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Bio-mathematical approaches to the study of human variation in mortality.

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Bio-Mathematical Approaches to the Study of Human
Variation in Mortality
Department of Anthropology, University at Albany, SUNY, Albany,
New York 12222
Age-specific mortality, Hazard modeling, Heterogeneity, Carcinogenesis, Mortality crossover
The recent development of a number of biologically interpretable mathematical models of human mortality has facilitated the study
of human variation in mortality patterns. This paper reviews the biological
basis of these models, describes the models themselves, and presents the
results of four anthropological applications of these models to the study of
human variation in mortality. The models examined include a multi-stage
model of carcinogenesis, the Gompertz, Gompertz-Makeham, and Siler models of the age patterns of total mortality, the fixed gamma distributed model
of individual heterogeneity with respect to mortality, and a stochastic compartment model useful for studying the covariates of mortality. The examples presented include applications of: 1)the multi-stage model to the study
of colon cancer, 2) the fixed frailty gamma distributed model of heterogeneity to the blackiwhite mortality crossover, and to a similar crossover
identified in historical data, and 3) the Siler model to document and classify
the international age patterns of mortality among contemporary nations
and with several prehistoric and one contemporary anthropological population.
The age patterns of human mortality result from the interaction of endogenous
and exogenous factors. The endogenous factors are biological characteristics inherent to the organism, while the exogenous factors are aspects of the environment, either physical, biotic, or social, that are external to the organism (Bourgeous-Pichat, 1951). Despite the fact that both endogenous and exogenous factors
are involved in the process of mortality, the various disciplines conducting research on human mortality have tended to examine one or the other. Gerontologists, for example, tend to look to endogenous factors for the explanation of aging,
while epidemiologists and sociologists usually examine exogenous characteristics
in a n attempt to explain the observed variation in mortality patterns among human populations. Gerontologists can control for environmental variation through
appropriate experimental methods. The social scientist, however, whose interest is
in variation at the population level and whose data are observational, does not
have this option. In practice, these researchers often simply assume t h a t individuals and populations are biologically identical with respect to the endogenous
factors affecting mortality. However, the individuals comprising a population are
certainly not identical, and there may even be variation among populations with
respect to these endogenous factors (Mourant, 1983; Overfield, 1985). Studies of
variation in the age patterns of mortality among European populations have uncovered consistent regional variations (Coale and Demeny, 1988). While these
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[Vol. 32, 1989
patterns were discovered 20 years ago, the causes of these variations, endogenous
or exogenous, are not yet understood (United Nations, 1983). Some of the international variation in mortality could be due to endogenous as well as exogenous
Recently, several social scientists have begun studying human variation in mortality patterns using biologically interpretable mathematical models of mortality
(Vaupel et al., 1979; Mode and Busby, 1982; Manton and Woodbury, 1983; Manton
and Stallard, 1984; Siler, 1983; Mode and Jacobson, 1984; Weiss and Chakraborty,
1984; Gage and Dyke, 1986). These approaches attempt to combine the endogenous
models of mortality developed by gerontologists, with the comparative study of
human mortality patterns used by social scientists. From the point of view of the
social scientist, the theoretical advantage of this approach is that the bio-mathematical models can be used to control for variation in endogenous factors, so that
the effects of the exogenous factors can be more clearly observed. From the point of
view of human biologists, with an interest in both the biological and social implications, the advantage is that both endogenous and exogenous factors can be examined together in order to understand the variation in human mortality that
results from endogenous causes, from exogenous causes, and from their interaction.
This review examines, from the human biological perspective, several biologically interpretable mathematical models of human mortality. In particular this
paper will 1) briefly review the physiological basis of mortality and theories of
aging that have provided the basis for most bio-mathematical models of mortality,
2) present simple descriptions of the models of mortality themselves, and 3) examine the implications of several applications of these models to the study of human
biology. A review of this length cannot be comprehensive. Not covered are the
statistical methods of applying these bio-mathematical models to data (see Lee,
1980; Allison, 1984; Gage, 1988a) or the methods used in epidemiology and demography that provide the necessary data (see Pollard et al., 1974; Lee, 1980;
Leslie and Gage, 1989).
Ideally, biologically interpretable models of mortality should incorporate the
pan-specific endogenous factors that influence mortality and account for individual variability if it is present, so that the effects of human variation in exogenous
factors, endogenous factors, and the interactions of these two factors, can be assessed at the population level. In this section the pan-specific characteristics of the
age patterns of human mortality will be presented along with a brief discussion of
three endogenous aspects of mortality, maturation, aging, and individual variability. This information will form the basis for evaluating the bio-mathematical models presented in the second section.
The human mortality curve
The pan-specific characteristics of the human age pattern of mortality, variously
called the hazard rate, force of mortality, or instantaneous death rate, is presented
in Figure 1 (Mildvan and Strehler, 1960; Brown and Forbes, 1974a,b; Economos,
1982).In fact, these same features are characteristic of most mammalian mortality
curves, provided that the differences in longevity are not considered (Caughly,
1977). The classic mortality curve declines rapidly and continuously throughout
the infant and childhood ages, remains relatively constant through the adolescent
and early adult years of life, and then begins to increase at an ever-increasing rate
during later life. At the oldest ages, the increase in mortality appears to slow and
may even level off. This last phase of mortality occurs quite late in life among
westernized human populations (90-95 years of age); however, it may account for
as much as 70% of the life span of some other organisms (Economos, 1979). Assessment of whether this slowing of the increase in mortality is a truly pan-specific
feature of human mortality will require data on a wider range of elderly popula-
Pig. 1. Graphical depiction of the age patterns of human mortality.
tions than has currently been examined. A final characteristic of some human
mortality curves is a hump that occurs during the early adult phase of life. However, statistical analysis suggests that this is not a consistent or pan-specific characteristic of human life tables (Gage and Dyke, 1986).
The biological process of aging, that is, the decline and senescence of individuals
as they approach the maximum human life span, has been examined at molecular,
cellular, and organismic levels (Warner et al., 1987). Nevertheless the process of
aging is not yet well understood. At least nine scientific theories or mechanisms of
aging have been proposed (Schneider, 19871, beginning with the waste product
theory (Carrel1 and Ebeling, 1923) and the wear-and-tear theory (Pearl, 1924).
While many theories of aging have been largely discredited, others still remain as
potential explanations of the process of senescence. In fact, the process of aging
may be the result of several mechanisms acting simultaneously, possibly on different levels (Economos, 1982; Schneider, 1987). For example, a multi-stage process, probably involving DNA damage and repair, operates a t the molecular and
cellular levels and may be responsible for the increase in cancer mortality with
age. A second process, operating at the organismic level, may contribute to the
generalized loss of physiological function or vitality with age leading to a generalized increase in mortality with age. This latter process can be explained by any
or all of the theories of aging. Different bio-mathematical models of aging have
been developed to describe each of these potential mechanisms of aging. The model
of cellular aging will be referred to as the multi-stage model, while the model of
aging a t the organismic level will be called the physiological model. The basic
biological rationale for each model is presented below.
Current theory indicates that at least some forms of carcinogenesis are a multistage, probably two step, process. This theory is based on 1)the observation that
many cancers are genetic in origin, and 2) histological examination of cancerous
and precancerous tissues. The discovery of oncogenes and familial risk factors for
cancers, such as polyposis coli (colon cancer) and xeroderma pigmentosum (skin
cancer), have demonstrated the involvement of genes with oncogenesis. The identification of abnormal hyperplastic lesions, or benign tumors, as the precursors of
[Vol. 32, 1989
0 HeartRate
A MaxV02
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Fig. 2. The decline with age of several physiological characteristics possibly associated with decreased
resistance to death, plotted as a percent of maximum value. Maximum heart rate and maximum oxygen
consumption from Dill et al. (1958), longitudinal data. Cell renewal rate from Grove and Kligman
(19833, cross-sectional data. Creatinine clearance from Shock (1985),longitudinal data fitted with piecewise linear regression.
cancerous cells indicates that carcinogenesis is a multi-stage process. Based upon
these findings, it is hypothesized that carcinogenesis results from two independent
mutational events possibly occurring at the same locus on two homologous chromosomes. The development of cancers, such as colon cancer, in individuals without
a family history of these cancers is thought to occur in the following manner: In the
first stage, a normal stem cell divides into two daughter cells, one of which is
normal while the other has sustained a mutation, at some small probability. The
mutated daughter cell could die, or could divide and give rise to two mutated
daughter cells, or could, at some small probability, give rise to one mutated daughter cell and one doubly mutated daughter cell sustaining a second mutation at the
same loci as the first but on the homologous chromosome. If cancer is a two-stage
process, then this doubly mutated cell loses its regulatory capacity and grows out
of control, giving rise to a malignant tumor (Moolgavkar and Knudson, 1981;
Kundson, 1987). The development of colon cancer in individuals with familial
polyposis coli is thought to require only the second mutational event, under the
assumption that these individuals inherit cells that have already experienced the
first mutational event.
In contrast, the physiological models of mortality assume that there is a general
decline in vitality or physiological capacity with age, causing a corresponding
increase in mortality. The mathematics of the models depend heavily on the nature
of the decline in physiological capacity with age. If the rate of decline in physiological capacity corresponds proportionally to the increase in mortality, then a
simple model may apply. However, if physiological vitality does not decline a t the
same rate that mortality increases, a more complex model is necessary. A comparison of the decline in physiological capacity of several systems is shown in Figure
2. The problem is that some physiological functions appear to decline more slowly
with advancing age, others may decline linearly with respect to age, that is a t a
constant rate, and still others decline at an accelerating rate with respect to age.
Only a few years ago, it was generally agreed that physiological capacity declines
linearly with age (Simms, 1946; Shock, 1960, 1974; Strehler, 1977). It is now
thought (Economos, 1982; Shock, 1985) that physiologic capacity probably declines
at an accelerating rate as age increases, similar to the rate of increase in mortality
between the ages of 30 and 80 years (Fig. 1).Economos and Shock argue that
mortality may be operating differentially on individuals with poor physiological
function biasing the rate of decline in these traits and making them appear to
decline more slowly than the true decline in physiological capacity.
Biological theories of the decline in infant and childhood mortality during maturation have not received as much attention as the biological basis of aging. Perhaps this is because infant and childhood mortality consists of a larger variety of
causes of death than senescent mortality and hence has been considered to be a
more complex biological process. Bourgeous-Pichat (1951)divided infant and childhood mortality into two types, those deaths caused primarily by endogenous
causes, such as the inborn errors of metabolism, and those caused predominately
by exogenous causes, that is the infectious diseases. Weiss (personal communication) has suggested a second class of endogenous childhood diseases, which he
terms the “generative diseases.” The generative diseases consist of the childhood
forms of the degenerative diseases, such as the childhood neoplasms, etc., which
appear to result from the ontogenic processes of maturation rather than the degenerative process of aging. Regardless of the details of the classification of these
diseases, most studies of mortality during early life have concentrated on identifying the environmental covariates of infectious disease mortality, since it is the
infectious diseases that account for the majority of mortality during this period of
development (see, for example, Trussell and Richards, 1985). These analyses generally ignore any of the endogenous factors that influence the rate of decline in
infectious disease mortality with age. As a result, there are no biologically based
mathematical models of the age patterns of mortality to describe mortality during
the first few years of life. Nevertheless, the decline in mortality during early life
(Fig. 1)is universally concave, suggesting that some underlying biological process
or processes determine the general pattern of the decline.
While it is not within the scope of this paper to develop a comprehensive biomathematical model of the decline in all the aspects of mortality during infancy
and childhood, a reasonable place to begin, with respect to infectious disease mortality, is with the ontogeny of the immune system. The typical age trends in the
development of cellular immunity (the lymphatic system) and humeral immunity
(IgG) of normal healthy children are shown in Figure 3. The rate of increase in
both cellular and humeral immunological competence declines with age during the
first few years of life. This is qualitatively similar to the decline in the rate of
decrease in mortality with age during the first few years of life (Fig. 1). Although
there is evidence for declining IgG, as well as other immunoglobulins a t the older
ages, there is no strong evidence of an increased susceptibility to disease. Apparently, the idiotypic network is capable of maintaining the functional aspect of the
immune system in combating infection well into old age (Siskind, 1987). While the
decline in the rate of increase in immunological competence with age appears, at
least qualitatively, similar t o the decrease in mortality during the first few years
of life, the exposure to infectious agents and the exogenous factors affecting immunological competence, such as breastfeeding and nutrition, must also be considered.
A complete model of the decline in mortality during infancy and childhood will
require combining the model of infectious disease mortality with models of the
inborn errors of metabolism and “generative” types of mortality.
Most bio-mathematical models of mortality assume that after accounting for age
and sex, a population is biologically homogeneous. Individuals of the same age,
however, are not identical (Mourant, 1983; Overfield, 1985). They may differ as a
result of any number of genetic, ontogenic and behavioral risk factors, such as
genetic polymorphisms, birthweight, and smoking habits. If heterogeneity is
present, but not controlled for, serious biases in the estimates of mortality can
[Vol. 32, 1989
Fig. 3. The development of the human immune system plotted as the percent of adult values. Lymphoid
includes growth of the thymus, lymph nodes, and intestinal lymph masses, from Tanner (1962); IgG data
from Oxelius (1979).
occur as a result of mortality selection on physiological capacity. The definition of
mortality selection differs slightly from the genetic definition of selection. Mortality selection incorporates not only genetic selection but generic selection on ontogenic, physiologic, and behavioral attributes, which are not necessarily genetic,
but which are risk factors for mortality. To minimize confusion between these two
concepts of selection, they will be referred to as genetic selection and mortality
selection in the discussion below. In any event, the process of mortality selection on
physiological capacity can cause biases in the estimation and interpretation of
mortality differences between populations in much the same way that comparing
crude birth rates can cause misinterpretations when the age structure of the populations are not considered. If the cause of the heterogeneity is observed, then
simple corrective measures can be taken to control for it. However, much heterogeneity is effectively unobservable. Consequently, correction for the effects of heterogeneity is a n important consideration in any comparative study of mortality.
Control of unobservable heterogeneity requires a model that accounts for the
effects of mortality on the average physiological capacity of the survivors. To model
the consequences of heterogeneity effectively, it is first necessary to understand
how the variability in frailty, which is the inverse of physiological capacity and a n
individual’s relative susceptibility to death, is distributed among individuals. The
simplest model is one in which individuals are endowed with a level of frailty a t
birth that remains constant relative to other individuals throughout life (Vaupel et
al., 1979). This is called the fixed frailty model. Two important assumptions underlie this model: 1) frailty is controlled by a single biological trait, and 2 ) increased frailty with respect to one cause of death simultaneously increases a n
individual’s risk to all other causes of death and vice versa. Current biological
evidence suggests that neither of these assumptions is likely to be correct.
In general, the many known genetic and biological risk factors are usually associated with specific causes of death. This suggests that many biological characteristics are encompassed within the concept of frailty. There are a few notable
exceptions. Smoking, a behavioral risk factor, appears to be associated with a wide
variety of degenerative diseases, including heart disease, lung cancer, emphysema
and others (US.Printing Office, 1964). Another possible exception may be the
HLA B8 allele, which characteristically declines in frequency as a cohort ages,
probably as a result of generalized genetic selection against the HLA A l B8 haplotype. Individuals with this haplotype appear to be frailer than average under
most circumstances, except to breast cancer, a condition to which these individuals
apparently have increased resistance (Williams and Yunis, 1978). This type of
trade off, increased frailty with respect to one disease and decreased frailty with
respect to another, however, may be a more general characteristic of genetic polymorphisms, so that frailty with respect to different causes of death may be negatively rather than positively correlated as assumed by the fixed frailty model.
Another example, the A blood group of the ABO system is thought to be associated
with increased risk of bacterial infection and carcinoma, while the 0 blood group
is associated with increased risk to viral infection, ulcers, and autoimmune diseases (Mourant, 1983).
Theories concerning the maintenance of allelic polymorphisms and the evolution
of senescence also suggest that frailty is not likely to be positively correlated across
causes of death or across the life span. First, genetic traits that increase generalized frailty prior to the end of the reproductive period should rapidly be eliminated
by evolution, reducing the heterogeneity in future generations and the effects of
heterogeneity on the age pattern of mortality, at least until the end of the reproductive period. Second, stable polymorphisms that are suspected risk factors and
presumably selected against genetically, such as the A blood type, must be maintained by some opposing force of genetic selection, possibly through heterosis,
variations in selection across the life cycle, or environmental heterogeneity (Hartl,
1980). Consequently, if variability in frailty is a function of genetic heterogeneity,
then it is possible that the effects of selection are approximately balanced, at least
until the end of the reproductive period.
Finally, experimental evidence on mice suggests that genetically and environmentally induced heterogeneity is incompatible with the concept of fixed frailty.
Charlesworth (1980) has argued that senescence may result from negative
pleiotropic effects of genes that are advantageous at an earlier age. If this is true,
and there is some experimental evidence that it is (Rose and Charlesworth, 1981),
then the individuals who tend to be frailer a t the younger ages but nevertheless
survive are likely to be more resistant at the older ages. Even environmentally
induced heterogeneity with respect to longevity, such as differential nutrition,
might produce opposing forces of mortality selection a t the various ages. It is well
known that in mice, dietary restriction delays the ontogeny of the immune response capacity, prolonging the life span at upper ages but reducing the resistance
to infectious disease at the younger ages (Walford et al., 1974). It is not clear if
these biological mechanisms are directly applicable to humans (Beall, 1987), although it is generally believed that for humans, undernutrition in children is
positively associated with deaths owing to the infectious diseases, and that overnutrition in adults is a risk factor for a variety of degenerative diseases.
In general, frailty is apt to consist of a number of independent biological traits
and is unlikely to be a fixed trait either with respect to different causes of death or
with respect to the life span. If frailty is maintained by a balance of two or more
forces of genetic selection, as must be the case with stable polymorphisms that are
known to be risk factors, then the forces of mortality selection could also compensate for each other and the age pattern of mortality might be largely unaffected.
There will still be effects on the age pattern of mortality if the balancing forces
operate in different directions at different ages. However, the overall impact upon
the age patterns of mortality is likely to be less extreme when the forces of mortality selection are balanced than when the forces of mortality selection operate in
a single direction. Additionally, if pleiotropic effects are responsible for senescence,
and if there is biological heterogeneity with regard to these traits, then frailty will
[Vol. 32, 1989
vary across the life span. All of these findings tend to question the utility of the
fixed frailty model. On the other hand, the concept of fixed one-dimensional frailty
might be a realistic assumption with regard to a particular cause of death or small
group of associated causes of death. For example, there is growing evidence that
many of the neoplasms share a common genetic etiology (Weiss and Chakraborty,
Ideally, mathematical models of mortality should be based upon biological principles. Unfortunately this is not always the case. Some mortality models are arbitrary statistical distributions that appear to describe certain data sets well or
have convenient mathematical properties but that are not necessarily based upon
biological principles. In fact, most mathematical models of mortality are somewhere in between the ideal and the arbitrary. The models presented below attempt
to incorporate biological principles into the study of mortality patterns. They include the models of aging, maturation, mortality selection, and a framework for
studying the physiological processes that influence mortality.
Models of aging
There have been many attempts to develop bio-mathematical models of mortality caused by aging. Some of these models have been discarded, along with the
theories of aging from which the models were derived. The models still used today
can be divided into multi-stage models of carcinogenesis, which are based on cellular processes, and physiological models, which are founded on the generalized
physiological decline in function with age, defined at the organismic level. The
physiological models can be further divided according to whether they assume that
the decline in physiological capacity is linear or is at an accelerating rate with
respect to age (Fig. 2).
The mathematical model of carcinogenesis presented here is a simple generic
multi-stage model used by Weiss and Chakraborty (1984). In this model it is
assumed that the mutations from one stage to the next occur a t a constant rate a.
In addition, it is assumed that the time from tumor formation to death is relatively
short, so that it is not necessary to include a term to describe the period of time
from onset of cancer t o death. If these assumptions are realistic, then it can easily
be shown that the hazard rate due carcinogenesis is
where h, is the hazard rate at age t, and b is the number of stages (Weiss and
Chakraborty, 1984). Instead of examining this form of the model, an approximation of the model that can be fit using linear regression has been employed in
practice (Weiss and Chakraborty, 1984). The age-specific hazard rate of the exact
model for various arbitrarily chosen values of b is shown in Figure 4.The hazard
rate owing to this model increases rapidly during adulthood, intended to represent
the increase in mortality with aging, but then begins to level off, similar to the
decline in mortality at the oldest ages shown in Figure 1. The maximum hazard
rate and rate of increase in the hazard are a function of a, the rate of somatic
mutation, while the age at which the hazard increases rapidly is controlled by b,
the number of mutations necessary. Whittemore and Keller (1978) and Whittemore (1978) have reviewed in detail the mathematical models of carcinogenesis,
including more complex models than the one described above.
The physiological models of aging have been reviewed by Strehler (1977) and
Economos (1982). One of the simplest physiological models of aging is that proposed by Gompertz (1825)in which he assumed that physiological capacity declines
with age as a negative exponential, similar to the decline in maximum heart rate
and oxygen consumption shown in Figure 2, and that the risk of death is propor-
Fig. 4. Graph of the hazard function of the multi-stage model of Weiss and Chakraborty (19841, at
various levels of a (the probability of a transition) and b (the number of stages).
tional to the amount of physiological capacity remaining to the individual. The
resulting model, commonly called the Gompertz equation, adequately fits human
mortality curves between the ages of 30 and 80 years. The hazard function of the
Gompertz model is
ht = a,ebst
where h, is the hazard rate at age t, a, is the initial rate of mortality due to aging,
and b, is the rate of increase in the initial rate of mortality with age (Brody, 1923;
Failla, 1960). The age pattern of mortality characteristic of the Gompertz model is
shown in Figure 5a. Two characteristics of the Gompertz model that may not be
realistic are: 1) the hazard rate, which continues to increase at an ever increasing
rate with age, and 2) the physiological capacity (Fig. 2), which does not appear to
decline as a negative exponential with age.
A second model (Simms, 1946; Jones, 1956) assumes that physiological capacity
declines owing to the accumulation of damage, and mortality is directly proportional to the amount of damage accrued. In this model, the rate of accumulation of
damage is assumed to be proportional t o the amount of damage that has already
been accumulated, so that the rate of decline in physiological capacity accelerates
with age. This theory also reduces to the Gompertz equation (Strehler, 1977).
While the Simms-Jones model is currently acceptable, since the physiological evidence appears to be consistent with an acceleration in the decline in physiological
capacity, it was rejected during the late 1960s and 1970s because at the time it was
generally believed that physiological capacity declined linearly with age (Economos, 1982; Shock, 1985).
During the 1960s and 1970s, a number of models were developed that reduced
exactly or approximately to the Gompertz equation, but that assumed a linear
decline in physiological capacity (Sacher and Trucco, 1962; Strehler and Mildvan,
1960; Brown and Forbes, 1974a, b; Economos and Miquel, 1978). To achieve Gompertzian mortality, that is exponentially increasing mortality when physiological
capacity declines linearly, these models had t o 1) assume that there is a fixed
physiological threshold below which death occurs, and 2) assume that either there
[Vol. 32, 1989
Fig. 5. Graphs of the hazard functions of the physiological models of mortality. a: The Gompertz model.
b The Gompertz-Makeham model. c: The negative Gompertz. d: The Siler model.
are random stochastic deviations around the linear decline in physiological function caused by environmental perturbations or that there is a distribution of physiological capacity at each age, that is, that individuals are heterogeneous. These
more complex models are of interest because not only do they predict an exponential increase in mortality between the ages of 30 and 80 years, they also predict
Figs. 5c, 5d
that the force of mortality will tend to level off at the oldest ages (Fig. 11, once the
average level of vitality of the cohort has fallen below the critical limit (Economos,
Currently, the multi-stage, Simms-Jones, and linear decline models can all be
justified from the available genetic and physiological data, a t least with respect to
certain causes of death. As mentioned above, more than one process may be oper-
[Vol. 32, 1989
ating simultaneously, suggesting that the process of aging may be a convolution of
two or more of these models. There is ample evidence that the multi-stage theories
describe the underlying process of carcinogenesis well, at least at the cellular level
(Weiss and Chakraborty, 1984). However, the physiological models are likely to be
more realistic with respect to other causes of death that are associated with the
process of aging (Strehler, 1977; Economos, 1982). The multi-stage and some of the
linear decline models have the advantage that they predict the slowing of the rate
of increase in mortality at the oldest ages. The Simms-Jones model has the advantage of simplicity. In any event, except for the multi-stage model all of the
remaining models can be reduced exactly or approximately to the Gompertz equation over the range of 30 to perhaps 80 years of age.
An improvement in the Gompertz mortality model was proposed by Makeham
(1860). He found that adding a second age-independent mortality hazard to the
Gompertz model improved the empirical goodness of fit of the model to human
mortality patterns between 30 and 80 years of age. The Gompertz-Makeham model
ht = a, + a,ebst
where a, is the new component of mortality, and the remaining term is the Gompertz model described in equation 2. Strictly speaking, the Makeham component
(a,) is not a model of aging. The constant hazard rate indicates that this component
of mortality operates independently of the physiological state of the individual and
is completely exogenous. The Makeham component is frequently interpreted as
accounting for accidental deaths andlor conditions that are so debilitating that
death would occur regardless of the physiological state of the individual, while the
Gompertz component of the model is considered to represent the effects of aging
and senescence. By adding his component to the Gompertz model, Makeham used
Bernoulli’s concept of competing hazards or causes of death (David and Moeschberger, 1978). Hazards are considered t o be competing when there are multiple
causes of death and the causes of death are independent of each other. Independence in this case means that individuals saved from one cause of death are subject
to the remaining causes of death in just the same manner as everyone else in the
population. This implies that there is no positive or negative correlation of an
individual’s frailty with respect to different causes of death or components of mortality. The age-specific mortality rates typical of the Gompertz-Makeham mortality model are shown in Figure 5b.
Models of maturation
No bio-mathematical models have been developed specifically to represent the
decline in mortality during maturation. Following the general strategy of researchers studying aging, an initial model of maturation might be based on the
assumption that mortality is proportional to the likelihood of infection and inversely proportional to the level of immune competence. As mentioned previously,
the distributions of the likelihood of infection can be obtained from the epidemic
models of Anderson (McLean, 1986). However, no mathematical function has been
developed to described the ontogeny of immunological competence as presented in
Figure 2. Empirical work has shown that this component of mortality may be
adequately represented by a Weibull or by a negative Gompertz distribution (Siler,
1979; Mode and Busby, 1982; Mode and Jacobson, 1984; Trussell and Richards,
1985; Gage and Dyke, 1986). The hazard function of the negative Gompertz is
where aiis the initial rate of mortality caused by this component of mortality, and
b, is the rate a t which this component declines with increasing age. The agespecific mortality rates typical of the negative Gompertz are shown in Figure 5c.
Models of the entire lifespan
The simplest competing hazard model of mortality intended to represent the
entire life span consists of the Gompertz-Makeham model representing adult mortality added to a negative Gompertz component representing infant and childhood
mortality (Siler, 1979). Since all three components are additive, they are considered to be competing, but non-interacting causes of death. The hazard function of
Siler’s model is
h, = aiePbLt
+ a, + usebst
where the parameters are defined in equations 2-4. The hazard function of the
Siler model is shown in Figure 5d. This model has been successfully applied to
human populations by Siler (1983), and Gage and Dyke (1986) and to primate
populations by Gage and Dyke (1989). Several more complicated competing hazard
models, have also been proposed (Theile, 1871; Mode and Busby, 1982; Mode and
Jacobson, 1984). All of these models contain a fourth component of mortality intended to represent the hump seen in the mortality curve of some human life
tables. Tests of the Siler model on numerous national life tables, however, indicate
that this hump is not a consistent feature of human mortality curves (Gage and
Dyke, 1986). None of these models predict the slowing of the rate of increase of
mortality a t the oldest ages that was shown in Figure 1.
A simple model of frailty
With the exception of the bio-mathematical model of aging by Economos and
Miquel(1978), all of the other biological theories of aging assume that individuals
are homogeneous. As discussed previously, this over-simplification could bias the
estimated age-specific mortality curve. When biologically, environmentally, or behaviorally induced heterogeneity can be observed or measured, such as is the case
with hypertension, the effects of heterogeneity on the trajectory of human mortality can be accounted for by either classifying the population into homogeneous
groups before analysis or by using a method that incorporates the heterogeneity as
a covariate, such as in the accelerated lifetime or proportional hazards models
(Lee, 1980). However, unobservable heterogeneity cannot be controlled for in this
manner and requires a model of heterogeneity. The simplest and most frequently
applied model for correcting for heterogeneity will be described here. More complex
models of heterogeneity have been developed by Heckman and Singer (1982), Manton and Stallard (19841, and Yashin et al. (1985). A few of these more complex
models have been applied to mortality data (Trussell and Richards, 1985; Manton
et al., 1986).
The simplest model of unobservable heterogeneity is a fixed frailty model in
which the level of frailty is assumed to produce a constant proportional hazard
across the life span (Vaupel et al., 1979).The proportional hazards assumption can
be written as
h, = h i - 2 ,
where h, is the observed mortality hazard, hi is the “true” mortality hazard for a
homogeneous population with a constant level of frailty, and z, is the average level
of frailty in the observed population a t age t (Vaupel et al., 1979). In practice it is
convenient to use the ratio of absolute frailty of the cohort at age t to the absolute
frailty of the cohort at birth so that zo is equal to 1.0, and h’, is the hazard rate for
a population with constant level of frailty equal to 1.0. Additionally, frailty is
measured so that an individual with a frailty of 2.0 is twice as likely to die as an
individual with a frailty of 1.0. If the zt are observable, then this is simply the
proportional hazard model when a non-parametric underlying hazard is assumed
and an accelerated lifetime model when a parametric underlying hazard model
such as the Gompertz-Makeham model is used. However, if heterogeneity is un-
[Vol. 32, 1989
observable, a model of how mortality selection modifies the average level of frailty
with age is also necessary.
The model of fixed gamma distributed frailty assumes that an individual’s level
of frailty is fixed at birth and that this level is gamma distributed among individuals. The gamma distribution is a convolution of several exponential distributions
with a probability density function of
where yt and k are the level and shape parameters, respectively and r ( k ) is the
gamma function. The gamma distribution was chosen for this purpose because
when the gamma is used, mortality selection does not affect the shape of the
distribution of frailty with age, and hence the effect on the age patterns of mortality is relatively simple. The age trajectory of frailty is
and equation 6 becomes
where s, is the proportion of individuals surviving to age t (Vaupel et al., 1979).
Bio-mathematical models of the age patterns of mortality can be incorporated into
this fixed gamma distributed frailty model simply by replacing h’, in equation 9
with h, of the Gompertz-Makeham, or one of the other models presented above. For
example the Siler model for a heterogeneous population becomes
where the survivorship function, s, is
(Siler, 1979; Gage and Dyke, 1986). The influence of mortality selection on the
observed age pattern of mortality for various values of k is shown in Figure 6.
This model, k, or more precisely, 2k, represents the dimensionality of heterogeneity (Manton and Stallard, 1981). A value of k = 0.5 suggests that frailty is
defined by a single dimension, while k = 1.0 suggests that two dimensions are
involved in frailty, and so on. Biologically, uni-dimensional frailty implies that
individuals that are frail with respect to one cause of death are also frail with
respect to all other causes. Multi-dimensional frailty, on the other hand, suggests
that individuals that are frail with respect to one cause of death are not necessarily
frail with respect to other causes, that is, the biological causes of frailty are independent and specific to one or more (but not all) causes of death. The results
presented in Figure 6 show that mortality selection has the largest effect if frailty
is uni-dimensional and very little effect if as many as 36 independent biological
traits are involved. The Siler model combined with the fixed gamma distributed
frailty model predicts that the rate of increase in mortality slows or even declines
at the oldest ages if frailty is a single biological trait. The theoretical discussion of
the biology underlying frailty suggests that frailty may consist of many independent traits, in which case frailty will have only minor effects on the age patterns
of mortality. The empirical work discussed below, however, suggest that frailty
may consist of a small number of dimensions there by having a significant effect on
human age patterns of mortality.
A framework for studying biological aspects of mortality
While the primary focus of the bio-mathematical models presented above is to
describe a biological process, aging, maturation, or mortality selection, the sto-
Fig. 6. Graph of the Siler hazard functions corrected for heterogeneity using the fixed gamma distributed frailty model a t various levels of k, where 2k represents the number of dimensions of frailty.
chastic compartment model of mortality proposed by Manton and Stallard (1984) is
a useful framework for studying the associations of mortality with other physiological characteristics of an individual. The simplest stochastic compartment
model of mortality is the two-compartment system shown in Figure 7a. This flow
diagram has an alive state and a dead state, where the transition rates between
the states, h,, are the hazard column of a life table. These hazard rates could be
measured using either a non-parametric procedure, an empirical life table for
example, or a parametric mortality model such as the Gompertz-Makeham or Siler
However, this simplistic compartment model is of little other than heuristic
interest. The value of stochastic compartment models is their use in explicitly
placing the mortality process within the context of other characteristics. An example of a model that incorporates diabetes as a morbid characteristic is shown in
Figure 7b. Here there are four states: well, diabetic, death due to diabetes, and
death due to other causes, with four sets of transition rates between them. In this
example, transition rates out of the morbid state depend upon the individual’s age
and current state. If the morbid state is observable, then the transition rates can
be measured directly from individuals in each state using the proportional hazards
or accelerated lifetime models depending upon whether a non-parametric or parametric hazard function, such as the Siler model, is used. If the morbid state is
unobserved then a frailty model must be incorporated.
These stochastic compartment models can be extended to represent more complex situations where there is interdependence among causes of death (for example, the association of death owing to heart disease and death owing to diabetes in
patients with diabetes). This type of model is a considerable improvement over
studying causes of death using the concept of competing risks, since the competing
risk models assume that causes of death are independent. The stochastic compartment models are an important innovation because they set the stage for studying
the associations between morbidity and mortality. In some respects, however,
these compartment models are similar to the multi-stage carcinogenesis model
described previously, which postulates a morbid state, a first mutation event,
followed by a second mutation event that rapidly leads to death.
[Vol. 32, 1989
Fig. 7. Digraphs of two stochastic compartment models. a: A simple two-state model. b A more complex
model incorporating a disease state. (Adapted from Manton and Stallard, 1984.)
Strategies of data analysis
While it is not within the scope of this paper t o describe the methods used to
apply the various models to actual data sets, brief descriptions of the two major
strategies currently used are presented below. The first strategy involves fitting
the models to standard life tables rather than to the raw mortality data. The
advantage of this approach is that censored and truncated data can be accounted
for during life table construction. The models can then be fitted to the life table
using a standard non-linear fitting package, such as that in SYSTAT or SAS, and
either least squares or maximum likelihood techniques (Gage and Dyke, 1986;
Gage, 1988a). The multi-stage, Gompertz, Gompertz-Makeham, and Siler models
can all be fitted using this approach without further mathematical derivations.
The combination Siler-fixed gamma distributed frailty model can in theory also be
fitted, although this has not yet been attempted.
The second strategy fits these models directly to longitudinal data collected on
individuals without first computing a life table. This is the strategy commonly
employed to fit the proportional hazards and accelerated lifetime models (BMDP,
SPSS, SAS, SYSTAT). This approach requires that the likelihood function be derived for each model with and without censoring and/or truncation. While likelihood functions for some applications are available in the literature, others are not.
The advantage of this approach is that the covariates of mortality can be studied
using smaller data sets than is possible using the life table method. Packaged
procedures are available for fitting a variety of biologically arbitrary statistical
functions, such as the exponential, Weibull, log-normal, and log-logistic, developed
by statisticians for reliability analysis. The biologically derived models are not yet
available in any of the common statistical packages.
The stochastic compartment models with observed states can be fitted using the
proportional hazards model or accelerated lifetime models, running a separate
model for each transition and using appropriate censoring. This process has been
described by Allison (1984). A second program, RATE, distributed by %ma1 can fit
stochastic compartment models directly.
This section presents three applications of the bio-mathematical models of general interest to anthropologists. These include the use of these models to 1) test the
biological assumptions that underlie them, 2) help interpret differences in mortality between populations, specifically mortality crossovers, and 3) statistically compare and classify the international variation in mortality patterns. This section is
not intended as a comprehensive listing of all of the work that has been done in this
area, even within anthropology. Discussion of topics such as the changes in level of
mortality, that is expectation of life, with respect to human biological and social
evolution are not considered here. This particular aspect of mortality has been
reviewed by Behm and Vallin (1982) for national populations and by Gage et al.
(1989) for anthropological populations.
Testing bio-mathematical models
Weiss and Chakraborty (1984) have tested the multi-stage model of carcinogenesis, equation 1,by comparing the fits of the model to the age incidence of colon
cancer in individuals having familial polyposis coli with the age incidence of colon
cancer among individuals without family histories of this disease. Familial polyposis coli is a disease in which a risk factor is inherited. Individuals with this trait
usually develop colon cancer during the second or third decade of life, considerably
earlier than the onset of colon cancer in the general population. One possible
explanation for this earlier onset of cancer is that individuals with familial polyposis coli inherit epithelial cells that have already undergone the first mutation
event or first few mutations of the multi-stage process that culminates with a
malignant tumor. The number of stages obtained from fitting the multi-stage
model to the onset of colon cancer in individuals without this trait should be
somewhat greater than in individuals with familial polyposis coli. Tests of this
kind that have been conducted by others (Ashley, 1969; Knudson, 1977) using the
linearized approximation of the multi-stage model have concluded that four to six
stages are involved in the process of carcinogenesis in normal individuals with two
to three fewer stages required for individuals with familial polyposis coli. Weiss
and Chakraborty (1984),however, obtain different results using the exact equation
for the multi-stage model and non-linear fitting methods.
They show that the exact model fits these data substantially better, and predicts
a greater number of stages necessary for development of colon cancer in the unaffected population than the approximate model: between 8 and 15 stages compared to 4 to 6 stages. More importantly, their study indicates that the exact model
estimates a larger number of stages, 13 to 40, for individuals with familial polyposis coli than for individuals without this disease. This is the reverse of the results
obtained with the approximate model and is inconsistent with the presumed genetic etiology of familial polyposis coli, where familial polyposis coli supposedly
represents a head start on the multi-stage process of carcinogenesis. Weiss and
Chakraborty (1984) conclude that despite the model's good fit to the data, it is
'FORTRAN Program RATE, available from DMA Corporation, P.O. Box 881, Palo Alto, CA 94302.
[Vol. 32, 1989
Fig. 8. Graphs of mortality and survivorship for a cohort that survived a dysentery epidemic (stressed
cohort) and a cohort that was not exposed to this epidemic in townships of Deerfield (a and b) and
Greenfield (c and d), Connecticut, showing the crossover in mortality and survivorship. Data from
Meindl and Swedlund (1977).
probably mis-specified and does not parameterize appropriately the biological process of carcinogenesis. Consequently, a new bio-mathematical model of the multistage biological process of carcinogenesis is required. Nevertheless, the simple
Figs. 8c, 8d
multi-stage model is empirically useful. Weiss and Chakraborty (1989)discuss how
the multi-stage model might be improved, based on results they obtain using the
simple multi-stage model on a variety of carcinomas in a variety of populations.
Mortality crossovers
A second application of these models concerns the detailed study of mortality
crossovers. A mortality crossover occurs when a disadvantaged cohort or popula-
[Vol. 32, 1989
tion has higher mortality early in life compared to an advantaged cohort or population, but the mortality rates of the disadvantaged population fall below those of
the advantaged population a t the older ages. Several explanations for mortality
crossovers have been postulated, including poor data, mortality selection, environmental effects, and underlying physiologicigenetic differences (Manton and Stallard, 1984; Coale and Kisker, 1986). Poor data may result in a mortality crossover
owing to differential underenumeration of individuals exposed to the risk of death,
differential underenumeration of the number of deaths, or differential misestimation of age, particularly at the older ages. Coale and Kisker (1986) have recently argued that all observed mortality crossovers are due to poor data.
The process of mortality selection could decrease the frailty of the survivors
producing a crossover in the hazard rate and a convergence, but not a crossover in
the survivorship curves. A crossover in survivorship would imply that above the
age where the crossover took place, stronger selection had operated on the originally advantaged population than on the originally disadvantaged population.
Environmental factors could also produce a mortality crossover, particularly if the
exposure t o various risks differed by age between the two populations. Finally,
mortality crossovers could reflect actual population differences in biologic, genetic,
or physiologic responses with age. To use mortality crossovers to study the interaction of biology and environment on mortality, it is important to first demonstrate
that the pattern is not simply a result of poor data. Two examples will be described
here: a mortality crossover of adjacent birth cohorts in historic Connecticut and the
black/white mortality crossover in the contemporary United States.
Meindl and Swedlund (1977) have reported a mortality crossover between
stressed and non-stressed cohorts of the white population residing in the Connecticut Valley in the late 16th and 17th centuries. The study is based on historical
birth and death registry data for individuals born between 1793 and 1814 and who
lived out their lives in the townships of Deerfield and Greenfield. Relatively severe
dysentery epidemics occurred in 1803 in Deerfield and in 1802 in Greenfield, and
consequently the period was divided into two birth cohorts, a stressed cohort born
between 1793 and 1803 representing individuals who were exposed between the
ages of birth and 10 years of age to these epidemics and an unstressed cohort born
between 1804 and 1814, who appear to have survived childhood under mortality
regimes more typical of the times. Given the relatively short time span between
the cohorts, it is a reasonable assumption that other factors associated with mortality including errors in vital registration were the same for both stressed and
non-stressed cohorts. While there is reason to believe that these data are not
completely accurate, there is no reason to believe that the data are better for one
cohort than the other. Thus, these historical data represent a relatively wellcontrolled natural experiment concerning the potential effects of an epidemic
on cohort survival. The cohort mortality and survivorship rates are shown in
Figure 8.
The mortality crossover in the hazard rate occurs by age 10 years in both townships, while mortality in the non-stressed cohort appears to be highest relative to
the stressed cohort during the childbearing years. The dynamics of survivorship
between the stressed and non-stressed cohorts differs from one township to another
partially as a result of the differences in severity of the epidemics. For Deerfield,
a crossover in survivorship occurs by 25 years of age, and survivorship of the
non-stressed cohort drops below that of the stressed cohort. For Greenfield the
survivorships appear to converge a t about 45 years of age, but do not display a
prominent crossover. Statistical examination of the survival distributions indicate
that, in general, mortality was higher in the non-stressed cohort than in the
stressed cohort after the age of 10 years (Meindl and Swedlund, 1977). The convergence of survivorship in Greenfield suggests that the differences in dynamics of
these two cohorts could be due to mortality selection. The crossover in survivorship
between the stressed and non-stressed Deerfield cohorts indicates that some additional mechanism is operating. Meindl and Swedlund suggest that the survivors of
the stressed cohort may have had enhanced immunological response as a result of
the dysentery epidemic. Reporting the causes of death associated with the mortality differential during the childbearing years might shed light on this issue. In any
event, the examination of the crossover has allowed Meindl and Swedlund to examine the effects of an environmentally induced event, dysentery, on the age
patterns of mortality and possibly to identify an interaction of an exogenous factor
with an endogenous factor (immunological competence).
The crossover of blacklwhite mortality has been observed in the United States,
at least since 1900 (Zelnick, 1969). However, since the differential between black/
white expectation of life is greater than that of the stressed and non-stressed
cohorts of Deerfield and Greenfield, the crossover takes place at a later age. Figure
9 contains the period estimates of mortality patterns in the United States for 1950.
In this case the crossover in the hazard rate occurs at the age of 75 years, while
survivorship converges but does not crossover, a t least by age 85 years. Cohortbased estimates of the crossover have been made by Manton and Stallard (1981,
1984) and by Wing et al. (1985) using the stochastic compartment model and
Gompertz, Gompertz-Makeham, and Weibull models.
The black/white crossover is more difficult to interpret than the crossover in
historical Connecticut because of differences in the quality of demographic data for
blacks and whites. The U.S. census tends to underenumerate the number of young
adult black males and the number of older adult black females. Additionally, there
appear t o be more problems with age misestimation in both census and registry
data among blacks than among whites (Rives, 1977). Researchers have attempted
to overcome this problem by 1) using data corrected for underreporting and then
estimating the amount of additional error that would be required to explain completely the black/white crossover (Kitagawa and Hauser, 1973; Manton and Stallard, 1981,1984), and 2) examining long-term epidemiological studies for evidence
of a black/white crossover, where error in the data is not likely (Wing et al., 1985).
Results of studies that have used these correction procedures or longitudinal data
all indicate that the blacWwhite mortality crossover is not completely an artifact
of bad data, as is argued by Coale and Kisker (1986).
Manton and Stallard (1981, 1984) have attempted to estimate the amount of
error that would be necessary to eliminate the blacklwhite crossover using the
parameter k of the fixed frailty gamma model as a measure. By definition 2k is the
number of biological traits involved in defining frailty. Their procedure consists of
empirically correcting the two life tables by various levels of k and comparing the
corrected life table for blacks with the corresponding life table for whites. The
results suggest that a relatively small value of k, between 0.5 and 1.0, is necessary
if mortality selection is the only cause of the blacWwhite crossover. The implication of this result is that if the entire mortality crossover is a result of mortality
selection, then frailty is a one- or two-dimensional characteristic. This result is at
odds with the biological theories discussed above. Either part of the crossover is
due to other environmental and/or biological causes, or the fixed frailty gamma
distributed model of heterogeneity is mis-specified. There is growing evidence that
there are physiological differences between blacks and whites that might differentially influence the age patterns of mortality (Overfield, 1985; Manton et al.,
1987; Andersen, 1987). However, it is not clear if these physiological differences
are genetically or environmentally induced. If mortality selection does explain any
of the blacWwhite crossover then the published life tables underestimate the black/
white mortality differentials in the United States.
International variation in mortality patterns
Model life tables were originally developed to facilitate life table construction
with insufficient or defective data. However, Coale et al. (1988) in developing their
model life tables identified regional variation in the age patterns of mortality
among European nations. Additional studies have extended these model life tables
to include the less-developed countries (United Nations, 1982; Clairin et al., 1980).
[Vol. 32, 1989
Fig. 9. Graphs of mortality (a) and survivorship (b) for blacWwhite mortality in the United States in
1950, showing the crossover in mortality and the convergence in survivorship. Cross-sectional data from
Preston et al. (1972).
In these studies, classification of life tables into the characteristic patterns of
mortality have depended upon a n original sample of types that was arrived at by
inspection (Coale et al., 1988). The Siler model, which translates the entire mortality curve into five parameters, makes it possible to classify life tables using the
methods of statistical taxonomy. Two applications of this technique are described
below: 1)the development of a new international system of model life tables and 2)
TABLE 1 . Description
Cluster No.
Cluster 1
Cluster 2
Cluster 3
Cluster 4
Cluster 5
Cluster 6
Cluster 7
international age patterns of mortality
Regional representation'
Central and South America, Eastern and
Southeastern Europe
Argentina, Chile, Canada, U.S. Greenland, France, Scotland,
West Germany, and Finland
Central African Countries
Sweden, Norway, Denmark, Japan, Taiwan, Ceylon,
Australia, and New Zealand
Southwestern Europe, Near East, and Eastern North Africa
Central America, Taiwan, Hong Kong, and Singapore
South Africa (white and black), Rhodesia, U S . , France,
and U.S.S.R.
Coale and Demeny
'Not all countries are necessarily listed.
q h e West pattern of mortality is equivalent to the average age pattern of mortality across all seven clusters
the comparison of two anthropological life tables with the international system of
model life tables.
Gage (1988a-c) has used the Siler model to document the extent of variation in
mortality patterns. A study of contemporary national populations was based on a
worldwide stratified sample of life tables dating from the period between 1945 and
1975. The life tables were fit to the Siler model (uncorrected for heterogeneity), and
the resulting parameter estimates were controlled for expectation of life and gender using multiple regression and classified into natural groupings using K-mean
cluster analysis (Gage, 1988b,c). A search for the optimum set of naturally occurring clusters resulted in a system of seven model life tables. The regional composition of these seven clusters and their relationships with the four regions of the
Coale et al. (1988) age patterns of mortality are presented in Table 1. The variation
in the age patterns of mortality among the clusters is shown in Figure 10.
Perhaps the most surprising result of this study is that post hoc statistical
analysis of the clusters using MANOVA indicates that adult mortality is a more
important factor than infant and childhood mortality in classifying life tables
based on the age patterns of mortality (Gage 1988b,c). The major differences in
adult mortality among contemporary nations falls into two characteristic patterns
when compared to the average pattern for the entire worldwide sample: 1) low
mid-adult mortality coupled with increasing mortality at the older ages (Fig. 10a)
and 2) high mid-adult mortality combined with declining mortality a t the older
ages (Fig. lob). The first pattern appears to be characteristic of European, Latin
American, North African, and some Asian populations. The second pattern is
characteristic of the remaining populations, including most of the European-derived populations residing outside of Europe (Table 1). The most extreme example
of the high mid-adult, but declining older age pattern of mortality is the cluster
containing all of the Central African nations (Cluster 3, Fig. lob). The remaining
African countries south of the Sahara display similar but less extreme examples of
the same pattern of mortality (Cluster 7, Fig. lob). What is particularly interesting about the differences between this African pattern and the average pattern of
mortality for the entire worldwide sample is that it is the same as the differences
in mortality patterns observed between blacks and whites in the United States as
described above. Whether these variations in national mortality patterns are the
result of bad data, mortality selection, or environmental and/or biological differences among populations remains largely unknown.
There are few comparative studies of mortality among the populations traditionally studied by anthropologists. Weiss's (1973)model life tables were developed
primarily as a tool for life table construction rather than a study of the human
variation in mortality among anthropological populations. However, Gage (1988a)
has studied three anthropological life tables using the Siler model to smooth and
[Vol. 32, 1989
Cluster 1
Cluster 4
Cluster 5
Fig. 10. Comparisons of the difference in q(t),the yearly probability of dying in the interval, among the
average mortality schedule for the entire worldwide sample and each of the clusters, while holding the
expectation of life constant. a contains the clusters with patterns similar to a Coale and Demeny (Coale
et al., 1988) model life table. b contains the clusters that have no analog in the Coale and Demeny
system. Adapted from Gage (1988b,c).
improve the results. The intent of this particular paper was methodological. Nevertheless, the age patterns of these three life tables can be compared with the seven
characteristic age patterns of mortality identified in national populations to determine how mortality in these populations corresponds to mortality among contemporary nations.
The anthropological life tables studied by Gage (1988a) include two paleo-pop-
ulations, Libben (Lovejoy et al., 1977) and Meinarti (Moore et al., 1975), and one
contemporary population, the Yanomama (Nee1 and Weiss, 1975). The parameter
estimates obtained by fitting the Siler model to these three life tables were used to
identify the cluster in the national model life tables that is most similar to each
anthropological life table based on the minimum multivariate distances. The two
paleo-life tables are very similar to each other, and both are closest to Cluster 5,
which corresponds most closely to the South regional pattern of the Coale et al.
(1988) model life tables. Comparisons of the age patterns of mortality of the Libben
life table with the average of the worldwide sample and the pattern of Cluster 5 are
presented in Figure l l a . In general, the age patterns of these paleo-populations
have lower infant mortality and much higher adult mortality than either Cluster
5 or the average of the worldwide sample a t similar expectations of life. While
Cluster 5 is the cluster most similar to the paleo-life tables, the age pattern of
mortality of these paleo-populations differs considerably from the age patterns of
any contemporary population. These discrepancies could represent errors in aging
skeletal materials and other defects in the data or could result from real environmental or biological differences between paleo-populations and contemporary populations.
The Yanomama life table, on the other hand, is most similar to Cluster 3, which
represents the Central African countries in the national system described above,
but has no analog in the Coale and Demeny model life tables. Comparisons of the
age patterns of Yanomama mortality with the worldwide average and Cluster 3
mortality patterns are presented in Figure l l b . In this case, the Yanomama age
pattern of mortality is reasonably similar t o Cluster 3 except that Yanomama
mortality is slightly lower during childhood and slightly higher a t the older ages.
Comparison of the Yanomama with the pattern of mortality for the two paleopopulations indicates that they differ more among themselves than they differ
from the seven worldwide age patterns of mortality. The multivariate distances
among all seven clusters and all three anthropological life tables indicate that the
paleo-life tables and Yanomama life tables are on opposite sides of the set of
worldwide model life tables. Once again, these differences could be the result of
either real environmental or biological differences between the contemporary and
paleo-populations or simply defects in the data.
Much work remains to be carried out before the endogenous and exogenous
factors influencing mortality are unraveled and a clear picture of the variation in
human mortality emerges. This review has attempted to describe the current state
of development of bio-mathematical models of human mortality, the strategies of
applying these models, some of the results, and the areas where research is most
needed. In this regard, the models described here are not always the most sophisticated available. Some of the models, such as the fixed frailty model, have been
theoretically generalized to age-dependent frailty models, and so on. References to
these more complex models have been included. However, the author is unaware of
any applications of these complex models to data. In fact, the necessary statistical
methods may not be available to apply some of the theoretical models. This review
has focused only on those models for which statistical methods of application are
The most obvious deficiency in bio-mathematical modeling of mortality is the
absence of a model concerning infant and childhood mortality. Mortality during
these ages is actively being researched by many workers using arbitrary underlying models, usually the negative Gompertz or the Weibull (Siler, 1979; Trussell
and Richards, 1985; Gage and Dyke, 1986). Given the predominance or infant
mortality that still exists in the less-developed countries today, development of a
bio-mathematical model of mortality for these ages deserves a high priority.
A second area that needs additional work concerns the biological basis of individual heterogeneity and the models of frailty. Vaupel, Manton, and their associ-
[Vol. 32, 1989
Cluster 3
Fig. 11. Comparisons of the difference in q(t),the yearly probability of dying in the interval between the
average mortality schedule for the entire worldwide sample and each of two anthropological life tables,
along with the closest cluster, while holding the expectation of life constant. a shows a comparison of
Cluster 5 with the Libben life table. b shows a comparison of Cluster 3 with the Yanomama life table.
Adapted from Gage (1988a-c).
ates (Vaupel et al., 1979; Manton and Stallard, 1984) have established the potential importance of mortality selection on the dynamics of the age pattern of
mortality. However, they have used a model (the fixed gamma distributed frailty
model), which maximizes the impact of frailty on the age patterns of mortality,
particularly when frailty is considered to be uni-dimensional, that is, a single
biological trait. These researchers have usually assumed that the dimensionality
of frailty is low 1 or 2 dimensions, because this is the level of frailty required to
eliminate convergence in mortality crossovers. This result, however, is based on
the assumption that all of the crossover is a result of mortality selection, a view
that seems unlikely (Manton et al., 1987). Additionally, there have been no attempts to independently estimate k. in a manner similar to the tests of the parameter b in the multi-stage model conducted by Weiss and Chakraborty (1984). The
only independent test of heterogeneity has been conducted by Trussell and Richards (1985) using a non-parametric distribution of frailty (Heckman and Singer,
1982) to analyze infant and childhood mortality. Trussell and Richards were unable to identify a bias due to mortality selection. Furthermore, the biological evidence suggests that the dimensionality of frailty may be much higher than the
dimensionality assumed by Vaupel, Manton, and their associates, which would
considerably reduce the potential impact of mortality selection on the age patterns
of human mortality. This is an important and as yet unresolved issue. If individual
heterogeneity does seriously impact upon the dynamics of mortality, then heterogeneity needs to be controlled for when comparing mortality among populations.
The comparison of the mortality crossover of stressed and non-stressed historical
cohorts was conducted before most of the models discussed in this paper were
available. It would be advantageous to reanalyze these data using the bio-mathematical models of frailty so that more sophisticated statistical analysis can be
carried out on the existence of the crossover, particularly with regard to survivorship. Nevertheless, this is one of the more interesting studies because it shows how
anthropological data can contribute to an understanding of the interaction of endogenous and exogenous factors, which ultimately control the age patterns of
human mortality.
The international comparison of mortality data presented in this work did not
attempt to control for unobserved heterogeneity. However, partial correction for
heterogeneity may have occurred by standardizing all of the fitted life tables by
expectation of life and by sex, provided that the effects of mortality selection are a
linear function of the parameters of the Siler model. This standardization was not
originally conducted to correct for unobserved heterogeneity, but to make it possible to compare the shape of the mortality curve among populations with different
expectations of life and between the sexes. Consequently, the patterns of mortality
presented in Figure 10 represent the observed variation among the populations,
and some of the differences observed may be due to mortality selection and not to
exogenous or endogenous differences in mortality factors among the populations.
Finally, more work needs t o be done concerning the age patterns of mortality of
anthropological populations. While the current results are intriguing they must be
verified by fitting the models to additional anthropological populations to determine if the current findings are representative of anthropological populations in
Preparation of this paper was supported by NIH grants HD22981 and HD25346,
which are gratefully acknowledged. I would like to thank Dr. Shelley Zansky for
her comments on the manuscript and Kathleen O’Connor and Elizabeth Bachman
for library research and for preparing the figures.
[Vol. 32, 1989
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