# How can I best calculate the expected amount of life lost over the next 50 years due to a constant yearly 1.5% risk of death?

Say that Person X has 50 years left to live until they die from cause Z. Cause Y is a constant 1.5% risk of death per year for the next 50 years. I want to determine what the expected amount of life lost is due to cause Y.

I've tried three approaches to determining this that all yield vastly different results. Is any of these valid? If so, which and why aren't the others?

1. I thought I'd be able to calculate the expected amount of life lost due to cause Y by:

(1-(1-0.015)^50)*25.5

Where (1-0.015) is the chance of surviving in a year, raised to the 50th power to calculate the chance of survival over the next 50 years, 1- this amount is to convert from the chance of surviving to the chance of dying, and multiplied by 25.5 as the average number of years remaining in that persons life over the 50 year timespan.

This yields a value of 13.52 expected years lost due to cause Y.

2. I also thought that this was equivalent to saying that you lose 1.5% of your life each year. So then, over the next 50 years you'd lose 50*0.015 = 0.75 years of life.

3. My last approach was to say that for each of the next 50 years, if the risk is realized, you lose not only that year, but also each of the following. Therefore the amount of time lost can be calculated by:

sum(0.015*(years remaining)) over the next 50 years = 19.13 years.

shown differently:

0.015*49+0.015*48 + ... + 0.015*1 = 19.13 years

With such big discrepancies, clearly at least two of these approaches are flawed. What's the best way to calculate this and why are the others flawed?

• Change the number from 1.5% to, say, 50%. In this case you're fairly certain of dying in the next five years or so and will lose almost all 50 years of life. How do your three approaches stack up against this common sense estimate? (They will all be obviously wrong, but by a great deal and in various directions and amounts, all of which will give you useful hints about the errors in those approaches.)
– whuber
Nov 10 '20 at 19:57

The probability of dying at year i where i is 1 to 50 is $$p(i) = 0.015 \cdot 0.985^{(i-1)}$$ the remaining probability, say p(51), is the probability of dying from cause Z $$p(51) = 0.985^{50}$$

Furthermore dying at year i removes (51-i) years from your life And the total loss is the sum of the products of the probabilities of death at year i and the cost in years for death at that i: $$L = \sum\limits_{i=1}^{50}{(51-i) \cdot p(i)} + (51-51) \cdot p(51)$$ $$L = \sum\limits_{i=1}^{50}{(51-i) \cdot 0.015 \cdot 0.985^{(i-1)}} + 0 \approx 15.18$$ As mentioned in the comments below this assumes a discrete (not compounded continuously) probability of death at the beginning of each year. You could assume the midpoint or compound continuously to get slightly different results.

• What are we to make of the fact that your $p(i)$ do not sum to $1$ as a valid set of probabilities ought? (They sum only to $0.53\ldots$)
– whuber
Nov 10 '20 at 19:20
• @whuber good point - the remaining probability is that they do not die from cause Y, so the years lost for those is 0 Nov 10 '20 at 19:59
• Exactly: it would be good to make that point explicitly in your solution, even though it doesn't change the numerical answer.
– whuber
Nov 10 '20 at 20:02
• Thanks! I'm glad to have the solution for this; it makes sense to me. It helps clarify why my attempt #3 was incorrect; I wasn't accounting for the chance of death up until that point, so it overestimated. I think approach #2 mainly failed to account for the remaining years of life lost at each point. Whereas #1 is using the likelihood of surviving all 50 years, which just doesn't make sense to use to multiply against the median yrs remaining. Does this sound right? Nov 10 '20 at 21:49
• I think you may also need to accommodate the age at which this cause of death is likely to occur, for example a disease such as COVID19 mainly affects the likelihood of death for the elderly who would die from some cause or other within the next few years, whereas a disease such as Spanish flue mainly affected young adults who would otherwise have had many years of life left. Nov 16 '20 at 10:53