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From the CDC (https://www.cdc.gov/nchs/fastats/deaths.htm):

Death rate: 863.8 deaths per 100,000 population
Life expectancy: 78.6 years

Now in a static situation I would expect that the death rate to be the reciprocal of the life expectancy or about 1,270 deaths per 100K which is about a 40% difference from the actual. Quite a lot. Is this because the population age profile is not static? The US median age is about 38 years and has increased by about 1 year over the last decade. Is this really enough of a variation to account for the 40% difference? I tried looking for the mean age to see if that statistic could shed more light on the subject but could not find any data.

I would like to understand this in more detail so any information is appreciated.

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    $\begingroup$ Mortality is not constant with age (nor across other variables). It's also changing over time. $\endgroup$ – Glen_b Oct 29 '20 at 6:23
  • $\begingroup$ Welcome to CV, DavidS! Accompanying @Glen_b's comment, the distribution of age is also not constant, and also varies from nation to nation, state to state, city to city, etc. Check out the population pyramid. $\endgroup$ – Alexis Oct 30 '20 at 18:31
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    $\begingroup$ Glen_b & Alexis: As Sextus Empiricus explains in his excellent answer below, it is the change of the mortality rates and population age distribution in time that is making the death rate deviate from from 1/life expectancy. The fact that mortality is age dependent and varies from state to state is not a factor here. $\endgroup$ – DavidS Oct 30 '20 at 20:21
  • $\begingroup$ @DavidS that mortality is not constant with age does play sort of a role (but I would agree it is not the driver behind the reasons why we see the large 40% discrepancy). The higher mortality for older ages amplifies the effects of variations in the population. In relatively young populations you have relatively a lot less deaths because those deaths occur at older age (but it is true that also with equal mortality among all ages you would still have effects of the population being younger than the survival curve suggests). $\endgroup$ – Sextus Empiricus Oct 30 '20 at 21:01
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    $\begingroup$ DavidS in your comment to @Glen_b and myself, I think you have not cottoned to the fact that "it is the change of the mortality rates and population age distribution in time that is making the death rate deviate from from 1/life expectancy" is exactly what were pointing out. :) For example, "The distribution of age is not constant" means the shape of the population pyramids shifts over time (for all the reasons Sextus Empiricus points out: births, deaths, immigration, and emigration). Just a thought. (And still welcome to CV! :) $\endgroup$ – Alexis Dec 12 '20 at 0:23
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In short

The discrepancy between death rate and the reciprocal of the life expectancy generally occurs when the age distribution of the population is not the same as the survival curve, which relates to a hypothetical population on which the life expectancy is based (and more specifically the population is younger than what the survival curve suggests). There can be several reasons that create differences between the actual population and this hypothetical population

  • The death rate per age group has dropped suddenly/fast and the population is not yet stabilized (not equal to the survival curve based on the new death rates per age group)
  • The population is growing. If every year more babies are born than the previous year, then the population will be relatively younger than the hypothetical population based on what a survival curve suggests.
  • Migration. Migration is often occurring with relatively younger people. So the countries with positive net immigration will be relatively younger and the countries with negative immigration will be relatively older.

Life expectancy

The life expectancy is a virtual number based on a hypothetical person/population for which the mortality rates in the future are the same as the current mortality rates.

Some example using data (2014) from the Dutch bureau of statistics https://opendata.cbs.nl/statline/#/CBS/nl/dataset/7052_95/table?dl=98D9

death rates, survival rate, and probable age of death

  • graph 1 shows (current) death rate for age $i$ $$f_i$$
  • graph 2 shows survival rate for age $i$ (for a hypothetical population that will experience the death rate for age $i$ as it is for the people that are currently of age $i$) $$s_i = \prod_{j=0}^{j=i-1} (1-f_j)$$
  • graph 3 shows probability for dying at age $i$ $$p_i = s_i f_i$$

Note that $p_i$ is a hypothetical situation.

Death rates

In the above example, the hypothetical population will follow the middle graph. However the actual population is not this hypothetical population.

In particular, we have much less elderly people than what would be expected based on the survival rates. These survival rates are based on the death rates in the present time. But when the elderly grew up these death rates were much larger. Therefore, the population contains less elderly than the current survival rate curve suggest.

The population looks more like this (sorry for it being in Dutch and not well documented, I am getting these images from some old doodles, I will see if I can make the graphs again):

example

So around 2040 the distribution of the population will be more similar to the curve of the survival rate. Currently, the population distribution is more pointy, and that is because the people that are currently old did not experience the probabilities of dying at age $i$ on which the hypothetical life expectancy is based.

How death rates are changing

In addition, there is a slightly lower birth rate (less than 2 per woman), and so the younger population is shrinking. This means that the death rate will not just rise to 1/life_expectancy, but even surpass it.

This is an interesting paradox. (As Neil G commented it's Simpson's paradox)

  • On the one hand the death rate is decreasing in each separate age group.
  • On the other hand the death rate is increasing for the total population.

Note this graph interactive version on gapminder

change in death rate

We see that in the past decades the death rates have dropped quickly (due to decrease in death rate) and now are rising again (due to stabilization of the population, and due to decrease in birth rate). Most countries follow this pattern (some started earlier some started later).

Simulation

In this question the answer contains a piece of R-code that simulates the survival rate curve for a change of the risk-ratio of death for all ages.

Below we use the same function life_expect and simulate the death rate in a population when we let this risk ratio change from 1.5 to 1.0 in the course of 50 years (thus life expectancy will increase and the inverse, the death rate based on life-expectancy, will decrease).

What we see is that the drop in the death rate in the population is larger than what we would expect based on the life expectancy, and is only stabilizing at this expected number after some time when we stop the change in risk ratios.

Note, in this population we kept the births constant. Another way how the discrepancy between the reciprocal of the life expectancy and the death rate arrises is when the number of births is increasing (population growth) which causes the population to be relatively young in comparison to the hypothetical population based on the survival curve.

simulation example

### initial population
ts <- life_expect(base, 0, rr = 1.5, rrstart = 0)
pop <- ts$survival
Mpop <- pop

### death rates
dr <- sum(ts$death_rate*pop)/sum(pop)
de <- 1/(ts$Elife+1)

for (i in -100:200) {
  ### rr changing from 1.5 to 1 for i between 0 and 50
  t <- life_expect(base, 0, rr = 1.5-max(0,0.5*min(i/50,1)), rrstart = 0)
  
  ### death rate in population
  dr <- c(dr,sum(t$death_rate*pop)/sum(pop))
  
  ### death rate based on life expectancy
  de <- c(de,1/(t$Elife+1))
  
  ### update population
  pop <- c(1,((1-t$death_rate)*pop)[-101])
  Mpop <- cbind(Mpop,pop)
}

### plotting
plot(de * 100, type = "l", lty = 2, lwd = 2, ylim = c(1.10,1.4),
     xlab = "time", xaxt = "n", ylab = "rate %")
lines(dr * 100, col = 2)
legend(0,1.10, c("death rate in population", "death rate based on life expectancy"),
       lty = c(1,2), lwd = c(1,2), col = c(2,1),
       cex = 0.7, xjust = 0, yjust = 0)
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    $\begingroup$ This is a beautiful answer. I guess you probably know, but the paradox is called Simpson's pardox. $\endgroup$ – Neil G Oct 30 '20 at 6:28
  • $\begingroup$ @NeilG I believe that it is slightly different from Simpson's paradox, which is also about the difference between groups versus whole, but arrises differently. Simpson's paradox is about differences in trends, differences in regression. The paradox here arrises due to differences in weights to compute the average. $\endgroup$ – Sextus Empiricus Oct 30 '20 at 6:36
  • $\begingroup$ You might be using a narrow definition. Judea Pearl defines Simpson's paradox as the reversal of a statistical relationship with the consideration of additional confounders. In your example, the statistical relationship is death rate change over time, and the additional confounder is the age group. $\endgroup$ – Neil G Oct 30 '20 at 6:50
  • $\begingroup$ @NeilG you are right, it is Simpson's paradox (not even in the narrow definition). I realize now that this paradox can arrise in different ways. In this case it is four variables, one of them is groupsize. In the other case, the reversal of correlation, it only requires three variables and has not group size involved. (and my associations only looked at this last case). $\endgroup$ – Sextus Empiricus Oct 30 '20 at 8:18
  • $\begingroup$ How does migration affect this? $\endgroup$ – gerrit Oct 30 '20 at 8:49
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There is a simpler way to understand this than the (detailed and correct) explanation in another answer.

The life expectancy now depends on the death rate in the past.

As an extreme example, suppose that some disaster infected every person in a country with a medical condition which was incurable and had a 50% fatality rate.

The annual death rate would therefore be 50,000 per 100,000 population (ignoring deaths from other causes, for simplicity).

But the life expectancy would not be 2 years, because almost everyone in the country has already lived for more than 2 years.

The only situation where the numbers are reciprocals of each other is the unlikely situation that all the factors affecting birth rate, death rate, and age-related mortality have remained constant for the lifespan of the current population.

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    $\begingroup$ The life expectancy now depends on the current death rate (which is indeed depending on estimates from past data but only the very recent past). If this catastrophic event is killing all people evenly (independent of age), then the next year the deathrate and life expectancy can be each others reciprocal again. The effect that we are observing is not due to catastrophic events (it does not explain the 40% difference) which changes the population. The mismatch is because the population did not change. We had a quick drop in death rates, to which the population is not yet stabilized. $\endgroup$ – Sextus Empiricus Oct 30 '20 at 7:04

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