4. Conclusions

The incorporation of unobserved frailty in the analysis of mortality at adult ages is of considerable theoretical and empirical interest. For instance, the estimation of relative frailty models sheds light on questions of whether a convergence in the mortality rates of different subpopulations, or a ‘flattening’ of the mortality curve with age, can be explained by a selection process towards low-frailty individuals in heterogeneous populations.

While frailty models have been primarily applied to old and oldest-old mortality, recent evidence from twin studies suggests that the consideration of unobserved heterogeneity among individuals in their susceptibility to death is also relevant for the traditional age range from, say, 40–100 years. The application of relative frailty models to these ages, however, is often hampered by the fact that the observed mortality pattern across adult ages is well-described by a Gompertz curve. In this case the estimation of the usual frailty models, which are based on Gompertz or similar hazard functions, fails because the observed pattern lacks the typical ‘flattening’ of the mortality curve that results from the selection towards low-frailty individuals in a heterogeneous population.

In this paper we propose a modified DeMoivre hazard function which is suitable for the estimation of frailty models of mortality for adult ages, e.g., between 40–95 years. The hazard conditional on a constant frailty in this specification increases faster than exponential. The observed hazard, on the other hand, can capture a broad range of mortality patterns that are commonly encountered in the analysis of mortality at adult ages.

We apply the Gamma-distributed relative frailty model with a modified DeMoivre hazard to male and female mortality in Bulgaria during 1992–93. The two characteristic features of this mortality pattern are a substantial difference in the level of adult mortality between males and females, and a considerably steeper increase of mortality with age for females than for males. Our analyses show that a substantial part of this differential increase of mortality can be explained by a differential selection process in the male and female population. Since the level of mortality is higher for males than for females, the male population faces an earlier and stronger selection towards low-frailty individuals, and the resulting differences in the frailty composition of the population at older ages can explain the convergence between male and female mortality.

This finding of our analyses is robust across different specifications for the hazard function. Moreover, sensitivity analyses and a comparison with nonparametrically estimated hazard functions show that the modified DeMoivre hazard function leads to plausible and relatively robust estimates.

In summary, this paper provides a new substantive and methodological approach to the understanding of mortality at adult and old ages. First, we propose a new hazard function that implies, conditional on frailty, a faster than exponential increase of mortality with age. This model is therefore suitable for the estimation of frailty models at ages between 40–100 years, i.e., the age range where Gompertz or logistic models often yield a reasonable description of the observed mortality pattern. Second, we argue that unobserved heterogeneity provides a plausible explanation of the adult mortality pattern in Bulgaria — and possible also other countries — during the early 1990s. Our estimations using the modified DeMoivre hazard function suggest that the stronger selection process towards low-frailty individuals in the male population, caused by an overall higher level of mortality, may constitute a primary mechanism leading to the convergence of male and female mortality at higher ages. This finding implies that the convergence is not necessarily caused by a differential process of aging across sexes, but is merely a consequence of the different levels of mortality, and a subsequently different selection process, for males and females.