Sep 11, 2012

"Life Expectancy" Doesn’t Measure How Long You’re Expected to Live

The US Government estimated its population had a life expectancy of 78.5 years in 2009. If you type “life expectancy” into Google, it will spit back the World Bank estimate of 78.2 years in 2010. You’ve likely read numbers close to these in textbooks and articles.

But what do these numbers actually mean? You might guess from the first that someone born in the United States in 2009 could be expected to live about 78.5 years. This is not the case! It actually measures how long someone would be expected to live if every year of their life was spent in 2009. In other words, there is no accounting for progress that decreases mortality rates. And that’s on purpose. It is what is known as a “period life expectancy”.

Period life expectancies are used to track the general health of a population. With them you can easily compare one country to another. You can also monitor general population health over time. But the number you want if you’d like to know how long people will actually live is known as a “cohort life expectancy”. It measures how long someone born in a particular year (a cohort) can be expected to live. It is also not in the US Government yearly mortality report for 2009. The reason is that we won’t know it until everyone born in 2009 is dead! That will hopefully take a long long time. So, while we accurately know the period life expectancy for 2009, we do not yet know the cohort life expectancy for 1909. There are estimates of course, including one of about 83.9 years in 2009 from the Social Security Administration. I would like to offer, however, my own estimate of 86.4 years in 2009 and give the reasoning behind it. In particular the method of calculation reveals why life expectancy is likely still in the 80’s and not in the 120’s or 200’s like some would hope.

Life expectancy is calculated from something called a life table, which is just a big list of the probabilities of dying for someone of a certain age in a certain year. Below is plotted the one for 1940 and one for 2000. The period life expectancy for 1940 and 2000 would be calculated from exactly this data.

The cohort life expectancy for 1940 would only use a single value from this table, namely the one for age 0. The value for age 1 would come from the 1941 life table, and the value for age 60 would come from the 2000 life table. By the time 2000 came around, the chances of dying at age 60 had fallen from 2.2% to 1.0%, a decrease of over 50%! If we plot the 1940 life table against the known cohort data, you can see that they’re quite different, and the the difference grows every year.

To complete the picture we need to estimate what age-specific death rates someone born in 1940 will face in the coming years. As I’ve written before, the mortality rate for any age decays exponentially, the rate of progress is relatively stable for any age, and we’ve made a lot more progress for younger ages than older ages. I’ve plotted the average yearly progress for each age below.

To estimate the cohort mortality rate we can take the period rate and discount it by the age-specific progress over how ever many years it will take someone to reach that age. As an example, the average progress for a 60 year old is 1.25% per year, which means that a 60 year old next year will be about 1.25% less likely to die than a 60 year old this year. Compounding this over 60 years yields an estimated decline of 53%, taking the 1940 value of 2.2% to an estimated 1.05%. Pretty close to the 1.0% value actually seen in 2000. The plots below shows the 1940 period mortality data, the 1940 cohort mortality data that’s available, and the 1940 cohort mortality data estimated using this method. The first truncates at the end of the known cohort values, and the second includes all ages up to 100. The estimates are pretty darn close.

One interesting aside is that the period and estimated cohort mortality curves reunite in old ages. While more distant ages have more time for progress to compound, they have less progress per year than younger ages. In other words, mortality might decrease at 3.3% per year for a 5 year old, but you only have 5 years of compounded progress until you reach that age. At this point the probability of dying will be (1 - 3.3%)^5 = 85% of what it was at your birth, down 15%. Mortality for a 60 year old decreases at a more modest 1.3% per year, but over 60 years it can fall quite a bit. In this case (1 - 1.3%)^60 = 47% or about 53% of what it was in the year of your birth. Eventually, however, the yearly rate of progress falls so far that the compounded rate of progress starts falling. This happens at about 72 years old, and it falls continuously from then on. The net effect of this is what is called the rectangularization of the life curve, which produces a lot more people living to old age, but no one living dramatically longer than before.

Applying this method to the 2009 data produces the below curve. It starts more rectangular than the 1940 curve and gets even more rectangular.

It is from this data, using a method very much like the one that the US Government uses, that I calculated the 2009 cohort life expectancy of 86.4 years, some 9.9% longer than the official 2009 estimate above.

Whether you believe this number or not, it is actually intended to measure how long someone born in 2009 is expected to live. Almost every discussion I have seen on this topic, including the Scientific American article linked above, mistakenly interpret period life expectancy as cohort life expectancy. If life expectancy numbers have ever seemed low to you, it is because they are.

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