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Monday, February 25, 2013

Taeuber's Paradox and the Life Expectancy Brick Wall

A friend and I were talking the other day about mortality, morbidity, and other cheerful topics, and I happened to mention to him Keyfitz's classic 1977 paper on Taeuber's Paradox (for which there is, at the time of this writing, no Wikipedia page, mercifully), which in turn sprang from the somewhat counterintuitive finding that if cancer were eliminated as a cause of death, it would yield an increase in life expectancy of only a little more than 3 years. Maybe not everybody will find this result counterintuitive. It's by no means certain that Conrad Taeuber himself did. But Nathan Keyfitz did, and I did, and my friend Jeff did; and so, for purposes of this discussion, that's a quorum.

Cancer is not one disease, of course. Like "heart disease," it's a multiplicity of unspeakably terrible ailments. Nevertheless we count it as one disease in discussions of mortality in this country, so that we can point at it and say "Cancer is the Number Two cause of death in America," and then presidents can declare war on it, $10 billion a year in taxpayers' money can be set aside for research on it (approximately $500 billion in 2012 dollars spent since Nixon declared war), a $50-billion-a-year commercial industry of toxic therapies (some of which cost $10,000 a month) can be built around it, and meanwhile delusional goofballs like Ray Kurzweil can talk of achievable immortality (with arguments that don't even come close to passing the straight-face test) when there's no cancer cure in sight. (I don't consider transplanting my brain into silicon to be the same as achieving immortality, incidentally.)

It might do the Kurzweils of the world some good to spend a little time pondering the fact that roughly $20,000 in anti-cancer research money has been spent for every single person in the U.S. who has died of cancer in the last 40 years; and yet after 40 years, cancer is still the No. 2 cause of death in America; and after it's gone, after it's cured once and for all, this bane of human existence, this No. 2 Cause of Death, we will have extended human life a grand total of (drum roll, please) 3.3 years (loud cymbal-crash).

One reason eliminating such a significant cause of death has such a miniscule impact on life expectancy is that other causes of death rush in to fill the void. If you're 75 years old, suddenly eliminating cancer as a cause of death still leaves you with all the other killer diseases that make 75-year-olds go tits-up. It's more complicated than that, of course. One thing you have to consider is that eliminating a disease of later life has much less effect on life expectancy than eliminating an early-in-life disease. If you can prevent a fatal disease of childhood, the contribution to average life expectancy is much greater than if you can cure a disease that only befalls 90-year-olds. This is why life expectancies rose so sharply in the first years of the 20th century (and why we're not likely to see such a surge repeated any time soon). Starting in the early 1900s, killer diseases of early childhood (and early adulthood) began to abate one by one.

Bottom line, the calculation of Potential Gain in Life Expectancy (PGILE) is far from straightforward, because you need to know the mortality rate for the illness-in-question for every year of a person's life, and depending how that curve shapes out, you get a final PGILE number that's bigger or smaller than you might have guessed based on the illness's overall ranking in national causes of death.

Back in 1999 (but unfortunately not since then), the Centers for Disease Control, using 1990 Census data (and other data of the time), published information on the potential gain in life expectancy to be expected if various categories of death were eliminated. The numbers are shown in the table below.

CARDIOVASCULAR: All cardiovascular diseases 6.73
CANCER: Malignant neoplasms, including neoplasms of lymphatic and hematopoietic tissues, AIDS, etc. 3.36
Diseases of the respiratory system 0.97
Accidents and "adverse effects" (health-care-induced deaths) 0.92
Diseases of the digestive system 0.46
Infectious and parasitic diseases 0.45
Firearm deaths 0.4
Certain conditions originating in the perinatal period 0.33
Suicide 0.3
Homicide and "legal intervention" (law-enforcement and penal-system-induced deaths) 0.29
Diabetes mellitus 0.27
Congenital anomalies 0.2
Alcohol-induced deaths 0.17
Drug-induced deaths (medicinal and recreational drug overdoses) 0.1
Sudden infant death syndrome 0.1
Nephritis, nephrotic syndrome, and nephrosis 0.1
Alzheimer’s disease 0.05
Urinary tract infection 0.04
Non-metastatic neoplasms, and "neoplasms of uncertain behavior and unspecified nature" (medical mysteries, basically) 0.04
Parkinson’s disease 0.03
Senile and presenile organic psychotic conditions 0.03
All others 1.96
TOTAL 17.3
Data taken from U.S. Decennial Life Tables for 1989-91, U.S. Dept. of Health and Human Services, Centers for Disease Control and Prevention, National Center for Health Statistics, Volume 1, Number 4.

What the table says is (for example) if we could eliminate cardiovascular disease as a cause of death in America, average life expectancy would go up by 6.73 years. Again, it seems a trifle odd that if you eliminate the No. 1 cause of death in America, a disease category that kills roughly one in three people, life expectancy goes up only 8.6%. But (again), this is partly a reflection of the fact that cardiovascular ailments are (for the most part) not child-killers; they're diseases of middle age and old age. And if you don't die of heart disease, there are plenty of diseases of old age that will still kill you.

So in my own morbid way, I thought it might be a fun exercise if, utilizing CDC's data, we were to total up the potential-gain-in-life-expectancy numbers for all causes of death, to see where we stand in terms of life expectancy if we eliminate all causes of death. (Yes yes, I know I know, the numbers can't just be considered strictly additive, but this is a Gedankenexperiment, so cut me some Gedanken Laxheit.) It turns out the total possible gain in life expectancy from prevention of all causes of death is a rather modest 17.3 years, putting a theoretical limit on U.S. life expectancy of 78.2 + 17.3 == 95.5 years.

I don't for a moment hold this up as any kind of rigorous result. But I do think there is qualitative support here for the general notion that any quest for immortality that's based on mere elimination of current causes of death is fundamentally misguided. The individual PGILE numbers (as much as their sum) hint strongly at the idea that to extend human life significantly will mean doing far more than merely preventing the preventable causes of death (even if we consider the top 15 causes of death all "preventable" in one way or another, which of course many of them are not).

Exactly what that means, I'll leave as an exercise for the reader -- and will expound on in a separate blog. Assuming, of course, I live that long.


  1. This line of reasoning seems deeply flawed.
    PGILE numbers presumably reflect the extra time people would get on average before something else on the list kills them.
    This has the additional implication that if you fix one of the things on the list the PGILE numbers for all the other ones will increase.
    Thus adding them up is completely meaningless.

    1. I agree that adding them up is meaningless. I did it as a discussion-starter. Don't take things too literally around here.

      Nonetheless, the small gains from each item suggest to me we're not going to extend human life expectancy dramatically any time soon, with current approaches. I'll talk about "other approaches" in another blog, soon.

    2. > the small gains from each item suggest to me we're not going to extend human life expectancy dramatically any time soon, with current approaches

      I think that's a stretch, the main thing it seems to say is that several things on that list are conspiring to wipe out everyone beyond a certain age and we are probably going to have to fix all of the members of that group to make significant progress.
      This is what my silly thought experiment below shows. My two categories each have a PGILE of 0, but add them together and you get a PGILE of ~infinity.
      However if we assume that the problematic group is the top 3 then I'd be inclined to agree that it seems unlikely that this is going to happen in time to save Kurzweil.

  2. Lets have some reductio ad absurdum fun.
    Imagine our descendants have cured / fixed / prevented all death and have effectively infinite life spans. However they are too lazy / immature to deal with the resulting over crowding issues, so they decided instead to have the state execute everyone when they reached the end of their "natural" life span of 95.5 years. To keep things interesting they also decide to flip a coin for each person and based on the flip either hang them or shoot them.
    Now zip forward a couple of generations to a point where they've forgotten what was so bad about over crowding. These guys are thinking it might be nice to live a little longer. So they dig through the internet archive for material on mortality and come across your article. First up they need to generate the table, we have 2 causes of death, hanging and shooting, what are the PGILE numbers? Well their mortality tables indicate that hanging only affects people aged 95.5 and 100% of people aged 95.5 who don't die of hanging die instead of shooting. Presuming the same will happen to the hanging victims after we prevent that the expected increase in life span is 0 and so the PGILE number is 0. Similar reasoning leads to a PGILE number of 0 for shooting as well.
    And at this point they sum it all up putting a theoretical limit on life expectancy of 95.5 + 0 + 0 == 95.5 years. And conclude that to extend human life significantly will mean doing far more than merely preventing this hangy shooty business.

  3. The main reason this is flawed of course is that you are talking about the *average* life expectancy. If curing cancer does increase somebody's life expectancy by 50 years (and some not at all), why is that bad? If my calculations are correct, cancer patients are surviving 23.7 years longer. This is assuming a 7 billion population and 12 million new cancer cases per year, over a 78.2 year lifespan the world sees slightly less than one billion cancer cases. That's about 1 in 7.

    1. Ditto.

      1) Overall life expectancy of the total population is a useless statistic on which to base the usefulness of clinical medicine. What would be more interesting would be how many in the upper quarter of the bell curve would live longer?

      2) "Curing" cancer is not the standard. We won't ever "cure" cancer as mutations to cells happen every day in our lives. What we are doing is reducing the size of tumors and putting specific tumor growth into remission. And yes, if you can put it into remission until after you are hit by a bus, what is the trouble with that (i.e. that person's CVD did not 'cause' their death nor did the 'cancer'). We want to increase time of survivial, aka all you guys who will die at age 95 with your prostates all messed up.

  4. Here's another thought experiment.

    Let's split causes of death into two categories: age-related, for causes of death which affect the elderly far more than the young, and attritive, for causes of death which affect young and old alike.

    Now, pretend we fixed all age-related causes of death. It'd seem to follow that the mortality rates for the elderly would drop to the mortality rates for people in their 20s - approximately 0.15% a year for men and 0.05% a year for women.

    We may have eliminated just three or four of the top ten causes of death from your chart, but nevertheless, the results would be dramatic. Mean life expectancy for men at birth becomes 1,515 years; for women, 4,557.

    The strangest effect is that life expectancy for adults becomes a constant, regardless of age; even a 50,000 year old woman could reasonably expect to live, on average, another 4,600 years. This distorts mean life expectancy somewhat, with a few extremely long-lived individuals balancing out hundreds who die young. Median life expectancy is perhaps more useful. It shows half of men living to 472 and half of women to 1,383.

    Anyway. I conclude that the combined PGILE for eliminating just some of the items on the chart is around 3,000 years.

    1. > Median life expectancy is perhaps more useful. It shows half of men living to 472 and half of women to 1,383.

      A human half like, like with radioactive decay. Then we can play fun games like figuring out how old you are by asking what portion of your childhood friends are still alive.

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  6. They can't be simply additive since removing all known diseases causing death will by definition remove death as we know it.

    For example, say I've sent for two exterminators, Bill and Ted, to kill a rabbit. Bill promises to come by sometime this week to do his job, Ted also promises to kill the rabbit within the week.

    So the life expectancy of the rabbit is less than one week.

    Say we terminate Bill. The rabbit's fate isn't much changed, and its life expectancy is still less than one week.

    Say we terminate Ted. Still no change to the rabbit's luck.

    But if we terminate both Bill and Ted, then suddenly the rabbit's life expectancy has increased miraculously by over 100x (they survive naturally for around 3 years).

    Same if we eliminate all causes of death, human life expectancy could perhaps miraculously increase by 100x, who knows?


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