# Somebody is wrong on the Internet

In my post below on the increasingly bad effects of income inequality on life expectancy, someone who claims to be an associate professor of public health at an Ivy League school takes me to task for not understanding the difference between period and cohort life expectancy. They then offer this purported quote from a purported book they authored:

So what do we mean by life expectancy? The usual meaning, called “period life expectancy” is a fictional concept – it tells us something about the present, but isn’t really a prediction of the future. “Cohort life expectancy” is an actual description of reality in the past: It’s the average age of death of people born in a given year, but in order to calculate it, everybody born in that year must have died already. So when we talk about life expectancy 100 years ago, we’re really talking about cohort life expectancy, and we’re comparing it to period life expectancy in more recent years. That isn’t wrong, but it’s a little bit complicated. In fact, during eras of increasing life expectancy, period life expectancy will underestimate the eventual average age of death of the present population, so if anything the change has been even more dramatic than it appears.

The first two sentences of this aren’t exactly wrong, although I would call period life expectancy a statistical statement rather than a fictional concept. They do make the valid point that period life expectancy isn’t, statistically speaking, necessarily or even reasonably a *prediction*, which as I’ve pointed out is something that’s constantly misrepresented by the popular media in stories discussing period life expectancy.

But then things really go off the rails.

*“Cohort life expectancy” is an actual description of reality in the past.*

No, cohort life expectancy is a statistical prediction of the average age at death of people born at a particular time (usually a particular year).

*It’s the average age of death of people born in a given year, but in order to calculate it, everybody born in that year must have died already.*

This is incoherent on its face. You can’t calculate a future probability for an event — the death of everyone in a cohort — that has already happened. Dead people don’t have life expectancies, so you can’t calculate the life expectancy of people born in the USA in 1900, since those people are all dead now, like the stars of 1950s Hollywood movies (this is an allusion).. When we calculate the average age at which people born in the USA in 1900 died, we’re not calculating their life expectancy, we’re just calculating how long they lived, rather than calculating a set of future probabilities given various assumptions, which is what life expectancy, both in its cohort and period versions, is.

So when we talk about life expectancy 100 years ago, we’re really talking about cohort life expectancy, and we’re comparing it to period life expectancy in more recent years.

In my experience, when people talk about life expectancy 100 years ago, 99.9% of the time they’re talking about period life expectancy at birth. For example you can find lots of statements such as that life expectancy in the US in 1900 was 47 years, or that life expectancy dropped by seven years between 1917 and 1918 because of the Spanish flu. These statements are *always* about period life expectancy at birth. Indeed it’s fairly difficult to find statistics on cohort life expectancy a century ago, or 20 years ago for that matter, because very few people are even familiar with the concept.

*In fact, during eras of increasing life expectancy, period life expectancy will underestimate the eventual average age of death of the present population, so if anything the change has been even more dramatic than it appears.*

Ok this statement is true.

Period life expectancy and cohort life expectancy are both important and useful concepts, but they’re useful in different ways. LGM commenter Phil Koop explains the difference:

There are two measures of life expectancy. Period life expectancy applies current age-specific hazard rates to the future periods of people now alive. Cohort life expectancy adjusts those future age-specific hazard rates according to past trends (the rate of medical advances, for instance.)

The former procedure is well-defined and the numbers won’t change retrospectively, but isn’t really trying to measure how long you are expected to live. The latter procedure tries to get at the question of interest, but as you say, is uncertain because nobody can know for sure what those future hazard rates will be.

This trade-off runs through all of risk measurement and risk management. One thing we have learned from bitter experience is that throwing up your hands and saying “ah well, nobody can really know the answer to this” is the worst possible course. “If you choose not to decide, you still have made a choice.”

Here’s a concrete example. When calculating period life expectancy at birth, it’s assumed for the purposes of this calculation that current population-wide age-specific mortality rates will remain the same throughout the lifetime of the cohort. So when calculating the life expectancy of a male infant born in the USA in 2019, for instance, we assume that the annual mortality risk for 70 year old males in 2019 (2.2%) will be the same in 2089, when that child, if still alive, will be 70.

But in fact age specific mortality rates are almost always falling. For example, 15 years earlier, the annual mortality risk for a 70 year old male was 27% higher than it was in 2019. Cohort life expectancy calculations attempt to calculate how* the trend *in the change in age specific mortality rates will affect mortality going forward, in contrast to period life expectancy, which assumes no change going forward, in order to produce a number that isn’t an estimate of changing probabilities, but a statement of a current statistical fact.

It’s possible to calculate cohort life expectancy at any point in a cohort’s life span, and indeed agencies such as the Social Security Administration calculate it for people born this year, and even in future years. Of course the older the cohort is, the more accurate the estimate is likely to ultimately prove to be. For example, period life expectancy for males the year Joe Biden was born was 62.6 years, but cohort life expectancy for that cohort is now calculated to be 70.1 years, with almost no uncertainty remaining regarding what the average age at death will be for people born in that year. For male infants born this year, SSA estimates a life expectancy of 83 years, while period life expectancy at birth for males born this year is going to be around 75 years. The difference in the two numbers comes from the assumption that age specific mortality rates will continue to fall more or less along the trend line that they’ve fallen over the past century. (This assumption could prove problematic for reasons that the doomerist members of the LGM commentariat can easily imagine).

Calculating cohort life expectancy can be very complicated and challenging, but the concept itself is both fairly straightforward and very practically useful — after all, what we really want to know for purposes of public policy is how long people will actually live, as opposed to how long they would live if nothing were to ever change from the status quo, which is what period life expectancy — which again is what almost everyone thinks of as “life expectancy” — tells us.