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 crossovers are also observed between other
populations that are subject to quite different mortality levels at adult, but not necessarily
at old and oldestold 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 sexratio 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 malefemale 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 mortalityadvantage at adult ages, females in Bulgaria are
subject to a substantially more rapid increase in the level of mortality by age. The lifetable
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 lifetable aging rate
(LAR) reflects some known sexspecific deviations from the fitted Gompertz model in the left
graph of Figure 1. In particular, the female lifetable aging rate is increasing between age
40–75, which has been attributed to a postmenopausal mortality increase that is due to lower
evolutionary selection forces at postreproductive 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 oldestold ages. The female lifetable aging rate is thus clearly bellshaped,
while the male lifetable 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 lowfrailty 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 malefemale convergence of mortality is attributed to factors such as a
postmenopausal acceleration of mortality for females [Horiuchi 1997], sexdifferences in
metabolism, hormonal levels and other fundamental biological aspects [Hazzard 1986, Hazzard and ApplebaumBowden 1989, Waldron 1985],
genetic differences related to the female ‘advantage’
of having two X chromosomes [Waldron 1985], potential systematic behavioral and
psychological sexdifferences in coping with stress and the aging process itself [Baltes et al. 1999], and
agerelated 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 lifetable 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 lifetable 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 frailtycomposition 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 ageincrease 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 lowfrailty 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 gammadistributed frailty with mean one and variance
s^{2}, and show
that the observed hazard and survival curve at age x, denoted
(x) and
(x); are equal to

(1) 
and

(2) 
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  s^{2} 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 s^{2}, 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
 (3) 
shows that the slope of the observed mortality pattern is also affected by the selection towards lowfrailty
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
s^{2}m(x)/(1  s^{2} 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 s^{2} and the
mean frailty
(x) = (1  s^{2} 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) = ae^{bx},
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 lowfrailty 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 lifetable 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 lifetable
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
relativefrailty 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
lifetable aging rate as shown in Figure 2. This decline of the lifetable 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 lifetable 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 relativefrailty 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 postmenopausal increase in the
lifetable aging rate and a decline at old and oldestold ages, while the male pattern exhibits a
modest increase in the lifetable aging rate until about age 80. The observed lifetable aging rate
in the relativefrailty Gompertz model, however, exhibit a markedly different pattern. The
lifetable 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 lifetable aging rate
declines as the population becomes increasingly selected towards lowfrailty individuals. The
convexconcave pattern of mortality change observed in Figure 1, hence, is not implied by the
relativefrailty Gompertz model in Figure 2 that exhibits a monotonously declining lifetable 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 relativefrailty 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
lowfrailty individuals is thus absent and the male and female mortality pattern is characterized by
systematically different mortality curves with different lifetable 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 convexconcave pattern of the
mortality hazard can result from the selection process in the population towards lowfrailty
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 earlylife 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 malefemale 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 lowfrailty 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 oldestold 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.
