However they spin it the ONS estimate of only 8% adults unvaccinated compromises their conclusions on vaccine safety.

By Prof. Norman Fenton

Spoiler: In May 2022 the ONS estimated just 8% of adults in England were unvaccinated. But this was based on their (biased) subset of the England population. If their estimate is correct and other independent estimates for the whole population are also correct then between 69% and 99.6% of adults missing from their sample are unvaccinated.

The Office for National Statistics (ONS) vaccine mortality surveillance reports (the latest being [1]) are based on a subset of 39 million England residents that excludes all those not registered with a GP and not registered in the 2011 census. Hence, since the population of England is about 56 million it is missing a lot of people who are not at all representative of those in the ONS sample.

For the report [1] we know that as of May 2022, the ONS estimated that in their sample there were:

• 5,279,949 unvaccinated

• 33,511,404 vaccinated

This means that 13.6% were unvaccinated in May 2022. However, we also know that the ONS was estimating only 8% of adult (i.e. those aged 18+) were unvaccinated at this point.

If we remove those aged under 18 and account for the fact that a higher proportion of that age group are unvaccinated, we can see how the ONS arrive at their 8% figure. Specifically, if we assume 50% of the unvaccinated were aged less than 18 and that 10% of the vaccinated were aged less than 18 then:

That gets us to the 8% adult unvaccinated proportion claimed by the ONS for their report.

But we also know that there are approximately 8,000,000 England residents aged 18+ who are missing from the ONS sample (specifically, the total number of England residents aged 10+ missing from the ONS sample is approximately 10,000,000 and approximately 80% of these are aged 18+).

Let:

Then:

and the total number of adults overall is:

The proportion z of unvaccinated in the whole population is

Independent sources indicate that, in May 2022, z was at least 20% (i.e. z is at least 0.2). Specifically, the UKHRA estimated 20% [2] and the ICM survey used in the BBC documentary ‘Unvaccinated’ found 26% [3].

Given such a z we are interested in knowing the proportion of unvaccinated in those missing from the ONS sample. Let α be the proportion of unvaccinated in those missing from the ONS sample. Then we know:

Hence, by equations (1) and (2) we have:

So, if z = 0.2 we get:

Therefore:

Hence, if the ONS estimate of 8% adult unvaccinated in their sample is correct and if there are at least 20% adult unvaccinated in the whole of England, it follows that at least 69% of the adults missing from the ONS sample are unvaccinated. But what if the 26% adult unvaccinated reported in the ICM survey for the BBC documentary was the true proportion for adult unvaccinated? Then substituting z = 0.26 above we get:

In other words, this would mean 99.6% of the adults missing from the ONS sample are unvaccinated. So, if the ONS estimate of 8% adult unvaccinated for their sample is correct then, based on other independent estimates of the adult unvaccinated in the whole population, this would mean that between 69.0% and 99.6% of the 8,000,000 adults missing from the ONS sample were unvaccinated.

Is this feasible?

It would mean the ONS sample is not at all representative of the whole England population. It is much more likely that the ONS estimate is too low. Yet the ONS are now claiming that, as of end of August 2022 the number of unvaccinated has dropped even further.

Their latest report states: “of those aged 12 years and over 93.6% had received a first dose of a COVID-19 vaccine”. Hence their latest estimate is that just 6.4% of those aged 12 and over are unvaccinated and this would mean even less (about 5%) of those aged 20% are unvaccinated.

It is highly unlikely that many of the 8,000,000 aged 20+ missing from the ONS sample (none of whom are registered with a GP) have had the vaccine since May 2022. This means the ONS sample is now even more unrepresentative of the England population.

Conclusion

Either the ONS is underestimating the proportion of unvaccinated in its sample or the sample is so unrepresentative of the whole population that any inferences made using the ONS data are worthless.

Either way, the ONS estimate of proportion unvaccinated must not be used for any comparisons of vaccine efficacy or safety of the whole England population.

Norman Fenton, Professor of Risk Information Management at Queen Mary University of London and a Director of Agena, a company that specialises in risk management for critical systems.

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[1] Office for National Statistics, “Deaths by vaccination status, England - Office for National Statistics”, 2022 [Accessed: 06-Oct-2022].

[2] UK Health Security Agency, “Weekly national Influenza and COVID-19 surveillance report”, 2022.

[3] ICM, “Exploring attitudes towards Covid-19 vaccinations (for STV) - icmunlimited”, 2022. [Accessed: 06-Oct-2022].