Unfortunately, a common trait among journalists, politicians and actually, lets face it, most of the population, is that they’re not very good at math. An unfortunate side-effect of this is that we make very poor arguments from data.

We don’t want to be making illogical arguments, even if we make them for an otherwise good cause. Hence it behoves me to point out a bad argument that was made many times against the efficacy of The Vaccine.

The argument is this:

the portion of people in hospital who are vaccinated is higher than the vaccination rate, therefor the vaccine makes things worse, not better.

I can’t count the number of times I’ve heard it. Unfortunately, it includes an implicit mathematical error, which I will now seek to demonstrate. Of necessity I am going to simplify the problem, and all numbers that I am about to use are hypothetical, so please don’t quote them.

Suppose we have a hypothetical virus. Everyone has equal chance of getting the virus, and if they do, they have a 14% chance of landing in hospital.

However, in reality, we can break this down further. A subset (20%) of the population are more vulnerable to the virus, because they are old, or unhealthy, or perhaps both. This group actually account for more of the hospitalisations because they are 10 times more likely to end up in hospital!

This is summarised in the following table:

GroupProportion of population Hospitalisation rate Hospitalised
  Vulnerable people20%x50%=10%
  Everyone else80%x5%=4%
  TOTAL100%  =14%

Some genius invents a vaccine, which people start injecting into themselves. Eventually, across the population, 75% of people are vaccinated. Again, however, if we break this down, the rotund/geriatrics were a lot more worried about the virus and 95% of them got vaccinated, while only 70% of everyone else bothered.

Supposing the vaccine is effective such that it halves the hospitalisations in every group? What are the results? The population-wide hospitalisations would reduce from 14% down to 7.85%, which is good!

GroupProportion of population Hospitalisation rate Hospitalised
  Unvaccinated25%x6.8%=1.7%
  Vaccinated75%x8.2%=6.15%
  TOTAL100%   7.85%

But already we observe a paradoxical outcome. The hospitalisation rate for vaccinated people is higher than for unvaccinated people. Even though 75% of the whole population are vaccinated, 78% of people in hospital are vaccinated. How can this be?

The reason is simple, a higher portion of the vaccinated people come from the vulnerable portion of the population. If I break the results down further into ‘vulnerable’ and ‘everyone else’, you will see the hospitalisation rate still halved for the vaccinated in each group:

GroupProportion of population Hospitalisation rate Hospitalised
Vulnerable:     
  Unvaccinated1%x50%=0.5%
  Vaccinated19%x25%=4.75%
 SUB- TOTAL20%   5.25%
Everyone else:     
  Unvaccinated24%x5%=1.2%
  Vaccinated56%x2.5%=1.4%
  SUB-TOTAL80%   2.6%

How can we generalise this error, to make sure we avoid it? It’s all about variables. A big danger when drawing mathematical conclusions is to look at only two variables – in this case vaccination and hospitalisation, and ignore all other variables. This is easiest for our brains, but when we do this, we implicitly assume that all the other variables are ‘controlled’ or ‘independent’. In this case, there was another variable, vulnerability, which was not independent.

Rory Sutherland, Spectator UK’s Wiki Man, previously demonstrated this same effect here: https://www.spectator.co.uk/article/what-data-does-not-tell-us. This issue is also famously relevant in the gender wage gap issue. (The wage gap claim is based on measuring two variables, gender and wage, but neglecting a large range of other variables that are correlated to gender, but are nevertheless distinct variables. Further discussion of that here: https://www.forbes.com/sites/karlynborysenko/2020/03/31/great-news-ladies-the-gender-pay-gap-is-a-myth/)

So if you are a journalist or a politician, and you are thinking perhaps of drawing a mathematical conclusion of some kind – maybe you’re going to give a speech in parliament, or pour rice into a bowl to demonstrate how small 0.3% is or something like that… perhaps consult with a mathematician first. Just an idea. Because these errors are very easy to make.

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