One or more rates of ageing? The extended gamma-Gompertz model (EGG)
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One or more rates of ageing? The extended gamma-Gompertz model (EGG) Giambattista Salinari1
· Gustavo De Santis2
Accepted: 12 May 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract Hidden heterogeneity poses serious challenges to survival analysis because the observed (aggregate) and the unobservable (individual) hazard functions may differ markedly from each other. However, the recent discovery of the so-called “mortality plateau” (i.e., the approximately constant value when mortality levels off, at very old ages) has brought new insights and pushed researchers towards the use of the gammaGompertz mortality model. Among the assumptions of this model, two are particularly relevant here: the shape, not the level, of the individual hazard function is a constant and so is the rate of ageing, i.e., the relative increase in mortality risks as people get older. The latter, however, does not pass empirical tests: the rate of ageing seems to vary (albeit only slightly) by age, gender, birth cohort and country. In this paper, we propose a new model (EGG, or extended gamma-Gompertz) which overcomes this limitation by allowing the rate of ageing to increase gradually with age before converging to a constant value, as in Gompertz. While preserving all the fine theoretical and empirical properties of its simpler predecessor, the EGG model adapts better to empirical reality, i.e., in this paper, the mortality profile of the cohorts born between 1820 and 1899 in five countries with high-quality data. The advantages of this more refined mortality model are discussed. Keywords Mortality models · Gompertz · Gamma-Gompertz · Frailty · Selection bias
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Gustavo De Santis [email protected]
1
Dipartimento di scienze economiche e aziendali, University of Sassari, Via Muroni, 25, 07100 Sassari, Italy
2
DiSIA - Dip. di Statistica, Informatica, Applicazioni, University of Florence, Viale Morgagni, 59, 50134 Florence, Italy
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G. Salinari, G. De Santis
1 Introduction Since the first formulation of the proportional hazards model (Cox 1972), the concept of hazard function has acquired a pivotal role in survival analysis. This is due both to its interpretability (the instantaneous risk of experiencing death1 on part of those who survived to time t) and its mathematical tractability, which, among other advantages, permits researchers to overcome the problems that derive from data incompleteness, such as censoring and truncation (Hanagal 2011). Two broad ways of introducing covariates in hazard models can be distinguished (Finkelstein and Esaulova 2006). In additive and proportional models, the effect of a given covariate (e.g., being a smoker or doing regular gym) is to shift the entire hazard (or log-hazard) function up- or downwards. Instead, in accelerated failure time models, covariates are supposed to change the slope of the hazard function, i.e. how it varies with age (Kalbfleisch and Prentice 2002). Sometimes, a direct look at the data may suggest the best model, but this approach is justified
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