Wednesday, October 27, 2021

Mortality by the numbers

The following post talks about life expectancy, which inevitably means talking about people dying, and mostly-inevitably doing it in a fairly clinical way.  If that's not a topic you want to get into right now, I get it, and I hope the next post (whatever it is) will be more appealing. 

Maybe I just need to fix my news feed, but in the past few days I've run across at least two articles stating that for most of human existence people only lived 25 years or so.

Well ... no.

It is true that life expectancy at birth has taken a large jump in recent decades.  It's also true that estimates of life expectancy from prehistory up to about 1900 tend to be in the range of 20-35 years, and that estimates for modern-day hunter-gatherer societies are in the same range.  As I understand it, that's not a complete coincidence since estimates for prehistoric societies are generally not based on archeological evidence, which is thin for all but the best-studied cases, or written records, which by definition don't exist.  Rather, they're based on the assumption that ancient people were most similar to modern hunter-gatherers, so there you go.

None of this means that no one used to live past 25 or 30, though.  The life expectancy of a group is not the age by which everyone will have died.  That's the maximum lifespan.  Now that life expectancies are in the 70s and 80s, it's probably easier to confuse life expectancy with maximum lifespan, and from there conclude that life expectancy of 25 means people didn't live past 25, but that's not how it works.  For example, in the US, based on 2018 data, the average life expectancy was 78.7 years, but about half the population could expect to still be alive at age 83, and obviously there are lots of people in the US older than 78.7 years.  The story is similar for any real-world calculation of life expectancy.

A life expectancy of 25 years means that if you looked at everyone in the group you're studying, say, everyone born in a certain place in a given year, then counted up the total number of years everyone lived and divided that by the number of people in your group, you'd get 25 years.  For example, if your group includes ten people, three of them die as infants and the rest live 10, 15, 30, 35, 40, 50 and 70 years, that's 250 person-years.  Dividing that by ten people gives 25 years.

No matter what particular numbers you use, the only way the life expectancy can equal the maximum lifespan is if everybody lives to exactly that age.  If some people in a particular group died younger than the life expectancy, that means that someone else lived longer. 

Sadly, the example above is likely a plausible distribution for most times and places.  Current thinking is that for most of human existence, infant mortality has been much higher than it is now.  If you survived your first year, you had a good chance of making it to age 15, and if you made it that far, you had a good chance of living at least into your forties and probably your fifties.  In the made-up sample above, the people who made it past 15 lived to an average age of 45.  However, there was also a tragically high chance that a newborn wouldn't survive that first year.

Life expectancies in the 20s and 30s are mostly a matter of high infant mortality, and to a lesser extent high child mortality, not a matter of people dying in their mid 20s.  For the same reason, the increase in life expectancy in the late 20th century was largely a matter of many more people surviving their first year and of more children surviving into adulthood (even then, the rise in life expectancy hasn't been universal).

In real environments where average life expectancy is 25, there will be many people considerably older, and a 24-year-old has a very good chance of making it to 25.  The usual way of quantifying this is with age-specific mortality, which is the chance at any particular birthday that you won't make it to the next one (this is different from age-adjusted mortality, which accounts for age differences when comparing populations).

At any given age, you can use age-specific mortality rates to calculate how much longer a person can expect to live.  By itself, "life expectancy" means "life expectancy at birth", but you can also calculate life expectancy at age 30, or 70 or whatever.  From the US data above, a 70-year old can expect to live to age 86 (85.8 if you want to be picky).  A 70-year-old has a significantly higher chance of living to be 86 than someone just born, just because they've already lived to 70, whether or not infant mortality is low and whether the average life expectancy is in the 70s or 80s or in the 20s or 30s.  They also have a 100% chance of living past 25.

Looking at it from another angle, anyone who makes it to their first birthday has a higher life expectancy than the life expectancy at birth, anyone who makes it to their second birthday has a higher life expectancy still, and so forth.  Overall, the number of years you can expect to live beyond your current age goes down each year, because there's always a chance, even if it's small, that you won't live to see the next year.  However, it goes down by less than a year each year, because that chance isn't 100%.  Even as your expected number of years left decreases, your expected age of death increases, but more and more slowly as you age.

Past a certain point in adulthood, age-specific mortality tends to increase exponentially.  Since the chances of dying at, say, age 20 are pretty low, and the doubling period is pretty long, around 8-10 years, and the maximum for any probability is 100%, this doesn't produce the hockey-stick graph that's usually associated with exponential growth, but it's still exponential.  Every year, your chance of dying is multiplied by a fairly constant factor of around 1.08 to 1.09, or 8-9% annual growth, compounded.  Again from the US data, at age 20 you have about a 0.075% chance of dying that year.  At age 87, it's about 10%.  At age 98, it's about 30%.

This isn't a law of nature, but an empirical observation, and it doesn't seem to quite hold up at the high end.  For example, CDC data for the US shows a pretty plausibly exponential increase up to age 99, where the table stops, but extrapolating, the chance of death would become greater than 100% somewhere around age 110, even though people in the US have lived longer than that.

It's been predicted at some point, thanks to advances in medicine and other fields, life expectancy will start to increase by more than one year per year, and as a consequence anyone young enough when this starts to happen will live forever.  Life expectancy doesn't work that way, either.  There could be a lot of reasons for life expectancy in some population to go up by more than a year in any given year.

Again, the important measure is age-specific mortality.  If the chances of living to see the next year increase just a bit for people from, say, 20 to 50, life expectancy could increase by a year or more, but that just means that more people are going to make it into old age.  It doesn't mean that they'll live longer once they get there.

The key to extending the maximum lifespan is to increase the chances that an old person will live longer, not to increase the chances that someone will live to be old.   If, somehow, anyone 100 or older, but only them, suddenly had a steady 99% chance of living to their next birthday, then the average 100-year-old could look forward to living to about 169.  This wouldn't have much effect on overall life expectancy, though, because there aren't that many 100-year-olds to begin with.  

What are the actual numbers, once you get past, say, 100?  It's hard to tell, because there aren't very many people that old.  How many people live to a certain age depends not only on age-specific mortality, but on how many people are still around at what younger ages.  This may seem too obvious to state, but it's easy to lose track of this if you're only looking at overall probabilities.

Currently there's no verified record of anyone living to 123 and only one person has been verified to live past 120.  No man has been verified to live to 117, and only one has been verified to have lived to 116.  Does that mean that no one could live to, say, 135?  Not necessarily.  Does it mean that women inherently live longer than men?  Possibly, but again not necessarily.  Inference from rare events is tricky, and people who do this for a living know a lot more about the subject than I do, but in any case we're looking at handfuls out of however many people have well-verified birth dates in the early 1900s.

Suppose, for the sake of illustration, that after age 100 you have a steady 50/50 chance of living each subsequent year.  Of the people who live to 100, only 1/2 will live to 101, 1/4 to 102, then 1/8, 1/16 and so forth.  Only 1 in 1024 will live to be 110 and only 1 in 1,048,576 -- call it one in a million -- will live to 120.

If there are fewer than a million 100-year-olds to start with, the odds are against any of them living to 120, but they're not zero.  At any given point, you have to look at the ages of the people who are actually alive, and (your best estimate of) their odds of living each additional year.  If there are a million 100-year-olds now and each year is a 50/50 proposition, there probably won't be any 120-year-olds in twenty years, but if there does happen to be a 119-year-old after 19 years, there's a 50% chance there will be a 120-year-old a year later.  By the same reasoning, it's less likely that there were any 120-year-olds a thousand years ago, not only because age-specific mortality was very likely higher, but because there were simply fewer people around, so there were fewer 100-year-olds with a chance to turn 101, and so forth.

In real life, a 100-year-old has a much better than 50% chance of living to be 101, but we don't really know if age-specific mortality ever levels off.  We know that it's less than 100% at age 121, because someone lived to be 122, but that just indicates that at some point there's no longer an exponential increase in age-specific mortality (else it would hit 100% before then, based on the growth curve at ages where we do have a lot of data).  It doesn't mean that the mortality rate levels off.  It might still be increasing to 100%, but slowly enough that it doesn't actually hit 100% until sometime after age 121.

It may well be that there's some sort of mechanism of human biology that prevents anyone from living past 122 or thereabouts, and some mechanism of female human biology in particular that sets the limit for women higher than for men.  On the other hand, it may be that there aren't any 123-year-olds because so far only one person has made it to 122, and their luck ran out.

Similarly, there may not have been any 117-year-old  men because not enough men made it to, say, 80, for there to be a good chance of any of them making it to 116.  That in turn might be a matter of men being more likely to die younger, for example in the 20th-century wars that were fought primarily by men.  I'm sure that professionals have studied this and could probably confirm or refute this idea.  The main point is that at after a certain point the numbers thin out and it becomes very tricky to sort out all the possible factors behind them.

On the other hand, even if it's luck of the draw that no one has lived to 123, there could still be an inherent limit, whether it's 124, 150 or 1,000, just that no one's been lucky enough to get there.

Along with the difference between life expectancy and lifespan, and the importance of age-specific mortality, it's important to keep in mind where the numbers come from in the first place.  Life expectancy is calculated from age-specific-mortality, and age-specific mortality is measured by looking at people of a given age who are currently alive.  If you're 25 now, your age-specific mortality is based on the population of 25-year-olds from last year and what proportion of them survived to be 26.  Except in exceptional circumstances like a pandemic, that will be a pretty good estimate of your own chances for this year, but it's still based on a group you're not in, because you can only measure things that have happened in the past.

If you're 25 and you want to calculate how long you can expect to live, you'll need to look at the age-specific mortalities for age 25 on up.  The higher the age you're looking at, the more out-of-date it will be when you reach that age.  Current age-specific mortality for 30-year-olds is probably a good estimate of what yours will be at age 30, but current age-specific mortality at 70 might or might not be.  There's a good chance that 45 years from now we'll be significantly better at making sure a 70-year-old lives to be 71.  

Even if medical care doesn't change, a current 70-year-old is more likely to have smoked, or been exposed to high levels of carcinogens, or any of a number of other risk factors, than someone who's currently 25 will have been when they're 70.  Diet and physical activity have also changed over time, not necessarily for the better or worse, and it's a good bet they will continue to change.  There's no guarantee that our future 70-year-old's medical history will include fewer risk factors than a current 70-year-old's, but it will certainly be different.

For those and other reasons, the further into the future you go, the more uncertain the age-specific mortality becomes.  On the other hand, it also becomes less of a factor.  Right now, at least, it won't matter to most people whether age-specific mortality at 99 is half what it is now, because, unless mortality in old age drops by quite a bit, people today are unlikely to live to be 99.

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