Sensitivity of life disparity with respect to changes in mortality rates

Abstract

Abstract This article is concerned with sensitivity analysis of life disparity with respect to changes in mortality rates. A relationship is derived that describes the effect on life disparity caused by a perturbation of the force of mortality. Recently Zhang and Vaupel introduced a threshold age, before which averting deaths reduces disparity, while averting deaths after that age increases disparity. I provide a refinement to this result by characterizing the ages at which averting deaths has an extremal impact on life disparity. The results are illustrated using data for the female populations of Denmark in 1835, and for the United States in 2005. (ProQuest: ... denotes formulae omitted.) 1. Introduction In keeping with Keyfitz's idea that everybody dies prematurely, since every death deprives the person involved of the remainder of his expectation of life (Keyfitz 1977:61-68), the measure e[dagger] for the average life expectancy lost due to death has been widely studied. It first appeared in Mitra (1978) and was developed further by Vaupel (1986) and recently in Vaupel and Canudas-Romo (2003), Zhang and Vaupel (2008) and Shkolnikov et al. (2009). Zhang and Vaupel (2009) initiated a new direction of analysis, studying the impact on e[dagger] of a concentrated decrease in mortality at age a. Life disparity is measured by life expectancy lost due to death (1) ... where e(x) is the remaining life expectancy at age x, (2) ... l(x) = exp (-H(x)) is the probability of survival to age x, H(x) = ∫x0 µ(y) dy is the cumulative hazard function and µ(x) is the age-specific hazard of death. Since nobody lives forever, it is generally assumed thatH is strictly increasing, attaining all non-negative real numbers. The function ... is the life table distribution of deaths. Goldman and Lord (1986: equations (6), (14)) and Vaupel (1986: equations (4), (5)) independently showed that life disparity (1) is the product of the life expectancy at birth and the entropy of the life table (3) ... In this article, I will formally derive and discuss a relationship concerning the effect of a change in mortality at some age on life disparity. 2. Relationship Let φ(a) represent the change of e[dagger] caused by a reduction in mortality at age a (with a precise definition thereof via a limit of derivatives given in the proof). The main relationship in this article consists of two equivalent formulae ((4), (5)) for the function φ. Theorem 1 The function φ(a), representing the change of e[dagger] caused by a reduction in mortality at age a, satisfies the relationship (4) ... and, equivalently, (5) φ(a) = l(a)(H(a)e(a) - e(a) + e[dagger](a)), where e[dagger](a) denotes life expectancy lost due to death among people surviving to age a, (6) ... Furthermore, φ has the following three properties: (i) Monotonicity Let a, the age of cumulative hazard unity, be defined via H(a) = 1. Then φ is strictly increasing on [0, a] and strictly decreasing and strictly positive on [a, ∞), having a global maximum of φ(a) = exp(-1)e[dagger](a) at a = a and a local minimum of φ(0) = e[dagger](0)-e(0) at a = 0. More precisely, (7) ... (ii) Curvature Let a* be defined by H(a*) = 2. Then φ is strictly concave on [0, a*] and strictly convex on [a*, ∞). More precisely, ... (iii) Asymptotic Behaviour ... Zhang and Vaupel (2009) showed that, if the life table entropy (cf. (3)) satisfies e[dagger]/e(0) a[dagger], in which this effect is positive. If e[dagger]=e(0) = 1, then a[dagger] = 0, and the effect is positive at all ages other than zero. …