A New Look at Entropy and the Life Table (original) (raw)
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Two New Mathematical Equalities in the Life Table
Canadian Studies in Population
This paper discusses known mathematical equalities and inequalities found within life tables and proceeds to identify two new inequalities. The first (theorem 1) is that at any given age x, the sum of mean years lived and mean years remaining exceeds life expectancy at birth when age is greater than zero and less than the maximum lifespan. The second inequality (theorem 2) applies to the entire population and shows that the sum of mean years lived and mean years remaining exceeds life expectancy at birth. Illustrations of the two inequalities are provided as well as a discussion. Résumé Cet article examine les égalités et les inégalités mathématiques connues dans les tables de mortalité et procède à l'identification de deux nouvelles inégalités. La premiére (théorème 1) est que,à tout âge x donné, la somme d'années moyennes vécues et d'années moyennes restantes dépasse l'espérance de vie à la naissance lorsque l'âge est supérieur à zéro et inférieur à la durée de vie maximale. La deuxième inégalité (théorème 2) s'applique à l'ensemble de la population et montre que la somme d'années moyennes vécues et d'années moyennes restantes dépasse l'espérance de vie à la naissance. Des illustrations des deux inégalités sont fournies ainsi qu'une discussion. Keywords Carey's Equality Theorem. Life years lost. Life expectancy at birth. Mean years lived. mean years remaining. variance in age at death Mots-clé théoréme d'égalité de Carey. années de vie perdues. espérance de vie à la naissance. nombre d'années vécues. nombre d'années restantes. variance de l'âge au décès
Statistics in Medicine, 1988
How much unique information is contained in any life table? The logarithmic survivorship (l?) columns of 360 empirical life tables were fitted by a weighted fifth degree polynomial, and it is shown that SIX parameters are adequate to reproduce these curves almost flawlessly. However, these parameters are highly intercorrelated, so that a two-dimensional representation would be adequate to express the similarities and differences among life tables. It is thus concluded that a life table contains but two unique pieces of information, these being the level of mortality in the population which it represents, and the relative shape of the underlying mortality curve.
Experimental Gerontology, 2010
Historical human mortality curves display 5 phases, differing in dimensions with population, time and circumstance. Existing explanatory models describe some but not all of these, and modelling of entire curves has hitherto necessitated an assumption of multiple distributions.
Linking the population growth rate and the age-at-death distribution
Theoretical Population Biology, 2012
The population growth rate is linked to the distribution of age at death. We demonstrate that this link arises because both the birth and death rates depend on the variance of age-at-death. This bears the prospect to separate the influences of the age patterns of fertility and mortality on population growth rate. Here, we show how the age pattern of death affects population growth. Using this insight we derive a new approximation of the population growth rate that uses the first and second moments of the age-at-death distribution. We apply our new approximation to 46 mammalian life tables (including humans) and show that it is on par with the most prominent other approximations.
A note on the compression of mortality
The rapid increase of human longevity has brought up important questions about what implications it may have for the variability of age at death. Earlier works reported evidence of a historical trend of mortality compression. However, the period life table model, which is widely used to address mortality compression, produces an artificially compressed picture of mortality as a built-in feature of the model. We base our study on examining the durations of exposure of birth cohorts (also as compared to period mortality schedules) to selected levels of mortality observed at old age. We also address the problem in a more conventional fashion, by examining the distribution of ages at death (in period tables and cohorts) above and below the mode. Overall, mortality has been significantly decompressing already since the 1970s. This finding contradicts with most previously reported results. The decompression of old-age mortality may indicate further optimistic prospects of ever-decreasing mortality. Mortality may well not be concentrated in the future within a narrow age interval but more dispersed along age groups, though at ever later ages on average.
Expectation of life at old age predicted from a single death rate: Models and applications
2017
This paper introduces empirical relations between the death rate at a given age and the remaining life expectancy at that same age. The relations prove to be of prediction accuracy exceeding that of the common alternative, extrapolation of the death rates into older ages based on data at younger ages. Being close in accuracy to models by Horiuchi, Coale and Mitra, the proposed models may be of use in cases when the latter models may not be applied because of either lack of data on old-age mortality or violation of the underlying assumptions, such as population stability. Combining the proposed models with constrained extrapolations of old-age mortality will be a useful tool in estimating and projecting old-age mortality, completing life tables for young cohorts and extending model and empirical life tables to old age.
At Modal Age at Death, the Hazard Rate is Determined by its Derivative
Institut für Demographie - VID, 2021
The relation between the hazard rate and its derivative at modal age at death, an equivalent to which has been featured by Pollard (1991), Canudas-Romo (2008), Thatcher et al. (2010), and Tuljapurkar and Edwards (2011), is presented as a handy tool in studying mortality compression in its period and cohort dimensions. Our analytical findings indicate birth cohorts to differ substantially from the period life tables with respect to the distribution of deaths around the mode. Empirical results support theoretical predictions and show that previously reported effect of compression of deaths above the mode might be a feature of period life tables and not of the cohort mortality schedules. Our results are also useful in computing the modal age at death and sensitivity analysis.