A Quiescent Phase in Human Mortality? Exploring the Ages of Least Vulnerability

Abstract

Demographic studies of mortality often emphasize the two ends of the lifespan, focusing on the declining hazard after birth or the increasing risk of death at older ages. We call attention to the intervening phase, when humans are least vulnerable to the force of mortality, and consider its features in both evolutionary and historical perspectives. We define this quiescent phase (Q-phase) formally, estimate its bounds using life tables for Swedish cohorts born between 1800 and 1920, and describe changes in the morphology of the Q-phase. We show that for cohorts aging during Sweden’s demographic and epidemiological transitions, the Q-phase became longer and more pronounced, reflecting the retreat of infections and maternal mortality as key causes of death. These changes revealed an underlying hazard trajectory that remains relatively low and constant during the prime ages for reproduction and investment in both personal capital and relationships with others. Our characterization of the Q-phase highlights it as a unique, dynamic, and historically contingent cohort feature, whose increased visibility was made possible by the rapid pace of survival improvements in the nineteenth and twentieth centuries. This visibility may be reduced or sustained under subsequent demographic regimes.

Appendix: Parameter Estimation Methods

Although the hazard function and the survival probability function are mathematically equivalent—that is, there is a strict one-to-one correspondence between them—hazard functions are typically difficult to estimate from empirical data because they tend to be very noisy. Thus, in most survival analysis contexts, the analyst works with the survival probability model that corresponds to the hazard in order to estimate the parameters by maximizing the likelihood of the observed events. However, life tables are unique in containing relatively complete information (in terms of person-years of exposure and number of events) on the event history for the population of interest. This makes the direct computation and estimation of continuous hazard functions more tractable relative to other analytic contexts.

We can obtain the Siler model parameters in two different ways: (1) directly fitting the Siler hazard to the empirically observed hazards from life tables via nonlinear least squares (NLS) estimation, and (2) maximizing the likelihood function for the probability of death, an approach known as maximum likelihood estimation (MLE).

In the direct approach, we obtain the parameters by minimizing the sum of squared deviations of model-predicted and observed log of survival probability: that is, we minimize (Σx _n [log(q(x _n )) – log(1 – e ^{μ( x n )})]²). This is the NLS approach. In the second approach, we write the binomial likelihood that corresponds to d(x) deaths of a total of l(x) individuals, where the probability of a single death is q(x) = 1 – exp(–μ(x)). This likelihood is a function of the parameters involved in μ(x). We maximize this likelihood for the observed life table data (d(x _n ) and l(x _n )) to obtain the Siler model parameters. This is MLE.

The two approaches for parameter estimation are asymptotically equivalent. Empirically, however, the parameter estimates and standard errors generated by each approach might differ slightly because of the finite sample size (especially for the oldest ages) and the discretization of continuous age into one-year intervals. Although maximum likelihood may intuitively be expected to yield parameter estimates with the best model fit based on the maximum likelihood estimator’s asymptotic properties, in specific (finite) samples representing whole populations where accounting for sampling is not a concern, the quality of parameter estimates from a directly fitted hazard may in fact be superior.

optimx (Nash and Varadhan 2011) is a unique tool that integrates several optimization algorithms available in R. More than a dozen algorithms including several Newton-type, gradient, and derivative-free algorithms are implemented under one simple call. optimx organizes the results of multiple algorithms according to the values of the objective function. The results are provided in a manner that facilitates the comparison of the performance of different algorithms in terms of objective function values, computational effort, and the quality of solution (i.e., whether the solution satisfies first- and second-order Kuhn-Karrush-Tucker conditions for local optimum). We employed optimx for both the NLS and MLE approaches.

We found that the NLS estimation approach is slightly better than the MLE approach in terms of better convergence of iterative algorithms for parameter estimation. Although the MLE estimates provided a better fit of the survival probability, the NLS approach provides a better fit to the observed mortality hazard (see Fig. 7), as determined by the closeness of the fitted values to the empirically observed values. Results and figure references in the main text therefore rely on the NLS estimates.

Fig. 7

The empirical mortality hazard trajectory (left) and survival probability (right) for the cohort of Swedish females born in 1889, along with trajectories fitted using maximum likelihood (MLE; blue dots) and nonlinear least squares (NLS; red dashes) methods. The NLS method provides a better fit for the hazard, but the MLE provides a better fit for the survival curve