Mortality Exercises 3: Person Years Remaining

In these exercises, we will calculate person-years lost from a sudden mortality crisis like Covid. Our approach will be to do the full life-table calculation and then compare this to the approximation involving \(H/A\) that we did in lecture.

More future reference (but not now), see Goldstein and Lee (2020) (https://www.pnas.org/content/117/36/22035.full)

A. Person years remaining in stationary pop with US life table

Here we’re going to sum up all of the person-years in a simulated but not-totally-unrealistic population

Notes: * We are using 2019 period life table from Human Mortality Database * Remaining person years is a cohort concept, but we are using period life expectancy for simplicity here. * We are using stationary age structure, not the observed * To give us numbers that are the right order of magnitude, we’re going to assume 3.5 million births. * We’re going to be a bit sloppy with distinction between l(x) vs L(x). For actual research, please be more careful.

Some preliminaries

Nx.stationary = lx * 3.5
sum(Nx.stationary) ## [1] 278.8149

[1] 278.8149

Now we’re ready to calculate remaining person years, which will be the number of people in each age group, multiplied by their life expectancy, summed over all ages.

Nx.stationary

  [1] 3.500000000 3.480559193 3.479202039 3.478401915 3.477775859 3.477289004 3.476836986
  [8] 3.476419791 3.476037406 3.475655063 3.475272762 3.474855754 3.474404052 3.473917670
 [15] 3.473327154 3.472597832 3.471660357 3.470410784 3.468849450 3.466630097 3.464169663
 [22] 3.461468664 3.458354744 3.455209073 3.451893664 3.448512465 3.444893427 3.441071717
 [29] 3.437151130 3.432994693 3.428671843 3.424080500 3.419324334 3.414472340 3.409320377
 [36] 3.404040024 3.398529943 3.392825206 3.386825217 3.380430163 3.373777271 3.366767107
 [43] 3.359670718 3.352254044 3.344552719 3.336435323 3.327638750 3.318135300 3.307831929
 [50] 3.296769233 3.284856503 3.272070512 3.258030815 3.242883712 3.226645228 3.208626515
 [57] 3.188954016 3.167849506 3.144902753 3.120467931 3.094242011 3.066457445 3.036340382
 [64] 3.004265068 2.970537665 2.935162942 2.897977177 2.858517027 2.816916842 2.773452596
 [71] 2.727766025 2.678596514 2.625740545 2.569709785 2.509498591 2.446168612 2.376913907
 [78] 2.303115954 2.224991218 2.143013398 2.055692900 1.963057032 1.864667902 1.761442311
 [85] 1.652720914 1.540047051 1.424018734 1.304725884 1.182183283 1.060226928 0.941983961
 [92] 0.831472518 0.722161102 0.615871444 0.515278869 0.421916846 0.339684840 0.267828874
 [99] 0.206518139 0.155514387 0.114212214 0.081701981 0.056861486 0.038461519 0.025262681
[106] 0.016102077 0.009954834 0.005968038 0.003469562 0.001956463 0.001070599

Q.1 How many person years remain per person?

(Fun fact: in stationary population average life remaining is the same as average age of pop! Life left = life lived)

B. Person Years Lost

Here we assume there is a “flash” of mortality, with intensity 1/3 of all-cause annual mortality. How many person years will be lost?

print(lost.py)

[1] 14.83438

What fraction of the original person-years of the population were lost?

print(lost.py.frac)

[1] 0.001278089

Q2. How many person years remain per person after the flash of mortality? How many years are lost per person? Is this a few years, months, weeks, days, or hours, …?

## approximately 3 weeks 
0.05320511*365

[1] 19.41987

C. Share of person-years lost (using our approximation)

Compare your answer to our approximation which tells us that the fraction of person years lost in this case should be about H * delta/A

We leave it to you to calculate \(H\) and \(A\) that correspond to the stationary population to the 2019 US life table

(Note: The values should be something close to \(H = 0.15\) and \(A = 40\). If you’re having trouble estimating the exact values, you can use these in order to answer the Qs)

H

[1] 0.1541554

Q3. How did we do? Are we close to life-table based calculation?

We did well = are very close!

Q4. On twitter, one of the workshop participants didn’t like this approach, saying that the tiny number was misleading – and said that it was like comparing the economic damage done by Hurricane Katrina in New Orleans to the GDP of the entire USA. It would seem small, but is a minimizing way to think about the actual damage of the Hurricane. What do you think?

I think it’s one perspective.

Note: When comparing past mortality crisis, Goldstein and Lee had similar concerns and ended up divided remaining-years-lost from each crisis by the remaining-years-lost in a corresponding non-crisis year. You can look at their Fig. 4 to see the results of this analysis. (https://www.pnas.org/content/117/36/22035.full)

Congratulations!

You’ve finished the last exercise of the day!