The paper by Dr. Lin suggests a new statistical model for fitting and estimation of mortality rates at advanced ages.  This model can also be described as phenomenological model, because it is not based on any fundamental mechanism or theory, but is simply an empirical fitting formula with 4 parameters.  For the purpose of theoretical justification for this formula, the author argues that a similar approach is applied in reliability engineering.  The idea to apply the reliability theory to explain the late-life mortality trends is indeed very appealing and it resonates strongly with our own studies on the same topic (Gavrilov, Gavrilova, 1991; 2001).  However, a clear distinction should be made between the fundamental reliability theory based on particular mechanisms of failure, and empirical fitting methods occasionally used in the practice of reliability engineering.  The model suggested in this paper represents the second, empirical approach of reliability engineering, although it might be possible that a theoretical justification for the suggested formula could be found later.

The discussed paper is using a somewhat non-traditional approach.  It relates hazard function with survival function (also called reliability function).  Taking into account that both functions are related to each other by definition, it raises questions on the interpretation of results for such analyses.  This topic may deserve some discussion because this is the second time when I encounter mortality analysis, which relates two variables linked to each other by definition.  The first case is a series of papers by Azbel (1999a; 1999b) who compared cumulative survival functions at different ages (related to each other by definition) and found remarkably strong relationship between two values of  l_x   (which is not surprising taking into account functional dependence between two values of survival probabilities).   The problem of interpretation for the results of such analyses may deserve a special discussion.

I would like to say a few words about parameters of reliability formula used by the author.  The formula used by the author is rather complicated, but two parameters of it look familiar: parameters B and C.  Parameter B looks like an analog of Makeham parameter in the Gompertz-Makeham formula.  To the author’s benefit, Dr. Lin analyzed changes of parameters in history using a set of Taiwan life tables.  According to this data, parameter B in history behaves very similarly to the Makeham parameter: it is high at the beginning of the 20th century, then it declines sharply and stabilizes in the second half of the 20th century.  As for the parameter C, it represents the limiting mortality at extreme ages (late-life mortality plateau).  The values of parameter C presented in the paper by Dr. Lin look too high – 2-3 per year.  With so high levels of mortality at advanced ages we could not expect not only the case of Jeanne Calment but even supercentenarians – persons survived by ages 110.  For example, using estimates of mortality at extreme ages proposed by Greenwood and Irwin (1939) we can calculate chances of centenarian to become a supercentenarian, which is equal to (0.5)¹⁰ = 0.001.  Thus, given the number of centenarians in developed countries, we can expect the emergence of supercentenarians in these countries.  On the other hand, the case of Jeanne Calment is a real challenge to the probability theory, because the chances of centenarian to reach age 122 are close to zero:  (0.5)²² = 2·10^-7.  In order to explain the case of Jeanne Calment we need to postulate that mortality is declining at very advanced ages.

Finally, paper by Dr. Lin compares results of fitting the Gompertz model and the author’s reliability model to mortality data.  To my opinion, it is not entirely fair to compare the 2-parameter Gompertz model with 4-parameter model proposed by the author.  I would suggest a comparison of the author’s reliability model with the Gompertz-Makeham model, which has age-independent term similar to the parameter B in the author’s formula.

Summarizing, I would like to say that we are now at the beginning of coming to a consensus on the mortality patterns over age 100.  This Symposium is definitely a step forward in developing such consensus.

Acknowledgments.  I would like to thank the organizers of this Symposium for the unique opportunity to take part in this very interesting scientific meeting. 

Azbel, M.Y. (1999a)  Phenomenological theory of mortality evolution: its singularities, universality, and superuniversality.  Proc. Natl. Acad. Sci. U.S.A.  96: 3303-7.

Azbel, M.Y. (1999b)  Empirical laws of survival and evolution. Proc. Natl. Acad. Sci. U.S.A.  96: 15368-15373.

Gavrilov, L.A. & Gavrilova, N.S. (1991) The Biology of Life Span: A Quantitative Approach, New York: Harwood Academic Publisher.

Gavrilov, L.A. & Gavrilova, N.S. (2001) The reliability theory of aging and longevity.  Journal of Theoretical Biology  213(4): 527-545.

Greenwood, M. & Irwin, J.O. (1939) The biostatistics of senility. Hum. Biol. 11: 1-23.

Contact address of the discussant:

Dr. Natalia S. Gavrilova, Center on Aging
NORC/University of Chicago
1155 East 60th Street
Chicago, IL 60637-2745
USA
Fax: (773) 256-6313, Phone: (773) 256-6359
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