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Discrete Barker Frailty and Warped Mortality Dynamics at Older Ages

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Demography

Abstract

We develop a discrete variant of a general model for adult mortality influenced by the delayed impact of early conditions on adult health and mortality. The discrete variant of the model builds on an intuitively appealing interpretation of conditions that induce delayed effects and is an extension of the discrete form of the standard frailty model with distinct implications. We show that introducing delayed effects is equivalent to perturbing adult mortality patterns with a particular class of time-/age-varying frailty. We emphasize two main results. First, populations with delayed effects could experience unchanging or increasing adult mortality even when background mortality has been declining for long periods of time. Although this phenomenon also occurs in a regime with standard frailty, the distortions can be more severe under a regime with Barker frailty. As a consequence, conventional interpretations of the observed rates of adult mortality decline in societies that experience Barker frailty may be inappropriate. Second, the observed rate of senescence (slope of adult mortality rates) in populations with delayed effects could increase, decrease, or remain steady over time and across adult ages even though the rate of senescence of the background age pattern of mortality is time- and age-invariant. This second result implies that standard interpretations of empirical estimates of the slope of adult mortality rates in populations with delayed effects may be misleading because they can reflect mechanisms other than those inducing senescence as conventionally understood in the literature.

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Notes

  1. Evidence from animal studies has confirmed several tenets of DOHaD and Barker theories (Bateson and Gluckman 2011), but empirical evidence from human populations is less favorable, inconsistent, and generally identifies effects of small magnitudes (Bengtsson and Lindström 2000; Lundborg 2008; Madsen et al. 2010; Painter et al. 2005; Roseboom et al. 2001). However, there is scarce, if any, evidence drawn from countries that experience the types of mortality decline and environmental and ecological conditions that we highlight here as conditions for observability of Barker effects.

  2. The difference between standard and Barker frailty traits is that the former is fixed at birth whereas, at least conceptually, the latter could be shaped by early postnatal experiences. None of the mechanisms that we describe at the outset can be entirely captured by a trait fixed at birth. Thus, the distribution at birth of δ must reflect both the at birth-distribution and prospective changes that result from subsequent heterogeneous individual experiences. It is thus a trait with a more complex genesis that cannot be well specified unless we know the impact of postbirth experiences on the at-birth trait or propensity. To circumvent this conceptual problem, one can think of the distribution of δ at birth as what would be observed at a very early age beyond which additional experiences do not affect the underlying quantity in the absence of mortality up to that age. In this case, the distribution of δ as the outcome of the mixing of the distribution of fixed-at-birth susceptibility and the (unconditional on survival) probability of subsequent exposures that promote delayed manifestations. Thus, δ acts as a net propensity, induced by in utero deprivation or subsequent exposures to express delayed manifestations.

  3. The assumption that Barker frailty leads to mortality excess only at adult ages downplays the consequence of Barker frailty in a secular mortality decline because the proportional gains in survival to older ages implied by an arbitrary rate of background mortality decline should be higher when there is no excess mortality at younger ages. As a consequence, our model understates the effects of Barker frailty.

  4. This simplified functional form for mortality decline avoids cumbersome algebra but leads to no loss of precision or generality.

  5. The expressions for P B (y, t) and its derivatives with respect to time t and age y are shown in Online Resource 1.

  6. By construction, \( \frac{\partial \ln \left({\upmu}_s(y)\right)}{\partial y}={\upbeta}_s \) is age- and time-invariant. The implication of this expression seems to have gone unnoticed in the literature (but see Vaupel and Missov (2014) for an analogous expression for continuous frailty). Even in the absence of Barker effects and with an age-invariant β s (y) at adult ages (as in a Gompertz baseline adult mortality pattern), the age derivative of the average mortality pattern cannot be constant (across ages or across time when there is a mortality decline). The regime of frailty assumed here will always induce an age-dependent slope smaller than the standard slope. This has important consequences for the study of old-age mortality in that the standard interpretation of an empirical slope estimated after fitting—for example, a Gompertz function to a cohort’s adult mortality rates—is probably always incorrect. As suggested by the expression, such estimate contains an age- and time-dependent downward bias. To avoid this bias, one needs to estimate a Gompertz model controlling for both age and the age-and time-varying negative term in the expression. To our knowledge, this has never been done in empirical studies. Elsewhere, we show that Barker effects and mortality decline will always induce a negative correlation between the levels of child mortality experienced by a cohort and the cohort’s adult mortality slope (Palloni and Beltrán-Sánchez 2016).

  7. An expression for \( {P}_B^1\left( y, t\right) \) is in Online Resource 1.

  8. Linear correlations shown in Table 2 clearly underestimate the actual differences in slopes between Barker and background mortality due to the nonlinear pattern, but they still show flatter curves under the Barker regime.

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Correspondence to Alberto Palloni.

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Palloni, A., Beltrán-Sánchez, H. Discrete Barker Frailty and Warped Mortality Dynamics at Older Ages. Demography 54, 655–671 (2017). https://doi.org/10.1007/s13524-017-0548-4

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