http://www.cdc.gov/nchs/data/dvs/MortFinal2007_Worktable310.pdf gives death counts, USA, year 2007, at each age 0-120. The file http://www.cdc.gov/nchs/data/dvs/MortFinal2007_WorkTable210R.pdf gives death rates in each 5-year age group, USA, year 2007. Also broken down by race, sex, and into 113 popular causes of death. Now I will try to give the all-races, all-sexes, all-causes year-2007 death rates from that file and also http://www.cdc.gov/nchs/data/dvs/LCWK7_2007.pdf for below 1 year old: age rate/100K/yr ln(rate) 0-1 675.1 6.51 1-4 28.6 3.35 5-9 13.7 2.62 healthiest age; monotonic increase from here on: 10-14 16.9 2.83 15-19 61.9 4.13 20-24 98.3 4.59 25-29 99.4 4.60 30-34 110.8 4.71 35-39 145.8 4.98 40-44 221.6 5.40 45-49 340.0 5.83 50-54 509.0 6.23 55-59 726.3 6.59 60-64 1068.3 6.97 65-69 1627.5 7.39 70-74 2491.3 7.82 75-79 3945.9 8.28 80-84 6381.4 8.76 85-199 12946.5 9.47 I attempted to draw a graph of this data, and the line ln(DeathRate)=(31/375)*Age+1.856 happens to agree with the data pretty well for ages 5-14 and 30-84. This confirms "Gompertz's law." I believe it also works well at ages 85-95. However, it does not work at ages <5 and 15-29. Perhaps you can produce better laws. I suspect ages 20-95 could be fit well by DeathRate = A + exp(B*age+C), which I think is called Gompertz-Makeham law. Also, I think there are papers out there about very old people's death rates perhaps also deviating from Gompertz law (ages>95), but the CDC does not seem to provide that data. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
I attempted to draw a graph of this data, and the line ln(DeathRate)=(31/375)*Age+1.856 happens to agree with the data pretty well for ages 5-14 and 30-84. This confirms "Gompertz's law." I believe it also works well at ages 85-95. However, it does not work at ages <5 and 15-29.
picture: https://dl.dropboxusercontent.com/u/3507527/DeathRates2007USA.png --I made several attempts to produce better laws. My first attempt was an already proposed formula: DeathRate = C + exp(A*Age+B) where I find A=0.071 B=2.626 C=6.09 has maximum |error|<0.308 for the ln(DeathRate) over the CDC 5-year age bins, for all Age>19. But it is no good for ages 0-19. DeathRate = C*Age^Q + exp(A*Age+B) with A=0.0828 B=2.29 C=15.7 Q=-1.88 has maximum |error|<0.469 for the ln(DeathRate) over the CDC age bins, for all integer Age>0. DeathRate = C/Age + Q + exp(A*Age+B) with A=0.0812 B=2.388 C=16.5 Q=-3.81 has maximum |error|<0.455 for the ln(DeathRate) over the CDC age bins, for all integer Age>0. (And I believe this last fit's parameters can be improved further, max |error|<0.450 is definitely achievable.) In the last of these formulae, the Q term stays fixed, the exp(A*Age+B) term increases exponentially with age, and finally the C/Age term decreases with age. The fact Q is negative kind of ruins the interpretation that it represents "acts of God," so I guess it instead represents "self repair" or "medical help" (?), see below. The C/Age term models the effect of "experience" as your brain and/or immune system "learn" to handle threats to your existence -- as time T goes by you learn to handle about T kinds of threats, leaving only a fraction of order 1/T of threats still able to kill you. The exponentially increasing Gompertz term perhaps could be explained as follows. Imagine your life consists of N "answers" for some very large initial value of N. Each year, any particular answer you own, is obliterated with some fixed probability p (all events independent). Also, each year, a random "challenge" is posed to you. You live if you still have the answer to that challenge. In this model, the number of still-existing answers tends to fall exponentially with time, ultimately making it exponentially unlikely you will survive the next challenge. The negative Q could also be regarded as representing a fixed ability to "invent" an answer you do not actually have.
It would be interesting to obtain data on infant deaths rates month by month throughout the first year of life. But I do not have it. I did manage to extend the CDC mortality rate table up past age 85 (ignoring the 201 death of unknown age seems to introduce safely neglectible error) using their death counts in year 2007. age rate/100K/yr ln(rate) 0-1 675.1 6.51 1-4 28.6 3.35 5-9 13.7 2.62 healthiest age; monotonic increase from here on: 10-14 16.9 2.83 15-19 61.9 4.13 20-24 98.3 4.59 25-29 99.4 4.60 30-34 110.8 4.71 35-39 145.8 4.98 40-44 221.6 5.40 45-49 340.0 5.83 50-54 509.0 6.23 55-59 726.3 6.59 60-64 1068.3 6.97 65-69 1627.5 7.39 70-74 2491.3 7.82 75-79 3945.9 8.28 80-84 6381.4 8.76 85-89 10067.3 9.22 90-94 13553.3 9.51 95-99 16539.7 9.71 100-104 18356.8 9.82 105-109 19236.5 9.86 110-114 19354.8 9.87 Fewer than 1 in a million American deaths are at ages>114. Gompertz's law indeed works well for ages 30-95, but if you manage to reach age 100 (achieved by fewer than 1% of American 2007 deaths) the death rate then seems to stop its exponential increase and become almost constant from then on. [ln(DeathRate) keeps increasing, but more slowly.] I have absolutely no idea why. The following formula apparently fits the 2007 USA death rate (per 100K per year) as a function of integer age x in years, to within a factor of 1.55 [equivalently maximum additive |error|<0.44 for fitting ln(DeathRate)] at every integer age 1-99: DeathRate = C/x + Q + exp(A*x+B) where A=0.0782, B=2.527, C=21.152, Q=-7.813. I have not attempted also to fit it for ages 100-115. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
It seems the stoppage of exponential growth in death rate with age (the "end" of Gompertz's law) at about age 100 has been noticed before. Furthermore, it has been disputed. Apparently the dispute is based on the claim that the ages of very old people are commonly misreported. Even if the misreporting is a random both-signed perturbation with zero mean, that still will cause the illusion of an "end of Gompertz's law." Some authors claim Gompertz really continues to work out to the oldest ages in both humans and other species. Others claim: that may depend on which species. Apparently some species do exist in which Gompertz-stoppage really does occur (invertebrates like fruit flies especially) and some exist in which it seems to keep working (mice). Deciding who is right is not easy for me. The stoppage (if genuine) could be "explained" by the following modification of my silly "challenges/answers" model. Imagine there are two kinds of answers, durable ones and non-durable ones. They get obliterated each year (or day, or whatever) with two fixed small probabilities -- the durable ones with a smaller probability of obliteration, but durable answers are initially rare. Each year (or day, or whatever) you are "challenged" and if you still have the "answer" to the challenge you are allowed to live. This model predicts the exponential death rate increase, but eventually the non-durable answers have been thinned so much that the durable ones outmass them. That causes a switchover to a slower Gompertzian growth constant in the exponential. It seems to me reasonable that such a thing ought to happen: if you accepted the challenges/answers model in the first place, then you should accept that some answers must be more durable, whereupon automatically you are logically forced to predict this. If the transition really does not exist in humans then I'll just slither out of that by claiming the constants in my picture are such that we don't see it; I have no trouble fitting reality either way. Re Infant mortality (defined as deaths of babies <1 year old) the worst place to be a baby is Washington DC and the best is Massachusetts (in the USA)... in 2009 USA there were 4.18 deaths per 1000 in the first 28 days of life ("neonatal"; 28 days is 1/13 years), and 2.21 in the rest of the first year, showing the death rate is 22.7 times greater during the neonatal period. Further, in the "early neonatal" period (first 25% of it) there were 3.23 deaths/1000 in year 2010, versus 0.82 in the rest of the neonatal period. This is 15.7 times greater death rate. So there initially is a very severe decline in death rate. It's almost like a 1/age^2 power law singularity, except that formula would yield infinite integral so cannot be correct...
actuarial tables used in life insurance seem to reflect the stoppage (i recall). i remember that a fixed probability of 1/6 of living another year at all ages 95 and above was used in one life insurance (or maybe annuity) quotation i was given long ago. there was probably more cold-hearted approximation than actual scientific investigation involved in these numbers though On Monday, February 16, 2015, Warren D Smith <warren.wds@gmail.com> wrote:
It seems the stoppage of exponential growth in death rate with age (the "end" of Gompertz's law) at about age 100 has been noticed before. Furthermore, it has been disputed. Apparently the dispute is based on the claim that the ages of very old people are commonly misreported. Even if the misreporting is a random both-signed perturbation with zero mean, that still will cause the illusion of an "end of Gompertz's law." Some authors claim Gompertz really continues to work out to the oldest ages in both humans and other species. Others claim: that may depend on which species. Apparently some species do exist in which Gompertz-stoppage really does occur (invertebrates like fruit flies especially) and some exist in which it seems to keep working (mice). Deciding who is right is not easy for me.
The stoppage (if genuine) could be "explained" by the following modification of my silly "challenges/answers" model. Imagine there are two kinds of answers, durable ones and non-durable ones. They get obliterated each year (or day, or whatever) with two fixed small probabilities -- the durable ones with a smaller probability of obliteration, but durable answers are initially rare. Each year (or day, or whatever) you are "challenged" and if you still have the "answer" to the challenge you are allowed to live. This model predicts the exponential death rate increase, but eventually the non-durable answers have been thinned so much that the durable ones outmass them. That causes a switchover to a slower Gompertzian growth constant in the exponential.
It seems to me reasonable that such a thing ought to happen: if you accepted the challenges/answers model in the first place, then you should accept that some answers must be more durable, whereupon automatically you are logically forced to predict this. If the transition really does not exist in humans then I'll just slither out of that by claiming the constants in my picture are such that we don't see it; I have no trouble fitting reality either way.
Re Infant mortality (defined as deaths of babies <1 year old) the worst place to be a baby is Washington DC and the best is Massachusetts (in the USA)... in 2009 USA there were 4.18 deaths per 1000 in the first 28 days of life ("neonatal"; 28 days is 1/13 years), and 2.21 in the rest of the first year, showing the death rate is 22.7 times greater during the neonatal period. Further, in the "early neonatal" period (first 25% of it) there were 3.23 deaths/1000 in year 2010, versus 0.82 in the rest of the neonatal period. This is 15.7 times greater death rate. So there initially is a very severe decline in death rate. It's almost like a 1/age^2 power law singularity, except that formula would yield infinite integral so cannot be correct...
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also i think it might have switched to 1/2 at age 100. On Monday, February 16, 2015, Thane Plambeck <tplambeck@gmail.com> wrote:
actuarial tables used in life insurance seem to reflect the stoppage (i recall). i remember that a fixed probability of 1/6 of living another year at all ages 95 and above was used in one life insurance (or maybe annuity) quotation i was given long ago. there was probably more cold-hearted approximation than actual scientific investigation involved in these numbers though
On Monday, February 16, 2015, Warren D Smith <warren.wds@gmail.com <javascript:_e(%7B%7D,'cvml','warren.wds@gmail.com');>> wrote:
It seems the stoppage of exponential growth in death rate with age (the "end" of Gompertz's law) at about age 100 has been noticed before. Furthermore, it has been disputed. Apparently the dispute is based on the claim that the ages of very old people are commonly misreported. Even if the misreporting is a random both-signed perturbation with zero mean, that still will cause the illusion of an "end of Gompertz's law." Some authors claim Gompertz really continues to work out to the oldest ages in both humans and other species. Others claim: that may depend on which species. Apparently some species do exist in which Gompertz-stoppage really does occur (invertebrates like fruit flies especially) and some exist in which it seems to keep working (mice). Deciding who is right is not easy for me.
The stoppage (if genuine) could be "explained" by the following modification of my silly "challenges/answers" model. Imagine there are two kinds of answers, durable ones and non-durable ones. They get obliterated each year (or day, or whatever) with two fixed small probabilities -- the durable ones with a smaller probability of obliteration, but durable answers are initially rare. Each year (or day, or whatever) you are "challenged" and if you still have the "answer" to the challenge you are allowed to live. This model predicts the exponential death rate increase, but eventually the non-durable answers have been thinned so much that the durable ones outmass them. That causes a switchover to a slower Gompertzian growth constant in the exponential.
It seems to me reasonable that such a thing ought to happen: if you accepted the challenges/answers model in the first place, then you should accept that some answers must be more durable, whereupon automatically you are logically forced to predict this. If the transition really does not exist in humans then I'll just slither out of that by claiming the constants in my picture are such that we don't see it; I have no trouble fitting reality either way.
Re Infant mortality (defined as deaths of babies <1 year old) the worst place to be a baby is Washington DC and the best is Massachusetts (in the USA)... in 2009 USA there were 4.18 deaths per 1000 in the first 28 days of life ("neonatal"; 28 days is 1/13 years), and 2.21 in the rest of the first year, showing the death rate is 22.7 times greater during the neonatal period. Further, in the "early neonatal" period (first 25% of it) there were 3.23 deaths/1000 in year 2010, versus 0.82 in the rest of the neonatal period. This is 15.7 times greater death rate. So there initially is a very severe decline in death rate. It's almost like a 1/age^2 power law singularity, except that formula would yield infinite integral so cannot be correct...
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-- Thane Plambeck tplambeck@gmail.com <javascript:_e(%7B%7D,'cvml','tplambeck@gmail.com');> http://counterwave.com/
-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/
forget that.... too much coffee On Monday, February 16, 2015, Thane Plambeck <tplambeck@gmail.com> wrote:
also i think it might have switched to 1/2 at age 100.
On Monday, February 16, 2015, Thane Plambeck <tplambeck@gmail.com <javascript:_e(%7B%7D,'cvml','tplambeck@gmail.com');>> wrote:
actuarial tables used in life insurance seem to reflect the stoppage (i recall). i remember that a fixed probability of 1/6 of living another year at all ages 95 and above was used in one life insurance (or maybe annuity) quotation i was given long ago. there was probably more cold-hearted approximation than actual scientific investigation involved in these numbers though
On Monday, February 16, 2015, Warren D Smith <warren.wds@gmail.com> wrote:
It seems the stoppage of exponential growth in death rate with age (the "end" of Gompertz's law) at about age 100 has been noticed before. Furthermore, it has been disputed. Apparently the dispute is based on the claim that the ages of very old people are commonly misreported. Even if the misreporting is a random both-signed perturbation with zero mean, that still will cause the illusion of an "end of Gompertz's law." Some authors claim Gompertz really continues to work out to the oldest ages in both humans and other species. Others claim: that may depend on which species. Apparently some species do exist in which Gompertz-stoppage really does occur (invertebrates like fruit flies especially) and some exist in which it seems to keep working (mice). Deciding who is right is not easy for me.
The stoppage (if genuine) could be "explained" by the following modification of my silly "challenges/answers" model. Imagine there are two kinds of answers, durable ones and non-durable ones. They get obliterated each year (or day, or whatever) with two fixed small probabilities -- the durable ones with a smaller probability of obliteration, but durable answers are initially rare. Each year (or day, or whatever) you are "challenged" and if you still have the "answer" to the challenge you are allowed to live. This model predicts the exponential death rate increase, but eventually the non-durable answers have been thinned so much that the durable ones outmass them. That causes a switchover to a slower Gompertzian growth constant in the exponential.
It seems to me reasonable that such a thing ought to happen: if you accepted the challenges/answers model in the first place, then you should accept that some answers must be more durable, whereupon automatically you are logically forced to predict this. If the transition really does not exist in humans then I'll just slither out of that by claiming the constants in my picture are such that we don't see it; I have no trouble fitting reality either way.
Re Infant mortality (defined as deaths of babies <1 year old) the worst place to be a baby is Washington DC and the best is Massachusetts (in the USA)... in 2009 USA there were 4.18 deaths per 1000 in the first 28 days of life ("neonatal"; 28 days is 1/13 years), and 2.21 in the rest of the first year, showing the death rate is 22.7 times greater during the neonatal period. Further, in the "early neonatal" period (first 25% of it) there were 3.23 deaths/1000 in year 2010, versus 0.82 in the rest of the neonatal period. This is 15.7 times greater death rate. So there initially is a very severe decline in death rate. It's almost like a 1/age^2 power law singularity, except that formula would yield infinite integral so cannot be correct...
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/
-- Thane Plambeck tplambeck@gmail.com <javascript:_e(%7B%7D,'cvml','tplambeck@gmail.com');> http://counterwave.com/
-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/
It makes a lot of sense that the people who have survived to 90 or 95 would be the hardiest, and less likely to be subject to the statistical laws that claimed their cohorts. --Dan
On Feb 16, 2015, at 10:33 AM, Thane Plambeck <tplambeck@gmail.com> wrote:
actuarial tables used in life insurance seem to reflect the stoppage (i recall). i remember that a fixed probability of 1/6 of living another year at all ages 95 and above was used in one life insurance (or maybe annuity) quotation i was given long ago. there was probably more cold-hearted approximation than actual scientific investigation involved in these numbers though
As you well know, this is an extremely slippery slope, down which many gambling addicts have fallen. Testing the difference between skill (or hardiness) and luck is extremely difficult. In fact, the vast majority of "wealth managers" live in the shadows between skill & luck. (Hence Woody Allen's wonderful line from "Midsummer Night's Sex Comedy" -- "I'm a money manager; I manage your money until it's gone.") At 12:15 PM 2/16/2015, Dan Asimov wrote:
It makes a lot of sense that the people who have survived to 90 or 95 would be the hardiest, and less likely to be subject to the statistical laws that claimed their cohorts.
participants (4)
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Dan Asimov -
Henry Baker -
Thane Plambeck -
Warren D Smith