Life with hazard ratios

DYNOMIGHT best topics follow about If you read anything about health or longevity, you’ll soon find yourself in a world of hazard ratios. Some study might say that eating more fiber might change your risk of dying by a factor of HR = 0.90. Another might say that occasional smoking might change it by HR = 1.30. But how much should you care about that? Is HR = 0.90 or HR = 1.30 a lot? What if you don’t want to eat more fiber? What if you like smoking? Instead of staring at a ratio1, a more sensible thing to do is think about life expectancy.2 But is it possible to convert a hazard ratio to a change in life expectancy? You might reason as follows: Baseline life expectancy is around 75 years. And HR = 0.90 corresponds to a 10% decrease in mortality. So perhaps that hazard ratio corresponds to something like 7.5 extra years of life expectancy? Unfortunately, that’s completely wrong. To see why, imagine that humans only die by playing Russian roulette. They start playing this once per day at the age of 75, with a revolver containing two bullets and six chambers. If you were to remove one of those two bullets, that would drop the person’s risk of death by HR = 0.5. (One bullet versus two.) But life expectancy would barely change, because even with just one bullet, almost nobody would survive for any significant amount of time past 75. For contrast, imagine again that humans only die via Russian roulette, but now they do this once per day from birth with a revolver with 2 bullets and 54,786 chambers. (Newborns emerge and instinctively reach for this gigantic gun.) You can show that these people also live 75 years on average. But now, if you remove one of the bullets, life expectancy doubles, because when someone is spared, it takes a long time before they get unlucky again.3 Neither of those is a good model for humans. We’re somewhere between the two, with heart disease and so on instead of revolvers and risks slowly rising as we age instead of suddenly starting at age 75 or staying constant throughout life.

But you get the point: If you want to convert a hazard ratio for some intervention to a change in life expectancy, the impact depends on how “spread out” baseline mortality risk is over time. Baseline life expectancy is simply not enough information. That’s one problem. Here’s another: What even is a hazard ratio? The technical definition is something like: The hazard ratio at a given time is the rate of an event in the treatment group divided by the rate of that event in the control group. Hazard ratios are often confused with their more beloved siblings, relative risks. Say you run a trial for 10 years and at the end, 10% of the control group died and 8% of the treatment group. Then the relative risk is RR = 0.8, nice and simple. But relative risks have problems, most notably that if you run a long enough trial, then no one will be alive at the end no matter the intervention, meaning RR = 1.0. That’s not helpful. Intuitively, you can think of the hazard ratio at age 40 as sort of like the relative risk for people between the ages of 39.99 and 40.01. In real life, interventions have different hazard ratios at different ages. Chemotherapy tends to have better results in younger patients who are more able to endure the side-effects. Having a slightly higher BMI (25-30 rather than 20-25) is associated with an increased risk of mortality in young people, but a decreased risk in the elderly. You may remember from 2020 that COVID’s mortality risk had a different age curve than baseline mortality, meaning the hazard ratio of getting COVID was different at different ages. This is important, because hazard ratios at different ages have different impacts on life expectancy. A hazard ratio of 0.9 at age 80 prevents more deaths than at age 20, because baseline mortality is higher at 80. But at the same time, if you save the life of a 20 year-old, they have more years in front of them. Beyond that, the hazard ratios at different ages interact: If some intervention decreases mortality at younger ages, that allows more people to reach older ages, increasing how much hazard ratios matter at older ages.4 If we knew the hazard ratio at all ages, we could account for those dynamics.

But we don’t, because when estimating hazard ratios, people almost always assume that the hazard ratio is constant.5 We’re quasi-forced to do this because there’s not enough data to estimate a whole time-series of ratios. That’s why papers contain single numbers like HR = 0.90. So even though Intervention A (say, more fiber) and Intervention B (say, light jogging) might have the same hazard ratio in a paper, those numbers could be the product of different underlying age-dependent effects, meaning those interventions could conceivably lead to vastly different changes in life expectancy. So is this all hopeless? Are single hazard ratio numbers just too far removed from what we care about to tell us anything meaningful? Surprisingly, no. It’s mostly OK. If we were a different species, it might be hopeless. But for modern humans in rich countries, mortality happens to be distributed in a way that produces a sort of lucky coincidence: When people estimate constant hazard ratio numbers, they’re implicitly sorta-kinda taking a weighted average of hazard ratios at different ages. And those weights happen to (sorta-kinda) reflect how much changes in mortality at different ages change. So, I will argue, even if the true intervention has a varying effect, it’s sorta-mostly OK to just take a hazard ratio from a paper and convert it to a change in life expectancy using this curve: If a paper showed that eating more fiber produces a hazard ratio of HR = 0.75, that corresponds to an increase of around 3.7 years. If a paper says that occasional smoking produces a hazard ratio of HR = 1.25, that corresponds to a decrease of around 2.9 years. This isn’t exact. If the intervention is better (or less bad) for older people this will tends to overestimate the increase (or underestimate the decrease) in life expectancy. If the intervention is worse (or less good) for older people, it will tend to underestimate the increase (or overestimate the decrease) in life expectancy. But as long as the hazard ratio doesn’t vary too much by age, it’s probably not off by more than around 30% in either direction. The easy case Say there’s some intervention (eating more fiber or whatever) that multiplies your risk of dying at age t by a factor of HR(t).

Then it can be shown that this changes life expectancy by approximately ΔL ≈ ∑ₜ ΔHR(t) × P(t) × L(t). Here, P(t) is the baseline probability of dying at age t. For males in the United States, it looks like this: Meanwhile, L(t) is conditional life expectancy at age t. That’s the average number of additional years left for someone who reaches age t. For males in the United States, it looks like this: Finally, ΔHR(t) is the decrease in hazard at age t. You can think of that as just ΔHR(t) = 1 - HR(t). Though if you’re OK with logarithms, there’s a somewhat better approximation that uses logarithms, which I’ve quarantined in a footnote.6 Let’s start with the easy case. What if your intervention has the same effect on mortality at all ages, so HR(t)=HR is just a constant? Then, the above equation simplifies into ΔL ≈ ΔHR × L̄, where L̄ = ∑ₜ P(t) × L(t). This makes sense! Again, P(t) is the baseline probability of dying at age t and L(t) is conditional life expectancy at age t. These are constant, so when you add them up, L̄ is just a number. For males in the United States, it happens to be 12.93 years. This quantity has a specific meaning: The average remaining life expectancy for US males when they die. That sounds a bit odd, but think of picking a random death and asking how many additional years people who reach that age live on average. That number is 12.93 years. So, if an intervention has a constant hazard ratio, the mean change in life expectancy for US males is just ΔL ≈ ΔHR × 12.93 years. Now we’re getting somewhere! If you prevent a fraction ΔHR of deaths, then you increase life expectancy by ΔHR times 12.93 years. Now remember the naive calculation we started with: Life expectancy for US males is 75.8 years. You might hope that if eating more fiber drops your risk of death by 10%, that would save 7.58 years.

Sadly, the above equation says that a 10% drop in risk only increases life expectancy by around 1.293 years—only 0.17 times as much. This is essentially the observation Keyfitz made in his 1977 paper, “What Difference Would It Make if Cancer Were Eradicated?” Cancer is responsible for 18 percent of deaths, so does that mean eradicating it would increase lifespan by 18 percent, or around 13.6 years? Nope, Keyfitz says, it’s only 2.3 years. If a cure for cancer were discovered and made available today, 350,000 cancer deaths would be avoided in the next year. The overall death rate would be lower by nearly 18 percent. If the cure were quick and inexpensive, a large fraction of the country’s hospital beds and medical personnel would be released for treatment of other ailments. Patients would be spared untold suffering. Such an implicit analysis underlies government proposals for eradication of cancer. The argument is sound for first effects on mortality but wholly misleading for the long term. The first effects would soon be offset by more mortality from diseases other than cancer. As a result of the cancer cures, the population would include a higher proportion of people subject to other causes of death. […] At the extreme, it might be said that everyone dies of something sooner or later, so that, when the effects of the eradication of cancer had shaken down, the same number of deaths would occur as before, and the only benefit would be the substitution of heart and other diseases for cancer. A cure for cancer would only have the effect of giving people the opportunity to die of heart disease. Cheerful stuff! We can also write our approximation in terms of baseline life expectancy as ΔL ≈ ΔHR × 0.17 × 75.8 years, which makes explicit that 12.93 years is only 0.17 times as large as a naive estimate using baseline life expectancy. The discount factor of 0.17 is sometimes called the “Keyfitz entropy”.