### Rebuttal of the Criticism of our Longevity Study

Greetings,

Recently we were asked for our opinion on the new criticism of our study "Mortality measurement at advanced ages: A study of the Social Security Administration Death Master File", published by the North American Actuarial Journal, 2011, 15(3): 432-447. This criticism of our study was presented in the paper "Measurement of Mortality among Centenarians in Canada" by Ouellette and Bourbeau in the Society of Actuaries 2014 Living to 100 Monograph.

We have read this paper with great interest. Ouellette and Bourbeau make claims about "refutation" of our published findings while more careful analysis of the results presented in their paper leads to the opposite conclusion. Below we present evidence that paper by Ouellette and Bourbeau is a misrepresentation and misinterpretation of our findings, rather than their refutation.

The authors believe that they "refuted" our results published in NAAJ (Gavrilov, Gavrilova, 2011) while in reality the results by Ouellette and Bourbeau confirm our findings published in NAAJ, 2011 and presented at the 2014 Living to 100 Symposium (Gavrilova, Gavrilov, 2014).

For example, Ouellette and Bourbeau argue that their

The apparent slow down of mortality after age 109 years presented in Figure 1 of Ouellette and Bourbeau is based on 7 deaths only, with no proof of statistical significance. Our published computer simulation study (Gavrilova, Gavrilov, 2014) found that age-specific death rates (used by Ouellette and Bourbeau) are biased downward after age 110 years, thus producing a spurious mortality deceleration. This is exactly what Ouellette and Bourbeau observe in their study (mortality deceleration starting at age 109 years).

In our study (Gavrilov, Gavrilova, 2011) we analyzed properties of Nelson-Aalen hazard rate estimate, while Ouellette and Bourbeau incorrectly wrote that we studied the age-specific death rates:

In our 2011 paper we did not work with death rates but specifically analyzed the Nelson-Aalen yearly and monthly hazard rate estimates and demonstrated that yearly estimates are biased downward after age 90 (compared to more accurate monthly estimates). Death rates (used by Ouellette and Bourbeau), on the contrary, are not strongly biased downward up to age 110 years (as we demonstrated by computer simulations in a more recent 2014 study), therefore the results of their exercises with different lengths of age intervals were quite predictable. In our paper presented at the 2014 Living to 100 Symposium we conducted a computer simulation study, which found that yearly estimates of death rates are not biased up to age 110 years and after age 110 years death rates are biased downward. This is exactly the pattern that Ouellette and Bourbeau report in their paper.

It seems that Ouellette and Bourbeau incorrectly confused Nelson-Aalen estimate of hazard rate with probability of dying. This empirical estimate of hazard rate is obtained from the cumulative hazard rate in the following way:

We wrote about these differences between hazard rates and probabilities of dying in our book "The Biology of Life Span" in 1991. In our book we wrote at page 40:

However, some researchers still confuse these two different measures. In their paper Ouellette and Bourbeau calculated Nelson-Aalen estimates of hazard rate (they erroneously called them as

Also the sample of supercentenarians used by Ouellette and Bourbeau is too small (7 females only) for making robust conclusions about the shape of mortality trajectories after age 110 years.

Gavrilov L.A., Gavrilova N.S. 2011. Mortality measurement at advanced ages: A study of the Social Security Administration Death Master File. North American Actuarial Journal, 2011, 15(3): 432-447. PMCID: PMC3269912.

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3269912/

Gavrilova N.S., Gavrilov L.A. 2014. Mortality Trajectories at Extreme Old Ages: A Comparative Study of Different Data Sources on U.S. Old-Age Mortality. In: 2014 Living to 100 Monograph [published online - August, 2014, http://livingto100.soa.org/]. The Society of Actuaries, 2014, 23 pages. PMCID: PMC4318539. URL: https://www.soa.org/Library/Monographs/Life/Living-To-100/2014/mono-li14-3a-gavrilova.pdf

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4318539/

Gavrilov L.A., Gavrilova N.S. The Biology of Life Span: A Quantitative Approach, NY: Harwood Academic Publisher, 1991, 385p.

Ouellette, N. and Bourbeau, R. 2014. Measurement of Mortality among Centenarians in Canada. In: 2014 Living to 100 Monograph [published online - August, 2014, http://livingto100.soa.org/]. The Society of Actuaries, 2014.

======= THE END ===========

Recently we were asked for our opinion on the new criticism of our study "Mortality measurement at advanced ages: A study of the Social Security Administration Death Master File", published by the North American Actuarial Journal, 2011, 15(3): 432-447. This criticism of our study was presented in the paper "Measurement of Mortality among Centenarians in Canada" by Ouellette and Bourbeau in the Society of Actuaries 2014 Living to 100 Monograph.

We have read this paper with great interest. Ouellette and Bourbeau make claims about "refutation" of our published findings while more careful analysis of the results presented in their paper leads to the opposite conclusion. Below we present evidence that paper by Ouellette and Bourbeau is a misrepresentation and misinterpretation of our findings, rather than their refutation.

The authors believe that they "refuted" our results published in NAAJ (Gavrilov, Gavrilova, 2011) while in reality the results by Ouellette and Bourbeau confirm our findings published in NAAJ, 2011 and presented at the 2014 Living to 100 Symposium (Gavrilova, Gavrilov, 2014).

For example, Ouellette and Bourbeau argue that their

*"results refute recent findings suggesting that proper hazard rate estimation … leads to a steady death rate increase over age, up to age 106 and possibly beyond (Gavrilov and Gavrilova 2011, 2014)"*(Ouellette and Bourbeau, p.12). However, their own data, presented in Figure 1b, show a steady death rate increase over age, up to age 106, confirming our conclusion (Gavrilov, Gavrilova, 2011).The apparent slow down of mortality after age 109 years presented in Figure 1 of Ouellette and Bourbeau is based on 7 deaths only, with no proof of statistical significance. Our published computer simulation study (Gavrilova, Gavrilov, 2014) found that age-specific death rates (used by Ouellette and Bourbeau) are biased downward after age 110 years, thus producing a spurious mortality deceleration. This is exactly what Ouellette and Bourbeau observe in their study (mortality deceleration starting at age 109 years).

In our study (Gavrilov, Gavrilova, 2011) we analyzed properties of Nelson-Aalen hazard rate estimate, while Ouellette and Bourbeau incorrectly wrote that we studied the age-specific death rates:

*"The similarity in the age-trajectory of death rate series for various lengths of the age interval was, however, recently challenged by Gavrilov and Gavrilova (2011, 437; see their figure 1 for one-year vs. one-month age intervals), despite that one would expect comparable trends given the definition of the death rate"*(Ouellette and Bourbeau, 2014, p.7-8).In our 2011 paper we did not work with death rates but specifically analyzed the Nelson-Aalen yearly and monthly hazard rate estimates and demonstrated that yearly estimates are biased downward after age 90 (compared to more accurate monthly estimates). Death rates (used by Ouellette and Bourbeau), on the contrary, are not strongly biased downward up to age 110 years (as we demonstrated by computer simulations in a more recent 2014 study), therefore the results of their exercises with different lengths of age intervals were quite predictable. In our paper presented at the 2014 Living to 100 Symposium we conducted a computer simulation study, which found that yearly estimates of death rates are not biased up to age 110 years and after age 110 years death rates are biased downward. This is exactly the pattern that Ouellette and Bourbeau report in their paper.

It seems that Ouellette and Bourbeau incorrectly confused Nelson-Aalen estimate of hazard rate with probability of dying. This empirical estimate of hazard rate is obtained from the cumulative hazard rate in the following way:

*"The Nelson-Aalen estimator... provides an efficient means of estimating the cumulative hazard function H(t). In most applications, the parameter of interest is not H(t), but rather its derivative h(t), the hazard rate."*(Klein, J.P., Moeschberger, M.I. Survival Analysis. Techniques for Censored and Truncated Data. Springer, 1997, p.152, section 6.2 "Estimating the Hazard Function"). In contrast to probability of dying, the Nelson-Aalen formula for hazard rate has length of age interval in denominator, so it is not dimensionless:*"Hazard rates are rates; i.e., they have units 1/t."*(Cleves, M., Gutierrez, R., Gould, W., Marchenko, Y. An Introduction to Survival Analysis Using Stata. 2nd ed. Stata Press, College Station, 2008, p.15). In contrast to probabilities, hazard rates can be multiplied, summed up and expressed by different units of measurement.We wrote about these differences between hazard rates and probabilities of dying in our book "The Biology of Life Span" in 1991. In our book we wrote at page 40:

*"Values for the probability of death depend on the length of the age interval (delta x) for which they are calculated. ... Any conversion of the values for the probability of death from one age interval to another, with the aim of comparing results, must be carried out in conformity with the algebra of probability theory, and not by simple multiplication or division of the figures."*However, some researchers still confuse these two different measures. In their paper Ouellette and Bourbeau calculated Nelson-Aalen estimates of hazard rate (they erroneously called them as

*'incorrectly "annualized" probabilities of death'*) and found that these estimates for more narrow age intervals are more accurate, which confirms our earlier results (Gavrilov, Gavrilova, 2011).Also the sample of supercentenarians used by Ouellette and Bourbeau is too small (7 females only) for making robust conclusions about the shape of mortality trajectories after age 110 years.

**References**

Gavrilov L.A., Gavrilova N.S. 2011. Mortality measurement at advanced ages: A study of the Social Security Administration Death Master File. North American Actuarial Journal, 2011, 15(3): 432-447. PMCID: PMC3269912.

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3269912/

Gavrilova N.S., Gavrilov L.A. 2014. Mortality Trajectories at Extreme Old Ages: A Comparative Study of Different Data Sources on U.S. Old-Age Mortality. In: 2014 Living to 100 Monograph [published online - August, 2014, http://livingto100.soa.org/]. The Society of Actuaries, 2014, 23 pages. PMCID: PMC4318539. URL: https://www.soa.org/Library/Monographs/Life/Living-To-100/2014/mono-li14-3a-gavrilova.pdf

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4318539/

Gavrilov L.A., Gavrilova N.S. The Biology of Life Span: A Quantitative Approach, NY: Harwood Academic Publisher, 1991, 385p.

Ouellette, N. and Bourbeau, R. 2014. Measurement of Mortality among Centenarians in Canada. In: 2014 Living to 100 Monograph [published online - August, 2014, http://livingto100.soa.org/]. The Society of Actuaries, 2014.

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