Methods for Comparing the Mortality Experience of Heterogeneous Populations

Abstract

Methods are presented which produce Maximum Likelihood Estimates (MLE) of the degree of heterogeneity in individual mortality risks under a variety of assumptions about the age trajectory of those mortality risks. With these estimates of the degree of population heterogeneity it is possible to adjust comparisons of mortality risks across populations for the effects of population heterogeneity, differential mortality selection, and different age trajectories of the force of mortality. These methods are demonstrated by applying a variety of standard assumptions about the age trajectory of the force of mortality to the analysis of a broad range of cohort mortality data for the U.S. and Swedish populations. The estimates of the degree of heterogeneity, produced under all of the selected force of mortality models, consistently indicated a considerable degree of heterogeneity in mortality risks.

An empirical analysis of the importance of controlling for unobserved heterogeneity when estimating the income-mortality gradient

“…6 Using the Kuhn-Tucker approach of Baker and Melino (2000), it can be shown that applying a NonParametric MLE estimator (Heckman and Singer 1984) to a left-truncated sample of durations causes a nonidentification of the probability distribution function of unobserved heterogeneity. 7 For further discussion of the (need for) functional form assumptions when modeling mortality risk, see Hougaard (1984) and Manton, Stallard, and Vaupel (1981;1986). younger ages, which requires the functional form assumption of a Gompertz specification for the age-dependency of mortality risk.…”

“…The consequences of ignoring such unobserved heterogeneity in the age-mortality gradient has received much attention in the literature following a seminal paper by Vaupel, Manton, and Stallard (1979) that refers to unobserved heterogeneity as "frailty". Although Vaupel, Manton, and Stallard's (1979) work sparked many studies -including Congdon (1994), Hougaard (1984, Manton, Stallard, and Vaupel (1981;1986) and Vaupel and Yashin (1985) -these are mostly concerned with life tables for heterogeneous populations. 3 This research does show that, as predicted by statistical theory, the positive age-mortality gradient is steeper once unobserved heterogeneity is taken into account.…”

Unobserved population heterogeneity

“…The resulting gamma-Gompertz-Makeham (ΓGM) fixed-frailty model is defined by a conditional hazard µ(x, z) = z ae bx + c , which for k = λ = 1/γ results in the following population hazard (Manton, Stallard, and Vaupel 1981)μ…”

The Deviant Dynamics of Death in Heterogeneous Populations

“…Harris and Singpurwalla (1968) and Mann, Schafer, and Singpurwalla (1974) developed methods for taking into account differences in reliability among machines and equipment. Zeckhauser (1 975, 1977, 1980a,b;Zeckhauser and Shepard 1976) pioneered the analyses of heterogeneity in human mortality and morbidity; Woodbury and Manton (1977), Keyfitz and Littman (1980), Stallard (1979, 1981a,b), and Vaupel, Manton, and Stallard (1979a;Manton et al 1981) have made further contributions.…”

“…If some assumption is made about the distribution of relative-risk (e.g., that it is gamma distributed) and about the relationship of p,(x) to p2(x) (e.g., that one is a constant multiple of the other), then estimates of the variance in heterogeneity can be calculated. Vaupel et al (1979b) and Manton et al (1981) applied this method to various cohorts of the four populations of male and female Swedes and US whites. The results suggest that for these populations, the variance in heterogeneity is roughly one.…”