The Kaiser Family Foundation put out data recently showing data on breakthrough covid cases by state. The KFF simply gave the percent of vaccinated individuals in each state that have experienced a breakthrough case since Jan 1. That is not particularly helpful from a scientific point of view, because the risk for vaccinated people is not very meaningful without knowing the risk for unvaccinated people. Vaccine efficacy is a measure of the reduction of risk, and if you don't know what you are comparing your risk to, you don't know how much reduction you have.
But it gave me an idea. If these states are putting out numbers of breakthrough cases relative to numbers of vaccinated people over time, I should be about to do a raw calculation of efficacy of the vaccines against the delta variant, by looking up data about the numbers of cases reported during recent weeks when most cases are from delta.
I chose California, because they have a large population, are reporting breakthrough cases weekly, and saw a spike in cases from the delta variant in July.
Their website only gives the running total of breakthrough cases, but it can be run through the wayback machine to give older numbers. That gives me 32580 breakthrough cases (14 days since completion) in CA from 6/31 to 8/1.
To find non-breakthrough cases (117369), I simply looked up the total number of cases and subtracted out the breakthrough cases.
The website giving the breakthrough cases gives the number of fully vaccinated people in CA for the beginning and end of this time period. I took an average: 20.7 million.
To find the number of people not fully vaccinated (18.8 million) I subtracted the number of fully vaccinated people from the population of the state.
So to do the final calculation, I need the risk for each group. 0.158% of vaccinated and 0.624% of unvaccinated people were infected during this time period.
The efficacy of a vaccine is: (unvaxed risk - vaxxed risk)/(unvaxed risk). It comes out to 75% efficacy. Now, it's important to realize that this number will be off if one group or the other is less likely to get tested. People who are unvaccinated might be hesitant to seek medical attention in general. Or people who are vaccinated might think that they just have allergies because their symptoms are milder. There are also some people in the "non-breakthrough" group who have a single dose of an RNA vaccine, or immunity from prior infection. That will tend to bring the efficacy number down. But it seems like a reasonable number, given that most Californians who are vaccinated were vaccinated months ago, and will have some decline in antibody levels.
My estimate is that 75% efficacy against the delta variant correlates with an antibody level of about 550 on the Roche scale, which is plausible for the general population vaccinated 3-6 months ago.