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"By the spring of 2021, many hospitals were reporting that their COVID patients were largely unvaccinated."

For the US, at least, I'm not sure that data was available to draw that conclusion. The system for registration of COVID and other vaccination in the US is Byzantine and classification of the COVID vacciantion status throughout much, if not all of the pandemic was unreliable. https://kffhealthnews.org/news/article/huge-gaps-in-vaccine-data-make-it-next-to-impossible-to-know-who-got-the-shots/

On the other hand, federal registration of population rates of vaccination was probably near complete or even overestimated leading to relatively accurate or even overestimation of the rates of COVID vaccination.

This gives rise to what I would refer to as the two-sets-of-books bias. It is likely that, while the CDC surveyed the state registries for the relative rates of hosptalization/deaths in the unvaccinated vs vaccinated, the "denominator data" of the local population rates of vaccination were derived from the more complete federal COVID vaccination database. This is stated explicitly in a CDC COVID-NET publication: https://www.medrxiv.org/content/10.1101/2021.08.27.21262356v1.full.pdf (Supplementary figure 3).

This bias may explain, in part, the claims of the CDC of 66% efficacy of COVID vaccination against all-cause mortality. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8553028/pdf/mm7043e2.pdf

Summarizing: the claim of efficacy of COVID vaccination severe disease and death is a central tenet of public health authorities, but it is less certain than commonly believed.

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Excellent thank you for this digestible summary.

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