# How many lives does a doctor save? - Part 2 - Diminishing marginal returns

In the first post, I worked out an upper bound for the *average direct health impact* of a doctor in the UK, and found it amounted to producing about 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. The first 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 in the next post). 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 (1). 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 QALYs** 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 (2).)

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 (3).)

### Approach 3: WHO disability adjusted life years

A lot of comments in the previous post were worried 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.

See part 1 which finds and upper bound

See part 3 on replaceability

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*References and Notes*

(1) Full working: £69 952/year * 43 years / £25 000/QALY = 120 QALYs

(2) 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.(1)

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)
```

(3) 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
```