The shortness of human life constitutes its limit

TL;DR

This paper affirms the existence of a practical limit to human lifespan around 120 years, clarifies misconceptions in previous analyses, and proposes precise terminology to distinguish between absolute and effective lifespan limits.

Contribution

It defends the evidence for a human lifespan limit against critiques, clarifies the interpretation of statistical analyses, and introduces terminology to better describe lifespan boundaries.

Findings

Human lifespan unlikely to greatly exceed 120 years

Critiques of earlier EVT analyses are unfounded

Proposes clear terminology for lifespan limits

Abstract

In this paper, we affirm our earlier findings of evidence for a limit to human lifespan. In particular, we assess the analyses in extreme value theory (EVT) performed by Rootz\'en and Zholud. We find that their criticisms of our work are unfounded and that their analyses are contradicted by several other papers using EVT. Furthermore, we find that even if we completely accept the conclusions about late-life human mortality reached by Rootz\'en and Zholud, their results do not actually contradict the findings presented in our original paper: whether unbounded or not, human lifespan is unlikely to greatly exceed 120 years, and the improbability of longer survival---whether it is exactly zero or merely astronomically small---acts as a de facto limit. In order to eliminate the confusion surrounding the issue, we propose the adoption of the term "limit" to denote the age at which the chance of survival is exactly zero and the term "effective limit" to denote the age at which the change of survival falls below a given threshold. Once this distinction is made, it can be demonstrated that the final result of Rootz\'en and Zholud is essentially a recapitulation of the main conclusion of our paper. Ultimately, much of the controversy surrounding the issue of a limit to human lifespan can be avoided by carefully reading the literature and applying statistics to practical human scales.

Tables

Table 1. Table 1: Values of L e subscript 𝐿 𝑒 L_{e} and ϵ italic-ϵ \epsilon inferred from other publications.

$L_e$	$ϵ$	Data source	Citation
114.9	0.5	IDL/GRG	dong_evidence_2016
125	0.0001	IDL/GRG	dong_evidence_2016
131.21	$\ll 0.025$	Belgian National Population Register	gbari_extreme_2017
128.73	$\ll 0.025$	IDL/HMD	feifel_who_2017
115.7	0.5	Statistics Netherlands	einmahl_limits_2017
123.7	0.025	Statistics Netherlands	einmahl_limits_2017
115	0.5	“Best-guess” MRAD	milholland_best-guess_2017
128	?	IDL/GRG	rootzen_human_2017

Full text

1 Introduction

In their paper rootzen_human_2017 , Rootzén and Zholud assert that there is no limit to human lifespan and also level several criticisms at our paper dong_evidence_2016 in which we reported evidence for such a limit. Harsh words, such as “unfounded”, “inappropriate”, “wrong” and “misleading” are used, but the severity of their language belies the fact that their paper is the one that is misleading by misrepresenting our findings and attacking us on tenuous, incorrect and unfounded bases (Section 2). A close examination of the work of Rootzén and Zholud will reveal that their analysis is inadequate to conclusively demonstrate that human lifespan is unlimited (Subsection 3.1). Furthermore, several other analyses have come to conclusions the opposite of Rootzén and Zholud (Subsection 3.2). Finally, even Rootzén and Zholud’s own work suggests that there is a de facto limit to human lifespan, one that is not much higher than the one we found in our original paper (Section 4). In one very narrow sense—one that we did not propose at all in our paper—human lifespan may be unlimited. But, in a much more practical and reasonable sense, human lifespan, given the current paradigm of medical research, is limited.

2 Refuting the criticisms of Rootzén and Zholud

In Section 4 of their paper, Rootzén and Zholud summarily dismiss the the results presented in our paper poiting to a limit to human lifespan dong_evidence_2016 in four brief paragraphs111Rootzén and Zholud also cursorily mention our findings that improvements in mortality decrease with age and argue that “ a slower rate of improvement is still an improvement, and, if anything, it contradicts the existence of a limit”, but this assertion is never backed up by any evidence or reasoning. A slower rate of increase does not imply that the increase will continue forever, nor does it imply that there is no limit. By Rootzén and Zholud’s logic, an examination of some partial sums of the series $\sum_{i=1}^{n}2^{-i}$ would show a slowing rate of increase as $n$ increases, which “if anything … contradicts the existence of a limit”. However, it has been mathematically proven that geometric series in this form converge (i.e., $\lim_{n\to\infty}\sum_{i=1}^{n}2^{-i}=1$), so there is in fact a finite limit to the sum. Furthermore, Rootzén and Zholud do not address the fact that that there has been not merely a deceleration of improvement but an absence of any detectable improvement whatsoever in supercentenarian mortality.. These paragraphs only address Figure 2 of our paper and do not address the findings presented in Figure 1 and Extended Data Figures 1, 2, 3, 4, 5 and 6 of our paper. In these four crucial paragraphs, Rootzén and Zholud begin with a falsehood, move on to a potential self-contradiction, make a nonsensical statement, and finally conclude by fundamentally misunderstanding the thesis of our paper.

In the first paragraph, Rootzén and Zholud state that the number of supercentenarian deaths in the International Database of Longevity (IDL) varies from 0 to 42. However, this statement is false: it is plain to see from Rootzén and Zholud’s Figure 6 and Figure 7 that the minimum number of deaths in any year is 1.

Rootzén and Zholud also assert in the first paragraph of Section 4 that the yearly maximum reported age at death (MRAD) “shows the same pattern” as the number of deaths; unfortunately, they do not provide any correlation statistics that could be used to evaluate that assertion. In the next paragraph, they mention a different line (representing $110+$ mean of maximum $n_{t}$ exponential variables) that they also assert, without statistical substantiation, “agree[s] well” with the MRAD, but this line differs greatly from the line representing the number of deaths (most visible in the interval 1990–2000). It would seem contradictory that the line representing number of deaths, which fluctuates in magnitude several-fold over short periods of time (for example, leaping from 5 to 35 in the early 1990s), and the line representing $110+$ mean of maximum $n_{t}$ exponential variables, which is devoid of such wild fluctuations (barely ambling from around 113 to around 115 over the same interval), could both strongly correlate with the MRAD. Perhaps the graphs are misleading, and both variables, counterintuitively, manage to correlate well with the MRAD, but Rootzén and Zholud provide no statistical evidence that they do.

In the third paragraph, Rootzén and Zholud say that our findings result from “inappropriate combination of data from different time periods”. This does not make sense: we merely used all the data from the entire time interval of the IDL (and, separately, the GRG). Presumably, they meant to say that there was an inappropriate combination of data from different countries since that is the issue they touch upon at the end of the previous paragraph. However, considering each country separately still finds evidence for a trend break, contrary to Rootzén and Zholud’s assertion. In Japan, all of the data is post-breakpoint, so there is no evidence for or against, although the number of supercentenarian deaths does increase more rapidly than the MRAD, suggesting the two have become decoupled. In the USA, the increase of the MRAD continues unabated, providing some evidence against a trend break; but note also the sharp jump in the number of supercentenarian deaths in the early 1990s and the comparatively shallower increase in MRAD, again undermining Rootzén and Zholud’s thesis that the MRAD is driven by number of supercentenarian deaths. In England and Wales, the slope of the trend decreases from 0.7 to 0.2. Rootzén and Zholud do not define what evidence is sufficient to indicate a trend break, prompting the question: if a more than 3-fold decrease in the slope is not evidence for a trend break, then what is? Finally, in France the slope goes from 0.2 to –0.3, indisputable evidence of a trend break. To summarize, of the four countries considered, one of them is invalid, one arguably does not have a trend break, and two almost certainly do. Furthermore, all four contain fluctuations in the number of supercentenarian deaths not reflected in the MRAD. Therefore, the balance of evidence favors the conclusions presented in our original paper.

The final paragraph dismisses the evidence of a limit in the 2nd through 5th highest reported ages at death with a repetition of the nonsensical accusation of “inappropriate combination of data from different time periods” and no substantiating evidence. Given the weakness of the reasoning employed by Rootzén and Zholud regarding the MRAD, it is likely that this assertion regarding the 2nd through 5th highest deaths is not supported by the evidence, but their thesis is ultimately impossible to evaluate due to the absence of information provided in the paper. Section 4 of Rootzén and Zholud’s paper terminates with a fundamental misinterpretation of the thesis of our paper. The main observation of our paper was that the MRAD had stagnated since the mid-1990s and, despite advances in medicine, fluctuated around 115, without further increase; this stagnation around the average of 115 constitutes the “limit” referred to in our paper. As a corollary, if one desires to know the absolute highest age to which any human could reasonably expect to live, then we also calculated that an MRAD of more than 125 would be expected only once every 10,000 years; this value may also be considered a “limit” to human longevity. In short, the term “limit” is overloaded, and can refer to distinct, yet related, concepts depending on context. In their paper, Rootzén and Zholud introduce a third definition of “limit”: the point at which the chance of survival is equal to zero. They contend that since the distribution of MRAD ages is unbounded, there is no “limit” to human lifespan. On its own, there is nothing fundamentally wrong with this contention, but the failure of our paper to conform to the arbitrarily altered definition of “limit” formulated by Rootzén and Zholud cannot be considered an error on our part. Indeed, as we will see, the definition of “limit” used by Rootzén and Zholud is not very useful, as it would apply even to situations in which the probability of survival is so small as to be negligible. From a very philosophical perspective, it is possible to argue that due to the inherently stochastic nature of the universe implied by quantum physics, it is impossible to ever assign a probability of zero to any event, so the conclusion reached by Rootzén and Zholud is not novel. Rootzén and Zholud build their paper on a more immediately applicable starting point—demographic data—but if their goal was to derive the conclusion of no limit to lifespan based on that foundation alone, they have made several omissions and questionable decisions that undermine the strength of their analyses.

3 Problems with the analyses of Rootzén and Zholud and

literature contradicting their results

Having made our reply to the criticisms Rootzén and Zholud leveled against our paper, we move on to several of our own criticisms of their paper. The conclusion reached by Rootzén and Zholud—that human life is “unlimited”—is extraordinary, even if qualified by “but short”. Such a strong conclusion requires strong evidence, but that which is presented by Rootzén and Zholud is not up to the task. Furthermore, the conclusions reached by Rootzén and Zholud put them at odds with multiple other papers, not just our own. These contradictions must be resolved before the contention of an unlimited human lifespan can be accepted.

3.1 Problems with the analyses of Rootzén and Zholud

Much of the work in Rootzén and Zholud’s paper is of good quality, but there are several areas where they use statistics inappropriately or come to conclusions that do not necessarily follow from the premises given. These deficiencies do not discredit the entire paper, but they significantly weaken the strength of the findings presented therein. Indeed, it is possible that each could be addressed and a confirmation of Rootzén and Zholud’s analysis found. Therefore, we present these issues in the spirit of a scientific dialogue, in the hopes that further investigation will advance our knowledge, either for or against Rootzén and Zholud’s conclusions.

First, the data used by Rootzén and Zholud appear to support a stagnation of supercentenarian mortality, with no improvement in supercentenarian survival in recent decades, something that would be consistent with a limit to human lifespan and not a continual improvement towards breaking that limit. In Table 3 of their paper, Rootzén and Zholud present the results of statistical tests in several sets of data for differences in supercentenarian mortality between the first half of each set (generally spanning from the 1960s until the 1990s) and the second half (generally from the mid-to-late 1990s until the mid-to-late 2000s). In every single case, they find no evidence of a change in mortality, with the lowest $p$-value being 0.18, not even close to significant. This finding essentially confirms our earlier resultsdong_evidence_2016 , i.e. a lack of improvement in supercentenarian mortality indicates that human lifespan has stopped increasing, and is essentially a restatement of the finding presented in Figure 3c of our original paper.

Second, Rootzén and Zholud have chosen an inappropriate null hypothesis and incorrectly interpreted the results of their statistical tests. Rootzén and Zholud begin with the assumption that the force of mortality after age 110 is constant, and that human lifespan therefore follows an exponential distribution. Then, they provide several $p$-values, in Table 5 of their paper, which fail to reject this null hypothesis. However, it is basic knowledge regarding $p$-values that a failure to reject the null hypothesis does not mean that the null hypothesis is true. Furthermore, the assumption of a constant force of mortality is not an appropriate null hypothesis. From early adulthood onwards, mortality increases relentlessly; it is far more parsimonious to assume that this increase will continue rather than that it will stop in its tracks at old age. Indeed, by making a constant force of mortality (which corresponds to an “unlimited but short” lifespan) their null hypothesis, Rootzén and Zholud have committed the fallacy of assuming that which was to be demonstrated. Finally, despite having tilted the scales in favor of a constant force of mortality, Table 2 of Rootzén and Zholud’s paper indicates that the GRG dataset, the data with the greatest temporal and geographic reach, strongly rejects the hypothesis of an exponential distribution. Thus, the main premise of Rootzén and Zholud’s paper, that the force of mortality after age 110 is constant, is not supported by the data and statistics presented.

3.2 Contradictions with the literature

Having established that Rootzén and Zholud failed to demonstrate a constant force of mortality, we should not be too harsh. After all, the task they have set out for themselves is impossible: showing that the force of mortality is constant would require showing that there is no change in mortality, which would require proving a negative. A more realistic goal would be to calculate a confidence interval for the force of mortality after age 110 in order to establish an upper bound for mortality. That has been done by othersmodig_how_2017 , and the upper bound of the resulting interval rapidly approached 1. Although the confidence intervals were wide, leading the authors of that paper to conclude that mortality after age 100 plateaued, the fact that the estimate of mortality also rapidly increased past 0.5 after age 110 and rapidly approached 1 indicates that the balance of evidence is actually in favor of a limited lifespan as defined by Rootzén and Zholud. Furthermore, they found that centenarian mortality has not improved, a result that strongly supports our findings of a limited human lifespan based on MRAD. As the authors of that paper concluded: “the maximum lifespan, measured as the age of the oldest person to die, is currently not increasing”.

Rootzén and Zholud have used EVT in their attempt to resolve the question of whether human lifespan is limited, reaching the conclusion that it is unlimited. Three other recent papers have examined the same question using EVT and come to the opposite conclusion. The firstgbari_extreme_2017 examined data from 46,666 Belgians who died at 95 or older and found an “ultimate age” of 115 for men and 123 for women, a result very similar to our own. The second paperfeifel_who_2017 analyzed data from the International Database on Longevity (IDL) and the Human Mortality Database (HMD). The authors of that paper write that “due to its small sample size, the hypothesis of an infinite lifespan could not be rejected for the IDL,” the dataset used by Rootzén and Zholud, but when they supplemented it with data from the HMD, “we found significant evidence for a finite lifespan in the combined data set and obtained reliable point estimates for the maximum attainable age” of 125–128 years in females. Finally, a papereinmahl_limits_2017 based on precise measurements of the ages at death of 285,000 residents of the Netherlands found “compelling statistical evidence” for a finite human lifespan in both men and women. They found an average annual endpoint to the lifespan distribution of 114 and 116 years and a maximum estimated upper endpoint of 125 and 124 years in men and women respectively222The authors of that paper also found a lack of a trend in with time in these values; although they imply that the lack of a trend would contradict our paper, this implication is based on a misinterpretation of our results, which found an initial upward trend, followed by a lack of a trend (and not necessarily a decrease, as indicated by the insignificant $p$-value) during recent decades. Instead, the lack of a trend is a confirmation of our findings that there has essentially been a limit in place the whole time, which was finally reached in the mid-1990s. . These estimates are very close to the estimates of average MRAD, 115 years, and maximum MRAD, 125 years, which we arrived at in our original paper. To summarize, all three of these other EVT papers confirm our results and contradict those of Rootzén and Zholud.

To conclude this section, we would like to use Rootzén and Zholud’s concluding sentences to tie together the two previous strands: the gaps in their paper and its variance with the literature. In the penultimate and final sentences of their paper, Rootzén and Zholud concede that “the IDL data set only includes 9 persons who lived longer than 115 years, so data is sparse above this age” but assert that “these 9 data points agree well with our conclusion”. No substantiation is given for this assertion, and the very sparsity of such data is itself suggestive of a limit to human lifespan, but even if it were true that those 9 data points did support Rootzén and Zholud’s conclusion, they must surely be outweighed by the contradictory studies based on over 300,000 data points.

4 Shortness of life as a de facto limit: theoretical

considerations and empirical results

As we have shown, there is considerable evidence against Rootzén and Zholud’s conclusion that the force of mortality remains constant after age 110. However, there still remains a slim possibility that they are correct. It is also possible that the force of mortality could increase with age and asymptotically approach 1 without ever reaching it, such as in a sigmoid function (Figure 1), a scenario not considered by Rootzén and Zholud. In any case, let us assume that Rootzén and Zholud are correct there is no limit, $L$, beyond which the probability of survival is 0. Would this contradict our results? And would it be a finding of importance? The answer to both questions is: no.

First, a hard limit, with a chance of mortality equal to 1, was never part of our original paper. As a commentary on our paper observedolshansky_ageing:_2016 , our paper’s thesis is that “there is no fixed limit beyond which humans cannot live, but that there are, nevertheless, limits on the duration of life”; the “limit” referred not to an age of certain death but the stagnation of advances in human maximum lifespan and the corollary emergence of an age beyond which survival, although not impossible, is prohibitively unlikely. To refer directly to the text of our paper: “we found that the probability of an MRAD exceeding 125 in any given year is less than 1 in 10,000”—not zero. In light of this, Rootzén and Zholud’s statement that “it is likely that the record age …will be shorter than 128 years” is neither contradictory to our conclusions nor particularly novel, but rather a slightly different estimate of the number: stated informally (we shall formalize it below), the age which MRAD is unlikely to exceed in the foreseeable future.

The main thesis of Rootzén and Zholud’s paper, that there is no “limit” by their definition to human lifespan is not particularly meaningful or novel. Indeed, it is not even necessary to consult any demographic data in order to arrive at the conclusion that there is no age with a zero chance of survival. According to quantum physics, anything is possible, even if extremely unlikely. So, the odds of a brain spontaneously appearing out of thin air due to quantum fluctuations have been calculatedlinde_sinks_2007 to be $1$ in $10^{10^{50}}$. Since aging is due to physical changes, i.e. the accumulation, depletion or modification of macromoleculeslopez-otin_hallmarks_2013 , it is possible that quantum fluctuations could spontaneously transform a supercentenarian’s body to that of a 20 year old, making them a shoo-in for a new longevity record. Calculating the exact probability of this event occurring, and the question of whether the number of zeros in its decimal representation would exceed the number of atoms in the observable universe, we leave as an exercise for future investigators.

In case the scenario described above strikes readers as too outlandish, we propose a more realistic occurrence based on Rootzén and Zholud’s model of “unlimited” lifespan: assuming that Rootzén and Zholud are correct and mortality after age 110 follows the “each year a coin is tossed” model, and rounding333Observant readers will note that the abstract of Rootzén and Zholud’s paper is contradicted by its body: in the former, the probability of dying is given as 47%, while in the latter that value is given as the probability of survival. While mixing up life and death is a major error, the proximity of both probabilities to 50% limits its effects. up the probability of survival to 50%, the odds of a supercentenarian living to age 150 are $1$ in $2^{40}$ or just under 1 in a trillion. So, on the one hand we are told that “human life is unlimited” and on the other hand the same model gives probabilities of survival that are so low that many would consider them next to impossible. At a certain point, discussing the probability of a single individual living to a given age under Rootzén and Zholud’s “unlimited” lifespan paradigm becomes akin to discussing the probability of receiving a single molecule of the active ingredient from a homeopathic remedy.

Our point is that the finding that human lifespan is “unlimited” on the basis that there is no age with a zero chance of survival may be mathematically true and it may be of theoretical or even recreational interest, but it is of little practical significance or use. Insisting that a “limit” to lifespan consist of an age with 100% mortality gives the term “limit” too narrow and restrictive a definition, one we did not use in our original paper and one that makes any statements regarding its existence or nonexistence almost completely inconsequential. Instead of discussing the limit, $L$, of lifespan as the age with a zero chance of survival, we propose that future discussions of extreme longevity focus on the effective limit, $L_{e}$ of lifespan, the age beyond which it is extremely unlikely any individual of the species will survive. The value of $L_{e}$ will depend on many factors, the most pressing of which is the definition of “extremely unlikely”. Therefore, $L_{e}$ can be stated formally by qualifying it with $\epsilon$, the desired definition of “extremely unlikely”. For example, our original paper calculated $L_{e}^{\epsilon=0.0001}=125$, i.e. the probability of any human surviving past 125 is 0.0001 or 1 in 10,000. In their paper, Rootzén and Zholud essentially calculate $L_{e}$ to be 128, but do not provide the value of $\epsilon$ they used. The values of $L_{e}$ and $\epsilon$ can be inferred from other publications (Table 1). Generally, $L_{e}^{\epsilon=0.5}$ is around 115, while values of $L_{e}$ for lower values of $\epsilon$ are in the mid-120s.

The value of $L_{e}$, for a given value of $\epsilon$, is not constant. It is determined by two factors: the population of the species (more individuals means more chances for one of them to reach an extreme age) and the chance of survival at each age leading up to $L_{e}$. In our original paper, we came to the conclusion that $L_{e}$ had basically reached its limit—here, meaning that $L_{e}$ had remained for years and would remain in the absence of unprecedented technological breakthroughs at its current level—based on the tacit assumption that the human population would not greatly increase and our explicit demonstration that late-life survival had stagnated. Rootzén and Zholud speculate that future research may further extend human lifespan, and we do the same in the conclusion of our original paper, stating that “there is no scientific reason why such efforts could not be successful”. However, the gains from such efforts have not been realized. There is currently no cure for aging, nor is there even a product approved to be marketed as a treatment for aging. Perhaps one day human lifespan will become unlimited, but right now the chances of mortality at old age act as an effective limit to lifespan.

5 Conclusion

The premise and even the title of Rootzén and Zholud’s paper are self-contradictory: “human life is unlimited — but short”. We resolve this contradiction by positing that the shortness of human life constitutes its limit. For a particular, narrow definition of “limit” it can indeed be said that there is no limit to human lifespan, but for a more useful definition there is one. If one finds it justified to say that human life is “unlimited” because there is a constant 47% chance of survival each year, one must also find it justified to say that roulette winnings are “unlimited” because there is a constant 48.7% chance of doubling one’s money by betting on black every time. It is easy to see the problem with relying on the latter reasoning for a moneymaking strategy, so why would anyone accept the former reasoning for a longevity strategy?

The use of an overly restrictive definition of “limit” (as detailed in Section 4) is not the only issue with Rootzén and Zholud’s paper. They also make unfounded criticisms of our findings of evidence for a limit to human lifespan (refuted in Section 2) and the evidence even for their modest conclusion is weak, with both statistical issues undermining their findings (enumerated in Subsection 3.1) and larger studies having contradictory results (reviewed in Subsection 3.2). Thus, the contention that “human life is unlimited” is most likely to be false.

Much of the controversy surrounding this issue could have been avoided by carefully reading our paper and using the context of the word “limit” as we used it to correctly infer its meaning in that paper. Imposing an inappropriate definition of the word does not contribute to the literature, as it seeks to rebut our paper for a thesis it does not actually advance.

However, there is some value to Rootzén and Zholud’s work, as it has drawn attention to the necessity of explicitly formulating a definition of “limited” that is useful for discussions of human lifespan. Although the evidence available suggests that, contrary to Rootzén and Zholud’s conclusion, the risk of mortality does not remain constant after age 110, there also is no age at which mortality is certain. In that sense, then, human life is unlimited: for an infinitely large cohort or for an infinitely many cohorts, there is no maximum age beyond which no individual will live. However, for finitely-sized cohorts, observed over a sensible interval of time (10,000 years, being a few times longer than the duration of recorded civilization, is probably the absolute maximum that can be seriously considered) the age-related elevation of mortality acts to constrain the highest age that is likely to be observed, with estimates of that constraint clustering around 125. In order to distinguish between these two concepts, we proposed the term “effective limit” to indicate the age at which the probability of a single individual surviving is negligible (falling below a given threshold). We hope that this new term will be useful in facilitating clear and meaningful discussion of longevity in the future.