Question about period life table

Howdy. I’m a health actuary, and a physician friend of mine is looking for a bit of help with life expectancy for an article he’s working on. I would greatly appreciate a bit of help from a life actuary.

He wants to adjust US life expectancy on this table Actuarial Life Table to be more comparable to that of other countries by removing the effects of between country differences for things like suicide and motor vehicle accidents. He has the probability of dying in the next year from each of the causes he’s interested in, by age and gender for the US and OECD countries.

I would like to get the adjusted life expectancy, but I don’t think it’s a simple thing to calculate from the table and the additional data he has, or is it? Part of me says there’s a selection effect - the people who died in a given year are different from those who didn’t in ways that would affect their expected future lifetime if both had lived. On the other hand, the factors he wants to adjust out, for the most part, reflect differences in cultural factors between the countries, and not health system factors.

I would appreciate any help you can provide regarding how this can be done from the information he has. Or if it cannot be done and there’s not an acceptable estimation procedure, please explain a bit about why it cannot be done. If you have any recommendations for moving forward with this project, that would be great! I consulted the SoA site and all of my actuarial and healthcare books and could not find anything helpful.

It would not be a very difficult question, but as an absolutely critical element, before starting he needs to be able to reproduce the expected lifetimes shown in the table from the death probability shown in the tables (quite closely if not perfectly). If he tries but cannot, ask about that here (showing his work) and someone could likely help.

If he can to that, then he would do pretty well with the other countries by just replacing each US probability of dying adjusted by the difference of probability of dying from motor vehicle and smoking, and of course the base mortality difference by country…

One important factor that he won’t have is that mortality rates vary by time as well as age, and if you looked at say the past 100 years they have generally been improving. So that table says a US male age 55 in 2017 has a .077627 probability of dying. The calculations (I expect) also assume that a US male age 25 in 2017 will have a .007627 probability of dying when he is age 55 in 2047. Maybe pretty good, maybe not. For his comparability studies, probably good enough if mortality changes at the same rate in all the countries he’s interested in, which may not be true.

The main thing he needs to understand is not to put too much faith is his results, especially any exact number of life expectancy in any country. But if he cannot fairly closely reproduce the US expectations of life, there is something wrong in his methodology and he should not put any faith in his calculations. (The calculations are not difficult. He should be able to reproduce the US expections pretty well, with a little help. Could be famous last words: I have not attempted to reproduce them.)

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But: why does he want to adjust for the overall country difference and the difference due to dying from motor vehicle and smoking?

Let’s say the US rate at age 60 is x and the foreign rate is y overall, and that there is a difference of z in deaths due to smoking.

Why does he want to use anything other than y for the foreign country? Is he going to adjust the US expectations too, and then say “here is the expectation of life in the US if no one smoked and no one committed suicide; here is the expectation of life in this other country if no one smoked and no one committed suicide?”

If his intent is to get results if no one smoked, then there is an important data and interpretation concern: does he have deaths due to smoking, or deaths among people who smoke?

My understanding is that he is trying to compare the healthcare systems of the US and other OECD countries. We always hear about how much worse US life expectancy is than other countries but without removing the effect of factors that are very different between the countries yet have little to do with health or healthcare, the comparisons are not meaningful comparisons of healthcare systems.

The US has many more motor vehicle deaths, homicide deaths and suicide deaths than other countries. The idea is to remove the effects of those differences and then see how much different the US looks vs other OECD countries.

He wants to remove the “excess” if you will, of the US over other OECD countries for those factors I mentioned from the probability of death (by age and sex, for recent data) to get adjusted life expectancy. Then he’d compare to life expectancy in the other OCED countries. The differences that remain are attributable to factors related to the healthcare system if we’ve removed the effects due to factors outside the healthcare system.

The idea is not to get to “if no one committed suicide in either the US or the other countries” but get to “if people committed suicide at the same rates in the US as in other OECD countries” and so forth for the other factors.

My first post may not have been clear. My apologies for that. If you would be willing to discuss with us on a zoom call, both of us would be appreciative.

You and he should try to duplicate the life expectancy calculations first. If he can’t do that (on his own or after further coaching), we won’t get anywhere. If necessary we could have a zoom session just on that, but I’m not going further (on applying adjustments) until he gets to where he he can duplicate (reasonably well) the unadjusted calcs. I’m retired, so my availability for zoom would be pretty good.

There may be lots of people here who would be better than me at interpreting the results and recommending factors to adjust for.

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We don’t know how to duplicate the life expectancy calculations.

life table

Multiplying the prob of death in year 0 and the number in the cohort at the beginning of the year, we see that 608 males die in year 0 so we have 99,392 males at the beginning of year 1.

Similarly, 0.000425 * 99,392 = 42 die in year 1 and 99,350 remain.

But what is the math for life expectancy? It doesn’t seem like I can get 75.69 life expectancy for males in year 1 from the life expectancy of males in year 0 and the probabilities of death. It seems like it’s a function of the health and situations of those one-year-olds during that year.

Thank you for your help (and patience!)

You are starting near the wrong end of the table. Let’s thing about age 116.

Start by building this table

Screen Shot 2022-05-10 at 8.41.21 PM

Can you see where that comes from?

If so, then there are two ways to proceed:
One way would be to build this table:

Then cross multiply (prob die then) * (avg yrs lived) and sum
Screen Shot 2022-05-10 at 8.47.04 PM

.7982, rounded to two decimals, is the .80 shown as the Life Expectancy for a male age 116.

Note that with that approach you need to include a row for age 120, to get the extra 1/2 year lived by the people who each 120, even though none of them survive to 121.

Another way of getting to the same thing is just total a column in our first table

Screen Shot 2022-05-10 at 8.53.25 PM

That total is called the Curtate Life Expectancy, and doesn’t count the fractional year lived after the birthday. Assuming that they live 1/2 year on average after their birthday, their Life Expectancy (or Complete Life Expectancy) is .298211845+.5=.798211845, same as we calculated the first way.

Note that nothing here gives you any direct way to calculate the Life Expectancy at any age other than 116, except repeating the entire process from the start with the new age, one at a time.

Don’t despair: there is a relatively simple way to do it, that we could discuss tomorrow. But make sure he understands this way first, since without this he’ll have no idea what the other formulas mean.

Also, note that for these methods you’ve got to have mortality rates all the way up to the age when almost everyone dies. Mortality rates only at the ages when most are alive don’t help much, certainly do not give you the “right” answer for life expectancy. So he should consider whether he has high age data, and the adjustments, for the other countries too.

Don’t worry. All will not be lost if he doesn’t, and he may be able to draw some useful conclusions just dealing with the more common ages. But start by getting him to understand what Life Expectancy means and how we would like to calculate it, and have him think about what ages he has data for.

I’ll check back tomorrow.

(Though whatever conclusions he eventually draws about relative lifetimes, I’m not sure how he will be able to extend that to comparisons of health care systems.)

One of the big issues is period life expectancy vs. cohort life expectancy, too.

And then life expectancy from which age – maybe start from an advanced age, because the heaviest users of healthcare are old. So from age 65 might be a good comparison age to start from (and period v cohort would have less differences, but they would still exist)

FWIW, there may already be research papers where they attempt to compare effects of healthcare quality, etc., on lifespan.

I understand wanting to do your own research, but given the differences in lifestyles between countries (especially stuff like differences in smoking rates and obesity) and the impact that has on health outcomes separate from how good the healthcare system is, this gets to be confounding key factors it seems to me. It’s not just a matter of drug use, gun ownership, and driving like maniacs.

Thank you!!

How is he planning on adjusting for the massive behemoth reason for differing life expectancies that trumps all others by a huge margin? That is to say, how is he going to adjust for differences in obesity rates and how they affect mortality?

The best healthcare system in the world can’t force you to stop eating donuts or get off the couch and start exercising. And collectively we Americans are way less healthy than our European counterparts due to (largely) bad choices made at the individual level and (to a lesser extent) cultural differences. Corn subsidies don’t help either.

That’s not the fault of our healthcare system.

Oh, if you’re looking at infant mortality, the fact that we count way more live births than other countries do also renders comparisons worse than meaningless. But if you start your comparison at age 1 or later then you won’t have any affect from the inconsistent data there since AFAIK we all agree on the definition of an already-born alive person.

The babies we save that other countries would let die are less healthy than average and are a drag on our overall mortality rates, but the impact there is tiny. Whereas the impact of the babies that we unsuccessfully attempt to save that other countries won’t count as a live birth is quite significant.

That is an issue that had occurred to me as well, maybe in part since (due to my bad choices) it probably applies to me more than you. I hadn’t thought about the infant issues at all, though they are important if you try to include infants in the analysis.

There are complications even adjusting for smoking, such as should you remove all deaths from smoking. Some health care systems would be more effective than others at treating people who have had complications of smoking. Do you want to remove that effect from the comparison.

As with covid deaths, there may be differences among countries in how deaths from smoking vs deaths of smokers are counted. Smoking is, I expect, almost never listed as the cause of death. It does have a significant effect on mortality.

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In case he’s still interested, here’s the relatively simple approach this version is easier to understand:
(.006081)(.5)+(1-.006081)(75.69)=(76.23), where .006081 is the probability of dying, 1-.006081 is the probability of living, 75.69 is the life expectancy at 1, and 76.23 is the life expectancy at 0 [all for males]

In words, those who die before 1 live .5 years before age 1 on average (and obviously none after 1 since already dead); those who survive to age 1 live 1 year before age 1 and live an average of 75.69 thereafter.

Here’s an alternative: how to get the expectancy at 1 from the expectancy at 0. It’s just an algebraic manipulation of the first.


Neither of those is useful for starting with the low ages in other countries. You need to know the expectancy at 1 or the expectancy at 0, and you know neither. (Or even if you know the base expectancy in either, you can’t use that starting point to get expectancy adjusted for smoking, etc.)

But starting at the high ages, you’re set, because future years beyond the table are so small if they even exist.

Since it isn’t shown, we suspect the table intends the expectancy at 120 is 0.5 (everyone will die before 121, so anyone alive at 120 lives another .5 years on average). If so, expectancy at 119 is
(.879126)(.5)+(1-.879126)(1+.5)=.6209 Oops. That is not exactly right. But you’re still extremely close

Probably they assumed some people lived even longer, even though not shown in the table. (As we might guess from what we see in the table. .797 chance of dying at 117. .837 at 118 (.040 higher). .879 at 119 (.062 higher). Would it really jump all the way to 1.00 by 120? Without knowing how quickly you reach the point that everyone dies, you can’t get the expectancies exactly, but you come very close.

I would imagine he’d say I’ve adjusted for factors A, B, C and D, to make it clear that he did not adjust for obesity. It’s also possible that I’ve not explained his thought process very accurately… I’m sure he’s considered obesity - he’s a family physician after all!

Dangerous assumption. Is his goal to learn or to demonstrate that the American medical system sucks?

An awful lot of people have the latter goal, and conveniently forgetting to take underlying health differences into account is a very easy way to (either dishonestly or incompetently) accomplish that goal.