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##### Abstract

Driven by technological progress, human life expectancy has increased greatly since the nineteenth century. Demographic evidence has revealed an ongoing reduction in old-age mortality and a rise of the maximum age at death, which may gradually extend human longevity1, 2. Together with observations that lifespan in various animal species is flexible and can be increased by genetic or pharmaceutical intervention, these results have led to suggestions that longevity may not be subject to strict, species-specific genetic constraints. Here, by analysing global demographic data, we show that improvements in survival with age tend to decline after age 100, and that the age at death of the world’s oldest person has not increased since the 1990s. Our results strongly suggest that the maximum lifespan of humans is fixed and subject to natural constraints.

##### Authors

Xiao Dong;  Brandon Milholland;  Jan Vijg

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### Evidence for a limit to human lifespan

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### Evidence for a limit to human lifespan

Figure 6 of the paper does not represent the GRG data it is based on.

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• ###### Daniel Wells | 3 years, 3 months ago

Could this be explained by the original authors using a slightly different definition of MRAD? Instead on the y-axis they seem to plot age at death of oldest person alive. Hence for the period 1990-1995 there are 'missing' points because everyone who died in this period was still younger than the at-the-time oldest person alive. Additionally the two deaths in the same year could also be explained by the oldest person dying, then the next oldest person dying in the same year. See https://twitter.com/dgoldenberg/status/783785617532395520 for a visual explanation. It seems this could explain the location and number of points in the graph, although not why they used a different definition for the supplementary figure compared to the main text figure.

• ###### Jonas Schöley | 3 years, 3 months ago

The figure caption states: "The yearly maximum reported age at death from the GRG database". In the main text Dong etal. define "the yearly maximum reported age at death (MRAD)". But maybe figures got mixed up. I'll check your hypothesis.

• ###### Jonas Schöley | 3 years, 3 months ago

Thank you Daniel. Yes, thats it. Based on GRG table C (http://www.grg.org/Adams/C.HTM) we get:

So this is an issue of using different statistics than reported.

The two in a single year are probably an error, because none of the other cases where two or more record holders day in the same year were reported.

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# Reanalysis of the evidence for a limit to human lifespan

In this analysis I look at figure 2 specifically which argues that the maximum age of death has plateaued.

I downloaded the data from the International Database on Longevity at the Max Planck Institute for Demographic Research. The terms of the data access do not permit third party sharing so the raw data is not uploaded to GitHub but you can download it yourself if you want to rerun the following analyses.

First I load the data into R, tidy up some of the columns, and subset to the same individuals used the in paper. (Not sure why they didn’t just use all 668 rather than just 534). Here is the breakdown by country:

GBR 66

JPN 78

USA 341

FRA 49

Now let’s recreate figure 2A.

The authors of the paper fitted two separate regression lines to this data arguing that after 1995 there was a change in the trend (a seemingly arbitrary choice of breakpoint - the choice of a broken vs linear trend has been analysed elsewhere).

You can see from the confidence intervals on the regression lines that the gradient for the second segment is actually consistent with being the same as the first segment. In the paper the authors calculate a p-value of 0.27 for the gradient of the second segment (null hypothesis = 0) and conclude “no further increases were observed”. They apply the same reasoning in a reply to a post-publication review on publons “The latter is not significant, so we conclude that the MRAD is essentially flat”. However, you can not accept the null hypothesis based on p > 0.05, you can only reject a null hypothesis. In this case a p-value of greater than 0.05 suggests that there is not enough data to conclude that the gradient is different from 0 (perhaps the null hypothesis should really by that the gradient is the same as the first segment, although the p-value is still non significant). The 95% confidence interval for the second segment gradient is −0.83 to +0.20 which includes the point estimate of the first segment gradient of 0.15 (using non rounded age values here).

CI: (2.5 %, 97.5 %): (-0.827, 0.196)

First segment point estimate: 0.153

P-value when H0 = 0.1533: 0.068

However this analysis is quite sensitive to the choice of breakpoint. In the above mentioned review response the authors re-analysed the data and found that a breakpoint of 1999 was a better fit. Although the package used (“segmented”) fits a continuous piecewise regression, I will continue using the method above to illustrate a different choice of date anyway. Replotting the regression lines using this breakpoint shows the confidence intervals more clearly supporting the downward trend and the p-value (with H0 = 1st segment gradient) is now significant at the 0.05 threshold. The upper 95% confidence interval is now −0.2 suggesting a downward trend (rather than a plateau).

First segment gradient point estimate: 0.193

CI: (2.5 %, 97.5 %): (0.095, 0.290)

Second segment gradient point estimate & confidence intervals: -0.687

CI: (2.5 %, 97.5 %): (-1.16 -0.21)

P-value for second segment when H0 = 0.193: 0.0040

# Higher order maximums (2nd, 3rd etc)

The authors note that due to the fact that each of these data points is just a single individual the apparent plateau they observe could be due to random fluctuation. To strengthen their argument they looked at the 2nd highest reported age at death, 3rd highest etc and claimed that these series showed the same pattern. However the data points were only plotted for the 1st MRAD and only cubic smoothing splines for the remaining. Fitting a cubic spline could be misleading / overfitting and each series should probably be processed in the same manner as figure 2A if one is to conclude that they show the same pattern. Below I plot each series individually so the actual data is visible. The cubic splines show downward trends towards the end although with increasing uncertainty and linearity. Similarly with the linear regressions the gradient of the second segments are lower than the first segments although with increasing consistency between the two (note variable y-axis).

# Mean age of death

In another alternate approach the authors looked at all individuals in the dataset to calculate mean age of death and concluded that the annual average age of supercentenarians had not increased since 1968 (the start of the dataset). I recreate their plot below but with the addition of error bars representing the standard error of the mean for each point in order to visualise the uncertainty in the values.

You can see that for the earlier points there are no error bars, this is because there is only a single data point for those years. It is therefore quite misleading to give each mean equal weighting by fitting a cubic spline to point estimates of the means alone.

A perhaps fairer approach is to recreate the graphs but using the whole dataset (note the dataset does not include anyone who died younger than 110). In this form the uncertainty in the first and last few years is much clearer, and the dip pattern fitted above is much less convincing. I would argue that a linear regression fits the data just as well and this gives an increase of ~ 0.04 years per year.

# Sample Sizes

In the study the authors analysed maximum reported age of death (MRAD) over different years but the data for each year was from a different combination of countries and hence the sample size varies. One therefore might expect that the MRAD could change solely due to variation in the sample size (we are more likely to see high maximums when there is more data). Here I investigate the effect of using different sample sizes on the MRAD.

To get an equation for the distribution of age at death we can fit a generalised extreme value distribution to data from the UK Office of National Statistics (which fits much better than a normal distribution).

location: -86.6

scale: 9.83

shape: 0.037

We will also need to estimate the sample size (number of deaths) for each year in each country. For this I multiplied the world bank crude death rate by population size. We can then see how the total sample size varies over time in the original papers analysis.

The trend is similar to the regression lines they fit and so any bias from sample size would result in an overestimate in their favour for the gradient of both of the regression lines. However the effect of sample size on MRAD is probably not linear - maybe the population sizes used are large enough that the MRAD is effectively independent. With the sample size and an equation for the distribution of age at death we can now calculate the probability distribution of MRAD (more formally the nth order statistic) for different sample sizes. First let us look at the distributions of MRAD for the estimated minimum and maximum sample size used in the study.

This shows we might expect a difference of over a year in the MRAD due to the change in sample size alone (dashed lines indicate mode). We can also look at how the modal MRAD changes over many different sample sizes.

The modal MRAD increases sharply at first and then starts to plateau once the sample size increases to millions of deaths. The dashed lines indicate the estimated minimum and maximum sample sizes used in the study. A double log distribution fits this curve well for reasonable sample sizes (>20).

We can also plot the difference from the mean MRAD for each year in the study based on changing sample size alone.

Hence the sample sizes used would probably have a noticeable although small effect on the MRAD and a correction would slightly weaken the authors conclusions by reducing the gradient of both regression lines. Even though the effect is moderate it would have been nice to see an analysis of this type reported in the study.

Whether or not there is a genuine limit rather than a temporary fluctuation would be clearer if there was more than 7-10 years of data beyond the breakpoint, given it is now a decade on perhaps there is new data available, for example from the USA Death Master File.

# Reproducibility

This analysis used readily available data and the code used is available at https://github.com/daniel-wells/human-lifespan-limit

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# A cohort is not representative of humanity

In the freshly published research letter [1], Dong, Milholland, and Vijg (DMV) reported that they found strong evidence for a limit to human lifespan. Analyzing data from International Database on Longevity [2], they found that the yearly maximum reported age at death (MRAD, i.e. age at death of the world’s oldest person died in a specific year) stopped increasing from the mid-1990-s reaching a plateau at around 115 years. Even though the authors acknowledge that the data on “the supercentenarians <…> are still noisy and made of small samples”, they feel safe to conclude that “the results strongly suggest that the human lifespan has a natural limit”. I argue that the results and conclusions of the study are likely to be caused by just a data artifact, and that they are hardly generalizable for the humanity.

The authors chose to divide the study period at the year 1995, which is an arbitrary decision. Yet, this decision imposed a strong effect on the results and conclusions. The conclusions are based primarily on the basis of liner regression trends for the two sub-periods (figure 1, lines 1 and 2). The main conclusion is derived from the negative slope for the second sub-period (figure 1, line 1), which is largely explained by the high outliers (1997 and 1999) and low outliers in the first and the last two years (1995, 2006 and 2007). When those outliers are omitted, the slope for the second sub-period flattens greatly (regression coefficients for lines 2 and 3 are -0.36 and -0.11).

### Figure 1. Reported age at death of supercentenarians

NOTE. The yearly maximum reported age at death (MRAD). The lines represent the functions of linear regressions. All data were collected from the IDL database (all 15 countries included, 1968–2007, n = 668). Unlike DMV, I use all the death records from IDL available on 9 October 2016, not just the data for France, Japan, UK, and US. Unlike DMV, I take MRAD values as age-at-death in days divided by 365.25 to convert into years, not the rounded to years values. The above mentioned data decisions of DMV are not explained and justified in the paper, and, in my view, are not optimal. The outliers are identified with Cook’s distance in two steps. Cook’s distances are 0.57 and 0.10 for the observations 1997 and 1999 in the model 2. After the removal of the high outliers, Cook’s distance for the observations 1995, 2006, and 2007 are 1.50, 0.19, and 0.46, correspondingly.

With the outliers omitted, there is only a tiny difference between the trend lines for the first sub-period and the whole period (figure 1, lines 1 and 4, regression coefficients are 0.15 and 0.12). Seeing how volatile the data are, it seems too hurriedly to drive humanity-wide conclusions based on the presented type of analysis. Imagine, we would now have these data just until 1991. The similarly arbitrary division of the study period at the year 1981 would have shown that the growth in MRAD had stopped and even reverted (figure 1, line 5). We would have then concluded that there was strong evidence for “the limit of human lifespan” at around 113.5 years. Yet, the following one and a half decade would have proven us to be misinterpreting the development of MRAD.

However, the haste of conclusions based on highly vulnerable linear regression estimates is not the only caveat of the presented paper. I believe, it is essentially incorrect to draw conclusions about such an ecumenical concept as human lifespan limit based on so sporadic and erratic data. Most likely, the slowdown in MRAD in 2000-s is a cohort effect. Namely, the difference behind the top-survivors’ data analyzed here is the increase in old-age cohort age-specific mortality rates that took place in the United States in the cohorts born in 1880-s as compared to the cohorts born in 1870-s (figure 2).

### Figure 2: Cohort mortality rates in the United states

NOTE. Data are cohort age-specific mortality rates (CASMR) from Human Mortality Database [3]. The lines represent the average CASMR over two groups of birth cohorts, ten 1-year birth cohorts each: born in 1870-s and 1880-s. See the similar graphs for France, Great Britain, Japan, and Sweden in Extended Data Figure 2.

The US had exceptionally low old-age mortality in the cohorts of 1870-s. That is why the majority of MRAD supercentenarians in 1980-s and 1990-s were from the US (see extended data table 1). For some reasons (which are beyond the scope of the present letter), maximal longevity of the US 1880-s births cohorts tuned out worse than that of the 1870-s birth cohorts (with the average difference in mortality rates of 0.05 at the ages 100-110; p-value < 0.0001). Higher mortality in succeeding cohorts as compared to the preceding cohorts is a relatively rare case; as we may see from the Extended Data Figure 2, similar rise in old-age mortality did not happen in France, Great Britain, Japan, or Sweden. So, quite a local effect of the increase in the US cohort mortality was uplifted by DMV to report the presence of a natural limit for human longevity.

Summing up, the presented evidence does not clearly “suggest that the maximum lifespan of humans is fixed and subject to natural constraints”. I believe, we better put faith in the results of demographers (e.g. Vaupel [4] or Vallin & Meslé [5]), who draw much more optimistic projections based on the population-wide analyses. The dynamics of human mortality show that increased percentages of human populations reach more and more mature ages; eventually the 115-year “limit” will fall, and, in some years, we will likely speculate over the 120-year limit. And so on.

### REFERENCES

1. Dong, X., Milholland, B. & Vijg, J. Evidence for a limit to human lifespan. Nature (2016). doi:10.1038/nature19793
2. Maier, H. et al. Supercentenarians (Springer, 2010). doi:10.1007/978-3-642-11520-2_2
3. The Human Mortality Database (http://www.mortality.org, 2016).
4. Vaupel, J. W. Biodemography of human ageing. Nature 464, 536–542 (2010).
5. Vallin, J. & Meslé, F. The Segmented Trend Line of Highest Life Expectancies. Population and Development Review 35, 159–187 (2009).

### Extended Data Figure 2: Cohort mortality rates in France, Great Britain, Japan, and Sweden

NOTE. Data are cohort age-specific mortality rates (CASMR) from Human Mortality Database (<http://www.mortality.org>). The lines represent the average CASMR over two groups of birth cohorts, ten 1-year birth cohorts each: born in 1870-s and 1880-s.

### Extended Data Table 1. Yearly maximal reported age at death of supercentenarians

NOTE. The table provides the information on yearly maximum reported age at death (MRAD) extracted from the whole the International Database on Longevity (<http://www.supercentenarians.org>, all 15 countries included, 1968–2007, n = 668) obtained on 9 October 2016. Color background delimitates 1870-s birth cohorts from 1880-s birth cohorts.

INFORMATION ON THIS TEXT
This letter was submitted to Nature in the form of Brief Communications Arrising on 2016-10-12. It was rejected on 2016-10-26. I hope, better justified critics from scientific community will be published in Nature soon. Meanwhile, I encourage the authors to publish their comments on my letter.

REPRODUCIBILITY
The R code to reproduce the analysis and figures presented here cam be found by the link: https://github.com/ikashnitsky/a-cohort-is-not-representative-of-humanity

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• ###### Brandon Milholland | 3 years, 3 months ago

Kashnitsky questions our finding of a limit to human lifespan [1]. His objections relate to our analysis of trends of yearly maximum reported age at death (MRAD) from the International Database of Longevity and do not address our other supporting evidence, including our findings of declining improvements in late life mortality, the plateau of the age with greatest improvement in survival, MRAD trends from the Gerontology Research Group, trends of the 2nd to 5th highest RAD from the IDL and trends in the average RAD of supercentenarians.

To demonstrate that the analysis in our Fig. 2a (Model 2 in Kashnitsky’s Fig. 1) is artifactual, Kashnitsky discarded points which are outliers according to Cook’s distance to a linear regression model; these discarded points account for nearly 40% of the data after 1995. After this manipulation of the data, Kashnitsky argues that the trend is best fit by a single linear correlation (Model 4), although this claim is difficult to evaluate because Kashnitsky does not provide any goodness-of-fit statistics for his models. Also, this finding is based on an omission of data without any biological reason, so it appears that Kashnitsky is changing the data to fit his model, rather than vice-versa. Finally, even after altering the data in this way, Kashnitsky still finds a lack of increase in MRAD after 1995 (Model 3), inadvertently providing robust support for our finding that MRAD has plateaued.

In model 5, Kashnitsky brings readers a historical scenario in which he invites us to imagine what would occur if an analysis similar to ours were performed in the year 1991, with the data split at 1981. He found that the potential lifespan limit, as the average MRAD, is around 113.5 years and argued the result is different compared to our estimate (114.9 years). In fact, this result does not contradict ours but supports it, because it falls within the 95% CI (113.1 – 116.7) which we provided.

Next, Kashnitsky contends that the exceptionally small mortality rate of the 1870s cohort in the US, combined with the higher mortality rate of the 1880s US cohort (Kashnitsky’s Fig. 2 & ED Fig. 2) may affect the MRAD (Kashnitsky’s ED Table 1). However, there are several problems with this argument. First, Kashnitsky does not examine cohort mortality in other decades to demonstrate that old age mortality continues to decrease, rather than the 1880s mortality in the US representing a return to a permanent steady state after an abnormal decrease in the 1870s. Second, Kashnitsky does not explain how this cohort effect in a single country would affect the global MRAD. Finally, Kashnitsky does not explain why, if the 1880s cohort has abnormally high mortality, MRADs from the 1890s cohort in both the IDL and GRG data display no significant increase. In summary, Kashnitsky has failed to provide support to his argument that “the results and conclusions of the study are likely to be caused by just a data artifact, and that they are hardly generalizable for the humanity.”

1 Dong, X., Milholland, B. & Vijg, J. Evidence for a limit to human lifespan. Nature 538, 257-259, doi:10.1038/nature19793 (2016).

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### Evidence for a limit to human lifespan

Dear Brandon Milholland,

Thank you for your interesting comments on your paper "Evidence for a limit to human life span", Dong et al., Nature 2016.

May I ask some questions regarding your Figure 1d, which presents the relationship between calendar year and the age that experiences the most rapid gains in survival:

How do you measure the gains in survival in this particular case: as a difference between the numbers of survivors over time, or as a difference between the LOGARITHMS of the numbers of survivors over time? Or something else?

What time intervals do you use, when you measure the gains in survival over time: just one year time interval (data for two close calendar years, say years 1981 and 1982, for example)? Or do you use longer time intervals?

P.S.: By the way, you may enjoy our related published study, which explains why the chances for longevity records are much smaller than they were assumed earlier: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4342683/

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• ###### Brandon Milholland | 3 years, 4 months ago

The gain in survival is measured as the slope of the linear regression of the log-transformed number of survivors per 100,000.

In Figure 1d, the data used is for the 100 year interval preceding the date (e.g., if the x-coordinate is 1940, then the regressions were performed on the data from 1840-1940; then the age with the highest gain was used to determine the y-coordinate of the point).

• ###### Leonid Gavrilov | 3 years, 4 months ago

Dear Brandon Milholland,

Many thanks for your helpful reply! What is the evidence that the dependence of log-transformed data for survivors is indeed a strictly linear function of time over the entire century?

Figure 1b and the Extended Data Figure 2 contain no data points (just graphic lines only) to prove that this relationship is indeed linear. Have you tried a segmented regression to check for possible breakpoints in the linear regression for this particular case?

Did you observe any curvature for these trajectories, and in what direction: were they concave up or concave down?

• ###### Brandon Milholland | 3 years, 4 months ago

They were linear, without any sign of curvature or breakpoints. This was confirmed by the goodness-of-fit statistics (R>.85), (p<2.2e-16).

• ###### Ilya Kashnitsky | 3 years, 3 months ago

Dear Brandon,

Goodness-of-fit statistics is not enough to conclude that there is no curvature in the data.

R code:

library(ggplot2)
library(dplyr)

# generate data with curvature
df <- data.frame(x = 1:100) %>% mutate(y = sqrt(x))

# plot
ggplot(df, aes(x,y)) +
geom_point() +
stat_smooth(method = 'lm', color = 'red') +
theme_minimal()

summary(lm(df, formula = y~x))


### Summary statistics of the liner model fitted to the evidently curved data

Call:
lm(formula = y ~ x, data = df)

Residuals:
Min      1Q  Median      3Q     Max
-1.7984 -0.2555  0.1324  0.3597  0.4406

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.719268   0.090610   30.01   <2e-16 ***
x           0.079116   0.001558   50.79   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.4497 on 98 degrees of freedom
Multiple R-squared:  0.9634,    Adjusted R-squared:  0.963
F-statistic:  2580 on 1 and 98 DF,  p-value: < 2.2e-16


### Conclusion

The goodness-of-fit statistics of a liner trend fitted to the data generated by y = sqrt(x) is 0.96 (compared to your 0.85).

Please, provide a better proof of no curvature in the data.

• ###### Ilya Kashnitsky | 3 years, 3 months ago

Sorry, markdown is parsed poorly.

Please, see the same comment properly parsed here http://txti.es/161027-comment-to-bm/images

• ###### Leonid Gavrilov | 3 years, 3 months ago

Dear Brandon Milholland,

Thank you for your brief reply to my earlier question (what is the evidence that the dependence of log-transformed data for survivors is indeed a strictly linear function of time over the entire century):

--"They were linear, without any sign of curvature or breakpoints. This was confirmed by the goodness-of-fit statistics (R>.85), (p<2.2e-16)."

What particular country, sex (male or female?), age group and time interval did you analyze by the goodness-of-fit statistics to get this R>.85, (p<2.2e-16)?

Also taking into account a prior valid comment by Ilya Kashnitsky here (see above), have you tried a more powerful segmented regression to check for possible breakpoints in the linear regression?

If the number of survivors to a particular age is indeed increasing strictly exponentially over time for the entire century with absolutely no deviation, as you suggest, that would be a major demographic discovery and a revolution in population forecasting. Hence some more details and proof are needed to justify this statement.

• ###### Edouard Debonneuil | 3 years, 3 months ago

It would be interesting to do a mascarade article with nematodes: "Evidence for a limit to nematode lifespan": similar biased statistics would support the adult lifespan of lab c elegans to converge towards 20 days, whereas in practice some people like Hugo Aguilaniu make those animals live 300 days.

• ###### Leonid Gavrilov | 3 years, 3 months ago

Correction to the Nature article "Evidence for a limit to human lifespan"

The following sentence in the legend for Extended Data Figure 4 needs to be corrected:

"The rate of change is the slope of the line calculated by an EXPONENTIAL regression, that is, b in the equation y = a + bx, where x is AGE and y is the logarithm of the number of survivors to that age per 100,000."

Two corrections are needed:

EXPONENTIAL --> LINEAR AGE --> TIME

The corrected sentence should be:

"The rate of change is the slope of the line calculated by LINEAR regression, that is, b in the equation y = a + bx, where x is TIME and y is the logarithm of the number of survivors to that age per 100,000."

I have discussed these corrections with one of the authors of the Nature article, @Brandon Milholland, who agreed with them:

• ###### Martin J Sallberg | 3 years, 2 months ago

Any good post mortem must contain an investigation of what went wrong. In this case, it is obvious that it is the extreme fragmentation of the most respected journals that prevents relevant empirical data from being used to falsify bunkum. This phenomenon is, on a general level, explained with ideas on how to solve it at http://scientific-method.wikia.com/wiki/Scientific_Method_Wiki

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### Comments about Vijg Letter and Olshansky commentary in Nature 5 October 2016, JWV

This publication is another travesty in a century-long saga of asserted looming limits to average and maximum human lifespan. It is disheartening how many times the same mistake can be made in science and published in respectable journals.

A century ago it was believed that average lifespan—life expectancy—would never exceed 65. As evidence to the contrary poured in, the limit was raised and raised again. Olshansky pegged it at 85. Japanese women today, however, can expect to live more than 87 years.

A century ago the maximum span of life was believed to be about 105. Again this limit was increased as people exceeded it. Vijg and Olshansky set it at 115 even though the current record holder, Jeanne Calment, lived 122.45 years: she is dismissed as an “outlier”.

In this sorry saga, those convinced that there are looming limits did not apply demography and statistics to test hypotheses about lifespan limits—instead they exploited rhetoric, deficient methods and pretty graphics to attempt to prove their gut feelings. The publications are essentially propaganda, not scholarly research.

Vijg’s travesty and Olshansky’s commentary on it in the same issue of Nature are further dismal examples. The material was published and is getting publicity because it seems plausible to many people that average and maximum lifespans cannot increase much more. The main evidence is summarized in colorful graphs that are problematic.

• It is claimed that life expectancy is plateauing, approaching a looming limit, but the Figures in Vijg, including Fig. 1a for France and subsequent Figures for Japan, Italy and other large countries with high life expectancies, do not support this. They show a continuing rise in life expectancy albeit, in some cases, at a somewhat slower rate than in some earlier periods. There is no evidence that the slower rate will become an even slower rate and then zero.
• The age at which the most rapid progress is being made in increasing survival is shown to be high—above 100 in recent years—and rising to higher and higher ages. It is claimed that this age plateaued after 1980 but again this is not supported by the graphs. The most important country for the analysis is Japan, a country with a large population and the world’s life expectancy leader. In Japan there is no plateau. Nor is there one for France and Italy, two other countries with large populations and high life expectancies, although there is some deceleration in the rate of increase. Again, there is no evidence that there will be further deceleration leading in the near future to a plateau.
• Data on the maximum recorded age at death are simplistically and without any statistical justification fit by two lines—a rising line and after 1995 a declining line. More powerful methods, including methods from Extreme Value Theory, should have been used to test whether the data imply a decline in maximum lifespan.

Like analogous, disproven publications over the past 100 years, Vijg et al. and Olshansky add nothing to scientific knowledge about how long we will live. The publications are advocates’ arguments based on selective use of data, with one-sided conclusions not supported by the data.

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• ###### Brandon Milholland | 3 years, 4 months ago

We respectfully disagree with Dr. Vaupel in his criticisms on our paper "Evidence for a limit to human life span", Dong et al., Nature 2016.

Our paper is entitled "Evidence for a limit to human lifespan"--not life expectancy. This distinction is important. We agree with Dr. Vaupel that life expectancy will likely continue to increase for the foreseeable future. However, maximum lifespan appeared to have reached its limit. Of course, life expectancy must be lower than maximum lifespan, so it too will reach its limit, but at its current rate of increase of around 0.2 years per year, it will be over a century before it reaches that limit in most countries, even longer if it decelerates as it approaches its ceiling.

How about the main claim of our article, the proposed limit to human lifespan? First, it should be evaluated on its own merits. Previous predictions may have been wrong, but that does not mean ours automatically is also wrong. As for Jeanne Calment, she is not dismissed as an outlier, but was included in our analyses. Her age at death of 122 exceeds our limit of 115, but that figure is not meant as a hard limit but rather the likely value of the world MRAD (maximum reported age at death) in any given year. The MRAD might be higher in some years, as it happened to be in 1997, but higher values are improbable and likely to be followed by a regression to the mean of 115 (hence the downward trend since Calment's death). Calment's exceptional longevity did prompt us to attempt to calculate the ultimate limit to human lifespan: we calculated that the MRAD would exceed 125 only once every 10,000 years. Since this time frame is longer than all of human history to date, we consider our estimate of the outer reaches of human lifespan to be a fairly liberal one.

Below we respond to each of Dr. Vaupel's points:

• Figure 1a and the corresponding graphs in Extended Data Figure 1 do show increasing life expectancies, but, as we have explained, this does not undermine the thesis of the article.

• As for the age at which the most rapid progress is being made, i.e. our Figure 1d, the plateau seems fairly obvious to us. In France, looking at the years prior to 1980, the age with the greatest increase in survival goes up every few years, but since then it has only gone up rarely. In females it went up to 103 in 2004, went back down to 102 the next year, and then up to 103 in 2007, so it looks like it is not really going up but perhaps fluctuating between 102 and 103; regardless, there hasn't been an increase for a decade (the most recent data available is for 2014). As for males, it increased to 101 in 2008, but before that it has been at 100 since 1981. This represents an increase of one in an interval of over 30 years; contrast that with the 30-year interval 1950-1980 where it increased 12 years, from 87 to 99. It is possible that the ages with the highest gain might creep up another year or two, but their slow rate of increase over the past few decades seems to indicate that they have either reached or will soon reach their limit. There is a similar situation in Italy, where the age with the greatest gain has remained stagnant since 2004 (in females) and 2002 (in males); and in Japan, where the corresponding years are 2004 and 1998.

• If the MRAD were increasing continuously without limit, then splitting the data into two parts should result in linear regressions with positive slopes in both parts. Instead, we find that the first half has a positive slope, while the second half has a negative slope. The latter is not significant, so we conclude that the MRAD is essentially flat and does not vary in a manner correlated with time after the 1990s. As an additional validation, we re-analzyed the data from Figure 2a using a segmented regression (R package “segmented", reference “https://www.researchgate.net/publication/234092680_Segmented_An_R_Package_to_Fit_Regression_Models_With_Broken-Line_Relationships”). The package automatically detect breakpoints. In the first model, segmented regression, the MRAD shows a linear increase between 1968 and 1998 (one year after Calment’s death) of 0.20 per year; then it decreases linearly between 1999 and 2006 by 0.69 per year. This model has an adjusted R-squared of 0.41 and an AIC of 140.7. In the second model, linear regression of the entire dataset, the MRAD shows a linear increase of 0.12 per year, with an adjusted R-squared of 0.27 and an AIC 146.2. With a higher R-squared and a lower AIC, the first model, segmented regression, is clearly a better fit and more appropriate model for the data.

The set of MRAD values fell within a fairly narrow range, so an analysis using extreme value theory does not seem warranted. Nonetheless, if other researchers believe that applying other statistical techniques to these data could yield additional insights, they should feel free to publish their findings.

Xiao Dong, Brandon Milholland and Jan Vijg

• ###### James F Fries | 3 years, 4 months ago

This is a really interesting topic. The discussions around this subject, however, have become polarized and repetitive. It’s a continuation of the same argument and overreactions from the same researchers.

115 is probably a closer number the the lifespan limit. People aren’t really living longer - at least not in the last 40 to 50 years. A few people live to 115, and many more to 110 years old. That says something pretty solid. If somebody is not only out there on the edge of a gaussian curve, but is seven to eight years farther than anyone else, it defies the laws of statistics. It doesn’t mean it couldn’t happen, but it is unusual. Jeanne Calment may be a very clever case of age exaggeration. Either way, you can’t really say anything scientific about a single outlier.

• ###### Leonid Gavrilov | 3 years, 4 months ago

Dear Brandon Milholland,

Thank you for your interesting comments on your paper "Evidence for a limit to human life span", Dong et al., Nature 2016.

May I ask some questions regarding your Figure 1d, which presents the relationship between calendar year and the age that experiences the most rapid gains in survival:

1. How do you measure the gains in survival in this particular case: as a difference between the numbers of survivors over time, or as a difference between the LOGARITHMS of the numbers of survivors over time? Or something else?

2. What time intervals do you use, when you measure the gains in survival over time: just one year time interval (data for two close calendar years, say years 1981 and 1982, for example)? Or do you use longer time intervals?