A matrix approach to the statistics of longevity in the gamma-Gompertz and related mortality models

Abstract

BACKGROUNDThe gamma-Gompertz model is a fixed frailty model in which baseline mortality increases exponentially with age, frailty has a proportional effect on mortality, and frailty at birth follows a gamma distribution. Mortality selects against the more frail, so the marginal mortality rate decelerates, eventually reaching an asymptote. The gammaGompertz is one of a wider class of frailty models, characterized by the choice of baseline mortality, effects of frailty, distributions of frailty, and assumptions about the dynamics of frailty. OBJECTIVETo develop a matrix model to compute all the statistical properties of longevity from the gamma-Gompertz and related models. METHODSI use the vec-permutation matrix formulation to develop a model in which individuals are jointly classified by age and frailty. The matrix is used to project the age and frailty dynamics of a cohort and the fundamental matrix is used to obtain the statistics of longevity. RESULTSThe model permits calculation of the mean, variance, coefficient of variation, skewness and all moments of longevity, the marginal mortality and survivorship functions, the dynamics of the frailty distribution, and other quantities. The matrix formulation extends naturally to other frailty models. I apply the analysis to the gamma-Gompertz model (for humans and laboratory animals), the gamma-Makeham model, and the gamma-Siler model, and to a hypothetical dynamic frailty model characterized by diffusion of frailty with reflecting boundaries.The matrix model permits partitioning the variance in longevity into components due to heterogeneity and to individual stochasticity. In several published human data sets, Caswell: A matrix approach to the statistics of longevity in heterogeneous frailty models heterogeneity accounts for less than 10% of the variance in longevity. In laboratory populations of five invertebrate animal species, heterogeneity accounts for 46% to 83% of the total variance in longevity. IntroductionThe gamma-Gompertz (hereafter, G-G) model is one of a class of models that investigate the effects of hidden heterogeneity -heterogeneity that is either unobservable or unobserved -on mortality rates. It calculates various consequences of that heterogeneity for the dynamics of cohorts (e.g. Vaupel, Manton, and Stallard 1979;Vaupel and Yashin 1985;Yashin, Iachine, and Begun 2000;Vaupel 2010;Wienke 2010;Missov 2013). I refer to this heterogeneity as frailty, making no assumptions about its causes. The goal of this paper is to present a matrix formulation that permits easy computation of all the properties of the G-G and other frailty models. Frailty models can be characterized by their components:1. the baseline mortality rate, 2. the effects of frailty on the baseline mortality rate, 3. the dynamics of individual frailty over time, and 4. the initial distribution of frailty.The baseline mortality rate in the G-G model is the Gompertz model, in which mortality increases exponentially with age t, µ(t) = ae bt .Frailty affects mortality as a proportio...

Lifetime reproduction and the second demographic transition: Stochasticity and individual variation

“…Our approach makes it possible to explore the addition of heterogeneity to the model. This can be achieved by developing multistate models that include more details in the reproductive process (Caswell 2014b) or by linking the results to Markov chain models incorporating heterogeneous frailty (Caswell 2014a). Sensitivity analysis of these models will show how the statistics of LRO respond to changes in the parameters of the mortality and fertility schedules .…”

“…Individual stochasticity (Caswell 2009) is the random variation among individuals in the outcomes of applying identical vital rates. Individual stochasticity is inherent in any set of mortality and fertility rates, and given those rates, we can calculate the consequences of these stochastic events (Caswell 2009(Caswell , 2011(Caswell , 2014aCaswell and Kluge 2015). Individual stochasticity has been found to be a major contributor to variance in LRO in many species (Caswell 2011;Tuljapurkar, Steiner, and Orzack 2009;Steiner and Tuljapurkar 2012).…”

“…To calculate the consequences of this heterogeneity requires a model that incorporates the heterogeneity, as frailty models do in the study of mortality (e.g., Vaupel, Manton, and Stallard 1979;Caswell 2014a). The calculated variance due to individual stochasticity can serve as a null model for comparison with observed measures of variance (Tuljapurkar, Steiner, and Orzack 2009;Steiner and Tuljapurkar 2012).…”

Extreme climate events and individual heterogeneity shape life‐history traits and population dynamics

“…In addition, interindividual variation in the outcomes of life-history traits also arises from individual stochasticity (Caswell 2009(Caswell , 2011(Caswell , 2014; referred to as dynamic heterogeneity in Tuljapurkar et al [2009], Orzack et al [2010], and Steiner et al [2010]). Individual stochasticity reflects that individual survival and reproductive success can vary throughout an individual's life by random variation only.…”

Demography and the statistics of lifetime economic transfers under individual stochasticity

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