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A Biometrics Invited Paper with Discussion: The Natural Variability of Vital Rates and Associated Statistics
Abstract
The first concern of this work is the development of approximations to the distributions of crude mortality rates, age-specific mortality rates, age-standardized rates, standardized mortality ratios, and the like for the case of a closed population or period study. It is found that assuming Poisson birthtimes and independent lifetimes implies that the number of deaths and the corresponding midyear population have a bivariate Poisson distribution. The Lexis diagram is seen to make direct use of the result. It is suggested that in a variety of cases, it will be satisfactory to approximate the distribution of the number of deaths given the population size, by a Poisson with mean proportional to the population size. It is further suggested that situations in which explanatory variables are present may be modelled via a doubly stochastic Poisson distribution for the number of deaths, with mean proportional to the population size and an exponential function of a linear combination of the explanatories. Such a model is fit to mortality data for Canadian females classified by age and year. A dynamic variant of the model is further fit to the time series of total female deaths alone by year. The models with extra-Poisson variation are found to lead to substantially improved fits.
... Our model describes the evolution of the age of the oldest living person under the following assumptions. We assume that the births of individuals follow a Poisson process with time-dependent intensity [26]. The lifespans of individuals in the population are independent and their distribution may depend on the date of birth. ...
... The explanation of the left-hand side of (25) is that the person died at time t i at age y i had age E at time t i − y i + E. The previous data point ensures that this person has already reached age z i−1 hence we condition their lifetime distribution on this fact. The transition probabilities in (26) are obtained similarly to (2) with the difference that a person at age u with u ∈ [x, y] at time t had age E at time t − u + E. ...
Human longevity leaders with remarkably long lifespan play a crucial role in the advancement of longevity research. In this paper, we propose a stochastic model to describe the evolution of the age of the oldest person in the world by a Markov process, in which we assume that the births of the individuals follow a Poisson process with increasing intensity, lifespans of individuals are independent and can be characterized by a gamma-Gompertz distribution with time-dependent parameters. We utilize a dataset of the world's oldest person title holders since 1955, and we compute the maximum likelihood estimate for the parameters iteratively by numerical integration. Based on our preliminary estimates, the model provides a good fit to the data and shows that the age of the oldest person alive increases over time in the future. The estimated parameters enable us to describe the distribution of the age of the record holder process at a future time point.
... The standard approach to estimate mortality models was presented by Brillinger (1986), where a Poisson distribution is assumed for the number of deaths ...
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.
... By modeling death counts across discrete age intervals as outcomes of a multinomial experiment, we align our approach with the categorical nature of age-specific mortality data. This method accommodates the variability of death counts across different ages and accounts for the fixed total number of deaths, providing an alternative to the commonly used Poisson assumption for modeling death counts (Brillinger, 1986). ...
We present a novel method for estimating the probability distribution of the modal age at death - the age at which the highest number of deaths occurs in a population. Traditional demographic methods often relies on point estimates derived from parametric models or smoothing techniques, which may overlook the inherent variability and uncertainty in mortality data. By contrast, our approach models death counts across age intervals as outcomes of a multinomial distribution, aligning with the categorical nature of mortality data. By applying a Gaussian approximation, we make the computation of modal age probabilities feasible. While this probabilistic method offers a robust approach to analyzing mortality data, we acknowledge its limitations, particularly the assumption of independent deaths, which may not hold during events like epidemics or when social factors significantly influence mortality.
... To estimate the parameters, we apply a Bayesian procedure (Gelman et al. 2013) and assume a Poisson distribution for the death counts D ijk , where the multi-index ijk represents age i in year j for country k (see, for example, Brillinger 1986). The Bayesian estimates are obtained by the mode of the posterior distributions, also known as the maximum a posteriori probability (MAP) estimate (Patricio and Missov 2023). ...
... The previous data point ensures that this person has already reached age z i−1 hence we condition their lifetime distribution on this fact. The transition probabilities in (26) are obtained similarly to (2) with the difference that a person at age u with u ∈ [x, y] at time t had age E at time t − u + E. ...
Human longevity leaders with remarkably long lifespan play a crucial role in the advancement of longevity research. In this paper, we propose a stochastic model to describe the evolution of the age of the oldest person in the world by a Markov process, in which we assume that the births of the individuals follow a Poisson process with increasing intensity, lifespans of individuals are independent and can be characterized by a gamma–Gompertz distribution with time-dependent parameters. We utilize a dataset of the world’s oldest person title holders since 1955, and we compute the maximum likelihood estimate for the parameters iteratively by numerical integration. Based on our preliminary estimates, the model provides a good fit to the data and shows that the age of the oldest person alive increases over time in the future. The estimated parameters enable us to describe the distribution of the age of the record holder process at a future time point.
... Crude mortality rates per 10 million persons were calculated as the number of deaths divided by the population at-large multiplied by 10 million. For years with death counts below 16, CDC WONDER reported the crude rate results as unstable due to having poor reliability (Brillinger 1986). Despite this limitation, the mortality rates were calculated for inclusion in the regression models and averages. ...
Climate change has been linked to an increased prevalence of lightning activity associated with hazardous storm formation. An increased frequency in lightning activity in the United States (US) would presumably increase the risk and incidence for lightning-related injuries and fatalities. Using cause of death data available from the US Centers for Disease Control and Prevention (CDC) via their WONDER online databases and the National Weather Service, the results showed a steady decline in the lightning-related mortality incidence and incidence rates in the US from 1979–2022. In the early 1980s, U.S. death rates were near four per 10 million, and are now near one per 10 million. Death rates in the American South are higher, but have also experienced improvement, having fallen from near six per 10 million in the early 1980s to two per 10 million in recent years. A limitation of this study is a lack of data on lightning-related injuries. Overall, the steady decline in lightning-related deaths in the US likely is related to a multitude of factors including early storm reporting, improved risk communication, improved public education regarding the hazard, increased urbanization and mechanization of outdoor work, improvements in emergency medical care, among other weather-related risk reduction strategies. These results highlight the success of risk reduction strategies for reducing lightning-associated mortality in the US while the threat of hazardous lightning has likely continued to increase.
Purpose
The growth in life expectancy (LE) slows down recently in several high-income countries. Among the underlying dynamics, uneven progress in LE across social groups has been pointed out. However, these dynamics has not been extensively studied, partly due to data limitations. In this paper, we explore this area for the 2010 decade using recent French data.
Methods
We utilize the recent change in French census mortality follow-up data (EDP) and apply P-spline models to estimate LEs across five occupational classes (OCs) and indicators of lifespan heterogeneity (edagger) within these OCs, for seven triennial periods (2011-2013 to 2017-2019).
Results
First, we found a similar ranking of OCs along the LE gradient over time and across sexes, from manual workers to higher-level OCs. Noteworthy, the lowest LE in women overlaps with the highest one in men drawing a sex-OC gradient. Second, we observe varying progress of LEs. In women, LE increases in higher-level OCs meanwhile it levels off in manual workers, so that the OCs gap widens (up to 3.4 years in 2017-2019). Conversely, in men LE stalls in higher-level OCs and increases in manual workers so that the gap, which is much larger than in women (+5.7 years in 2017-2019), is tending to narrow. Finally, the lifespan homogenizes in OCs only when LE is low.
Conclusion
Overall, the limited LE progress in France results from LE stalling in the middle of the sex-OC gradient, though LE increases at both ends. At the lower end, LE progress and lifespan homogenization suggest that laggards benefit recently improvements achieved earlier in other OCs. At the upper end, LE progress may come from a vanguard group within higher-lever OC, benefiting new sources of improvements. These findings underscore the need for further research to explore the diverse mortality dynamics coexisting in the current health landscape.
Background
Prostate cancer is the most common cancer among men in the United States, yet modifiable risk factors remain elusive. In this study, the authors investigated the potential role of agricultural pesticide exposure in prostate cancer incidence and mortality.
Methods
For this environment‐wide association study (EWAS), linear regression was used to analyze county‐level associations between the annual use of 295 distinct pesticides (measured in kg per county) and prostate cancer incidence and mortality rates in the contiguous United States. Data were analyzed in two cohorts: 1997–2001 pesticide use with 2011–2015 outcomes (discovery) and 2002–2006 use with 2016–2020 outcomes (replication). The reported effect sizes highlight how a 1‐standard‐deviation increase in log‐transformed pesticide use (kg per county) corresponds to changes in incidence. Analyses were adjusted for county‐level demographics, agricultural data, and multiple testing.
Results
Twenty‐two pesticides showed consistent, direct associations with prostate cancer incidence across both cohorts. Of these, four pesticides were also associated with prostate cancer mortality. In the replication cohort, each 1‐standard‐deviation increase in log‐transformed pesticide use corresponded to incidence increases per 100,000 individuals (trifluralin, 6.56 [95% confidence interval (CI), 5.04–8.07]; cloransulam‐methyl, 6.18 [95% CI, 4.06–8.31]; diflufenzopyr, 3.20 [95% CI, 1.09–5.31]; and thiamethoxam, 2.82 [95% CI, 1.14–4.50]). Limitations included ecological study design, potential unmeasured confounding, and lack of individual‐level exposure data.
Conclusions
The results of this study suggest a potential link between certain pesticides and increased prostate cancer incidence and mortality. These findings warrant further investigation of these specific pesticides to confirm their role in prostate cancer risk and to develop potential public health interventions.
شهد العقد الاخير من القرن العشرين اهتماما كبيرا بنماذج بواسون , الذي يعد احد نماذج العد التي تتعامل مع المتغير متقطع كمتغير معتمد. لقد واجه الباحثون عدد من التحديات لتطبيق هذا النموذج في عالم البيانات الحقيقية. اذ أظهرت بعض البيانات عدم تجانس عالي مما سبب ظهور مشكلة فرط التشتت. أُقتُرِح نموذج شبه بواسون ونموذج ذي الحدين السالب كنماذج بديلة لنموذج بواسون, بوجود مشكلة فرط التشتت. سعت البحث الى دراسة المقارنة بين نموذج بواسون الكامل ونموذج بواسون بعد حذف عدد من المتغيرات غير المهمة , مع نموذج شبه بواسون وذو الحدين السالب بعد اختيار افضل المتغيرات , تم تنفيذ ذلك من خلال بيانات مرضى الثلاسيميا.
A great deal of research has been carried out into expected values of q x or into expected number of deaths. Little attention has however been paid to the random variations in mortality rates or to the random variations in the number of deaths. Research in this direction might very well further our knowledge of the mortality process.
Overall mortality rates of sub-populations are frequently adjusted for factors like sex and age to make them more directly comparable. In this paper we are concerned with the problem of adjusting various early infant mortality rates of the Local Authorities of England and Wales for variations in their circumstances. For example, the proportion of infants of low birth weight (which is well correlated with perinatal mortality rate) shows a strong North-South contrast, and this needs to be taken into account. The usual direct and indirect methods of adjustment are not suitable here since we intend to correct for a relatively large number of factors. Instead we employ regression analysis, the residuals in effect being the corrected mortality rates. In most regression analyses chief interest lies in the estimated model, residuals often being discarded as worthless once the fit has been judged adequate and error estimated. In this paper by contrast they are of prime importance and we go to considerable lengths to ensure that the "order of merit" table that they produce is robust. With a stepwise procedure to ensure that we correct only for "worthwhile" explanatory variables out of our list of 86, Bath and Carlisle are identified as having extremely high corrected perinatal mortality rates and Hull and Hartlepool exceptionally low. Lists of outstanding Authorities change with the mortality rate used quite strongly. In a few cases we find that Authorities with low corrected rates for one grade of mortality (late neonatal, say) have high corrected rates for another (postneonatal, for example). "Explanation" levels are generally high, sometimes reaching 90 per cent or more. As for "important" regressor variables, it is found that those describing social and economic circumstances dwarf those representing medical services for all of the mortality rates used, though to be fair, these are inadequately represented in our data set. As far as perinatal mortality rate is concerned, the proportion of low weight infants, not surprisingly, turns out to be all-important.
