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Why Do Life Spans Differ? Partitioning Mean Longevity Differences in Terms of Age-Specific Mortality Parameters

^a Department of Ecology, Evolution, and Behavior, University of Minnesota, St. Paul

Scott D. Pletcher, Max Planck Institute for Demographic Research, Doberaner Strausse 114, D-18057 Rostock, Germany E-mail: pletcher{at}demogr.mpg.de.

Jay Roberts, PhD

	Abstract

Top Abstract Methods Data Analyses Discussion Appendix ENDIX References

 
Populations typically differ in mean life spans because of genetic, environmental, or experimental factors. In this paper methods are presented that clarify the relationship between differences in the longevity of two populations and differences in their underlying age-specific patterns of mortality. Data are examined from rodent and fruit fly (Drosophila melanogaster) experiments that investigated the longevity effects of a variety of environmental and genetic manipulations, including temperature, dietary restriction, laboratory selection for increased longevity, and severe inbreeding. Analyses suggest that longevity differences mediated by temperature and dietary restriction result predominantly from differences in the rate of increase in mortality with age. Increases in longevity through laboratory selection result primarily from a reduction in baseline mortality and not a slowing of the rate of aging. Although the methods are applied primarily in the context of simple mathematical models of mortality (e.g., the Gompertz model), they are quite general and can be applied to mortality models of arbitrary complexity. Mathematica protocols ("notebooks") and computer software have been developed to perform all the analyses discussed and are available from the first author.

INTEREST in the determinants of longevity pervades the life sciences. Demographers analyze the human variation in life span (we use life span and longevity interchangeably; both refer to age at death) with the goal of predicting future dynamics of the global population, whereas evolutionary biologists and ecologists investigate the variation in longevity among different species and among populations in different ecological settings. Often it is not the longevity of a particular population of organisms that is of interest, but rather the difference between two (or more) populations. For example, demographers often focus on the difference in longevity between males and females and between different racial or ethnic groups. Biologists might examine the influence of genetic, ecological, or experimental factors on longevity.

In nearly all comparative studies of longevity, there is something to be gained by examining differences in age-specific death rates in addition to the traditional method of comparing average longevities. First, the influence of experimental treatments on life span can be localized to specific ages of effect ⁽¹⁾. Second, mathematical models of age-specific mortality can be used to efficiently summarize data about the ages at death of all the individuals within a population ⁽²⁾. Third, these same models may provide a simple way to generalize survival differences between populations and test hypotheses concerning the mechanism by which differences in life span are generated ⁽³⁾.

Researchers in the biological and social sciences have shown interest in determining the relationship between longevity and mortality rates. In an attempt to understand life-span differences between male and female bean beetles, Tatar and Carey ⁽⁴⁾ divided life span into two stages (early life and late life) and assessed the contribution of mortality in each stage to the difference in mean longevity. Human demographers have long been interested in relationships between the prolongation of human longevity and changes in the age pattern of mortality. They have, for example, computed differential contributions of age-specific death rates to life-span changes in a number of countries ⁽⁵⁾. Such age-specific contributions could, in principle, be summarized as the effects of a few parameters of a mathematical model that fits the observed data well; yet neither demographers nor biologists have quantified the longevity effects of changes in a small number of biologically interpretable mortality parameters.

A number of mathematical models have been proposed for summarizing the relationship between age and mortality rates ⁽²⁾⁽⁶⁾. When such models are fit to mortality data, standard statistical analyses focus on detecting "significant" differences between populations in the parameters of the model ⁽⁴⁾. As experimental techniques become more refined and sample sizes get larger, statistically significant differences between populations can often be detected in many different parameters of any specific mortality model ⁽⁴⁾. However, interesting biological questions are often left unanswered: How are the differences in life span generated? Is the majority of the gain in longevity a result of a proportional decrease in mortality at all ages, or does it result mainly from a decline in the rate of increase in mortality with age?

We address these questions by presenting a straightforward method of decomposing the difference in mean longevity between two populations into additive contributions from each of the parameters of a mathematical model of mortality. The techniques are illustrated by using extensive demographic data from fruit flies and rodents. We examine longevity effects of a variety of genetic and environmental factors, including inbreeding and selection for increased longevity in flies, and dietary restriction in rats. Although they are only fully developed for five common models, the decomposition techniques are sufficiently general to be applied to any reasonable mortality model.

The calculations described in this paper require a significant amount of numerical computation. For this reason, easy to use Mathematica ⁽⁷⁾ protocols ("notebooks") and computer software have been developed that perform all analyses. These resources are freely available from the first author.

	Methods

Top Abstract Methods Data Analyses Discussion Appendix ENDIX References

 
Measurements of Survival
In survival analysis, three functions of age are used to describe the survival or mortality characteristics of a population: the hazard function, the survival function, and the probability density function for ages at death. In nearly all cases of interest to biologists, these functions are well approximated as mathematical functions of a small number of parameters, . The hazard function, h(t ), describes the conditional failure rate over a very short interval of time, (t,t + t ) ⁽⁸⁾. Unlike the age-specific probability of death, the hazard function is not a probability measure and therefore can take on values from 0 to ⁽³⁾. The survivorship function, S(t ), describes the probability that an individual lives beyond age t, and the density function, f (t ), describes the distribution of ages at death.

The hazard, survivorship, and distribution functions are related as follows ⁽⁸⁾.

(1)

(2)

(3)

Thus, if any one of the three functions is given; then the others can be derived.

Decomposition of Mean Longevity
Mean longevity can be directly calculated from either the density function or the survivorship function. The mean age at death can be represented as

(4)

or

(5)

In both cases, V () will be a complicated function of the parameters, , of the mortality model.

Given a specific mortality model and its corresponding mean life-span function, V (), define the difference in mean longevity between two populations, V, by

(6)

where ⁱ is the vector of parameter values for a mortality model of interest from population i and is a vector of parameter differences between the two populations. If the mean life-span function is available in closed form, the longevity difference predicted by the model can be calculated exactly. Otherwise, numerical integration must be used along with 4 or 5.

Let = be a vector of parameter values for population i with respect to a specific mortality model with k parameters. If (a) changes continuously from ¹ to ², (b) V() is differentiable with respect to , and (c) the increments of the ⁱ are proportional such that there exists a function, g(x), satisfying the relationship

(7)

for all j, then the difference in mean longevity between two populations, V, can be decomposed into additive effects contributed by each parameter of the mortality model ⁽⁵⁾. Horiuchi and colleagues ⁽⁵⁾ point out the need for the proportional increments assumption and discuss the sensitivity of the decomposition method to deviations from it. For the available data, the decomposition technique appears to be relatively insensitive to violations in this assumption ⁽⁵⁾.

Because V () is a continuous function of , the total differential of V (), dV, is given by

(8)

where d_i is the difference in _i between the two populations (e.g., ¹_i - ²_i). 8 is defined for any set of d_i, but if the |d_i| are small, dV will provide an accurate approximation to V; 8 can be used to approximate 6. Because 8 is composed of separate terms for each parameter in the model, it can be used to quantify the contribution of each parameter to the total difference in mean longevity between two populations. The fractional contribution of _i to the difference in longevity can be expressed as

(9)

For cases in which the difference in mean longevity is large, the linear approximation used in 8 may have a large error associated with it. The absolute squared error is measured as

(10)

If is greater than 1% of the true difference in means, 8 can be considered inaccurate. Because in all cases described here the mean function is monotonic with respect to each parameter, this problem can be overcome by dividing the total difference in parameters into a large number of smaller intervals, evaluating 8 for each interval, and summing the results. With the use of this method an arbitrary degree of accuracy can be obtained.

Specific Mortality Models
Table 1 presents the hazard, survival, and probability density functions for the five mortality models examined in this article: the Weibull model and four models in the Gompertz family (Gompertz, Gompertz–Makeham, logistic, and logistic–Makeham). Each of these models is discussed in greater detail in the paragraphs that follow.

(11)

where

is the Euler gamma function. For reference, analytical derivatives of V with respect to and ß are presented in the Appendix.

The Gompertz family of mortality models ⁽³⁾ is based on the idea of a hazard that increases exponentially with age ⁽⁹⁾. Subsequent modifications to this original model have led to the Gompertz–Makeham ⁽¹⁰⁾, logistic ^(11)(12), and logistic–Makeham models. Each of these models assumes an exponential increase in hazard throughout much or all of the life span. The Makeham models include a term for age-independent mortality, and the logistic models allow for a deceleration of mortality rates late in life (Table 1 ).

The Gompertz model is characterized by two parameters, and , referred to as the initial mortality parameter and rate parameter, respectively (Table 1 ). The mean function is

(12)

where

is the incomplete gamma function. Analytical derivatives for the Gompertz model are provided in the Appendix.

The logistic frailty model is based on the assumption that all individuals in a population incur the same baseline hazard function, but there is heterogeneity among individuals for the parameters of that function. If one assumes a Gompertz baseline hazard and individual 's distributed according to a gamma distribution, then the logistic frailty model results ^(11)(13). The model is characterized by three parameters: , , and s (Table 1 ). The first two parameters have interpretations analogous to the Gompertz model, and s is a parameter quantifying the degree of heterogeneity in the population. Interestingly, this parameter also describes the amount of deceleration in mortality rates at advanced ages.

The Gompertz and logistic models assume that senescent (age-dependent) factors determine observed mortality rates and are expressed from birth. This assumption may not be appropriate in some cases. Low levels of random causes of death may dominate mortality rates early in life when "intrinsic" factors are negligible. Moreover, evolutionary theory suggests that the onset of aging should coincide with the age at first reproduction ^(14)(15). To account for these situations, the Gompertz–Makeham and logistic–Makeham models add an age-independent parameter to the hazard functions of the standard Gompertz and logistic models, respectively (Table 1 ). For the logistic, Gompertz–Makeham, and logistic–Makeham models, numerical integration of 5 is required to calculate the mean and partial derivatives for use in 6 and 8.

Numerical Methods
For cases in which numerical methods were required, Mathematica ⁽⁷⁾ was used for numerical integration (function Nintegrate with a working precision of 40 digits). When mean longevity was calculated by using 5, an upper limit of integration of 10,000 was used. Monte Carlo simulation was used to provide a check on the numerical techniques. For each specific mortality model (logistic, Gompertz–Makeham, and logistic–Makeham), ages at death were simulated for a population of 20,000 individuals and the mean age at death was calculated. This process was repeated 2000 times for each of five sets of parameters, and the distribution of means was examined for each set. Mean longevities produced by both numerical integration and Monte Carlo simulation were nearly identical (data not presented). Numerical first derivatives were obtained by using the method of finite differences ^(16)(17).

	Data Analyses

Top Abstract Methods Data Analyses Discussion Appendix ENDIX References

 
In this section we apply the decomposition techniques to demographic data from a series of experiments in fruit flies and rats. We investigate how differences in age-specific mortality rates contribute to the differences in longevity produced by environmental and genetic manipulations. The popularity of the Gompertz model in studies of longevity has led to a categorization of changes in mortality rates into changes in the initial mortality rate (the parameter in the Gompertz family of models) and changes in the rate of increase in mortality with age (the parameter in the Gompertz family ⁽¹⁸⁾. Thus, we concentrate on the Gompertz family of models and focus on the contributions of these two parameters to differences in mean longevity between experimental treatments.

Much of the data analyzed here is unpublished. Unpublished data for Drosophila were obtained in the Curtsinger laboratory by using described methods for collecting demographic data ⁽²⁾⁽⁷⁾. Published data ⁽¹⁹⁾ on dietary restriction in rats were obtained from Dr. B. P. Yu of The University of Texas Health Science Center. Maximum likelihood methods were used to determine the specific model that best fit the data of interest and to obtain estimates for the parameters of that model ⁽⁴⁾. Previously published data were reanalyzed by using maximum likelihood techniques. Goodness of fit was assessed for each population by comparing the predicted mean longevity from the fitted mortality model with the observed average longevity. If the difference between these values was greater than 0.1% of the observed value, the population was excluded from all analyses. In some cases, a suitable degree of accuracy in the decomposition [see 10] could not be obtained because of excessive computing requirements. These comparisons were excluded from the reported results.

When the two populations being compared were characterized by different mortality models (e.g., one was best fit by a Gompertz model and the other by a logistic), the decomposition was carried out with the larger, more general model. Because models in the Gompertz family are nested, smaller models are special cases of larger models. For example, the Gompertz model is a special case of the logistic model with = . A significant number of comparisons were based on models other than the standard Gompertz, but the contributions of the deceleration parameter, s, and the age-independent mortality parameter, c, contributed little to differences in longevity.

For each specific environmental or genetic manipulation, the fractional contribution of each mortality parameter [see 9] to the longevity difference between each pair of populations in the data set was determined. The median (and its standard error) of the distribution of these contributions is reported. Thus, a median fractional contribution of 0.6 implies that the parameter of interest contributed to greater than 60% of the longevity difference in half of the decompositions and less than 60% of the longevity difference in the other half. Negative contributions result if parameter differences most often decrease the difference in longevity between experimental populations.

Environmental Manipulations
Two well-known environmental manipulations that lead to changes in longevity are temperature shift ^(20)(21) and dietary restriction ^(22)(23). Adult fruit flies reared at 18°C live longer than flies reared at 24°C. Rats whose caloric intake is restricted live longer than those allowed to feed ad libitum. A number of biochemical mechanisms for these changes have been proposed ⁽²³⁾, and although many of these hypotheses focus on metabolic rate and predict changes in the rate of aging ⁽¹⁸⁾, few studies report how age-specific mortality rates are affected by these treatments.

The first complete life tables of flies reared at different temperatures were published by Alpatov and Pearl ⁽²⁰⁾. Using these data and unpublished data from a larger temperature experiment conducted in the Curtsinger laboratory, we compared the relative contributions of the initial mortality parameter and the rate parameter from the Gompertz family of models to differences in average longevity. For each data set, the best fitting mortality model was determined, and parameter estimates for that model were obtained by using the method of maximum likelihood (Table 2 ). Pairwise decompositions were carried out for each combination (e.g., 18 vs 24, 18 vs 29, and 24 vs 29) within each data set. The median fractional contributions of the initial mortality parameter () and the rate parameter () to longevity differences are presented in Table 4 below.

The investigation into the aging effects of dietary restriction originated with McCay and colleagues ⁽²²⁾, who showed that a reduction in food intake below ad libitum fed rats and mice increases the length of life. Since this discovery, an increase in longevity through caloric restriction has been a robust and reproducible result. We reanalyzed data from a previously published experiment on 230 rats, of which 115 were allowed to feed ad libitum and the other 115 were restricted ⁽¹⁹⁾. After 6 weeks of age, restricted rats were limited to 60% of the caloric intake of ad libitum rats ⁽¹⁹⁾.

Mortality model estimates for the two groups are provided in Table 3 , and fitted age-specific mortality trajectories for the two groups are presented in Fig. 1. Mortality rates in restricted rats are essentially Gompertzian (likelihood ratio test of Gompertz, H_0, vs logistic–Makeham, H_A; = ), but the ad libitum rats show more complex mortality dynamics (likelihood ratio test of Gompertz, H₀, vs logistic–Makeham, H_A; p < .0001). Mortality rates are very low early in life, increase rapidly through middle age, and decelerate later in life. As suggested previously in the literature ^(18)(23)(24), the decomposition methods show that the difference in the rate parameter () accounts for nearly all of the difference in longevity between the two groups, whereas the difference in the initial mortality rate () actually reduced the difference in longevity (Table 4 ). As noticed by Yu and colleagues ⁽¹⁹⁾, the mortality dynamics of the ad libitum rats result in most individuals' living to near the maximum observed length of life.

View larger version (18K): [in this window] [in a new window]   Figure 1. Fitted mortality trajectories for rats fed ad libitum and rats under dietary restriction. The best fit mortality model for the ad libitum rats is a logistic–Makeham model, and the best model for the restricted rats is a standard Gompertz one. Parameter values are provided in Table 3 .

Several authors have published work focusing on the extended longevity of laboratory selected Drosophila, but only two have examined age-specific mortality differences ^(32)(33). Curtsinger and colleagues ⁽³²⁾ suggest that the increased longevity of the long-lived lines selected by Luckinbill results from a change in the initial mortality parameter of the logistic model rather than a change in the rate parameter. Service and colleagues ⁽³³⁾ claim the majority of the longevity increase in the flies selected by Rose is due to a change in the rate parameter.

To avoid the problems associated with comparing results from different laboratories, we analyzed data published by Curtsinger and colleagues ⁽³²⁾ on the Luckinbill lines along with unpublished mortality data on the Rose lines, which were also collected in the Curtsinger laboratory. We used decomposition techniques to determine the fractional contributions of the initial mortality parameter and rate parameter to the difference in longevity between selected and control lines (Table 4 ). Pairwise decompositions were performed by comparing each control line to each selected line. The Luckinbill lines consist of two selected and two control lines, and the Rose lines consist of five selected and five control lines.

For both the Luckinbill and Rose lines, the initial mortality parameter () contributes more to the longevity increase generated by laboratory selection than the rate parameter (). For the Rose lines, the initial mortality parameter usually accounts for 70% (males) and 60% (females) of the longevity difference. The influence of this parameter is larger in the Luckinbill lines, where, in males, it usually accounts for 80% of the differences in longevity. In females, all of the increase in longevity can be attributed to differences in initial mortality, as long-lived flies actually have a higher rate parameter than control lines (as evidenced by the negative contribution from ).

A series of approximately 70 inbred lines was generated from the Luckinbill selected and control populations. Survival data on 40 of these lines (20 derived from the selected populations and 20 from control populations) were collected in the Curtsinger laboratory (A. Khazaeli, unpublished data), and mortality parameters were estimated in the usual manner ⁽⁴⁾. Overall, inbreeding resulted in a decrease in longevity—each set of inbred lines lived shorter than their outbred, parental populations, but longevity differences between inbred lines from the control populations and from the long-lived populations remained quite large (Khazaeli and Curtsinger, manuscript in preparation). Pairwise longevity decompositions were carried out by comparing each control line inbred with each selected line inbred (Table 4 ).

Interestingly, when compared to the outbred populations, a larger fraction of the longevity differences is due to the rate parameter. Although the contribution from the initial mortality parameter is substantial (40–50%), it is less than that observed in the outbred populations (80–100%). This may be due to selection among inbred lines for low mortality early in life.

A final series of decompositions was carried out on 114 recombinant inbred (RI) lines that were created from two control inbred and two selected inbred lines (Curtsinger and colleagues, manuscript in preparation). Construction of the RI lines involved crosses between selected and control inbred lines, followed by two generations of recombination and 20 generations of subsequent inbreeding. If longevity is influenced by many genes, we expect genetically based variation in longevity among the RI lines. Survival data on all 114 lines and the four parental lines were collected in the Curtsinger laboratory; and mortality parameters were estimated for each line. Because of the extremely large number of possible paired comparisons among 114 lines, a random sample of 3000 paired comparisons was examined. In addition to the criteria for acceptance already outlined (<0.1% error in fit and <1% error in the decomposition), only comparisons whose mean longevity difference was greater than 5 days were included. Of the 3000 decompositions, 1378 (males) and 1376 (females) decompositions satisfied the criteria for use. The average longevity difference for the decompositions and the median effect of each parameter are given in Table 4 .

The majority of longevity differences between RI lines is due to a difference in the rate parameters of the mortality models rather than the initial mortality parameters. In males, differences in rate parameters usually generate nearly 100% of the observed longevity differences (Table 4 ). The negative value for the initial mortality parameter indicates that longer-lived males tend to have a higher initial mortality. In females, approximately 99% of the longevity differences are due to differences in rate parameters, whereas initial mortality rates usually account for only approximately 4% of the difference (the c and s parameters in the Gompertz family contribute small, negative values to the difference such that the sum totals to unity). As suggested for the inbred lines, the large contribution of the rate parameter is likely to be due to selection for low early-life mortality.

	Discussion

Top Abstract Methods Data Analyses Discussion Appendix ENDIX References

 
The numerical and analytical methods developed in this paper allow differences in mean longevities between two experimental populations to be interpreted in terms of differences in particular parameters that describe age-specific mortality. We can, for example, determine whether an increase (or decrease) in mean longevity produced by some experimental manipulation (or ecological situation) is caused by a change in the baseline probability of death or by a modification to the rate of increase in mortality with age.

Decomposition techniques provide a unique insight into the underlying age-specific mortality changes that accompany differences in mean longevity generated by genetic and environmental manipulations. The hypothesis that longevity differences in Drosophila maintained at different ambient temperatures are caused by changes in the rate of aging is supported. The rate parameter of the Gompertz family of models accounted for nearly all of the difference in longevity of male Drosophila reared at three different temperatures (Table 4 ).

The data examined here suggest that dietary restriction in rats causes qualitative changes in the age-specific mortality trajectories. The increase in life span seen when rats were calorically limited resulted from a reduction in the rate of increase in mortality with age. Interestingly, the ad libitum fed rats show much lower mortality rates at early ages when compared with restricted rats. In humans, the major causes of death for children and young adults are infectious diseases, whereas for older adults degenerative diseases such as heart disease are primary killers (S. Horiuchi, personal communication). It may be that undernourished humans are more vulnerable to the risk of infectious diseases, but as they become older, their low cholesterol, low blood pressure, and low weight help them survive the risk of degenerative diseases. This may be true for rats, causing the mortality reversal seen in Fig. 1. A second possibility is that ad libitum feeding itself is pathological and may shorten life span.

The mean longevity differences in Drosophila created through laboratory selection are generated primarily by differences in the baseline rate of mortality rather than differences in the rate of mortality increase with age. This effect is especially pronounced in the lines selected by Luckinbill and colleagues ⁽²⁶⁾. During these selection experiments, only those flies that survived to very old ages (approximately 7 weeks) were allowed to reproduce. Thus, there was strong selection for reduced mortality rates at all ages prior to this time, and it is not surprising that longevity differences result from changes in the initial mortality parameter. Changing this parameter shifts the mortality risk at all ages by the same proportion. As a result, long-lived lines have proportionately lower mortality rates throughout life. Although we were unable to apply decomposition techniques to their data, Lin and colleagues ⁽³⁴⁾ suggest that a similar shift of mortality rates underlies the increased longevity of the methuselah mutation in Drosophila.

The increased influence of the rate parameter on differences in longevity of the inbred and RI lines may be due to either selection for low early-life mortality during the inbreeding process or mutation accumulation at loci with effects later in life or both. During inbreeding, cultures were maintained on a 2-week generation cycle. Selection on age-specific mortality rates is essentially zero after approximately five days posteclosion ⁽³⁵⁾. Lines with a high initial mortality are more likely to go extinct. Surviving lines would tend to have similar and low initial mortality parameters. Moreover, selection on the rate parameter is negligible for the range of parameter values observed for these populations (data not presented). Mutations in genes that influence the rate parameter would be allowed to accumulate, resulting in genetically based variation in life span. The creation of the RI lines through recombination and subsequent inbreeding would allow for a second round of selection on early mortality rates, and longevity decompositions among these lines indicate a large contribution on average from the rate parameter to longevity differences (Table 4 ).

What are the implications of these methods and observations for the evolutionary biology of senescence? Sacher ⁽¹⁸⁾ suggested that the initial mortality parameter, , measures the initial vulnerability to disease or physiological damage, and that it is related to the genetically determined vigor of the genotype. Thus, a decrease in can be interpreted as a delay in the onset of chronological aging (a right shift of the curve) or an increase in the initial quality of the organism (a down shift of the curve) ⁽¹⁸⁾. Distinguishing between these possibilities is difficult but may be possible by identifying the genes responsible for the selection response. A mapping of the quantitative trait loci that influence age-specific mortality traits is a first step (Curtsinger and colleagues, manuscript in preparation).

The rate of aging is often defined as the rate of exponential increase in mortality rates with age ⁽²⁴⁾. This definition corresponds to the rate parameter, , in the Gompertz family of mortality models. As a result, or the mortality rate doubling time (= ) is the most common descriptor of the aging process in the evolutionary and gerontological literature. Because it is the increase in mortality rates that evolutionary biologists find perplexing and that medical researchers are always trying to reduce, understanding how differences in the rate parameters of experimental populations contribute to their longevity differences is fundamental to progress in these areas.

Previous work on assessing the changes in age-specific mortality rates generated by experimental manipulations have produced some interesting results. Sacher ⁽¹⁸⁾ reports that single, high doses of ionizing radiation affected the initial mortality parameter of the Gompertz model—rates of aging () are relatively unchanged. Higher reproductive effort in the bean beetle Callosobruchus maculatus decreased life span by increasing mortality by an equal amount at all ages ⁽³⁶⁾. Again, the rate of aging was unaffected.

Nusbaum and colleagues ⁽³⁷⁾ claim that laboratory selection for late-life fertility produced long-lived lines of Drosophila that exhibit a slower rate of aging than their respective control lines. Applying decomposition techniques to their published results supports this conclusion (of a mean longevity difference of nearly 37 days, approximately 78% is due to differences in the Gompertz rate parameter). Unfortunately, sample sizes in this experiment were small, and it is likely the Gompertz model provides a poor fit to these data. An indication of the lack of fit of the Gompertz model is provided by the large underestimates of mean longevity predicted by the fitted parameter estimates. For long-lived populations the observed mean longevity was 86.8 days, compared with 72.6 days predicted by the fitted model. For the control lines, observed longevity was 49.6 days, whereas the fitted Gompertz model predicted a mean longevity of 35.5 days. A reanalysis of previous studies in this area using proper methods for choosing an adequate mortality model and longevity decomposition techniques would provide valuable insight into the types of changes in age-specific mortality generated by phenotypic and genetic manipulations and into the generality of these changes.

The methods developed here are quite general and easily extended to other mortality models besides those examined. With the use of more complicated models, precise questions can be addressed about the relationship between age-specific mortality and average longevity. For cases in which the parameters of the mortality model in question have biological interpretations, the decomposition process is useful for understanding how ecological differences affect the evolution of patterns of senescence and how experimental manipulations affect the biological process of aging.

	Acknowledgments

 
This work was supported by the National Institutes of Health Grants AG-0871 and AG-11722 to J. Curtsinger. We thank S. Horiuchi and J. Wilmouth for pointing out the need for the proportional increments assumption. S. Horiuchi, P. Abrams, M. Tatar, C. Neuhauser, and two anonymous reviewers provided valuable comments on the manuscript. M. Rose and L. Luckinbill generously provided their Drosophila stocks.

Received January 22, 1999

Accepted January 12, 2000

	Appendix ENDIX

Top Abstract Methods Data Analyses Discussion Appendix ENDIX References

 
Here we provide analytical formulae for the derivatives of the mean functions for the Weibull and Gompertz mortality models. These formulae are used in 8 to determine the contribution of the parameters of the mortality model to differences in mean longevity.

The Weibull model is characterized by two parameters, denoted by and ß, which represent the baseline mortality and the rate of increase in mortality with age, respectively (Table 1 ). The derivatives of the mean function, V, with respect to each are

(A1A)

and for ß

(A1B)

where

is the digamma function representing the logarithmic derivative of the standard gamma function.

The Gompertz model is also characterized by two parameters: the initial mortality rate () and the rate of exponential increase in mortality with age (). The corresponding derivatives of the mean function, V, are

(A2A)

(A2B)

where

is the incomplete gamma function.

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Top Abstract Methods Data Analyses Discussion Appendix ENDIX References

 

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