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Parameter estimates in model specifications with different combinations of variables, French birth cohorts 1883-1901, ages 105 and above
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Background: The debate about limits to the human life span is often based on outcomes from mortality at the oldest ages among longevity pioneers. To this day, scholars disagree on the existence of a late-life plateau in human mortality. Amid various statistical analysis frameworks, the parametric proportional hazards model is a simple and valuable...
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... fact, by imposing a nonnegative constraint on parameter b, the test becomes one-rather than twotailed and p-values are consequently halved, giving an even higher level of significance for the Gompertz slope parameter in all our model specifications. A comparison based on AIC is presented in Table 3, and it also shows that the model with the Gompertz slope, b, describes the data better (e.g., 10,878.00 vs. 10,904.63 for the model with gender effect only). ...Context 2
... small modification was made to the code of the cdfDT function to extract the cumulative hazard estimates needed for this visual inspection. As done for the parametric approach, we computed the cumulative hazard from both the constant and Gompertz hazards according to the relationship H(t) = -ln[S(t)] and using the parameter estimates in Table 3 for models including a statistically significant gender effect. Figure 1 presents estimates for the French female cohort, and it shows that the Gompertz hazard model (blue line) is closer to the nonparametric estimates (in red) than the fitted constant hazard (green line). ...Context 3
... investigated other model specifications, first to test the sensitivity of our parameter estimates and second to choose the best-performing model for reliable conclusions on birth cohort and gender effects. The results across all model specifications are reported in Table 3. Precisely, we performed likelihood ratio tests between models having different baseline functions without cohort or gender effects (p = 1.402e-07, see Table 2), between models with gender effect only (p = 8.778e-08), and between models with cohort effect only (p = 1.413e-07). ...Context 4
... results across all model specifications are reported in Table 3. Precisely, we performed likelihood ratio tests between models having different baseline functions without cohort or gender effects (p = 1.402e-07, see Table 2), between models with gender effect only (p = 8.778e-08), and between models with cohort effect only (p = 1.413e-07). We find that the Gompertz slope parameter, b, is positive and statistically different from zero regardless of the model specification (Table 3). According to the AIC, models including the Gompertz slope consistently perform better than those assuming constant hazard. ...Context 5
... fact, by imposing a nonnegative constraint on parameter b, the test becomes one-rather than twotailed and p-values are consequently halved, giving an even higher level of significance for the Gompertz slope parameter in all our model specifications. A comparison based on AIC is presented in Table 3, and it also shows that the model with the Gompertz slope, b, describes the data better (e.g., 10,878.00 vs. 10,904.63 for the model with gender effect only). ...Context 6
... small modification was made to the code of the cdfDT function to extract the cumulative hazard estimates needed for this visual inspection. As done for the parametric approach, we computed the cumulative hazard from both the constant and Gompertz hazards according to the relationship H(t) = -ln[S(t)] and using the parameter estimates in Table 3 for models including a statistically significant gender effect. Figure 1 presents estimates for the French female cohort, and it shows that the Gompertz hazard model (blue line) is closer to the nonparametric estimates (in red) than the fitted constant hazard (green line). ...Context 7
... investigated other model specifications, first to test the sensitivity of our parameter estimates and second to choose the best-performing model for reliable conclusions on birth cohort and gender effects. The results across all model specifications are reported in Table 3. Precisely, we performed likelihood ratio tests between models having different baseline functions without cohort or gender effects (p = 1.402e-07, see Table 2), between models with gender effect only (p = 8.778e-08), and between models with cohort effect only (p = 1.413e-07). ...Context 8
... results across all model specifications are reported in Table 3. Precisely, we performed likelihood ratio tests between models having different baseline functions without cohort or gender effects (p = 1.402e-07, see Table 2), between models with gender effect only (p = 8.778e-08), and between models with cohort effect only (p = 1.413e-07). We find that the Gompertz slope parameter, b, is positive and statistically different from zero regardless of the model specification (Table 3). According to the AIC, models including the Gompertz slope consistently perform better than those assuming constant hazard. ...Similar publications
Citations
... The Gompertz model has also been widely applied in detecting plateau periods [2]. Multi-country comparison studies have proven useful for analyzing specific aspects of mortality, considering economic and societal differences [3]. Building on these pioneering works, this study seeks to assess several mortality models by comparing Denmark and the Netherlands as representative examples of developed European countries, aiming to improve health risk management. ...
This study provides a comparative analysis of human mortality models for Denmark and the Netherlands from 1970 to 2008, using data from the Human Mortality Database (HMD). The analysis explores mortality trends across age groups and genders, highlighting key patterns and differences between the two countries. The Lee-Carter model is applied to analyze mortality dynamics, while the Gompertz model estimates age-specific mortality rates, both proving effective in capturing long-term trends. The study finds consistent differences between male and female mortality, with men experiencing higher mortality rates across all age groups in both countries. Mortality rates have declined significantly over time due to advancements in healthcare, socioeconomic improvements, and better living conditions. To project future mortality trends, the Random Walk with Drift model is used, offering insights into longevity and life expectancy for both populations. The results emphasize the need for ongoing public health improvements and policies to address the challenges posed by aging populations
... In the gamma-Gompertz-Makeham setting, the parameter σ 2 captures the unobserved individual heterogeneity (Vaupel, Manton, and Stallard 1979), which implies in a levelling-off on the population's mortality rates at the oldest ages (see, for example, Missov and Vaupel 2015;Barbi, Lagona, Marsili, Vaupel, and Wachter 2018). Therefore, the difference between the estimates of σ 2 under the Bell and the Poisson assumptions may be due to three main aspects of the mortality data: (i) the number of people alive at the oldest ages is small; therefore, the mortality rates at those ages present a high variability around the mortality plateau (Barbi et al. 2018), which the Poisson distribution cannot accommodate; (ii) the number of males alive at the oldest ages is much smaller than the number of females (see, for example, Alvarez, Villavicencio, Strozza, and Camarda 2021;Dang, Camarda, Meslé, Ouellette, and Vallin 2023), which also increases uncertainty on σ 2 ; and (iii) the postponement of mortality (Vaupel 2010) implicates on a postponement of the mortality deceleration, and since the deaths after age 110 are grouped at age 110, the leveling-off of the risk of dying cannot be seen in all the recent populations (Dang et al. 2023). Finally, the difference between the estimates for σ 2 observed in Figure 4 also leads to different mortality plateaus. ...
... In the gamma-Gompertz-Makeham setting, the parameter σ 2 captures the unobserved individual heterogeneity (Vaupel, Manton, and Stallard 1979), which implies in a levelling-off on the population's mortality rates at the oldest ages (see, for example, Missov and Vaupel 2015;Barbi, Lagona, Marsili, Vaupel, and Wachter 2018). Therefore, the difference between the estimates of σ 2 under the Bell and the Poisson assumptions may be due to three main aspects of the mortality data: (i) the number of people alive at the oldest ages is small; therefore, the mortality rates at those ages present a high variability around the mortality plateau (Barbi et al. 2018), which the Poisson distribution cannot accommodate; (ii) the number of males alive at the oldest ages is much smaller than the number of females (see, for example, Alvarez, Villavicencio, Strozza, and Camarda 2021;Dang, Camarda, Meslé, Ouellette, and Vallin 2023), which also increases uncertainty on σ 2 ; and (iii) the postponement of mortality (Vaupel 2010) implicates on a postponement of the mortality deceleration, and since the deaths after age 110 are grouped at age 110, the leveling-off of the risk of dying cannot be seen in all the recent populations (Dang et al. 2023). Finally, the difference between the estimates for σ 2 observed in Figure 4 also leads to different mortality plateaus. ...
We focus on the gamma-Gompertz-Makeham model, and derive useful structural properties for this mortality model. We provide the basic properties like moments, remaining life expectancy, single life annuity, among many others, in closed form, and so it eliminates the need of evaluating them through numerical integration directly. The estimation of the gamma-Gompertz-Makeham model parameters is performed by using the maximum likelihood method under the traditional discrete Poisson distribution, as well as under the recently introduced discrete Bell distribution, which is an interesting alternative to the usual Poisson distribution, mainly in the presence of overdispersion. We illustrate the performance of the gamma-Gompertz-Makeham model in a human mortality database, and compute, based on the Poisson and Bell distributions, the remaining life expectancy, single life annuity and single assurance at ages 30, 55, and 80 in France, Italy, Japan, and Sweden, from 1947 to 2020 males and females separately.
... This is demonstrated by two recent papers that come to opposite conclusions based on IDL data (extended to ages 105+). A 2018 paper found support for the plateau model based on the Italian data subset [11], whereas a paper in 2023 using French data concluded that a return to a Gompertz trajectory is a better fit [12]. We observe that in fact the plots of French and Italian mortality data presented in these two studies look very similar with error bars increasing in the critical domain above 105 years of age to obscure the conclusion. ...
... (1) Plateau: The first is a plateau model where ( ) becomes flat for age ≥ 105 years. This model was accepted in the study of Italian data [11] but rejected in the similar study of French data [12]. (2) Gompertz: The second model is a return to a Gompertz trajectory for age ≥ 105 years where ( ) would increase exponentially. ...
... (2) Gompertz: The second model is a return to a Gompertz trajectory for age ≥ 105 years where ( ) would increase exponentially. This model was rejected in [11] but found to be a better fit in [12]. Note that this Gompertz trajectory does not match the Gompertz parameters at earlier ages, so its theoretical basis is shaky. ...
Demographic projections of maximum age to the year 2100 have produced mixed results. Mortality plateau models used in some studies have tended to predict relatively high ages over 130 years by the end of the century. We use the latest validation and count of supercentenarians in the G12 countries to test models of ageing beyond 100 years and conclude that a linearly increasing hazard model is a better fit. Plateau models starting in the age range 105 to 110 years are strongly excluded. Baseline forecasts of expected maximum age can be calculated using global projections of centenarian numbers with working assumptions that no new medical advances or global disasters would change the outlook. The linear model then leads to more modest forecasts of maximum lifespan of around 124 years by 2100. Our models are statistically in tension with the claim that Jeanne Calment lived to the outlier age of 122 years in 1997. Her claim is examined in the light of new evidence including signature samples from around 1933 when we believe an identity switch with her daughter took place.
... When observed death rates are employed instead, there is significant variability in the estimate of the AUM at older ages. Clearly, employing the AUM index for detecting mortality deceleration, and eventually the existence of a mortality plateau, has great potential; our proposed indicator could thus contribute to the current debate about the mortality plateau at the oldest ages (see, e.g., Barbi et al. 2018;Dang et al. 2023;Gampe 2021;Newman 2018). The quantification of the statistical uncertainty associated with the estimated AUM estimate should be a critical aspect to inform such analysis, especially for small sample sizes, and we plan to pursue this in our future research. ...
... However, for U. S. cohort data, it was revealed that mortality adheres to the Gompertz law within wide age range of 80-106 years [45,46]. Similar results were recently obtained using French cohort data [47]. Other scholars have found that the extent of mortality deceleration varies among different countries [48,49]. ...
The most important manifestation of aging is an increased risk of death with advancing age, a mortality pattern characterized by empirical regularities known as mortality laws. We highlight three significant ones: the Gompertz law, compensation effect of mortality (CEM), and late-life mortality deceleration and describe new developments in this area. It is predicted that CEM should result in declining relative variability of mortality at older ages. The quiescent phase hypothesis of negligible actuarial aging at younger adult ages is tested and refuted by analyzing mortality of the most recent birth cohorts. To comprehend the aging mechanisms, it is crucial to explain the observed empirical mortality patterns. As an illustrative example of data-directed modeling and the insights it provides, we briefly describe two different reliability models applied to human mortality patterns. The explanation of aging using a reliability theory approach aligns with evolutionary theories of aging, including idea of chronic phenoptosis. This alignment stems from their focus on elucidating the process of organismal deterioration itself, rather than addressing the reasons why organisms are not designed for perpetual existence. This article is a part of a special issue of the journal that commemorates the legacy of the eminent Russian scientist Vladimir Petrovich Skulachev (1935-2023) and his bold ideas about evolution of biological aging and phenoptosis.
... The plateau model is the most conservative option and any other plausible models would only decrease her prior probability for reaching her claimed age, so this criticism is unwarranted. This year a study examining updated French statistics in the IDL database concluded that a plateau was not a good model above 105 years of age (11). A Gompertz model was described as a better fit. ...
In 1997 Jeanne Calment died at a claimed age of 122 years and 164 days. The authenticity of her age was validated by Michel Allard and Jean-Marie Robine who published popular books about her case. In 2018 Nikolay Zak presented evidence that Jeanne Calment's daughter Yvonne had assumed her mother's identity. In 2019 the original validators and their colleagues defended their work and tried to refute the points of evidence made by Zak. In this comment we examine their arguments and find that they do not hold up. We provide new damning evidence in favour of the identity switch hypothesis.
... 90 years have become numerous enough to matter at all and to make the analysis of mortality kinetics in such populations reasonably reliable, the apparent deceleration of the age-associated increase in death rate among the oldest old has been attracting relentless attention, e.g. (Vaupel et al. 1998;Weitz and Fraser 2001;Bebbington et al. 2012;Lai 2012;Barbi et al. 2018;Newman 2018;Gavrilov and Gavrilova 2019;Dang et al. 2023). The phenomenon, which is often associated with the so-called late-life mortality plateaus, may be explained assuming that either (i) the age-associated increases in the individual frailty decelerate at older ages and (ii) progressive ageassociated changes in the composition of a population are such that those who are frailer initially or/and become frailer because of aging more rapidly die out Abstract Much attention in biogerontology is paid to the deceleration of mortality rate increase with age by the end of a species-specific lifespan, e.g. after ca. ...
Much attention in biogerontology is paid to the deceleration of mortality rate increase with age by the end of a species-specific lifespan, e.g. after ca. 90 years in humans. Being analyzed based on the Gompertz law µ(t)=µ0e^γt with its inbuilt linearity of the dependency of lnµ on t, this is commonly assumed to reflect the heterogeneity of populations where the frailer subjects die out earlier thus increasing the proportions of those whose dying out is slower and leading to decreases in the demographic rates of aging. Using Human Mortality Database data related to France, Sweden and Japan in five periods 1920, 1950, 1980, 2018 and 2020 and to the cohorts born in 1920, it is shown by LOESS smoothing of the lnµ-vs-t plots and constructing the first derivatives of the results that the late-life deceleration of the life-table aging rate (LAR) is preceded by an acceleration. It starts at about 65 years and makes LAR at about 85 years to become 30% higher than it was before the acceleration. Thereafter, LAR decreases and reaches the pre-acceleration level at ca. 90 years. This peculiarity cannot be explained by the predominant dying out of frailer subjects at earlier ages. Its plausible explanation may be the acceleration of the biological aging in humans at ages above 65–70 years, which conspicuously coincide with retirement. The decelerated biological aging may therefore contribute to the subsequent late-life LAR deceleration. The biological implications of these findings are discussed in terms of a generalized Gompertz-Makeham law µ(t) = C(t)+µ0e^f(t).
... For instance, it could be argued that as Italy is one of the world countries where life expectancy is highest (currently, more than 80 years for men and slightly less than 85 years for women), this deceleration may be the inescapable consequence of approaching the limits of survival. However, whether these limits exist is still a matter of debate and in all cases they have not been identified yet (Linh et al 2023). Besides, there are countries where life expectancy is even higher than it is in Italy, Japan for example, and where no marked slowdown in the progress of life expectancy has been detected. ...
In several European countries, life expectancy has progressed little in the past two decades. In this paper, focused on Italy, we investigate whether “austerity” and health regionalization may have contributed to this outcome. We show that the succession of reforms to the Italian health system introduced since the 1990s closely corresponds to discontinuities in the evolution of regional life expectancies, halting or reversing their previous trend towards convergence. This holds for both sigma and beta convergence and for both genders, albeit earlier for females.
... Clearly, there is great potential in employing the AUM index for detecting mortality deceleration, and eventually the existence of a mortality plateau; our proposed indicator could thus contribute to the current debate about the mortality plateau at the oldest ages (see, e.g. Barbi et al., 2018;Dang et al., 2023;Gampe, 2021). The uncertainty related to the AUM estimate should be a critical aspect to inform such analysis, and we plan to pursue this approach in future research. ...
In recent years, lifespan inequality has become an important indicator of population health, alongside more established longevity measures. Uncovering the statistical properties of lifespan inequality measures can provide novel insights on the study of mortality developments.We revisit the "e-dagger" measure of lifespan inequality, introduced in Vaupel and Canudas-Romo (2003). We note that, conditioning on surviving at least until age a, e-dagger(a) is equal to the covariance between the conditional lifespan random variable Ta and its transformation through its own cumulative hazard function (hence generalizing a result first noted in Schmertmann, 2020). We then derive an upper bound for e-dagger(a). Leveraging this result, we introduce the "Average Uneven Mortality" (AUM) index, a novel relative mortality index that can be used to analyze mortality patterns. We discuss some general features of the index, including its relationship with a constant ("even") force of mortality, and we study how it changes over time.The use of the AUM index is illustrated through an application to observed period and cohort death rates as well as to period life-table death rates from the Human Mortality Database. We explore the behavior of the index across age and over time, and we study its relationship with life expectancy. The AUM index at birth declined over time until the 1950s, when it reverted its trend. The index generally increases over age and reduces with increasing values of life expectancy, with differences between the period and cohort perspectives.We elaborate on Vaupel and Canudas-Romo’s e-dagger measure, deriving its upper bound. We exploit this result to introduce a novel mortality indicator, which enlarges the toolbox of available methods for the study of mortality dynamics. We also develop some new routines to compute e-dagger(a) and σ_Ta from death rates, and show that they have higher precision when compared to conventional and available functions, particularly for calculations involving older ages.
The most important manifestation of aging is an increased risk of death with advancing age, a mortality pattern characterized by empirical regularities known as mortality laws. We highlight three significant ones: the Gompertz law, compensation effect of mortality (CEM), and late-life mortality deceleration and describe new developments in this area. It is predicted that CEM should result in declining relative variability of mortality at older ages. The quiescent phase hypothesis of negligible actuarial aging at younger adult ages is tested and refuted by analyzing mortality of the most recent birth cohorts. To comprehend the aging mechanisms, it is crucial to explain the observed empirical mortality patterns. As an illustrative example of data-directed modeling and the insights it provides, we briefly describe two different reliability models applied to human mortality patterns. The explanation of aging using a reliability theory approach aligns with evolutionary theories of aging, including idea of chronic phenoptosis. This alignment stems from their focus on elucidating the process of organismal deterioration itself, rather than addressing the reasons why organisms are not designed for perpetual existence. This article is a part of a special issue of the journal that commemorates the legacy of the eminent Russian scientist Vladimir Petrovich Skulachev (1935-2023) and his bold ideas about evolution of biological aging and phenoptosis.
