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Since the beginning of the twentieth century, important transformations have occurred in the age-at-death distribution within human populations. We propose a flexible nonparametric smoothing approach based on P-splines to refine the monitoring of these changes. Using data from the Human Mortality Database for four low mortality countries, namely Ca...
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... main difficulty encountered with this approach is the following. Often, the roughness of the age distribution of life table deaths is such that the age with the highest number of deaths M * does not clearly stand out (see Figure 1). Instead, there are several age candidates, a few years apart from each other, which leads to modal ages at death estimates that are also a few years apart from each other. ...
Citations
... M captures the postponement of mortality towards older ages (Canudas-Romo, 2008). We calculated M by year, sex, and income with the nonparametric approach put forward by Ouellette and Bourbeau (2011). Because it can be arduous to estimate M when non-smoothed or erratic trends are observed, Ouellette and Bourbeau suggest using a P-Spline approach to smooth the age-atdeath distribution and find the mode. ...
In Denmark and Sweden, statutory retirement age is indexed to life expectancy to account for mortality improvements in their populations. However, mortality improvements have not been uniform across different sub-populations. Notably, in both countries, individuals of lower socioeconomic status (SES) have experienced slower mortality improvements. As a result, a uniform rise in the statutory retirement age could disproportionally affect these low-SES groups and may unintentionally lead to a reverse redistribution effect, shifting benefits from short-lived low-SES individuals to long-lived high-SES individuals. The aim of this study is twofold: to quantify and contextualise mortality inequalities by SES in Denmark and Sweden, and to assess how indexing retirement age will affect future survival to retirement age by SES in these countries. We used Danish and Swedish registry data (1988–2019), to aggregate individuals aged 50 + based on their demographic characteristics and SES. We computed period life tables by year, sex, and SES to estimate the difference in survival across different SES groups. We then forecast mortality across SES groups to assess how indexing retirement age will affect survival inequalities to retirement age, using two forecasting models—the Mode model and the Li-Lee model. Mortality inequalities are comparable in Denmark and Sweden, even though the latter generally has higher survival. We also find that indexing retirement age to life expectancy will have two main consequences: it will reduce the probability of reaching retirement for all SES groups, particularly those of low SES, and time spent in retirement will be reduced, particularly for those of high SES.
... The modal age at death has been an increasingly used indicator of longevity in recent years (Ouellette and Bourbeau 2011), but the mode's history in demography traces way back. In his essay on the normal age at death, Lexis (1878) combines two concepts: Quetelet's notion of a "normal man" and the Gaussian (normal) distribution (Véron, Rohrbasser, and Mendelbaum 2003). ...
... Smoothing procedures have also been used to estimate the mode, such as penalized spline (P-spline). Although it is a nonparametric method, P-spline smoothing (Eilers and Marx 1996) yields a continuous force of mortality and aids in estimating the mode more precisely (Ouellette and Bourbeau 2011). P-spline smoothing approximates the observed death counts by polynomial pieces that are joined in knots (B-splines) and penalizes for the number of selected knots (Psplines). ...
... P-spline smoothing approximates the observed death counts by polynomial pieces that are joined in knots (B-splines) and penalizes for the number of selected knots (Psplines). The main advantage of P-spline smoothing over other statistical estimation methods is that it finds the optimal trade-off between parsimony and fitting (Eilers and Marx 1996;Ouellette and Bourbeau 2011). However, P-spline smoothing is computationally demanding, assumes death counts are Poisson distributed, requires choosing the number of knots to be penalized, and yields unsatisfactory fits at the boundaries of the age-at-death distribution (Horiuchi et al. 2013). ...
... Modal age death corresponding to the maximum value of the density has become a better longevity indicator in low mortality population. ( Canudas-Romo 2008 [2] ,Canudas-Romo 2010 [3], Ouellette and Bourbeauet [4],Horiuchi et al. 2013 [5] , Basellini and Camarda [6], Shang and Haberman [7] , Bergeron-Boucher et al. [8]). Survival curves dimensions are used in mortality analysis to determine highest normal life duration that exceeds modal age, death around the modal age and the proportion of survivors in population (Cheung et al. [9], Ebeling et al. [10]). ...
This research work seeks to analysis the mortality trend experienced in Kenya over the sample period 1950 to 2021 using a multidimensional modeling framework. Life table functions, namely; life expectancy, survival function and age at death distribution are applied to depict mortality characteristics. Life expectancy and survival rate have significantly improved. There has been a shift in mortality status from a high mortality population, to an intermediate stage and mortality risk factors have increased across age. Mortality concentration curve and index within the Lorenz curve and Gini coefficient framework are used to analyze the lifespan inequality. Lifespan inequality is high with negligible improvements over time. Gompertz force of mortality is then estimated, which is statistically significant at 5% level. Deaths at exact age 25 is about 35 per ten thousand, with the rate death rate increasing by 6.09% per year starting from age 25. Under the assumptions of stable population, where the mortality and fertility functions are independent of time, Malthusian parameter is estimated which is less than zero for selected years. Kenya is a shrinking population and death rate decrease with increase in Malthusian parameter. Finally, to model long-term mortality rate forecast, Lee-Carter model is estimated. The model is statistically significant at 5% level explaining 78.4% of the variations. Expected life expectancy at a given age is projected to increase, with life expectancy at birth in 2030 and 2071 being 65.6 and 70.5 years respectively.
... A change in the modal age can only occur with pulling forces, indicating mortality improvement at ages older than the mode (Kannisto 2000;Canudas-Romo 2010, Bergeron-Boucher et al., 2015. Notably, the modal age at death has shown an accelerated increase since the onset of the old-age mortality decline, gaining prominence as a key indicator of lifespan, particularly since longevity extension has become determined by adult and old-age mortality in the 21st century (Kannisto 2000(Kannisto 2001Bongaarts 2005;Cheung and Robine 2007;Canudas-Romo 2008Ouellette and Bourbeau 2011;Horiuchi et al. 2013, Bergeron-Boucher et al., 2015. ...
This article explores the demographic landscape of Türkiye from 1920 to 2020 using life tables, focusing on adult mortality trends and employing the modal age at death as a key metric. It emphasizes the shifts in mortality patterns, particularly in relation to life expectancy, and addresses the historical and contextual factors influencing Türkiye's demographic transition. The literature review underlines the significance of the modal age at death as an indicator for assessing mortality dynamics, offering insights into compression and delay in mortality. The study utilizes a comprehensive methodology, including the acquisition of infant mortality rates, life table construction, and the calculation of modal age at death and standard deviation. The results highlight a remarkable increase in life expectancy over the century, driven by improvements in healthcare and reductions in infant and child mortality. The analysis of the modal age at death reveals trends of mortality delay and compression, with the decline in the standard deviation calculated for the modal age at death indicating a gradual shift of mortality to older ages. The findings align with Türkiye's demographic transition stages, emphasizing the evolving health landscape and the importance of considering modal age at death alongside life expectancy for a nuanced understanding of the trends of adult mortality. This study bridges a significant gap in the existing research on Türkiye by utilizing the modal age at death to assess older age mortality trends.
... Other pertinent examples include the analysis of sequences of age-at-death distributions over calendar years, which is instrumental for the study of human longevity (Mazzuco & Scarpa, 2015;Ouellette & Bourbeau, 2011;Shang & Hyndman, 2017) and also the study of the distributions of correlations between pairs of voxels within brain regions that can be derived from fMRI Bold signals (Petersen & Müller, 2016), where such distributions may be observed repeatedly for the same subject in longitudinal studies. ...
Series of univariate distributions indexed by equally spaced time points are ubiquitous in applications and their analysis constitutes one of the challenges of the emerging field of distributional data analysis. To quantify such distributional time series, we propose a class of intrinsic autoregressive models that operate in the space of optimal transport maps. The autoregressive transport models that we introduce here are based on regressing optimal transport maps on each other, where predictors can be transport maps from an overall barycenter to a current distribution or transport maps between past consecutive distributions of the distributional time series. Autoregressive transport models and their associated distributional regression models specify the link between predictor and response transport maps by moving along geodesics in Wasserstein space. These models emerge as natural extensions of the classical autoregressive models in Euclidean space. Unique stationary solutions of autoregressive transport models are shown to exist under a geometric moment contraction condition of Wu & Shao [(2004) Limit theorems for iterated random functions. Journal of Applied Probability 41, 425–436)], using properties of iterated random functions. We also discuss an extension to a varying coefficient model for first-order autoregressive transport models. In addition to simulations, the proposed models are illustrated with distributional time series of house prices across U.S. counties and annual summer temperature distributions.
... It is the mean value of the lifespan distribution from the life table, and a comparison of life expectancies can be used to indicate the average number of years by which a group of individuals outlives another group. Other measures of central tendency, such as the modal and median age at death, have long been included in mortality analysis and have enriched demographers' understanding of the evolution of human longevity (Lexis 1878;Kannisto 2001;Cheung and Robine 2007;Canudas-Romo 2008, 2010Ouellette and Bourbeau 2011;Cohen and Oppenheim 2012;Horiuchi et al. 2013). Recent research on mortality differences between population subgroups has started to incorporate these additional central tendency measures beyond the traditional comparisons of life expectancy (Brown et al. 2012;Zarulli et al. 2012;Diaconu et al. 2022). ...
The study of the mortality differences between groups has traditionally focused on metrics that describe average levels of mortality, for example life expectancy and standardized mortality rates. Additional insights can be gained by using statistical distance metrics to examine differences in lifespan distributions between groups. Here, we use a distance metric, the non-overlap index, to capture the sociological concept of stratification, which emphasizes the emergence of unique, hierarchically layered social strata. We show an application using Finnish registration data that cover the entire population over the period from 1996 to 2017. The results indicate that lifespan stratification and life-expectancy differences between income groups both increased substantially from 1996 to 2008; subsequently, life-expectancy differences declined, whereas stratification stagnated for men and increased for women. We conclude that the non-overlap index uncovers a unique domain of inequalities in mortality and helps to capture important between-group differences that conventional approaches miss.
... Defined as the age at which the maximum number of adult deaths occurs in a synthetic cohort of individuals experiencing similar mortality conditions, this indicator is less sensitive to improvements in mortality conditions in children and young adults compared to life expectancy at birth, which is highly sensitive to premature mortality (Canudas-Romo, 2008Horiuchi et al., 2013;Ouellette et al., 2012). In low-mortality countries characterized by aging populations, many studies have been devoted to the analysis of this indicator and the dynamic of its evolution over time and space (Canudas-Romo, 2008Cheung et al., 2005;Kannisto, 2000Kannisto, , 2001bKannisto, , 2007Missov et al., 2015;Ouellette, 2011;Ouellette et al., 2013;Ouellette & Bourbeau, 2011;Thatcher et al., 2010). This research has paved additional avenues for analyzing older adult mortality, which is still poorly studied in high-mortality countries. ...
... the variability of deaths around or above M(Canudas-Romo, 2010;Horiuchi et al., 2013;Ouellette & Bourbeau, 2011). The location of M is defined as the level of normal or natural life 10 Intended for publication span as understood by Lexis. ...
Since independence, efforts to reduce mortality have focused on child and maternal mortality, which is considered too high. Subsequently, analyses of maternal mortality have been extended to adults in general (15 to 50 or even 60 years old). Beyond these ages, mortality studies remain poorly documented due to the lack of adequate civil registration. The numerous errors in age are sources of bias and make it difficult to produce estimates. The aim of this doctoral dissertation is therefore to contribute to improving the state of knowledge on the mortality of adults aged 50-79. The option of restricting the analyses to the under-80s is intended to limit the effect of age errors which can be extreme at very old ages. Starting with a review of existing methods, we examine the extent to which a judicious use of these methods can make it possible to estimate the levels and trends of mortality in older adults by exploiting different sources (censuses, household sample surveys, population surveillance systems) available for several countries in the sub-Saharan region.
... Thus, deaths were more dispersed in men than in women. Unlike women, men did not experience convergence and compression in mortality (Ouellette and Bourbeau 2011;Yadav and Perianayagam 2020, p. 350). The wide sex differentials in the adult mortality have been much apparent in both demographically advanced and backward states. ...
India has seen a reduction in infant and child mortality rates for both the sexes since the early 1980s. However, a decline in mortality at adult ages is marked by significant differences in the subgroups of sex and regions. This study assesses the progress of inequality in age at death with the advances in mortality transition during 36 years period between 1981–1985 and 2012–2016 in India, using the Gini coefficients at the age of zero (G0). The Gini coefficients show that in the mid-2000s, women outpaced men in G0. The reduction in inequality in age at death is a manifestation of the process of homogeneity in mortality. The low G0 is concomitant of high life expectancy at birth (e0) in India. The results show the dominance of adult mortality over child mortality in the medium-mortality and low-mortality regimes. Varying adult mortality in the subgroups of sex and variance in the mortality levels of regions are the predominant factors for the variation in inequality in age at death. By lowering of the mortality rates in the age group of 15–29 years, India can achieve a high e0 that appears at high demographic development and the narrow sex differentials in e0 and G0 in a short time. Men in the age group of 15–29 years are the most vulnerable subgroup with respect to mortality. There is an immediate need for health policies in India to prioritise the aversion of premature deaths in men aged 15–29 years.
... Other pertinent examples include the analysis of sequences of age-at-death distributions over calendar years, which is instrumental for the study of human longevity (Mazzuco and Scarpa 2015;Shang and Hyndman 2017;Ouellette and Bourbeau 2011) and also the study of the distributions of correlations between pairs of voxels within brain regions that can be derived from fMRI Bold signals (Petersen and Müller 2016), where such distributions may be observed repeatedly for the same subject in longitudinal studies that include fMRI brain imaging and where measurements are taken at regular time intervals. ...
Series of distributions indexed by equally spaced time points are ubiquitous in applications and their analysis constitutes one of the challenges of the emerging field of distributional data analysis. To quantify such distributional time series, we propose a class of intrinsic autoregressive models that operate in the space of optimal transport maps. The autoregressive transport models that we introduce here are based on regressing optimal transport maps on each other, where predictors can be transport maps from an overall barycenter to a current distribution or transport maps between past consecutive distributions of the distributional time series. Autoregressive transport models and associated distributional regression models specify the link between predictor and response transport maps by moving along geodesics in Wasserstein space. These models emerge as natural extensions of the classical autoregressive models in Euclidean space. Unique stationary solutions of autoregressive transport models are shown to exist under a geometric moment contraction condition of Wu and Shao (2004), using properties of iterated random functions. We also discuss an extension to a varying coefficient coefficient for first order autoregressive transport models. In addition to simulations, the proposed models are illustrated with distributional time series of house prices across U.S. counties and of stock returns across the S&P 500 stock index.
... Then, premature mortality designates the transition region between childhood and adult deaths. This identification of the adult modal age at death has been used to understand the development of mortality across the twentieth century (Bongaarts 2005;Cheung et al. 2005Cheung et al. , 2009Canudas-Romo 2008, 2010Cheung and Robine 2007;Horiuchi et al. 2013;Kannisto 2000Kannisto , 2001Ouellette and Bourbeau 2011;Wilmoth and Horiuchi 1999;Wilmoth and Robine 2003). Pearson (1897) evaluated the problem from a statistical point of view: taking Lexis' idea even further and considering the distribution of deaths to be composed of five functions with different degrees of skewness. ...
Premature mortality is often a neglected component of overall deaths, and the most difficult to identify. However, it is important to estimate its prevalence. Following Pearson’s theory about mortality components, a definition of premature deaths and a parametric model to study its transformations are introduced. The model is a mixture of three distributions: a Half Normal for the first part of the death curve and two Skew Normals to fit the remaining pieces. One advantage of the model is the possibility of obtaining an explicit equation to compute life expectancy at birth and to break it down into mortality components. We estimated the mixture model for Sweden, France, East Germany and Czech Republic. In addition, to the well-known reduction in infant deaths, and compression and shifting trend of adult mortality, we were able to study the trend of the central part of the distribution of deaths in detail. In general, a right shift of the modal age at death for young adults is observed; in some cases, it is also accompanied by an increase in the number of deaths at these ages: in particular for France, in the last twenty years, premature mortality increases.
