Discussion of the paper
"Oldest-Old Mortality Rates and the Gompertz Law:
A Theoretical and Empirical Study Based on Four Countries"
presented by Ching-Syang Jack Yue (Jack C. Yue)
at the International Symposium "Living to 100 and Beyond: Survival at Advanced Ages"
(Lake Buena Vista, Florida, January 17-18, 2002)

Discussant: Leonid A. Gavrilov
Center on Aging, NORC and the University of Chicago

This is an interesting study, which suggests and applies new rigorous statistical methods to test the validity of the Gompertz law at extreme old ages. The author also suggested an explanation why the Gompertz law fails at older ages, based on the idea of population heterogeneity.

It is important to discuss the scientific background related to this study because the significance of this work could be more fully appreciated in the context of previous findings.

The current situation with applicability of the Gompertz law to extreme old ages is paradoxical. On the one hand, it is well known for a long time that the Gompertz law is not applicable to mortality rates at advanced ages -- the observed mortality rates are always lower than predicted by the Gompertz model, and, not surprisingly, the actual numbers of survivors to extreme ages is always higher than predicted by the Gompertz law.  The picture below illustrates the mortality deceleration observed at advanced ages contrary to the predictions of the Gompertz law:
mortality at advanced ages

The scientific literature on the deviation of mortality rates from the Gompertz law at advanced ages is abundant, starting with the study of Benjamin Gompertz himself (Gompertz,1825; Makeham, 1867;  Brownlee, 1919;  Perks, 1932;  Greenwood & Irwin, 1939;  Mildvan & Strehler, 1960;  Strehler, 1960;  Economos, 1979,1980,1983,1985;  Gavrilov & Gavrilova, 1991). The most recent and detailed review of the scientific literature on this issue was made by Olshansky (1998).

Paradoxically, the Gompertz law and the Gompertz-Makeham law are nevertheless often applied to estimate the oldest-old mortality rates by extrapolation in order to "close" the life tables. When confronted with the question why these "wrong" formulas are used, the demographers/actuaries usually reply that this is not an important issue, because life expectancy at birth is not very sensitive to the way how exactly the life tables are closed. The same "wrong" formulas and related assumptions are often used for graduation (smoothing procedures) of the mortality trajectories at advanced ages. It is extremely important, therefore, to know exactly how a particular life table was closed and/or graduated, before using it for testing of any statistical models. If the Gompertz or the Gompertz-Makeham laws were already introduced into the data by the method of life table construction/graduation, these data would not be useful for statistical hypothesis testing.

To summarize the coverage of scientific background, we may conclude that the Gompertz law is known to be not applicable to the oldest-old mortality, unless the data are spoiled by artificial introduction of this law during extrapolation/graduation procedures.  The discussed study by Jack Yue confirms that the Gompertz law is indeed not applicable to the oldest-old mortality, and this conclusion validates the new statistical methods suggested by the author for testing the Gompertz law.

The next important question is:  why is the Gompertz law not applicable to the oldest-old mortality? The author of the discussed paper suggests that this may be the result of population heterogeneity at advanced ages. The same hypothesis was also suggested by other authors (Carnes, Olshansky, 2001). There is, however, one problem with testing this hypothesis. It may be not difficult to generate the mortality trajectories that will be close to the observed trajectories by assuming that population is a mixture of subgroups of people with different Gompertz parameters. The real problem here is whether there is any sense in such computational exercise. Is there any direct evidence indicating increased population heterogeneity at advanced ages?

The answer to this question is provided at the picture below. The graph illustrates the dependence of daughters’ lifespan (expressed as additional years of life gained/lost compared to the reference lifespan level in the same birth cohort) as a function of maternal lifespan. For more on methodological details of this study see (Gavrilova, Gavrilov, 2001).  Interestingly the dependence of daughters lifespan on maternal lifespan looks like consisting of two lines – one for shorter-lived mothers (died before age 85) with very weak dependence of daughter's lifespan on maternal lifespan and another – for longer-lived mothers (died after age 85) with extremely strong and steep dependence.

                         heritability of longevity


Daughters born to long-lived mothers may live 10 years longer on average if their mother reached age 100. This indicates that long-lived people are fundamentally different from other people in the sense that their children also live significantly longer lives. The breaking point at about age 85 for mothers indicates the age when deaths becomes much more selective in their timing and when population heterogeneity becomes an important issue.

Overall, the discussed paper contains interesting ideas and important findings on the oldest-old mortality, which correspond nicely with results obtained by other authors.


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Carnes, B.A., Olshansky S.J. (2001) Heterogeneity and its biodemographic implications for longevity and mortality. Exp Gerontol. 36(3):419-30.

Economos, A.C. (1979) A non-gompertzian paradigm for mortality kinetics of metazoan animals and failure kinetics of manufactured products. AGE 2, 74-76.

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Contact address of the discussant:

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