## How many lives does a doctor save?

January 17, 2015 § Leave a comment

[This is cross posted on 80,000 hours: 1 2 3. Particular credit to Ben Todd for basically writing the third section.]

Doctors have a pretty solid reputation as do-gooders. There are regular news stories about how advances in medical science promise to help more people than ever before. Many of us have had the experience of being ill, seeing our doctor, and being made better.

So it seemed a pretty good career move for a 17-year old wanting to make a difference. Like thousands of others, I applied to read medicine. This is what I wrote on my personal statement:

I want to study medicine because of a desire I have to help others, and so the chance of spending a career doing something worthwhile I can’t resist. Of course, Doctors don’t have a monopoly on altruism, but I believe the attributes I have lend themselves best to medicine, as opposed to all the other work I could do instead.

Was I right? Is medicine a good career choice for someone wanting to ‘make a difference’?

We are going to try and answer this question, and try to estimate the impact one can expect to make becoming a doctor in the UK (which will be pretty similar to the impact made anywhere else in the developed world). We’ll start by estimating the average direct impact of a doctor. To estimate this, we’ll look at the total impact of medicine and then divide it by the number of doctors. Then we’ll try and account for the fact that by becoming a doctor, I’ll only cause there to be additional doctors. Since the first doctor will add far more to health than the hundredth, my impact will be less than the average direct impact. Then, we’ll account for replaceability: that if I never went to medical school, someone else would have taken my place. We’ll close and see how much good one does by deciding to go into medicine.

# How much good does medicine do?

Life expectancy in the UK was around 47 in 1900; it is now 80. To what degree can medicine take credit for this vast improvement?

By the lights of most experts, not very much. During the last century the population in the developed world became better fed, wealthier, better educated, and living in safer and more hygienic environments, and these take the lion’s share of responsibility for our longer lives. Consider this graph from Mckinlay and Mckinlay of the medical effects on US mortality. Note how total mortality falls dramatically despite the small proportion of GDP spent on health, and that dramatic increases in health spending do not accelerate this downward trend:

Despite this, it is likely medicine has some effect: data from the Netherlands show that there are ‘bumps’ in life-expectancy of those with a given disease that coincide with the advent of improvements in that diseases treatment. We really want to know is how big that effect generally is: is medicine 40% responsible for our better health? 4%? 0.04%?

One approach from Bunker is to compile an ‘inventory’ of medical care (including both prevention and cure) for the commonest diseases: look at medical trials to see what life expectancy effects a given treatment has, extrapolate these effects to the wider population with the disease, and then add them all up. Bunker gives the following estimates:

- Medical care can be credited with 5 to 5.5 years of the increase in life expectancy since 1900…
- … But Iatrogenic (medicine-caused) disease reduces life expectancy by 0.5 to 1 year.
- Medicine also improves wellbeing: the average person has five more years free of disability.

Bunker’s method is likely to overestimate the impact of medicine. The impact of a treatment in a clinical trial is known to be much higher than its effect in everyday clinical practice. However, it is the only quantified estimate available, so let’s use it to try and work out how much good an individual doctor does.

# How much good does a doctor do? A Fermi calculation

We do not know what it would be like to ‘subtract away’ all medical care from a population, but the 1900 population is not a bad surrogate: at this time, medicine was barely funded, unavailable to many and primitive. So ‘amount of good done by medicine between 1900 until now’ approximates ‘amount of good done by medicine’.

We’ll assume the benefits of medicine primarily consist of benefits to our health, though of course there might be others. We’ll also assume that we’re talking about doctors practicing medicine (providing care); medical research is a potentially high impact avenue beyond the remit of the current discussion.

We can use Bunker’s estimates to perform a ‘back of the envelope’ calculation of how much impact on health an average doctor has. To do this we’ll be using the Quality Adjusted Life Year (QALY) as our metric: one QALY is one year of healthy life.[ref]For more on QALYs and measuring healthcare, see here.[/ref] According to Bunker, the average person gains about 5.25 years due to medicine, but loses 0.75 years due to medicine-caused disease. That makes for a net gain of 4.5 QALYs. But we also need to consider how much healthy life is added by treating disability. This is trickier, but a generous estimate is to equate the ‘5 free years of disability’ to 2.5 QALYs.[ref]Bunker only talks about ‘years made free of disability’, but how valuable this is depends on how bad the disabilities are: five years free of dementia is worth much more than five years free of knee pain, for example. The WHO has a table of how much given disabilities should ‘weigh’: a weight of 0.2 means five years free of this disability is worth about as much as one extra year of healthy life. As very few conditions (and none of the commonest ones) are weighed higher than 0.5, we can be confident that the ‘five life-years made free of disability’ will equate to no more than half that amount in QALYs.[/ref]

Totting those up means the total gain per person is seven QALYs. On average, each of us has seven more years of healthy life (either in length or in quality) thanks to our doctors.

Now let’s look at things on a population level. Multiplying up by the 62.6 million population of the UK means medicine adds 438.4 million years of healthy life to the UK. We know there are around 2.7 doctors per 1 000 population in the UK, making around 172 000 total. So the ‘share’ per doctor is:

438 487 000 / 171 824 =** 2552 QALYs**

# Taking stock

This doesn’t look too bad: intuitively you could think of it as saving about 90 lives. But it turns out this figure is an upper bound.

- We’ve used generous estimates for the amount of good done by medicine – most experts think it should be less than this.
- We’ve ignored the fact that the increase in life expectancy in old age is probably associated with increased disability, and so the 4.5 years of increased life expectancy should not be counted at ‘full value’.
- Doctors cannot take sole credit for the impact of medicine: what about nurses, scientists, cleaners, managers?

More importantly, the figure we’ve worked out does not tell us how much difference I might expect to make by becoming a doctor. What we need to work out is the additional good done by an extra person becoming a doctor that wouldn’t have happened otherwise.

This means taking into account two extra effects:

- Diminishing marginal returns – some tasks performed by doctors have more impact than others. If there were one fewer doctor, the highest impact tasks they perform would be given to someone else, so the total impact wouldn’t reduce proportionally with the number of doctors.
- Replaceability – if I don’t become a doctor, someone else will. So again, the difference I make is reduced.

We’ll discuss these effects below, but they will mean that the impact of me becoming a doctor is much less than our estimate. In any case, I can do much more good by giving wisely. If I donate 10% of my salary to AMF, I’ll protect about 20,000 people from malaria, saving about 6 times as many QALYs, and causing many other economic and education benefits.[ref]There are some caveats. AMF will try to spend its funds on the most cost-effective programs first, and (hopefully!) over time the ‘lowest hanging fruit’ of cheap ways of greatly improving people’s welfare will be taken. So money I give later in my life may have less impact than current day estimates (although similar effects will likely apply to the work doctors do as well). Also, there might be different ‘knock-on’ effects of helping richer people than helping poorer people, and the wealthy might be ‘worth more’ if they can add a lot more wealth which trickles down. Anyway, the working: Average doctor salary in the UK is £69 952 a year. Assuming I give 10% of that pre-tax income, and I work for 43 years (qualify at 25, retire at 68), total given over my lifetime will be: £69 952/year * 43 years * 10% = £300 793.60 AMF’s effectiveness is thought to be around: $25/QALY. That’s around £16. So: £300 793.60 / £16/QALY = 18 800 QALYs So around 6 times the (upper bound on) direct benefit of a medical career.[/ref] I can do far more good with my chequebook then I can expect to accomplish with my stethoscope.

# Diminishing marginal returns

So the upper bound for the average direct health impact of a doctor in the UK, is around 2600 QALYs. We can think of this, very roughly, as saving 90 lives. This doesn’t, however, show how much difference you make by becoming a doctor. Working this out requires a number of adjustments. One is that we need to work out the impact of additional doctors, instead of the average doctor.

There’s already about 200,000 doctors in the UK. By becoming a doctor, let’s suppose I increase the number of doctors to 200,001. And let’s assume that all doctors in the UK are equally skilled (we’ll relax this assumption later). The extra doctor won’t produce a benefit of 2600 QALYs. That’s because doctors perform a huge variety of tasks. Some of these do more for the UK’s health than others. The NHS (to some extent) prioritises its distribution of resources so that the most effective tasks get done first. This is part of the remit of the National Institute of Clinical Excellence. So, if there’s one extra doctor, the tasks they do will be less effective than those that are already being done. So we’d expect an additional doctor to have less impact than the 200,000 people who are already doctors. This is called diminishing marginal returns.

How can we take the figure for the average impact of a doctor and work out the impact of an additional doctor? One very rough way of estimating this is to look at the maximum the NHS is prepared to spend to save on QALY, and compare it to the average it spends. We know the maximum the NHS is willing to pay for a QALY is between £20 000 to £30 000. This suggests that extra money given to the NHS produces about 1 QALY per £25,000. Assuming that the NHS can freely spend between salaries and technologies and is fairly good at working out when it is more effective to spend money on either, then the marginal benefit of a doctor per year is their salary (£69 952) divided by the cut off (£25 000), making for 120 QALYs over a career.[ref]Full working: £69 952/year * 43 years / £25 000/QALY = 120 QALYs[/ref] However, this assumption is unlikely to be true. So let’s try some better approaches:

# Approach 1: Life expectancy versus number of doctors

Plotting the most recent values the world bank has (source), we get the following plot:

We see an ‘r’ shaped sort of trend: the life expectancy initially goes up briskly with an increase in doctors per capita, but this relationship levels off as the doctors per capita increases beyond 1 per thousand or so. So it looks like there are diminishing returns of adding more doctors. A similar picture emerges when other sorts of ‘investment’ in health are considered: see for example the plots of health spending per capita versus life expectancy, or GDP per capita versus life expectancy.

From here, we can begin to work out the marginal impact of adding ‘one more doctor’ to the UK: one adds a best-fit line to the data, and see what how much ‘gain’ there is in health when you move ‘one more doctor’ along that line from where the UK is now. This gives a final answer of **950 QALY**s per medical career, just over a third of our original estimate for the impact of a doctor. (I’ve included the working out in a footnote for the interested.)[ref]First, we need to find the best fit relationship between number of doctors and life expectancy. The best candidate for this is a hyperbolic curve: it seems plausible there will be a ceiling on how far life expectancy could rise through adding more doctors, even in the limit case of a population comprised entirely of doctors treating each other.

This graph is the same as the first, save we have shifted down 47 units – the amount we estimated earlier would be the baseline of no medicine, and so the values on the Y-axis are ‘added years’ of life expectancy. The hyperbola is given by the dashed blue line. Now we have our trend, we can work out the expected impact of moving the UK from its current doctors per capita (2.743 per 1000) to the value it would have with one extra doctor. The equation of our best fit line is given by:

Added life expectancy = 30.79459*(Doctors per capita)/(0.16801+Doctors per capita)

So plugging in the difference between our current doctors per capita and the ‘one more doctor’ case:

Marginal change = 30.79459*(2.743016)/(0.16801+2.73016) - 30.79459*2.743016)/(0.16801+2.743016) = 9.76877 * 10^-6 years.

So the marginal impact of one more doctor in the UK will raise UK life expectancy by just under one ten-thousandth of a year. Putting this change into added years of healthy life requires us to multiply by the population of the UK, as well as a correction factor due to our prior estimate that for every 9 years of lifespan medicine adds, it adds another 5 years of healthy life via freedom from disability.

Marginal QALY yield per doctor = 9.76877 * 10^-6 * 62,641,000 * 14/9 = 950 (2sf) [/ref]

This method of estimation isn’t perfect. It looks at all countries, whilst we might want to use data only from developed countries to look at the impact of doctors in the UK. Much like our previous post, wealth remains a potential confounder: both life expectancy and doctors per capita correlate with gross domestic product, and it might just be that richer societies are able to buy better education, hygiene, nutrition, and other things that really do the work of making their inhabitants healthier, and they coincidentally buy more doctors too.

# Approach 2: OECD health data and regression

We can try a different tack to try and accommodate these concerns. The OECD is a group of wealthy to fairly wealthy countries which maintain records of themselves along a variety of indicators. Amongst these are life expectancy, healthcare spending per capita, doctors per 1000 people, and gross national income per capita adjusted for purchasing power parity. We can regress these to a combined model to work out how large a contribution each of these factors make to improved life expectancy, and, once again, we can then work out how big an effect changing one of our variables (doctors per capita) by ‘one more doctor’ will change the life expectancy of the population. The final answer here is that the impact of one more doctor is around 670 QALYs. (Again, the full working in a footnote.)[ref]We can regress these data to a linear model, such that:

Life expectancy = k1 + k2*(GNIPPP) + k3*(Doctors per 1000) +k4*(Healthcare spending pc)

Where k1, k2, k3, and k4 are constants. The best fitting model (adjusted R-square 0.32, P=0.002):

Life expectancy = 75.336 + 0.0000291*(GDIPPP) + 0.433*(doc/1000) + 0.000886*(healthcare spending pc)

This model explains about a third of the variance (adjusted R-square = 0.32), suggesting the main determinants of health in wealthier countries are not wealth, nor spending on healthcare, nor number of doctors. However, of these three it is health spending that is the largest factor, and the effects of either GNIPPP or doctors per 1000 population are negligible – neither are statistically significant, and 95% confidence intervals for either variable cross zero. In other words, we cannot be that confident, on the basis of this analysis, that increasing the number of doctors per capita increases life expectancy at all.

Our best estimate of changing doctors per capita given by the 0.433 coefficient – our central measure. From this we can work out the marginal impact ‘one extra doctor has’ by the similar procedure to before:

0.433*0.0000160 extra doctors per capita = 6.91*10^-6 years in added life expectancy 6.91*10^-6 * 62 641 000 * 14/9 = 673 QALYs [/ref]

# Approach 3: WHO disability adjusted life years

What about quality of life, and not just length? Although I tried to account for this by Bunker’s estimates of how much good medicine does via removing disability, it would be nice to tackle the issue more explicitly.

The WHO keeps data on the burden of disability in a population, as DALYs per 100 000 people (a DALY is a measure of length and quality of life, it is the inverse of a QALY – more DALYs are a bad thing, as well as other differences summarized here). Plotting DALYs per 100 000 against doctors per 100 000 gives the following:

This graph looks like a mirror image of the life expectancy versus doctors per capita graph above. Although we cannot directly compare ‘DALYs averted’ to ‘QALYs gained’, using a similar technique to approach 1 (draw a line of best fit, work out how much gain one makes by moving ‘one more doctor’ along, and multiply appropriately) means each doctor averts 645 DALYs per career. This reassures us our figures are on the right track.

# Conclusion

These estimates are necessarily very rough, though it’s reassuring to find our three estimates in the same ball park. Splitting the difference between our three best estimates gives the impact of ‘one more’ medical career in the UK as about 760 QALYs, around a third of our estimate of the average doctor. Looking at the degree of noise in the data, I estimate the 95% confidence interval is about 600 – 920 QALYs.

The expected impact of becoming a doctor is now around 25 lives: still pretty good, but giving 10% to effective charities can produce a health benefit 25 times larger than that. This underlines the importance of thinking at the margin for those wanting to make the biggest difference they can. One should try to estimate not how much good a career does in general, but how much more good they can do if you get involved. In the case of first world medicine, it appears most of the highest priority interventions for improving health and wellbeing have already been done, and so the additional impact of one more doctor is not that large.

Our estimate, however, is still too generous. By becoming a doctor I won’t increase the number of doctors by one. Rather, it seems I’ll just take the job from someone else. I’ll be replaceable. We’ll look at this adjustment in the last post.

# Replaceability [by Ben Todd]

So, given diminishing returns, my going into medicine would save 600-920 QALYs, with my best guess at 760 QALYs. That’s roughly equivalent to saving 25 lives.

But even this is too generous. If I become a doctor, I won’t increase the total number of doctors by one. The NHS has a limited budget, so it can’t just hire every qualified person who applies; medical schools have limited places by law, and there are more applicants who are ‘good enough’ than there are places. If I become a doctor, then I’ve just displaced someone else who would have taken the job.

If we also assume that doctors are equally skilled, then by becoming a doctor you’d have zero impact. You’d simply displace someone equally good. Fortunately, it’s not as bad as this. First, doctors are clearly not equally skilled (as we’ll look at later). But more importantly, by becoming a doctor, you do, in one sense, increase the number of doctors.

# Supply-Demand Effect

If I become willing to work as a doctor, then I increase the supply of doctors. If more people are willing to be doctors, then the NHS can slightly decrease the wages for doctors.[ref]It might be easier to see the other way round. Imagine that tomorrow lots of doctors decided they didn’t want to be doctors anymore. In response, the NHS would increase wages and try to tempt them back.[/ref] If the wages are slightly lower, the budget can be used to hire slightly more doctors.

How this all balances out is studied by economists. If the labour market for doctors is in equilibrium, then increasing the supply of doctors by one doctor, will probably increase the number of doctors by about 0.6.[ref]Medicine is highly skilled, so I use a theoretical elasticity of labour demand of -0.5. I assume the elasticity of labour supply is about 0.75, which is an average figure for all jobs taken from this meta-study:

Evers, M., & R. de Mooji, D. v. (2005). What Explains the Variation in Estimates of Labour Supply Elasticities? CESIFO Working Paper No. 1633.

This analysis only applies to the extent that the market for doctors can reach equilibrium. It’s not likely to be in equilibrium due to distorting factors like regulated wages and medical education caps, but it’s not obvious how this affects the figure. If someone has better estimates of labour elasticities for doctors, I would be very interested in seeing them.[/ref]

In other words, if 10 new qualified people become willing to be doctors, then we’d expect 6 new positions to be created, and only 4 people who would have been doctors to fail to get a job (or a place at medical school). This means the impact of becoming a doctor is only 60% of our current estimate, i.e. about **450 QALYs**.

# Skill Level Effect

Finally, we have to correct this figure for the fact the doctors are not equally skilled. By becoming a doctor, you’ll change the average skill level, and this effect adjusts our impact upwards. When several people attempt to become doctors, the NHS attempts to select the best ones. This means that if you become a doctor, the 0.4 doctors you’d expect to displace from the pool should be less skillful. So, you’ll increase the average skill level. In the next section, I’ll try to put an estimate on this factor. It’s a little technical, so skip ahead to the summary if you don’t want the details.

In an ideal world, as someone enters medicine the NHS would replace the worst doctors in the pool. So, by becoming a doctor, your effect on the pool would be to remove the worst 0.4 of a doctor and replace them with yourself. The worst doctor is likely to have zero output. So, now let’s assume I’m an average doctor. If we add 10 doctors like myself, and lose 4 with zero output, then we’ve effectively gained 10 doctors of average output. So, we have to add another 4 doctors to our earlier estimate of 6. So then the supply-demand correction factor is cancelled out, and you have 100% of the impact of a marginal doctor.

This could easily be an underestimate. The worst doctor is likely to have negative impact. That’s not because they’re likely to directly do more harm than good, but because they might hamper the effectiveness of lots of team mates. In fact, if there are about 200,000 doctors, there’s reason to expect the worst to roughly counteract the good done by an average doctor i.e. to have an output of ‘-1’ marginal doctors.[ref]Assume that the ability of doctors is a random variable, X, that’s normally distributed. The population of doctors is a sample of size ‘n’ from X. If n > 10,000, then the expected minimum and maximum value of the sample can be given, to a very high degree of approximation by:

Expected max = b+0.5772*a stdev from mean

Expected min = -(b+0.5772*a) stdev from mean

Where:

a = (2*log(n))^(-0.5) b = (2*log(n))^0.5-0.5*(2*log(n))^(-0.5)*(log(log(n))+log(4*pi))

So, for a population of 200,000 doctors, the expected value of the minimum would be 4.5 standard deviations from the mean. The standard deviation is at least 45% of the mean (Ben’s blog post), so the expected value of the minimum is 100 – 4.5 * 45 = -102.5% of the mean[/ref]

That would mean the by becoming an average doctor I’d displace harmful doctors, doing more good than our previous estimate. In fact in this case, the skill level effect would add another 0.8 doctors worth of impact.[ref]If the worst doctors roughly counteract the benefit of an average doctor, then we can treat them as having impact of -1 doctors. So, if we add 10 doctors of average skill, and lose 4 doctors of -1 skill, then we’ve effectively gained 14 doctors of average skill. This means applying a correction of 8 doctors to the original 6 doctors figure.[/ref]

Unfortunately, the idea that I’ll knock out the worst doctor is a large idealisation. In reality, selection processes are not perfect. Moreover, there’s not always mechanisms to select across the entire NHS, and there’s limits on who can be made redundant. So, it’s very unlikely I’ll knock out the worst doctor, which means the correction we’ve just worked out is an upper bound.

It’s difficult to see how to estimate the efficiency of the NHS’s selection procedures. But let’s suppose, to get a rough idea, that instead of knocking out the worst doctor, I knock out doctors in the bottom decile. If you’re an average doctor, then your output is probably more than 80% higher than the bottom decile. Then the correction factor is about 0.2 additional doctors worth of impact.[ref]We add 10 doctors of average skill, and lose 4 of 55% skill, so we effectively gain about 8 doctors of average skill. This means we need a correction factor of 2 doctors[/ref] This is a very rough estimate, but it’s bounded between 0 and 0.8. Zero is the case when the NHS is unable to select according to ability at all.

If you’re a top-decile doctor instead of an average one, then the correction factor would be about 1. (The bounding range would be 0.8 to 1.6)

Also note that I’ve effectively assumed that doctor skill is uncorrelated with the potential of their position to create health benefits: on my model so far, a top decile doctor would have as much excess impact whether a GP, cardiothoracic surgeon, public health doctor or whatever else. Normally, the highest skilled people get put into the positions with the highest potential for impact (at least insofar as the selection process is efficient). It’s not clear this holds to a large extent in medicine. The most effective activities of the NHS are often unsexy areas like providing vaccines or management. The most skilled doctors might be placed in the most difficult areas, rather than those with the highest effectiveness. Further, many doctors don’t seem to seek to maximise their impact in job selection – there isn’t even publicly available information to help doctors do this. So, my assumption could be reasonable. If this is the case, then there could be opportunities for skilled, altruistically motivated doctors to seek out the positions with the most potential for impact.

If there was a tendency for the highest ability people to be put into the most effective positions, then it would increase the impact of becoming a doctor, especially for highly skilled doctors, because the impact of their additional skills would be better leveraged. This correction seems likely to be significantly less than a factor of 2.[ref]We’ve also ignored the tendency to pay worse doctors less. In a perfectly efficient market, you would be paid exactly in proportion to your ability to produce value. So if you were replaced by less skillful doctors, they would be able to pay for more of them to make up for the gap. In practice, this is a long way from the truth. Doctors salaries vary much less than their differences in output, and moreover, are highly regulated.[/ref]

# Summary – how much good do doctors really do?

- We started with an upper bound for the average direct impact of 2600 QALYs
- The impact of marginal doctors is only a third of the average direct impact, meaning the difference a doctor makes is around 760 QALYs, with a 95% confidence interval of 600-920 QALYs.
- Supply-Demand Effect: The number of doctors only increases by 0.6, not 1
- Skill-level adjustment: You’re more skilled than your replacements: If you’re of average skill, the number of doctors effectively increases by an extra 0.2, or in total 0.8, so your marginal impact is 600 QALYs. That’s roughly saving 20 lives.
- The skill-level adjustment is very rough. But it’s likely to be in the range of an extra 0 to 0.8, depending on how effectively the NHS selects doctors according to skill. So, if you’re an average doctor, crudely putting our two ranges together, your impact is in the range of 360 to 1300 QALYs, with a best guess of 600.
- Skill-level adjustment if you’re in the top decile of doctors: The number of doctors effectively increases by an extra 1.0, or 1.6 in total, so your impact is 1050 QALYs. The full range would be from 840 to 2000 QALYs.
- We’ve assumed doctor skill is uncorrelated to the potential of their position to have impact.

# Conclusion

So we’ve now got a much better estimate of how big a difference I will make working as a doctor: around 600 QALYs, or around 20 lives – one every two-and-a-bit years or so of my career. Giving 10% of my salary to effective causes that fight global poverty will add around 18 800 QALYs, a health impact about 30 times higher than what I do directly in my work.

I think my 17-year-old self would find that pretty galling. He’d signed up to medicine to save loads of lives, and he’d find it a bit of a downer to see this his entire medical career would likely do as much good as a £10 000 donation to the right charity. But that would be the wrong way of looking at things: instead, he should see that saving 17 lives is a vast amount of good, and being able to do 30 times more good on top of that is awesome.

It should also brighten up everyone else’s day too. One does not have to be a doctor or an aid worker or any other ‘archetypally moral’ career to make a vast difference; it is within reach of most people in the developed world. So let’s do it.

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