## 2. A Modified DeMoivre Hazard Function

In an early attempt to describe mortality patterns with a mathematical formula, Abraham DeMoivre [DeMoivre 1725, p. 4, cited in Keyfitz and Smith 1977, p. 273] hypothesized that ‘the number of lives existing at any age is proportional to the number of years intercepted between the age given and the extremity of old age’, i.e.,

where w is the maximum attainable age in the population. From this, the hazard rate, or the force of mortality at age x is defined as

 (4)
where mD(x) denotes the DeMoivre hazard function. The hazard rate in this example increases towards infinity as x approaches the maximum attainable age w. The existence of this maximum attainable age implies that the force of mortality increases faster than in the Gompertz model, especially as x approaches the maximum age w [for a related discussion see Zelterman 1992]. For the application to contemporary mortality patterns the hazard function mD(x) in (4) is not sufficiently flexible. We therefore propose a modified DeMoivre hazard given by

 (5)
where mMD(x) is the modified DeMoivre hazard function. The survival curve sMD(x) corresponding to the above hazard function is given by

 (6)
The parameter b in (5) and 6) needs to satisfy b > in order that the hazard and survival curves are meaningful. The hazard function mMD(x) defined in (5) has two limiting properties that render it a plausible and easily interpretable specification. First, the original DeMoivre hazard mD(x) emerges — up to a factor of proportionality — from the modified DeMoivre hazard mMD(x) when the product bw in (5) approaches one. This occurs, for instance, when b ®1/w for a fixed maximum attainable age w. Formally this property is represented as lim b®1/wmMD(x) = lim bw®1mMD(x) = mD(x). Second, as the maximum attainable age w becomes large, the modified DeMoivre hazard approaches the Gompertz hazard mG(x) = aebx. That is, the hazard mMD(x) in (5) has the limit lim w®¥mMD(x) = mG(x). The reason for this convergence to the Gompertz hazard is easily seen by noting that the limit lim w®¥(1 - )w = ex.

The interpretation of the parameters in the modified DeMoivre hazard is further facilitated by the fact that mMD(x)|x=0 = a and x=0 = x=0 = b, which implies that the modified DeMoivre hazard agrees with a Gompertz hazard with equal parameters a and b at the age x = 0.

In heterogeneous populations with unobserved frailty the observed mortality rate differs from mMD(x) because the population becomes increasingly selected towards low-frailty individuals with age. If we assume a Gamma-distributed relative frailty model, then the observed hazard rate and survival curve implied by the modified DeMoivre model, denoted MD(x) and MD(x), follow directly from equations (1) and (2) as

 (7)
and

 (8)
where mMD(x) and sMD(x) are the hazard function and survival curve in equations (5) and 6) for individuals with a constant frailty z = 1.

Figure 3 plots the modified DeMoivre hazard function and the implied life-table aging rate for different values of s2 representing different degrees of unobserved heterogeneity in the population. In this figure we have used a maximum attainable age w of 122.45 years, i.e., a w that corresponds to Madame Jeanne Calment age at death, and we used the modified DeMoivre hazard function to model mortality during ages 40–100. [Note 2] The dashed-dotted line in both graphs represents the values for the modified DeMoivre hazard without unobserved frailty, or equivalently, for individuals with a constant frailty z = 1. Conditional on a constant frailty z the modified DeMoivre hazard increases faster than the exponential Gompertz hazard and the life-table aging rate is also an increasing function of age. As the age approaches the maximum attainable age w, both the mortality hazard and the life-table aging rate approach infinity and the probability of surviving to ages larger than w is zero.

The existence of such an upper limit to life-span is in contrast to the Gompertz model that does not imply a maximum attainable age and it is certainly controversial in view of the recent debate about the limits to the increase in life expectancy and particularly to the biological limits of life-span [Gavrilov and Gavrilova 1991, Manton and Stallard 1996, Vaupel et al. 1998, Wilmoth 1997, Wilmoth et al. 2000]. An emerging consensus in this debate seems to be that if upper limits to life-span exist, they need to be seen in a dynamic perspective [Carey and Judge 2000] and almost certainly do not represent immutable biological limits that are insurmountable by medical or social progress in survival to very old ages. At the same time, in any given socioeconomic environment the observed human life-span is finite and no human being has been documented to survive above Madame Jeanne Calment’s age at death. Hence, while there probably does not exist an absolute biological limit to life-span, human life-span in any socioeconomic context may be limited and the changes of this upper limit to life in itself may constitute an interesting area of research [e.g., Carey and Liedo 1995, Wilmoth and Lundström 1996].

The mortality hazard in the modified DeMoivre model, conditional on a constant frailty z, increases faster than exponential and implies an increasing life-table aging rate with age. Despite this fact, the observed mortality pattern in a heterogeneous population can reflect a substantially different pattern. In Figure 3 we have therefore included the observed mortality hazard and life-table aging rate that is implied by the modified DeMoivre hazard with different degrees of unobserved heterogeneity in the population. Most strikingly, the observed mortality pattern does not necessarily represent the mortality pattern conditional on the frailty z [for an influential related discussion see Vaupel and Yashin 1985], but reflects properties that are characteristic of the Bulgarian mortality pattern in Figure 1 and also that of other countries. For instance, the observed life-table aging rate between ages 40–100 can be initially increasing, and the extent of this increase in the life-table aging rate depends on the degree of heterogeneity and the level of mortality. At the same time, the relative increase in mortality by age decreases at old and oldest-old ages. Once age approaches the maximum attainable life-span w, this decline in the life-table aging rate reverses again and the observed mortality rate and the life-table aging rate increase and ultimately approach infinity at w. While this last implication for extremely old ages is controversial in view of the discussion on the limits to human longevity, the modified DeMoivre hazard in combination with unobserved heterogeneity can represent observed age-patterns of mortality for the quite wide age-range from adult to old and oldest-old ages during which most deaths in humans occur.

In the age range below age 100, therefore, the modified DeMoivre hazard function mMD(x) has several properties that make it a plausible choice for estimating frailty models in mortality. For instance, nonparametric estimations of the hazard function, which are feasible in bivariate frailty model applied to twin data [Yashin et al. 1995], have revealed that the hazard for individuals with constant frailty z = 1 during adult ages is increasing substantially faster than the Gompertz hazard, while the observed mortality rates can be approximated by a Gompertz or Logistic hazard function. In addition, detailed multivariate follow-up studies document a differential selection in male and female cohorts with respect to physiological characteristics and functional abilities that indicate an important role of selection processes for understanding male-female differences in mortality [Manton et al. 1995]. On the basis of these and related findings, Caselli et al. [2000, p. 8] have concluded that the ‘correction for unobserved heterogeneity in demographic life tables may be needed not only for the oldest-old but also for the traditional interval of aging between 35 and 85 years of age, for which the observed trajectory of mortality appears to be well-described by a Gompertz curve.’

The modified DeMoivre hazard function in Figure 3 provides a possibility to estimate frailty models in the above situation. In particular, the observed hazard MD(x) as well as the baseline hazard mMD(x) in this Figure agree highly with the nonparametric estimates reported in Caselli et al. [2000] and Yashin et al. [1995]. Since nonparametric estimation is only feasible with special data, as for instance data on the mortality of twins or data with proportional-hazard covariates, the modified DeMoivre hazard introduced in this paper provides a suitable alternative for frailty modeling with vital statistics data. The hazard function introduced in this paper, therefore, allows the investigation of the ‘selectivity hypotheses’ for the convergence of male and female mortality in Bulgaria, and possibly also other countries, where relative-frailty Gompertz models with unobserved heterogeneity do not yield an accurate description of the observed mortality dynamics.

Two basic approaches exist for the estimation of the unknown parameters a, b, w and s2. First, we can choose a plausible value for w, such as Madame Jeanne Calment’s age at death. The remaining parameters a, b, s2 can then be estimated via a maximum likelihood using w = 122.45. This approach should be taken if mortality data at very old ages, say above age 110 that could shed light on the value of w, are not available or reliable. In our experience, this procedure yields a quite robust and plausible estimation, and the results of the parameters a and b are not very sensitive to the choice of w as long as it is chosen within a plausible age range, say 110 – 150 years.

The second possibility is to estimate all four parameters a, b, w and s2 using maximum likelihood estimation. Since w determines the convexity of the hazard function, and s2 influences the extent to which the increase in (x) is flattened due to the selection process towards low-frailty individuals, the joint estimation of all four parameters is not feasible when only one mortality pattern is observed. It is, however, feasible if mortality patterns of several subpopulations, e.g., by sex or educational attainment, are analyzed. An example for this estimation is given in Section 3. Quite naturally, an effective estimation of the life-span w requires reliable data at very high ages that may not be available in many countries. If the focus of the investigation is on adult and old-age mortality, e.g., as in our Bulgarian example on the age range 40–100, an estimation strategy that assumes a specific value for w and then conducts a sensitivity analysis may be preferable to the estimation of w from data that are censored at some upper age limit. If w is nevertheless estimated in these cases, our experience suggests that the resulting ‘best-fitting mortality curve’ is often characterized by a too low maximum age w, and a too high level of unobserved heterogeneity.

Independent of which approach regarding the specification of w is chosen, the parameters a, b, and s2 can be functions of characteristics of individuals or subpopulations. In particular, we implement the estimation of the modified DeMoivre mortality model with

where ya, yb, ys are vectors of covariates that influence the level of the parameters a, b, and s2. For instance, in the next Section we will estimate a model where some of the parameters can differ by sex. This dependence of the parameters on sex is incorporated by including a dummy for females in ya, yb and/or ys. While this specific analysis is a relatively simple dependence of the parameters a, b, and s2 on covariates, considerably more complex specifications are also feasible.

 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 http://www.demographic-research.org/Volumes/Vol3/8