# Response to Comment on “The plateau of human mortality: Demography of longevity pioneers”

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Science  19 Oct 2018:
Vol. 362, Issue 6412, eaav3229
DOI: 10.1126/science.aav3229

## Abstract

Beltrán-Sánchez et al. based their comment on misleading calculations of the maximum survival age. With realistic numbers of people attaining age 105 and the estimated plateau, the Jeanne Calment record is indeed plausible.

Beltrán-Sánchez et al. (1) based their comment on calculations for their quantity Tmax, projected for cohorts with n members alive at age 105. The authors call Tmax “maximum survival age,” but that is misleading. Tmax is not an age with low probability of being exceeded. In the model in use, the probability of Tmax being exceeded is more than 63%. The lifespan of the last survivor out of n is a random quantity. Tmax is the mode of a distribution with a long upper tail; it is a low-side estimate, not any kind of upper bound.

Specifically, the probability that at least 1 of n further lifespans exceeds x is 1 – [1 – exp(–mx)]n. With x = Tmax = (1/m) log(n), the probability is 1 – [1 – (1/n)]n > 1 – (1/e) = 0.632. That is, the probability is greater than 63%. To keep chances of longer lifespans below 10%, all ages in the upper part of table 1 of Beltrán-Sánchez et al. would need to be shifted upward by about 3.5 years, putting all the bounds for females above 121.

Under the model, percentiles of the distribution of maximum age do depend logarithmically on n. But n grows over time and may well be growing exponentially as cohorts accumulate, cohort sizes grow, and survival to age 105 improves.

Beltrán-Sánchez et al. mistakenly take the sample sizes 463 and 3373 from our paper [table 1 of (2)], which are period counts at all ages above 105 over 7 years, as if they were values of n, numbers reaching age 105 in a birth cohort. (Also, the upper part of their table 1 uses m = 0.629, taken from our table 2, for their calculation for females, not 0.475 as stated in their first paragraph, and not 0.645, the correct value from table 2 of our paper.)

Proper specification of n over time requires study of empirical data. With realistic trends in n and mortality plateaus such as we estimated, Lenart et al. (3) find that the lifespan of Jeanne Calment is not implausible. It is not surprising that her record still holds; although her record might not be broken for decades, nothing that could be called a limit to lifespan is in sight.

In agreement with us, Beltrán-Sánchez et al. are not claiming that there is a fixed limit to lifespan. Their model implies maximum ages increasing into the future at slow rates, just as we expect. Along with advances in survival at younger ages in growing populations, two considerations not featured might make increases in maximum ages faster, although still slow: Hazards beyond age 115 might be decreasing with age, as seen at extreme ages in Mediterranean fruit flies and other model organisms. The cohort trend is toward slightly improving hazards beyond age 105, which we report might be sustained.

Contrast what we are seeing at extreme ages with what a limit to lifespan could look like. It is possible to imagine evolutionary models in which genetic load unrestrained by natural selection at old ages drives hazards to infinity at some finite age in a “Wall of Death” (4). That turns out not to be true. It is possible to imagine the processes that are driving up aggregate cohort hazards with age faster and faster through the 50s, 60s, and 70s continuing unabated at extreme ages, reducing chances of survival by further orders of magnitude. That also turns out not to be true.

It is important not to lose sight of the substantive lesson of our study and others: The evolutionary, physiological, and environmental influences that govern lifespans are flexible, not rigid. There is plasticity to senescence. As the study of extreme longevity helps us to understand how evolution can allow flexibility in survival chances, we may be better able to bring genomic knowledge to bear on improving health and survival at the more modest ages to which more of us can aspire.

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