Holger Rootzén and Dmitrii Zholud write:

This paper studies what can be inferred from data about human mortality at extreme age. We find that in western countries and Japan and after age 110 the risk of dying is constant and is about 47% per year. Hence data does not support that there is a finite upper limit to the human lifespan. Still, given the present stage of biotechnology, it is unlikely that during the next 25 years anyone will live longer than 128 years in these countries. Data, remarkably, shows no difference in mortality after age 110 between sexes, between ages, or between different lifestyles or genetic backgrounds.

This relates to our recent discussion, “No no no no no on ‘The oldest human lived to 122. Why no person will likely break her record.'”

I’ve not looked at Rootzén and Zholud’s article in detail, nor have I tried to evaluate their claims, but their general approach seems reasonable to me. The rule of thumb of a 50% mortality per year at the highest ages is interesting.

That said, I doubt that we can take this constant hazard rate too seriously. My guess is that an empirically constant hazard rate is an overlay of two opposing phenomena: On one hand, any individual person gets weaker and weaker so I’d expect his or her conditional probability of death to increase with age. On the other hand, any group of people is a mixture of the strong and the weak, which induces an inferential (“spurious,” as Feller called it) correlation. Maybe these two patterns happen to roughly cancel out and give a constant hazard rate in these data.

Assume that most deaths are due to a series of accumulated insults of one type or another (ie death by a thousand cuts), then we would expect age-specific mortality to peak and then drop off. The slope vs age should decrease each year after the peak. For example if there were 40 events that need to happen and each independently occurs with probability .06 each year, the curve would look like this: https://image.ibb.co/nfDR58/age_specific_mortality.png

code for plot: https://pastebin.com/DEqEcPru

Obviously that is a really simplified model (and I tuned the parameters so it would look somewhat realistic) but I don’t think it is surprising that there is much less relationship between mortality rate and age in the extremely elderly.

Actually I reran it with 100k people and believe this is how to get the hazard ratio (# deaths/# alive of that age):

tab = table(sort(t_death))

hr = sapply(1:length(tab), function(i) tab[i]/length(which(t_death >= as.numeric(names(tab))[i])))

plot(names(tab), 1e4*hr, main = “Hazard Ratio”,

type = “l”, xlab = “Age”, ylab = “Deaths per 10k pop”, log =”y”)

The hazard ratio goes flat and then noisy at the end (due to small n):

https://image.ibb.co/hD4aho/age_specific_mortality2.png

Hazard Rate, not hazard ratio… Anyway,

I don’t understand what model exactly they are arguing against. From their paper I still don’t know:

Ok, so there is no way they had an actual model… I also see some “pro” quotes

Continuing on to that paper I see the quote:

Ok, so we are talking about lifespans of fruitflys and medflys now… However, I see in the first paper the main argument is similar to the one in the current paper “the chance of death for individuals may level off at older ages.”

I don’t get how this indicates the lifespan is “unlimited”.

The ‘unlimited’ bit is obviously kind of ridiculous, but I’m sure that what they have in mind is that the mortality rate never goes to 100%. According to their model, if we dignify it by that name, there is a finite (though miniscule) chance that someone could live to be 1000 years old.. or pick any other number.

The passing of Doug Altman should remind us that a staggering amount of time, talent and money has been wasted in medical research thanks to scandalously poor statistical practices. And it wasn’t just bad methodology. It was that a method for manufacturing “science” displaced curiosity, observation, reflection, serendipity and insight.

If recent rapid advances in cancer immunotherapy are indicative of what’s to come when gerontologists stop trying to divine the secrets supposedly hidden in sample means and start trying new approaches to sound old ideas (e.g. Coley’s toxins -> immunotherapy) who knows how long our kids might live?

This is just the equilibrium fixed point of “Whatever doesn’t kill me makes me stronger.”

“On the other hand, any group of people is a mixture of the strong and the weak, which induces an inferential (“spurious,” as Feller called it) correlation. Maybe these two patterns happen to roughly cancel out and give a constant hazard rate in these data.”

Is this to suggest that the “strong” in fact have a decreasing hazard rate after a certain age? in order to cancel out the uncontroversial increasing hazard rate and get a constant rate at the population level?

Anon:

I’m supposing that, for most people, the hazard rate is increasing, but that different people have different curves. The set of people still alive at age 120 are stronger (lower average hazard as a function of age) than the set of people alive at age 110. So you could get an approximately constant average hazard rate as a function of age, even if each individual person’s hazard rate is an increasing function.

There are four main issues with the paper from the post:

1. It makes several unfounded criticisms of our paper (https://dx.doi.org/10.1038/nature19793).

2. Its conclusions are not adequately supported by its analyses.

3. It is contradicted by several other papers in the literature.

4. It is premised on a narrow and impractical definition of “limited” (one that we emphatically did not use in our original paper). Even if the rest of their paper is assumed to be correct, the probability of survival under their model quickly becomes so small that past a certain point it is unlikely to ever observe a survivor over the course of a reasonable human timescale. There can be some debate about where to draw the line, but the minuscule odds of survival act as a de facto limit to human lifespan .

These issues are enumerated in detail in this preprint:

https://arxiv.org/abs/1803.04024

Thanks, this is the impression I got (but I only skimmed the paper), whatever distinction is being made between limited and unlimited seemed pointless.

Also, I liked your figure 1 but have some questions. What is the y-axis? Is it percent of total deaths, or # deaths/ # people of that age? What processes are being proposed to yield those curves? How would you simulate them?

Thanks! The y-axis is chance of dying, so it would be the same as #deaths/#people alive. Your simulation above is similar to the “hits” model proposed by Szilard (https://www.ncbi.nlm.nih.gov/pmc/articles/PMC222509/). Additional considerations would include: 1. Multiple layers of redundancy. A cell needs to receive a certain number of hits to “die” (doesn’t have to strictly correspond to cell death, could also stand for senescence, loss of function); a certain fraction of cells need to “die” for the individual to die. This would bring you in line with Szilard’s model. You could even elaborate on it further with heterogeneous populations of cells, corresponding to organs, with different rates of hits and different tolerances to number of hits 2. Relaxing the assumption that the rate of hits is constant. It is likely to increase over time, either on its own (reflecting some developmental process) or in proportion to the number of pre-existing hits. 3. Entirely stochastic model. Instead of considering an individual to automatically die after receiving a certain number of hits, give them a probability of dying that is some function of the number of hits/fraction of remaining cells/etc.

So have model been worked out that can yield those four curves in your Fig 1 while having plausible parameters?

It sounds like you are saying no, which makes me wonder why anyone thinks they are exploring the consequences of them.

There’s some fascinating stuff in that Szilard (1958) paper. To start with:

The common modern claim is something like:

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC222509/

Can anyone find a reference for this? I just tried 4 different sources and none provided a ref. Who assumed this and based on what? How widespread was this assumption in reality since someone 40 years earlier got it almost exactly correct?

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC222509/

This seems high to me relative to the numbers I am familiar with, but it is age of peak death. What is that today? I see for life expectancy it is something like:

https://www.cdc.gov/nchs/data/databriefs/db293.pdf

Can anyone cite a source with a comparison between age of peak death as estimated in 1958 vs age of peak death in 1958 and more recently?

I checked cdc wonder and got a most common age at death of 82 for males and 87 for females for 1999-2016. So it was misleading to compare that to life expectancy (mean age at death).

https://image.ibb.co/iOiMq8/USmortality.png

I enjoyed your paper and thought you had the better of the argument. Have you, perhaps elsewhere, elaborated on this contingency: “given the current paradigm of medical research”?

The FDA has finally gotten around to approving a trial of metformin specifically to see if it might have a positive effect on things thought to drive aging. The results were posted to cinicaltrials.gov last week: https://clinicaltrials.gov/ct2/show/NCT02432287 Maybe the paradigm is shifting.

Fascinating! Danazol is the other one: https://clinicaltrials.gov/ct2/show/NCT03312400.

Thanks! I’m working on an article now regarding the potential of a paradigm shift. Certainly, a paradigm shift is a big IF that could change things (but still an IF). I think rapamycin or another mTORC inhibitor is the best bet, but I’ll be watching the results from metofrmin with interest. Right now, it looks like there is only results from the first 6 weeks. Very preliminary, especially for treating aging. We shall see!

Some input to your issue above

“3. It is contradicted by several other papers in the literature.”

Somehow related Extremes also published in March 2017, Volume 20, Issue 1, pp 199–237: “A general estimator for the right endpoint with an application to supercentenarian women’s records” website: https://link.springer.com/article/10.1007/s10687-016-0260-6

Not discarding that this “human lifespan” topic is attached to some controversy, in that paper the conclusion goes for the “limited human life”; chapter 5 refers to the “Case study: supercentenarian women lifespan”.

I work in life insurance, and the constant hazard rate at extreme ages for different demographic segments is a well known phenomenon as far as I’m aware (see https://en.wikipedia.org/wiki/Compensation_law_of_mortality). The most compelling theory I’ve heard to explain this comes from reliability theory, where the basic idea is that humans are born with redundant, irreplaceable components that sustain their bodies. Once the redundancies have been fully exhausted (after age 110 or so for humans) then you’re just waiting for an opportunistic illness to finish you off. The probability of catching that opportunistic illness in any given year is roughly 50% at these extreme ages according to industry actuarial tables. So basically, in order to get to 122 from 110, you have to be on the right side of 12 straight coin flips, avoiding these illnesses that can deal you the final blow. Check out the work of Gavrilov and Gavrilova for more on this theory. Even if this is the right explanation, I suppose people can still live past 122 if medical advances can delay the age at which the redundancies of the healthiest amongst us are exhausted (or if they’re super luck with the coin flips).