YEARBOOK OF PHYSICAL ANTHROPOLOGY 32:185-214 (1989) Bio-Mathematical Approaches to the Study of Human Variation in Mortality TIMOTHY B. GAGE Department of Anthropology, University at Albany, SUNY, Albany, New York 12222 KEY WORDS Age-specific mortality, Hazard modeling, Heterogeneity, Carcinogenesis, Mortality crossover ABSTRACT 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 0 1989 Alan R. Liss, Inc. 186 YEARBOOK OF PHYSICAL ANTHROPOLOGY [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 causes. 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). THE PHYSIOLOGICAL BASIS OF MORTALITY AND THE THEORIES OF AGING 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- Gage1 HUMAN VARIATION IN MORTALITY 187 &e 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). Aging 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 YEARBOOK OF PHYSICAL ANTHROPOLOGY 188 c [Vol. 32, 1989 0 HeartRate A MaxV02 0 Cell Renewal A Creatinine 0.50 ! I I I 20 40 60 80 Age 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 Gage1 HUMAN VARIATION IN MORTALITY 189 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. Maturation 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. Heterogeneity 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 YEARBOOK OF PHYSICAL ANTHROPOLOGY 190 [Vol. 32, 1989 2.0 1.5 1.0 0.5 0.0 0 10 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 Gagel HUMAN VARIATION IN MORTALJTY 191 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 192 YEARBOOK OF PHYSICAL ANTHROPOLOGY [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, 1989). THE BIO-MATHEMATICAL MODELS 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- HUMAN VARIATION IN MORTALITY Gage1 193 a=.l,b=2 a=.l,b=8 a=.&b=32 a=.05,b=2 a=.05,b=8 a=.05,b=32 0 40 Age 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 194 YEARBOOK OF PHYSICAL ANTHROPOLOGY [Vol. 32, 1989 a 0 a0 40 Age b 1 Age 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 HUMAN VARIATION IN MORTALITY Gagel 195 C 0 20 40 60 80 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, 1982). 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- 196 YEARBOOK OF PHYSICAL ANTHROPOLOGY [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 is ht = a, + a,ebst (3) 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. Gage1 HUMAN VARIATION IN MORTALITY 197 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 , 16) 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- 198 YEARBOOK OF PHYSICAL ANTHROPOLOGY [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- HUMAN VARIATION IN MORTALITY Gagel 199 h Y 3 0 20 40 60 eo Age 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 models. 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. 200 YEARBOOK OF PHYSICAL ANTHROPOLOGY [Vol. 32, 1989 a ht Well Death b 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 Gage1 HUMAN VARIATION IN MORTALITY 201 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. SELECTED APPLICATIONS OF THE MODELS 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. 202 YEARBOOK OF PHYSICAL ANTHROPOLOGY [Vol. 32, 1989 a stressed non-stressed b Age 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 Gage1 HUMAN VARIATION IN MORTALITY 203 C d Age 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- 204 YEARBOOK OF PHYSICAL ANTHROPOLOGY [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 Gage1 HUMAN VARIATION IN MORTALITY 205 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). YEARBOOK OF PHYSZCAL ANTHROPOLOGY 206 [Vol. 32, 1989 a white black 0 40 60 Age b --- 0 0 20 40 60 white black so Age 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) Gagel HUMAN VARIATION IN MORTALITY TABLE 1 . Description Cluster No. Cluster 1 Cluster 2 Cluster 3 Cluster 4 Cluster 5 Cluster 6 Cluster 7 of 207 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 equivalent2 East None None North South None None '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 YEARBOOK OF PHYSICAL ANTHROPOLOGY 208 [Vol. 32, 1989 a 0.015 I 0.005 Cluster 1 Cluster 4 Cluster 5 -0.005 -0.015 0 20 40 60 b -0.060 0 20 40 60 m Age 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- Gage1 HUMAN VARIATION IN MORTALITY 209 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. CONCLUDING COMMENTS 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 available. 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 210 0.20 - 0.15 - *.** 3' . 0.10- 0.05 *., f *. 0.05 0.00 --- -0.05 Cluster 3 Yanomama -0.10 -0.15 0 20 40 60 80 Age 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). Gage1 HUMAN VARIATION IN MORTALITY 211 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 general. ACKNOWLEDGMENTS Preparation of this paper was supported by NIH grants HD22981 and HD25346, which are gratefully acknowledged. 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