Conditional probabilities and imperfect vaccines

Why is it plausible most future Covid-19 hospitalisations are among those vaccinated?

In recent reporting, BBC journalist Marianna Spring showed a sticker left by protesters:

Our main concern is the numerical claim.

A sticker reads:

“60–70% of hospital admissions and deaths are from people who have had 2 doses of the vaccines.”

That sticker provides a short link to “NHS SOURCE”. The document is not, in fact, from the National Health Service.

It is from the modelling sub-group advising the UK government. Despite the exact quote-marks, those words do not appear in the document. The sticker intends to paraphrase this part:

32. The resurgence in both hospitalisations and deaths is dominated by those that have received two doses of the vaccine, comprising around 60% and 70% of the wave respectively. This can be attributed to the high levels of uptake in the most at-risk age groups, such that immunisation failures account for more serious illness than unvaccinated individuals.

Given the nature of the protest, the implication is vaccines are thus ineffective. The figures do not refer to current admissions and deaths. Those numbers are about modelling the future.

That does not mean it is safer to go without a vaccine. (Image: SAGE SPI-M-O)

Why is it plausible most hospitalisations are among vaccinated people?

The answer lies in conditional probabilities and Bayes’ Theorem. Conditional probabilities are the chance an event happens, given another event already did.

We should avoid confusing the inverse. These two probabilities are different:

  • The probability of hospital admission, given the person has two vaccine doses.
  • The probability a person in hospital with the disease is vaccinated with two doses.

Suppose there were 1,000 people who — if infected — would get a serious illness. Assume a full vaccine course reduces the likelihood of hospital admission by 90%. Here, 950 people (95%) receive that vaccine. The remaining 50 go to hospital without it.

The vaccine is imperfect: among the vaccinated, 95 also go to hospital (as the assumed reduction is 90%). In this calculation, about two-thirds of those hospitalised after infection had vaccinations. That is different to the probability of hospitalisation after vaccination. That chance is 10% of its non-vaccinated probability.

We can write that paragraph in mathematical notation. Here, ℙ(V | H) means the probability they had a full vaccine (V), given they are in hospital (H):

In notation, for those who like that sort of thing.

This calculation depends on both vaccine coverage and effectiveness.

With wide vaccine coverage, a low proportion of failures can outnumber non-vaccinated people. As the modelling document states:

This is not the result of vaccines being ineffective, merely uptake being so high.

Probabilities can be confusing and counter-intuitive.

This blog looks at the use of statistics in Britain and beyond. It is written by RSS Statistical Ambassador and Chartered Statistician @anthonybmasters.

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