## Test the hypothesis that death rates are declining at advanced ages

### Ambition 2

Dramatic reductions in death rates have been achieved before age 90; the fundamental uncertainty is whether death rates can be substantially reduced at older ages including 100+. If not, then life expectancy might rise—in the populations doing best—from 87 to the early 90s but will not reach the late 90s. On the other hand, if age-specific death rates can be reduced at ages 100+, then life expectancy could increase to 100 or even higher.

Research beyond the frontiers of mathematical demography is needed. At advanced ages, the increase of death rates with age is affected by mortality selection: the frail tend to die first, leaving a more robust population of survivors. This selection results in slower increases in cohort death rates with age, leading, after age 105 or so, to a plateau of constant mortality for a cohort.

We will test the hypothesis that progress can be achieved in reducing death rates at advanced ages by showing the progress is being made, using the demographic theory of old-age mortality.

Our tentative results suggest that the rate of progress—for individuals—in reducing death rates at age 100 is substantially greater than the observed rate of progress. If the progress continues and if more and more people survive to age 100, then this implies that observed rates of progress will increase toward the individual rate and life expectancy at high ages will increase more rapidly than currently is the case.

At advanced ages, the increase of death rates with age is affected by mortality selection: the frail tend to die first, leaving a more robust population of survivors. This selection results in slower increases in cohort death rates with age, leading, after age 105 or so, to a plateau of constant mortality for a cohort. The plateau occurs when

- the decline in death rates resulting from the mortality of those facing high risks of death is counterbalanced by
- the increase in death rates resulting from the aging of the surviving population.

Note that mortality stagnates for the cohort but continues to increase for individuals.

Furthermore, the relative gap between more and less advantaged populations (e.g., females vs. males or people of high vs. low SES) tends to diminish (because of the higher mortality of the disadvantaged, leading to a relatively more robust surviving population). Understanding underlying patterns of mortality at advanced ages—and the prospects for progress in reducing death rates—requires further development of sophisticated demographic models that can control for mortality selection. A start has been made, especially concerning the theory of mortality at high ages, but the theory remains controversial and ambitious research is needed to reformulate the theory so that it can be applied to and tested on data and to innovate statistical methods to estimate the parameters of the models.

Before diving into such models and methods, it is worth studying observed trends across populations in mortality at age 100. Some preliminary findings are displayed in **Fig. 3**. We used unsmoothed death counts and population counts from the HMD supplemented by national statistical data for the most recent year or two. The data for Sweden confirm published reports of stagnation, but the results for France and Japan suggest that some progress can be made in saving lives even among centenarians. Note, for instance, that the death rate at age 100 for French women fell from more than 50% in 1950 to close to 30% in 2017. We will systematically study such trends, for males as well as females, across all countries in the HMD.

If age-trajectories of death rates after 80 and especially after age 90 are compared for two populations, then the trajectories almost always converge. That is, relative differences in death rates narrow with age. Such convergence is predicted by the theory of selection in heterogeneous populations, “frailty” theory. **Fig. 4** provides an example. Note that death rates for 2010–18 converge with age toward the rates for 1980-89 and that curves for Italy converge toward the curves for France. Also note that the death rates rise more slowly with age. Because the curves converge, death rates declined at a pace of 1.8% per year for France and 1.6% for Italy at age 90 but only at rates of 0.7% and 0.5% at age100.

Harnessing the demographic theory of selection in heterogeneous populations to estimate key parameters has proven difficult. When I developed frailty theory (with Manton and Stallard), we assumed that mortality follows a Gompertz curve and that frailty is Gamma distributed. There was no evidence this should be the case, which discouraged application of the model to data. Now it has been shown that the distribution of frailty (relative risk) converges toward a Gamma distribution at advanced ages almost regardless of its distribution at birth. Furthermore, at (or near) a mortality plateau, death rates for individuals have to increase exponentially (Gompertz) and frailty has to be Gamma distributed—any other mortality patterns or frailty distributions lead, in a relative-risk framework, to either increasing or decreasing mortality, not a plateau. This is a stunning result, a beautiful contribution of formal demography. Although some doubts remain about the leveling off of mortality and the assumption that individuals differ in their relative risks of death (the proportional-hazards assumption), I have decided that the next step is to boldly push ahead with developing frailty theory to model mortality patterns after age 90 over age, time and place—to uncover fundamental relationships. This is high-payoff research that might fail or prove problematic.