Conditional probabilities and imperfect vaccines III

Exploring a paradox of high vaccine coverage.

In the Washington Post, columnist Kate Cohen argues for two regular statistics:

I propose a running tally in bold type: covid deaths among unvaccinated vs vaccinated citizens. Two numbers, side by side.

This proposal may sound simple. Many public health analyses demonstrate effectiveness of approved Covid-19 vaccines against infection and hospitalisation. In a population without high coverage, most deaths from a disease will be among those not vaccinated.

Survival analysis suggests high effectiveness against hospital admission. (Image: Public Health England)

When coverage widens, immunisation failures can outnumber unvaccinated outcomes. There are numerous explanations of this phenomenon, including in my co-written column in The Observer.

Going forwards in the calculation

One way is to take the reader through each step of the calculation. Suppose we had vaccine coverage of 95%. The vaccine is 94% effective against hospitalisation with a disease.

Assume 1,000 people would go to the hospital without the vaccine. Vaccine coverage is 95%, so 950 of our cohort got their doses. In the whole group, 50 are not vaccinated, and so go to hospital. The effectiveness of an unused vaccine is zero.

The vaccine is 94% effective, and so reduces the likelihood of hospital admission by that proportion. Now, 94% of 950 is 57 — the number of people going to hospital despite vaccination.

Vaccines are effective, decreasing the total number of hospitalisations from 1,000 to 107. However, the vaccinated share of those hospital admissions is 53% (57 of 107).

Going backwards

Another way is to start from the vaccinated share. Using a coverage estimate, we can then recover an effectiveness figure. That figure may not exactly match estimates produced by public health statisticians, which should be our primary source for vaccine effectiveness analyses. It is a useful exercise from statistics in the public domain.

Say 40% of hospital admissions were among those vaccinated. Coverage is 86% of adults. If the vaccines did nothing, we would expect about 6.1 times as many vaccinated…