INSERT_TITLE_0 A Modified DeMoivre Hazard Function

1. Introduction

Male mortality exceeds female mortality rates at adult ages in many populations. This female advantage in survival often diminishes with age, and male and female mortality rates converge at higher ages [Carey and Judge 2000, Hummer et al. 1998, Manton et al. 1995, Waldron 1985]. Similar patterns of convergence or mortality cross-overs are also observed between other populations that are subject to quite different mortality levels at adult, but not necessarily at old and oldest-old ages [Gavrilov and Gavrilova 1991, Vaupel and Yashin 1985]. Consider for instance the Bulgarian male and female mortality pattern during 1992–93 in Figure 1(a). The striking aspect of this mortality pattern is on one hand the substantially higher mortality level for males, especially during adult ages, and on the other hand the differential increase in the force of mortality by age [see also Kohler 2000a,b]. The mortality sex-ratio in Figure 1(b) shows that males in Bulgaria around age 40 experience a mortality level that is almost 200% higher than that of females. This male-female difference diminishes to 25% around age 80 and it virtually vanishes at ages above age 90. This convergence in male and female mortality levels occurs because, despite their mortality-advantage at adult ages, females in Bulgaria are subject to a substantially more rapid increase in the level of mortality by age. The life-table aging rate [Carey and Liedo 1995, Horiuchi and Coale 1990], i.e., the relative increase of the mortality hazard per additional year of age, depicted in Figure 1(b) shows that the relative increase of female mortality with age is above the relative male mortality increase at all ages above age 45. [Note 1] In addition to the difference in absolute level, the life-table aging rate (LAR) reflects some known sex-specific deviations from the fitted Gompertz model in the left graph of Figure 1. In particular, the female life-table aging rate is increasing between age 40–75, which has been attributed to a post-menopausal mortality increase that is due to lower evolutionary selection forces at post-reproductive ages [Horiuchi 1997]. After age 75, the relative increase of mortality by age is declining and the mortality increase by age is slowing down at these old and oldest-old ages. The female life-table aging rate is thus clearly bell-shaped, while the male life-table aging rate is increasing, with some minor fluctuations, up to age 80. Afterwards the increase of male mortality by age is slowing down similar to the female pattern. Ignoring these age specific patterns and averaging across the whole age range 40–100 years, female mortality rates increase by approximately 10.8% per year of age (based on the estimates of the Gompertz model in Figure 1(a)), while male mortality rates increase by only 8.25% per year of age. This differential increase in mortality by on average 2.5 percentage points implies the strong convergence between male and female mortality rates at higher ages in Figure 1(a).

The important question in this context is whether the differential increase in the force of mortality per year of age is due to a differential aging process between males and females, or whether this difference can be attributed to a stronger selection of the male population towards low-frailty individuals that is caused by the higher overall level of male mortality. The knowledge which of these two factors is primarily responsible for the above convergence pattern is essential for the development of appropriate theories of aging and mortality change [Carnes et al. 1996, Vaupel and Yashin 1985]. In the former case, the male-female convergence of mortality is attributed to factors such as a post-menopausal acceleration of mortality for females [Horiuchi 1997], sex-differences in metabolism, hormonal levels and other fundamental biological aspects [Hazzard 1986, Hazzard and Applebaum-Bowden 1989, Waldron 1985], genetic differences related to the female ‘advantage’ of having two X chromosomes [Waldron 1985], potential systematic behavioral and psychological sex-differences in coping with stress and the aging process itself [Baltes et al. 1999], and age-related social and behavioral changes [House et al. 1990]. In the latter scenario, which emphasizes the process of differential selection, the higher mortality level for males — especially at adult ages — implies that the male population is more rapidly selected towards individuals with a relatively low risk of mortality. In a heterogeneous population, the differential strength of the selection process between males and females then leads to a slower increase in the observed male mortality as compared to the female mortality [Vaupel et al. 1979, Vaupel and Yashin 1985]. A convergence in the observed male and female mortality pattern can thus occur even when both sexes are subject to a mortality curve that differs — conditional on a constant frailty composition — only by a factor of proportionality. That is, the mortality convergence occurs even if male and female mortality are characterized by an age pattern that, conditional on the frailty level, exhibits an identical life-table aging rate and an identical relative increase in the mortality rates with age. In this selection hypothesis, therefore, the main difference in the ‘law of mortality’ for males and females is in the level of the mortality risk, i.e., a proportionally higher level of male as compared to female mortality. Differences in the life-table aging rate and patterns of mortality convergence between sexes are thus primarily attributed to changes in the frailty composition of population, instead of fundamental differences in the process of aging itself.

The investigation of whether the above ‘selection hypothesis’ can provide a plausible explanation for the male–female differences in the increase of mortality with age requires the estimation of mortality models with unobserved frailty. In a seminal analysis, Vaupel, Manton, and Stallard [1979] have introduced relative frailty models in which individuals in a population are heterogeneous with respect to their susceptibility to death. This relative risk of death, denoted by the frailty z, is unobserved on the individual level. Despite this unobservability, the mortality patterns can be adjusted for the distortions caused by the selection process due to differential mortality in heterogeneous populations. In particular, based on assumptions about the initial distribution of unobserved frailty in the population and its effect on the force of mortality, inferences can be made about the composition of the population with respect to frailty at some specific age x and the level of mortality that would have prevailed if there had been no changes in the frailty-composition of the population over time. Frailty models of mortality therefore allow the investigation of whether observed mortality patterns can be explained by selection processes within a population, or by differential selection across subpopulations. For instance, the analyses in this paper focus on the question of whether frailty models can provide a plausible explanation for the convergence between male and female mortality in Bulgaria and possibly other countries.

Relative frailty models assume that the mortality rate at age x of a person with frailty z equals zm(x), where m(x) is the mortality rate of individuals with z = 1. Individuals with z > 1 therefore experience a force of mortality that is proportionally higher than m(x) at all ages, while individuals with z < 1 experience proportionally lower mortality rates. The composition of a cohort with respect to the frailty z changes as a cohort grows older because the most frail individuals tend to die earlier than the least frail individuals. The increase of the observed mortality rates with age is therefore determined by two factors: (a) the age-increase in mortality holding frailty constant, which is reflected in m(x); and (b) the extent to which the cohort at age x becomes selected towards low-frailty individuals, which is reflected in the distribution of the frailty z in the population conditional on survival up to age x.

Vaupel et al. [1979] assume a gamma-distributed frailty with mean one and variance s2, and show that the observed hazard and survival curve at age x, denoted (x) and (x); are equal to


where m(x) and s(x) are the baseline hazard rate and the survival curve for individuals with a constant frailty of z = 1. Moreover, the mean frailty (x) of the population who is alive at age x equals (x) = = (1 - s2 log s(x))-1, which indicates that the mean frailty of the population who has survived to some age x decreases as the fraction of survivors to age x declines. Equations (1) and (2) indicate that the selection process in heterogeneous populations drives a wedge between the baseline hazard rate m(x), which pertains to individuals with constant frailty z = 1, and the observed hazard rate (x). The extent to which these rates differ depends on two factors: (a) the variance s2, i.e., the variation in unobserved frailty in the population at the beginning of the observation period, and (b) the extent to which the population has already been selected, which is indicated by the survival function s(x). Moreover, the derivative

shows that the slope of the observed mortality pattern is also affected by the selection towards low-frailty individuals in heterogeneous populations. On the one hand, the first term indicates the relative increase in the force of mortality holding frailty constant at z = 1. On the other hand, the second term s2m(x)/(1 - s2 log s(x)) measures the strength of the selection process at age x. This strength depends on the mortality level m(x), the overall variance of frailty in the population s2 and the mean frailty (x) = (1 - s2 log s(x))-1 of the population who has survived to age x. This selection process implies that the observed relative increase in the mortality rate x is slower than the relative increase of the baseline hazard m(x). This difference between the observed and ‘true’ increase of the mortality curve by age increases the larger is the final term in equation (3). A typical pattern of observed mortality rates, which is implied by the above frailty model, is depicted in Figure 2(a). In this Figure we have assumed that m(x) follows a Gompertz curve with log m(x) = aebx, which implies a linear increase in the logarithm of the mortality hazard with age. The arrow in this graph reveals the wedge between the observed mortality rate (x) and the underlying rate m(x) for frailty z = 1. As long as mortality is relatively low, the two curves for (x) and m(x) trace each other closely. As soon as mortality has increased to moderate levels, however, the selection process towards low-frailty individuals in the population becomes significant and (x) increasingly diverges from m(x). In particular, while log m(x) increases linearly with age due to the Gompertz specification, the observed pattern log (x) is markedly flatter. The observed mortality increases slower than linear on the log scale, and the slope of the observed mortality pattern becomes increasingly less than the slope of Gompertz hazard. While the life-table aging rate in Figure 2(b) is constant across all ages in the Gompertz model, the introduction of the unobserved heterogeneity leads to a marked decline of the life-table aging rate at higher ages due to an increased selection of the population towards less frail individuals.

A potential problem in estimating the above frailty model becomes apparent when comparing the mortality pattern observed in Bulgaria (Figure 1) with the typical mortality pattern implied by a relative-frailty Gompertz model (Figure 2). First, a Gompertz model fits the Bulgarian mortality in Figure 1 relatively well, and to a first approximation a standard Gompertz hazard function provides a quite good description of the Bulgarian male and female adult and old age mortality pattern for the age range 40–100 years. While this good empirical fit of the Gompertz model may initially seem very desirable, it poses considerable problems in the context of frailty models. The problem arises because the characteristic feature of frailty models is the ‘flattening’ of the mortality curve and a decline in the life-table aging rate as shown in Figure 2. This decline of the life-table aging rate is due to the fact that the population alive at some age x becomes increasingly more selective towards ‘healthy’ individuals. The resulting flattening of the mortality curve should be most pronounced for the population that faces the highest level of mortality, that is, in our example the male population. The male empirical pattern in Figure 1, however, does not reflect such a flattening of the male mortality curves or a marked decline of the male life-table aging rate until relatively old ages. Instead, a Gompertz model with a linear increase of log m(x) provides a very good fit across all adult and old ages in Bulgaria, especially for males, and a divergence between the Gompertz model and the observed pattern — similar to the one depicted in Figure 2(b) — is absent.

The case for an alternative to the relative-frailty Gompertz model is furthermore strengthened by the deviations of the observed mortality pattern from the Gompertz model in Figure 1. As discussed above, the Bulgarian female mortality pattern exhibits a post-menopausal increase in the life-table aging rate and a decline at old and oldest-old ages, while the male pattern exhibits a modest increase in the life-table aging rate until about age 80. The observed life-table aging rate in the relative-frailty Gompertz model, however, exhibit a markedly different pattern. The life-table aging rate in this model attains it highest values at relatively young ages before mortality has affected the frailty composition of the population, and the life-table aging rate declines as the population becomes increasingly selected towards low-frailty individuals. The convex-concave pattern of mortality change observed in Figure 1, hence, is not implied by the relative-frailty Gompertz model in Figure 2 that exhibits a monotonously declining life-table aging rate.

A Gompertz model with relative frailty therefore does not provide a good explanation for the mortality pattern in Bulgaria. Two conflicting hypotheses can be considered in order to explain this apparent inability of the relative-frailty Gompertz model to replicate and explain the Bulgarian mortality pattern in Figure 1 and the convergence between male and female mortality levels: (a) There are no unobserved differences in individuals’ frailty in the Bulgarian population. A selection towards low-frailty individuals is thus absent and the male and female mortality pattern is characterized by systematically different mortality curves with different life-table aging rates. This implies that males and females in Bulgaria are characterized by a differential aging process. (b) The Bulgarian population is characterized by unobserved frailty, and the ‘true’ hazard m(x) in equation (1), which applies to individuals with a constant frailty, is increasing faster than a Gompertz hazard so that the convergence between male and female mortality and the convex-concave pattern of the mortality hazard can result from the selection process in the population towards low-frailty individuals.

The argument given in (a), namely that the Bulgarian population is homogeneous with respect to frailty, seems rather unlikely and it contradicts most recent mortality research that points to important variations in the determinants of survival and longevity that are due to variation in genetic factors [Herskind et al. 1996, McGue et al. 1993] and early-life experiences [Barker 1992, Doblhammer 1999, Elo and Preston 1992, Horiuchi 1983]. The investigation of the second hypothesis (b) for Bulgaria, however, is difficult because the Gompertz hazard function, and most other commonly used hazard functions like Kannisto, Makeham, Logistic hazard functions [for a review of these models see Manton and Yashin 2000, Thatcher et al. 1998], do not allow a meaningful incorporation of unobserved frailty in order to explain the Bulgarian pattern in Figure 1(a). Moreover, the explanation of the male-female mortality convergence in Figure 1(a) using selection processes in heterogenous populations requires that the selective forces of mortality start operating already at adult ages. For instance, this relatively early onset of a selection of the population towards low-frailty individuals is supported by recent evidence from twin studies which suggest that unobserved heterogeneity is important for the estimation of mortality pattern at adult ages, and not only for mortality at old and oldest-old ages [Caselli et al. 2000, Iachine et al. 1998].

In this paper we therefore propose an alternative specification, a modified DeMoivre hazard function, that is suitable to investigate the hypothesis of whether the differential slopes of the mortality pattern between males and females in Bulgaria could merely be the result of a differentially strong selection process in heterogeneous populations.


INSERT_TITLE_0 A Modified DeMoivre Hazard Function

Frailty Modelling for Adult and Old Age Mortality: The Application of a Modified DeMoivre Hazard Function to Sex Differentials in Mortality
Hans-Peter Kohler, Iliana Kohler
© 2000 Max-Planck-Gesellschaft ISSN 1435-9871