms of the age-specific force of mortality u(x) in (1) in three equivalent ways: (2) λ(s) = u(x) if s = l(x); or λ(s) = u(a(s)); or λ(l(x)) = u(x): In words, the survival-specific force of mortality λ(s) at surviving proportion s equals the age-specific force of mortality u(x) at the age x where the life table l(x) = s. The domain of the age-specific force of mortality u is 0 ≤ x ≤ w while the domain of the survival- specific force of mortality λ is 0 ≤ s ≤ 1. We give below an explicit formula (9) for the survival-specific force of mortality at surviving proportion s. This formula is analogous to (1) for the age-specific force of mortality. The complete expectation of life at age x, e(x), is the average number of years remaining to be lived by those who have attained age x: (3) ... Inserting the definition (1) in place of u(y) in (3) and integrating by parts gives (4) . …