A Vaccination Fallacy

I have been struck by the number of people upset by the latest analysis of SARS-Cov-2 “variants of concern” byPublic Health England. In particular it is in the report that over 40% of those dying from the so-called Delta Variant have had both vaccine jabs. I even saw some comments on social media from people saying that this proves that the vaccines are useless against this variant and as a consequence they weren’t going to bother getting their second jab.

This is dangerous nonsense and I think it stems – as much dangerous nonsense does – from a misunderstanding of basic probability which comes up in a number of situations, including the Prosecutor’s Fallacy. I’ll try to clarify it here with a bit of probability theory. The same logic as the following applies if you specify serious illness or mortality, but I’ll keep it simple by just talking about contracting Covid-19. When I write about probabilities you can think of these as proportions within the population so I’ll use the terms probability and proportion interchangeably in the following.

Denote by P[C|V] the conditional probability that a fully vaccinated person becomes ill from Covid-19. That is considerably smaller than P[C| not V] (by a factor of ten or so given the efficacy of the vaccines). Vaccines do not however deliver perfect immunity so P[C|V]≠0.

Let P[V|C] be the conditional probability of a person with Covid-19 having been fully vaccinated. Or, if you prefer, the proportion of people with Covid-19 who are fully vaccinated..

Now the first thing to point out is that these conditional probability are emphatically not equal. The probability of a female person being pregnant is not the same as the probability of a pregnant person being female.

We can find the relationship between P[C|V] and P[V|C] using the joint probability P[V,C]=P[V,C] of a person having been fully vaccinated and contracting Covid-19. This can be decomposed in two ways: P[V,C]=P[V]P[C|V]=P[C]P[V|C]=P[V,C], where P[V] is the proportion of people fully vaccinated and P[C] is the proportion of people who have contracted Covid-19. This gives P[V|C]=P[V]P[C|V]/P[C].

This result is nothing more than the famous Bayes Theorem.

Now P[C] is difficult to know exactly because of variable testing rates and other selection effects but is presumably quite small. The total number of positive tests since the pandemic began in the UK is about 5M which is less than 10% of the population. The proportion of the population fully vaccinated on the other hand is known to be about 50% in the UK. We can be pretty sure therefore that P[V]»P[C]. This in turn means that P[V|C]»P[C|V].

In words this means that there is nothing to be surprised about in the fact that the proportion of people being infected with Covid-19 is significantly larger than the probability of a vaccinated person catching Covid-19. It is expected that the majority of people catching Covid-19 in the current phase of the pandemic will have been fully vaccinated.

(As a commenter below points out, in the limit when everyone has been vaccinated 100% of the people who catch Covid-19 will have been vaccinated. The point is that the number of people getting ill and dying will be lower than in an unvaccinated population.)

The proportion of those dying of Covid-19 who have been fully vaccinated will also be high, a point also made here.

It’s difficult to be quantitatively accurate here because there are other factors involved in the risk of becoming ill with Covid-19, chiefly age. The reason this poses a problem is that in my countries vaccinations have been given preferentially to those deemed to be at high risk. Younger people are at relatively low risk of serious illness or death from Covid-19 whether or not they are vaccinated compared to older people, but the latter are also more likely to have been vaccinated. To factor this into the calculation above requires an additional piece of conditioning information. We could express this crudely in terms of a binary condition High Risk (H) or Low Risk (L) and construct P(V|L,H) etc but I don’t have the time or information to do this.

So please don’t be taken in by this fallacy. Vaccines do work. Get your second jab (or your first if you haven’t done it yet). It might save your life.

6 Responses to “A Vaccination Fallacy”

  1. ” The probability of a female person being pregnant is not the same as the probability of a pregnant person being female.”

    As the actress said to the bishop. 🙂

    Tuesday will be two weeks since my second BioNTech vaccination. While I’ll probably do some other travelling in the summer, I might leave out the annual trip to England, because of the delta variant. Travelling itself, with the usual hygiene rules, should be OK, but a) the main reason would be to attend a festival, probably not a good idea even if it is not postponed again, and b) returning from such a region means two weeks of mandatory quarantine in Germany, even vor those fully vaccinated and/or with a negative test. Even with several negative tests at different times it isn’t possible to shorten this. (I’ll have to read up on why that is the case.)

  2. You need to make this explanation *much, much, much* simpler!

    So, try leaving out *all* the formulae, because most people switch off the moment a formula appears (as why wouldn’t they, it’s a form of jargon abbreviation).

    And if “proportions” are the same as “probabilities”, just stick to the *one* term throughout, please don’t muddle us by swapping from one to the other.

    Please have another go. It’s important that people who understand these things practice making them understood to evefryone else.

    • telescoper Says:

      It’s a point I have made over and over again that most people don’t understand probabilities. I’ve even written a book about this. I see that conditional probabilities and Bayes’ Theorem are on the GCSE maths syllabus, so perhaps this will gradually change.

      This is however a personal blog read by at most a few thousand people. Educating “everyone” is not my aim nor is it something of which I am capable. I write about what I find interesting at the time, that’s all. If you don’t like it, you can have your money back.

    • How’s this: if one is vaccinated, then the probability of infection is lower, but not zero. Thus, if almost everyone is vaccinated, then it could be that most of those infected have been vaccinated, despite the fact that, for any one person, the chance of infection is much lower if vaccinated.

      Similar confusion arises in other areas. For example, in contrast to most other forms of cancer, most cases of prostate cancer never need any treatment at all; people can live with it for decades. On the other hand, almost all men get prostate cancer at some point, so (depending somewhat on where in the world one is), it is usually the third-leading cause of cancer death in men (after lung and colon cancer—since those are due largely to smoking and bad nutrition (including drinking alcohol), respectively, if one foregoes those vices, then it is the most likely cause of death by cancer for a man). That is one reason too few men make use of the fact that a simple blood test can detect it with high probability (which would be a boon for other forms of cancer). (For those who think that PSA tests are a bad idea, I can probably rebut every claim.)

  3. … and in the limit where everyone has had their 2 jabs, then 100% yes one hundred per cent of those dying of the virus will have had both vaccine jabs. Woooooo! [rolls eyes and waves hands around].

    [Seriously, I bought a copy of Gigerenzer’s “Reckoning with risk” shortly after it was published about 20 years ago, and still find it helpful as a very friendly way of setting these things out. Basically his argument is that natural frequencies are easier for lay people to understand than probabilities, which can be scary.]

  4. Just recently made a video about exactly the same subject at:https://www.youtube.com/watch?v=6VLFsF_ZE7E. This was before I discovered your blog!

    Unfortunately I believe that there is no real way to stop the spread of these lies/conspiracy theories. But we must keep trying to at least reduce their transmission!

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