### Evidence for a limit to human lifespan

##### Content of review 1, reviewed on October 19, 2016

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/

##### Source

© 2016 the Reviewer (CC BY 4.0).

#### Brandon Milholland

##### 12:49 a.m., 20 Oct 16 (UTC) | Link

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

##### 3:25 p.m., 20 Oct 16 (UTC) | Link

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

##### 5:35 p.m., 26 Oct 16 (UTC) | Link

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

##### 8:17 a.m., 27 Oct 16 (UTC) | Link

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

##### 8:35 a.m., 27 Oct 16 (UTC) | Link

Sorry, markdown is parsed poorly.

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

#### Leonid Gavrilov

##### 6:59 p.m., 7 Nov 16 (UTC) | Link

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

##### 8:20 p.m., 18 Nov 16 (UTC) | Link

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

##### 6:35 p.m., 20 Nov 16 (UTC) | Link

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: