Journals of Gerontology: BIOLOGICAL SCIENCES Cite journal as: J Gerontol A Biol Sci Med Sci 2014 August;69(8):965–970 doi:10.1093/gerona/glt164
© The Author 2013. Published by Oxford University Press on behalf of The Gerontological Society of America. All rights reserved. For permissions, please e-mail: [email protected]. Advance Access publication October 22, 2013
How Long Must Humans Live? Bruce A. Carnes1 and T.M. Witten2 Reynolds Department of Geriatric Medicine, The University of Oklahoma Health Sciences Center. 2 Center for the Study of Biological Complexity, Virginia Commonwealth University, Richmond.
1
Address correspondence to Bruce A. Carnes, PhD, Reynolds Department of Geriatric Medicine, The University of Oklahoma Health Science, 1122 N.E. 13th Street, ORB 1200, Oklahoma City, OK 73117. Email: [email protected]
Key Words: Biological warranty period—Effective end of reproduction—Sex ratio—Mortality acceleration. Received June 5, 2013; Accepted August 22, 2013 Decision Editor: Rafael de Cabo, PhD
T
he key feature that distinguishes humans from the rest of the living world is not taxonomic, it is technological sophistication. Each new generation of humans is more technologically advanced than previous generations. Most of that technology has been used to buffer humans from the mortality risks posed by both the biotic and abiotic world. More recently, human innovation has turned inward to seek interventions that mitigate the mortality risks posed by the flaws and failures of our own biology. The successful reduction of extrinsic and intrinsic mortality risks has led to the unprecedented population growth and population aging that is responsible for the severe economic and health challenges threatening the social fabric of nations around the globe. People who live beyond age 100 are a source of fascination for both the general public and the scientific community. The pursuit of technologies that progressively extend the human life span consumes enormous amounts of scientific effort and money. This ongoing pursuit of longevity dividends raises serious questions. Will the longevity dividends be accompanied by an equal extension of health? Will they be shared equally or will the longevity disparities that already exist become even more disparate? (1) More importantly, is “How long can humans live?” even the correct question to pursue scientifically? In this paper, we suggest that a more appropriate scientific question is “How long must humans live?” From a biological perspective, we have known the answer to this question since Darwin published the Origin of Species (2). Not just humans but all organisms must live long enough to produce the offspring that will replace them—referred to as Darwinian fitness (3). This window of time has been called a biological warranty period (4), essential life span (5), or
adaptive lifespan (6). The names may vary, but the underlying conceptual framework is the same. It is a window of time sufficient for organisms to grow, develop, mature, reproduce, nurture, and provide postreproductive parenting and a grand-parenting period in humans (7). The hunter-gatherer data suggest that the adaptive lifespan of humans falls within the range of 60–70 years of age (6). Beyond that window is a period of extended operation where mortality rates climb rapidly and health and physical function decline precipitously. This is the theory; what follows is a continuation of our examination of biodemographic factors that may enable us to more accurately identify the age that defines the boundary between normal and extended operation. This is an important biomedical research endeavor because this boundary represents a transition from expected health and vigor to a period when health and vigor become progressively harder to maintain. Data Data for laboratory mice and humans will be used in this quantitative analysis. Three biodemographic factors are used to define the upper boundary of the warranty period: (i) age for the effective end of reproduction (EER), (ii) the age when the sex ratio is unity, and (iii) the age when mortality enters an acceleration phase. Mouse data came from a breeding colony of mice established at the Biomedical Research Division at Argonne National Laboratory and maintained from May 1953 to October 1969 (8). Reproductive data exist for 1,798 Dams that produced 12,312 litters for the five strains of laboratory mice used in this study. The fate of 1,843 mice from these litters that became control animals was monitored from 965
Downloaded from http://biomedgerontology.oxfordjournals.org/ at University of South Australia on August 11, 2014
Species are defined by biological criteria. This characterization, however, misses the most unique aspect of our species; namely, an ability to invent technologies that reduce mortality risks. Old animals are rare in nature, but survival to old age has become commonplace in humans. Science now asks how long can humans live, but we suggest a more appropriate question is: How long must humans live? Three lines of evidence are used to identify the biological equivalent of a warranty period for humans and why it exists. The effective end of reproduction, the age when the sex ratio is unity, and the acceleration of mortality reveal that approximately 50–55 years is sufficient time for our species to achieve its biological mandate—Darwinian fitness. Identifying this boundary is biomedically important because it represents a transition from expected health and vigor to a period when health and vigor become progressively harder to maintain.
966
Carnes and Witten
Results Effective End of Reproduction Age at menopause is considered the absolute end of female reproduction. In the mouse data, menopause could not be determined because females were discarded when their reproductive output fell below a consistent but subjective level chosen by the animal husbandry technicians. Given this missing information, we created a metric called the EER that was calculated in the following way. Dams whose last parity was also her first parity were deleted (ie, an abnormal termination). Parity records of Dams were also deleted when cumulative pup deaths exceeded cumulative pups weaned (point of reproductive failure/exhaustion). These deletion rules left 1,798 Dams distributed across five strains of laboratory mice whose age at reproductive exhaustion defines their EER. However, the EER of a Dam is not a warranty period because it does not include the “wean time” for the mice born in the last included parity (see parity elimination rules earlier). Wean time (age of independence) in this paper was estimated as half the Dam age at first parity. The warranty period includes the EER and the wean time. Thus, the approximate 1-year warranty period for the five mouse strains (see Table 1) was calculated as EER + (Dam age at first parity/2).
Our source of information for calculating an EER for humans comes from U.S. government National Vital Statistics Reports (and associated data files) on births and fertility (14) that provide counts for the number of children born to women whose age was 44. Cumulative births by age of mother (all races combined) were calculated (Table 2) and used to provide a defensible estimate of EER for humans. Like the mice, the birth data make it clear that the human EER occurs at ages well below the age of menopause (reproductive cessation). As with mice, the human EER does not include the postreproductive parenting required to successfully usher the last child born to sexual maturity or age of independence. Based on anthropological studies (15) of human development and rites of adulthood in pastoral and hunter-gatherer societies, the age of sexual maturity is approximately 16 and the age at first birth is 19 (16). Thus, the warranty period for humans is estimated to be the 35-year EER (based on cumulative fertility) + 19 (the age when the last child becomes a parent) or approximately 50–55 years because Table 2 shows that EER could easily fall within an age range of 32–35. In both mice and humans, EER augmented by either weaning time or postreproductive parenting and possible grandparenting beyond the last birth is viewed as the age when the biological warranty period expires and the period of extended operation begins. Sex Ratio What does sex ratio have to do with a warranty period? Sex ratio has a lot to do with the available pool of prospective mates. Too few or too many of one gender or the other creates not only demographic problems, but biological and social ones as well. Nature films and documentaries focus on the physical confrontations between males as they compete for access to females. Those confrontations are associated with a heightened risk of injury and even death. To compensate for this, the sex ratio at birth typically favors males. Humans may seem to be above this sexual selection scenario, but they are not. The highest risks from avoidable mortality start at the age of sexual maturity, peak around age 25 and extend to approximately age 35. Humans are a social organism where a male and female typically
Table 1. Summary Information Used to Estimate a Biological Warranty Period (period of normal operation) for the Five Mouse Strains Used in This Study Strain Name A strain BALB/c C3HF C57BL C57L
EER (d) (mean ± SE)
N
Age at First Parity (mean ± SE)
N
Warranty Period (d)
293 (3.6) 326 (4.0) 281 (4.6) 330 (3.3) 304 (6.1)
388 415 258 562 185
103 (0.8) 104 (1.0) 96 (1.1) 110 (1.1) 115 (1.6)
592 424 280 633 241
344 378 329 385 362
Notes. EER = effective end of reproduction. The warranty period for a specified mouse strain is defined as EER + (age at first parity/2).
Downloaded from http://biomedgerontology.oxfordjournals.org/ at University of South Australia on August 11, 2014
0 to 5 days, 6 to 15 days, 16 to 30 days, 31 to 100 days, and from 101 to 1,200 days (9). The Dams provide information on reproductive biology and the control mice provide the mortality information used to create an interval based life table for each mouse strain. Population files for the United States from 1933 to 2010 were obtained from the Human Mortality Database (HMD; 10) in order to examine sex ratio at birth and the age when the sex ratio is one. Life table statistics for the United States in 2006 were also downloaded from the HMD. Single year of age mortality data (2004–2006) were downloaded from the National Bureau of Economic Research (NBER) website (11) and partitioned into three categories: all-cause mortality (no partition), extrinsic mortality (deaths imposed by outside forces), and intrinsic mortality (deaths arising from biology)—see (12,13) for details.
967
How Long Must Humans Live?
Table 2. Summary Information on Cumulative Fertility Used to Justify Age 32–35 as the EER for Humans Age (y) 44 Total
1980 10,169 552,161 1,226,200 1,108,291 550,354 140,793 23,090 1,200 3,612,258
Cumulative % 0.28 15.57 49.51 80.19 95.43 99.33 99.97 100.00
1990 11,657 521,826 1,093,730 1,277,108 886,063 317,583 48,607 1,638 4,158,212
Cumulative % 0.28 12.83 39.13 69.84 91.15 98.79 99.96 100.00
2000 8,519 468,990 1,017,806 1,087,547 929,278 452,057 90,013 4,604 4,058,814
Cumulative % 0.21 11.76 36.84 63.64 86.53 97.67 99.89 100.00
2010 4,497 367,678 951,688 1,133,713 962,170 464,870 107,045 7,725 3,999,386
Cumulative % 0.11 9.31 33.10 61.45 85.51 97.13 99.81 100.00
cooperate—despite potential sexual conflict (17,18)—in order to produce and raise children. This reproductive strategy suggests that the age when the sex ratio = 1 ought to fall within the vicinity of the warranty age (50–55 years) estimated in the previous section. Population data files from HMD for calendar years 1933– 2010 were used to examine this hypothesis. As one can see (Figure 1), the sex ratio (males/females) at birth exhibits some erratic behavior and also reveals a clear trend. It does favor males, and it ranges between approximately 1.055 and 1.048 (using the spline function results—green line). That is a very narrow range of variation for a 77-year observation window, which suggests sex ratio, has been remarkably stable over time. HMD life tables start out with 100,000 people at age 0. In demography, this hypothetical population is called the “radix” and 100,000 is a normal convention. However, a sex ratio of 1.0 at birth ignores the observed sex ratio disparity that exists in their own population tables—see Figure 1. To create a biologically motivated analysis of sex ratio, the HMD radix, lx (number alive at beginning of age interval) and dx (deaths within the interval) components of the life tables for males were multiplied at every age by the calendar year sex ratio at birth found in Figure 1. The end result of this exercise is male and female life tables that can be used to examine age-specific sex ratios without distorting the HMD q(x) (conditional probability of death) schedule that is the core mortality metric of a life table. The absolute difference between the number of males and females was calculated at every age. The smallest difference within a calendar year identifies the age when the sex ratio most closely approximates 1.0—see Figure 2. Although numerous factors almost certainly affect the equality age, it is interesting that the steep drops coincide with World War II (1941–1945; 405,399 deaths) and the peak casualty years of the Vietnam War (1965–1971; 58,209 deaths accounting for 97.7% of the deaths) (19). We have no explanation for the observed decline in the decade of the 1980s. The increase of the equality age starts in the 1970s and accelerates through the 1990s to the present. This transition may reflect a new era of advances in medicine
Figure 1. The sex ratio (male/female at birth) between 1933 and 2010. Red lines connect adjacent data points and the green line is the product of a smoothing function used to reveal the broader trend.
Figure 2. A plot of the age when the sex ratio = 1.0 for calendar years from 1933 to 2010. Red lines connect adjacent data points and the green line is the product of a smoothing function used to visualize the broader trend.
and health care that improved life expectancy in general and may have benefited males more than females because females, on average, are healthier than their age-matched male counterparts (at least in these younger age ranges).
Downloaded from http://biomedgerontology.oxfordjournals.org/ at University of South Australia on August 11, 2014
Notes. EER = effective end of reproduction. Adding age of sexual maturity (independence) to that EER results in an estimate of 50–55 y for the human warranty period (EER + age of independence).
968
Carnes and Witten
Mortality Dynamics The calculated proportions of age and gender specific extrinsic and intrinsic deaths from the NBER data were used to partition the all-cause conditional probability of death, qall(t), from the HMD into extrinsic and intrinsic q(t) such that qall(t) = qext(t) + qint(t). Standard life table techniques (21) were then used to transform q(t) from age 15 to 109 into mid-interval estimates of the instantaneous failure rate, λ(t), which is also called the “force of mortality” or the “hazard rate”. Rather than “force”, rate or velocity is a more accurate terminology in this demographic context because they refer to the rate of change in mortality per unit of time where time can be either instantaneous or averaged over a specified time interval, as was done in this analysis using 1-year age intervals (22,23). We are interested in how “mortality” changes as a function of age. To examine this behavior, we convert the interval based life table estimates of hazard rates into a parametric equation amenable to the calculus of physics. The equation we use is the traditional Gompertz hazard rate function - a speed or velocity metric:
In order to estimate the Gompertz parameters, the natural logarithm was applied to λ(t): Ln (λ(t )) = Ln (α) + β × t This transform produces a simple linear equation that can be solved using ordinary least squares (25). Figure 3 shows the Gompertz acceleration of mortality in five strains of laboratory mice. The curves have been normalized to the strain-specific acceleration observed at 300 days of age, an age that falls within the warranty periods estimated in Table 1. As one can see, mortality acceleration during the warranty period (
© The Author 2013. Published by Oxford University Press on behalf of The Gerontological Society of America. All rights reserved. For permissions, please e-mail: [email protected]. Advance Access publication October 22, 2013
How Long Must Humans Live? Bruce A. Carnes1 and T.M. Witten2 Reynolds Department of Geriatric Medicine, The University of Oklahoma Health Sciences Center. 2 Center for the Study of Biological Complexity, Virginia Commonwealth University, Richmond.
1
Address correspondence to Bruce A. Carnes, PhD, Reynolds Department of Geriatric Medicine, The University of Oklahoma Health Science, 1122 N.E. 13th Street, ORB 1200, Oklahoma City, OK 73117. Email: [email protected]
Key Words: Biological warranty period—Effective end of reproduction—Sex ratio—Mortality acceleration. Received June 5, 2013; Accepted August 22, 2013 Decision Editor: Rafael de Cabo, PhD
T
he key feature that distinguishes humans from the rest of the living world is not taxonomic, it is technological sophistication. Each new generation of humans is more technologically advanced than previous generations. Most of that technology has been used to buffer humans from the mortality risks posed by both the biotic and abiotic world. More recently, human innovation has turned inward to seek interventions that mitigate the mortality risks posed by the flaws and failures of our own biology. The successful reduction of extrinsic and intrinsic mortality risks has led to the unprecedented population growth and population aging that is responsible for the severe economic and health challenges threatening the social fabric of nations around the globe. People who live beyond age 100 are a source of fascination for both the general public and the scientific community. The pursuit of technologies that progressively extend the human life span consumes enormous amounts of scientific effort and money. This ongoing pursuit of longevity dividends raises serious questions. Will the longevity dividends be accompanied by an equal extension of health? Will they be shared equally or will the longevity disparities that already exist become even more disparate? (1) More importantly, is “How long can humans live?” even the correct question to pursue scientifically? In this paper, we suggest that a more appropriate scientific question is “How long must humans live?” From a biological perspective, we have known the answer to this question since Darwin published the Origin of Species (2). Not just humans but all organisms must live long enough to produce the offspring that will replace them—referred to as Darwinian fitness (3). This window of time has been called a biological warranty period (4), essential life span (5), or
adaptive lifespan (6). The names may vary, but the underlying conceptual framework is the same. It is a window of time sufficient for organisms to grow, develop, mature, reproduce, nurture, and provide postreproductive parenting and a grand-parenting period in humans (7). The hunter-gatherer data suggest that the adaptive lifespan of humans falls within the range of 60–70 years of age (6). Beyond that window is a period of extended operation where mortality rates climb rapidly and health and physical function decline precipitously. This is the theory; what follows is a continuation of our examination of biodemographic factors that may enable us to more accurately identify the age that defines the boundary between normal and extended operation. This is an important biomedical research endeavor because this boundary represents a transition from expected health and vigor to a period when health and vigor become progressively harder to maintain. Data Data for laboratory mice and humans will be used in this quantitative analysis. Three biodemographic factors are used to define the upper boundary of the warranty period: (i) age for the effective end of reproduction (EER), (ii) the age when the sex ratio is unity, and (iii) the age when mortality enters an acceleration phase. Mouse data came from a breeding colony of mice established at the Biomedical Research Division at Argonne National Laboratory and maintained from May 1953 to October 1969 (8). Reproductive data exist for 1,798 Dams that produced 12,312 litters for the five strains of laboratory mice used in this study. The fate of 1,843 mice from these litters that became control animals was monitored from 965
Downloaded from http://biomedgerontology.oxfordjournals.org/ at University of South Australia on August 11, 2014
Species are defined by biological criteria. This characterization, however, misses the most unique aspect of our species; namely, an ability to invent technologies that reduce mortality risks. Old animals are rare in nature, but survival to old age has become commonplace in humans. Science now asks how long can humans live, but we suggest a more appropriate question is: How long must humans live? Three lines of evidence are used to identify the biological equivalent of a warranty period for humans and why it exists. The effective end of reproduction, the age when the sex ratio is unity, and the acceleration of mortality reveal that approximately 50–55 years is sufficient time for our species to achieve its biological mandate—Darwinian fitness. Identifying this boundary is biomedically important because it represents a transition from expected health and vigor to a period when health and vigor become progressively harder to maintain.
966
Carnes and Witten
Results Effective End of Reproduction Age at menopause is considered the absolute end of female reproduction. In the mouse data, menopause could not be determined because females were discarded when their reproductive output fell below a consistent but subjective level chosen by the animal husbandry technicians. Given this missing information, we created a metric called the EER that was calculated in the following way. Dams whose last parity was also her first parity were deleted (ie, an abnormal termination). Parity records of Dams were also deleted when cumulative pup deaths exceeded cumulative pups weaned (point of reproductive failure/exhaustion). These deletion rules left 1,798 Dams distributed across five strains of laboratory mice whose age at reproductive exhaustion defines their EER. However, the EER of a Dam is not a warranty period because it does not include the “wean time” for the mice born in the last included parity (see parity elimination rules earlier). Wean time (age of independence) in this paper was estimated as half the Dam age at first parity. The warranty period includes the EER and the wean time. Thus, the approximate 1-year warranty period for the five mouse strains (see Table 1) was calculated as EER + (Dam age at first parity/2).
Our source of information for calculating an EER for humans comes from U.S. government National Vital Statistics Reports (and associated data files) on births and fertility (14) that provide counts for the number of children born to women whose age was 44. Cumulative births by age of mother (all races combined) were calculated (Table 2) and used to provide a defensible estimate of EER for humans. Like the mice, the birth data make it clear that the human EER occurs at ages well below the age of menopause (reproductive cessation). As with mice, the human EER does not include the postreproductive parenting required to successfully usher the last child born to sexual maturity or age of independence. Based on anthropological studies (15) of human development and rites of adulthood in pastoral and hunter-gatherer societies, the age of sexual maturity is approximately 16 and the age at first birth is 19 (16). Thus, the warranty period for humans is estimated to be the 35-year EER (based on cumulative fertility) + 19 (the age when the last child becomes a parent) or approximately 50–55 years because Table 2 shows that EER could easily fall within an age range of 32–35. In both mice and humans, EER augmented by either weaning time or postreproductive parenting and possible grandparenting beyond the last birth is viewed as the age when the biological warranty period expires and the period of extended operation begins. Sex Ratio What does sex ratio have to do with a warranty period? Sex ratio has a lot to do with the available pool of prospective mates. Too few or too many of one gender or the other creates not only demographic problems, but biological and social ones as well. Nature films and documentaries focus on the physical confrontations between males as they compete for access to females. Those confrontations are associated with a heightened risk of injury and even death. To compensate for this, the sex ratio at birth typically favors males. Humans may seem to be above this sexual selection scenario, but they are not. The highest risks from avoidable mortality start at the age of sexual maturity, peak around age 25 and extend to approximately age 35. Humans are a social organism where a male and female typically
Table 1. Summary Information Used to Estimate a Biological Warranty Period (period of normal operation) for the Five Mouse Strains Used in This Study Strain Name A strain BALB/c C3HF C57BL C57L
EER (d) (mean ± SE)
N
Age at First Parity (mean ± SE)
N
Warranty Period (d)
293 (3.6) 326 (4.0) 281 (4.6) 330 (3.3) 304 (6.1)
388 415 258 562 185
103 (0.8) 104 (1.0) 96 (1.1) 110 (1.1) 115 (1.6)
592 424 280 633 241
344 378 329 385 362
Notes. EER = effective end of reproduction. The warranty period for a specified mouse strain is defined as EER + (age at first parity/2).
Downloaded from http://biomedgerontology.oxfordjournals.org/ at University of South Australia on August 11, 2014
0 to 5 days, 6 to 15 days, 16 to 30 days, 31 to 100 days, and from 101 to 1,200 days (9). The Dams provide information on reproductive biology and the control mice provide the mortality information used to create an interval based life table for each mouse strain. Population files for the United States from 1933 to 2010 were obtained from the Human Mortality Database (HMD; 10) in order to examine sex ratio at birth and the age when the sex ratio is one. Life table statistics for the United States in 2006 were also downloaded from the HMD. Single year of age mortality data (2004–2006) were downloaded from the National Bureau of Economic Research (NBER) website (11) and partitioned into three categories: all-cause mortality (no partition), extrinsic mortality (deaths imposed by outside forces), and intrinsic mortality (deaths arising from biology)—see (12,13) for details.
967
How Long Must Humans Live?
Table 2. Summary Information on Cumulative Fertility Used to Justify Age 32–35 as the EER for Humans Age (y) 44 Total
1980 10,169 552,161 1,226,200 1,108,291 550,354 140,793 23,090 1,200 3,612,258
Cumulative % 0.28 15.57 49.51 80.19 95.43 99.33 99.97 100.00
1990 11,657 521,826 1,093,730 1,277,108 886,063 317,583 48,607 1,638 4,158,212
Cumulative % 0.28 12.83 39.13 69.84 91.15 98.79 99.96 100.00
2000 8,519 468,990 1,017,806 1,087,547 929,278 452,057 90,013 4,604 4,058,814
Cumulative % 0.21 11.76 36.84 63.64 86.53 97.67 99.89 100.00
2010 4,497 367,678 951,688 1,133,713 962,170 464,870 107,045 7,725 3,999,386
Cumulative % 0.11 9.31 33.10 61.45 85.51 97.13 99.81 100.00
cooperate—despite potential sexual conflict (17,18)—in order to produce and raise children. This reproductive strategy suggests that the age when the sex ratio = 1 ought to fall within the vicinity of the warranty age (50–55 years) estimated in the previous section. Population data files from HMD for calendar years 1933– 2010 were used to examine this hypothesis. As one can see (Figure 1), the sex ratio (males/females) at birth exhibits some erratic behavior and also reveals a clear trend. It does favor males, and it ranges between approximately 1.055 and 1.048 (using the spline function results—green line). That is a very narrow range of variation for a 77-year observation window, which suggests sex ratio, has been remarkably stable over time. HMD life tables start out with 100,000 people at age 0. In demography, this hypothetical population is called the “radix” and 100,000 is a normal convention. However, a sex ratio of 1.0 at birth ignores the observed sex ratio disparity that exists in their own population tables—see Figure 1. To create a biologically motivated analysis of sex ratio, the HMD radix, lx (number alive at beginning of age interval) and dx (deaths within the interval) components of the life tables for males were multiplied at every age by the calendar year sex ratio at birth found in Figure 1. The end result of this exercise is male and female life tables that can be used to examine age-specific sex ratios without distorting the HMD q(x) (conditional probability of death) schedule that is the core mortality metric of a life table. The absolute difference between the number of males and females was calculated at every age. The smallest difference within a calendar year identifies the age when the sex ratio most closely approximates 1.0—see Figure 2. Although numerous factors almost certainly affect the equality age, it is interesting that the steep drops coincide with World War II (1941–1945; 405,399 deaths) and the peak casualty years of the Vietnam War (1965–1971; 58,209 deaths accounting for 97.7% of the deaths) (19). We have no explanation for the observed decline in the decade of the 1980s. The increase of the equality age starts in the 1970s and accelerates through the 1990s to the present. This transition may reflect a new era of advances in medicine
Figure 1. The sex ratio (male/female at birth) between 1933 and 2010. Red lines connect adjacent data points and the green line is the product of a smoothing function used to reveal the broader trend.
Figure 2. A plot of the age when the sex ratio = 1.0 for calendar years from 1933 to 2010. Red lines connect adjacent data points and the green line is the product of a smoothing function used to visualize the broader trend.
and health care that improved life expectancy in general and may have benefited males more than females because females, on average, are healthier than their age-matched male counterparts (at least in these younger age ranges).
Downloaded from http://biomedgerontology.oxfordjournals.org/ at University of South Australia on August 11, 2014
Notes. EER = effective end of reproduction. Adding age of sexual maturity (independence) to that EER results in an estimate of 50–55 y for the human warranty period (EER + age of independence).
968
Carnes and Witten
Mortality Dynamics The calculated proportions of age and gender specific extrinsic and intrinsic deaths from the NBER data were used to partition the all-cause conditional probability of death, qall(t), from the HMD into extrinsic and intrinsic q(t) such that qall(t) = qext(t) + qint(t). Standard life table techniques (21) were then used to transform q(t) from age 15 to 109 into mid-interval estimates of the instantaneous failure rate, λ(t), which is also called the “force of mortality” or the “hazard rate”. Rather than “force”, rate or velocity is a more accurate terminology in this demographic context because they refer to the rate of change in mortality per unit of time where time can be either instantaneous or averaged over a specified time interval, as was done in this analysis using 1-year age intervals (22,23). We are interested in how “mortality” changes as a function of age. To examine this behavior, we convert the interval based life table estimates of hazard rates into a parametric equation amenable to the calculus of physics. The equation we use is the traditional Gompertz hazard rate function - a speed or velocity metric:
In order to estimate the Gompertz parameters, the natural logarithm was applied to λ(t): Ln (λ(t )) = Ln (α) + β × t This transform produces a simple linear equation that can be solved using ordinary least squares (25). Figure 3 shows the Gompertz acceleration of mortality in five strains of laboratory mice. The curves have been normalized to the strain-specific acceleration observed at 300 days of age, an age that falls within the warranty periods estimated in Table 1. As one can see, mortality acceleration during the warranty period (
