Converting Mortality to Life Expectancy

Research papers that report differences in mortality between groups almost always report relative risk of all-cause mortality. A few papers report changes in absolute risk with a statement like “+10 per 1000 persons.” But how can we convert a number like that to what we actually care about: life expectancy?

Intuition suggests that if you cut the mortality rate in half, remaining life expectancy should double. That’s wrong, though. Let’s discuss what assumptions would make that true, then talk about what really happens. Skip the block quote if you don’t want to read math in a a bad font.

Let’s say your probability of dying in the next year were independent of your age. In that case, your probability of living at least to time t would be e^[-t/L], where L is some constant. Then, your probability of being dead before time t is (1-e^[-t/L]). To get the probability that you die at exactly moment t, take the derivative to get p(t) = (1/L)e^[-t/L]. Life expectancy is then the integral from 0 to infinity of (t/L)e^[-t/L], which is L. So the inverse of your life expectancy, L, would be the rate of your exponential decay in this hypothetical world.

The upshot is that if the chance of dying were the same every year, then longevity would change inversely with the mortality rate. Smoking would drop life expectancy in half.

That’s not how human bodies work, though. We experience senescence. Once we turn 11, our chance of dying every year creeps up. This fact breaks the simple link between mortality and longevity. You can calculate real life expectancy using the probability that a person dies in the nth year of life, given that they reach the start of that year. In this Google Sheet, I reproduce the calculation from the government’s actuarial life tables.

In the tab Half_Mortality_Rate, I check what happens when I drop the probability of dying at every year in half. For a male born today, that would only raise life expectancy from 76 to 84.6 years! So halving mortality only increases life expectancy by 11%.

Early in life, the probability of dying is near zero. A 10-year-old boy has less than a one in 11,000 chance of dying. If that stayed the same our whole lives, we’d live over 11,000 years. Dropping the mortality rate in half for 10-year-olds makes a trivial difference to total life expectancy. The difference only becomes material later in life as the probability of dying within a year gets closer to one. The SSA estimates that a 119-year-old male would have an 89% chance of dying before his next birthday. Dropping that in half dramatically increases remaining life expectancy, but still only by about a year.

It’s tempting to take away from this that the effects of anything other than smoking must be too small to worry about. Heavy smokers have more than a two-fold increase in all-cause mortality. Effect sizes from studies of exercise, sleep, or nutrition are always much smaller.

But wait! Before we decide if an effect size is too small to care about, we should compare the effort required to get it. Consider smoking. When I make a very simple change for the life lost to smoking and divide by the number of cigarettes a lifetime smoker would go through, I get an average of 10 minutes of life per cigarette. This more sophisticated calculation got an answer of 11 minutes. These are rough calculations, so I’d say we agree.

How about red and processed meat? My last post was about a series of recent studies of the harms of red and processed meat. The authors found that cutting out three servings per week of red or processed meat (just a bit less than median total consumption) reduces all-cause mortality by about 7%. They made the controversial claim that 7% is too small to care about, so they say that doctors should stop telling their patients to cut back on eating red or processed meat. But we don’t live and die by percentages. We live and die by minutes.

On another tab of the same Google Sheet, I take the male mortality rates and adjust them down by 7% starting at age 30. Life expectancy goes up by 0.79 years. Spread out over 46 years of eating three servings of red meat per week, that works out to a full hour of life lost per serving. That hardly sounds too small to care about to me.

How can it be true that one cigarette shaves off 11 minutes of life but a hamburger shaves off a full hour? The median smoker has over 20 cigarettes per day, while the median red meat eater has about half a serving per day. It might seem more relevant to compare the typical life in the day of a smoker to the typical day in the life of a red meat eater. Fair point. But I’m interested in motivating myself to take healthy actions, so I prefer to look at the average impact of each individual action. I don’t decide to eat a hamburger one bite at a time. I decide one hamburger at a time. Knowing that the average hamburger will reduce my life expectancy by one hour will definitely motivate me to avoid red meat.*

*I’m taking this calculation a bit too literally. The red meat results don’t show that every single serving of red meat shaves off an identical time from our lives. They show what correlates with a lifetime of thousands of servings of red meat. But I need something simpler to think about when I’m staring at a menu.