In September, 2021 the US Center for Disease Control and Prevention (CDC) estimated that fully vaccinated people were 11.3 times less likely to die a COVID-19-associated death than people who were not fully vaccinated. The CDC analysis makes no attempt to account for the likely correlation of people who practice safe behaviors in general with those who get vaccinated (one of many safe behaviors). Nor does the CDC account for the likelihood that those who practice safe behaviors in general would be far less likely to die a COVID-19-associated death than people who do not practice safe behaviors in general.
In this analysis, we take three examples. In Example 1, we make what the author believes to be reasonable assumptions regarding the correlation of people who practice safe behaviors in general and those who are fully vaccinated; and the (negative) correlation of people who practice safe behaviors in general with those who die a COVID-19-associated death. These assumptions are tailored to the United States where COVID-19 vaccines are readily available; but a significant part of the population suffers from vaccinophobia and/or eschews safe behaviors in general. In this case we find that the increased practice of safe behaviors in general reduces the likelihood of dying a COVID-19-associated death by a factor of 2.9 while attributing only a reduction by a factor of 3.9 to the efficacy of vaccine. In Example 2 we significantly increase the likelihood that fully vaccinated people practice safe behaviors in general. While less likely than the assumptions in Example 1, the author does not believe the assumptions in this example to be unreasonable. In this case we find that the increased practice of safe behaviors in general reduces the likelihood of dying a COVID-19-associated death by a factor of 6.6 while attributing only a reduction by a factor of 1.7 to the efficacy of the vaccine. In Example 3, we also decrease the likelihood that those who practice safe behaviors in general will die a COVID-19-associated death by a factor of ten. This assumption appears highly unlikely to the author and is included for pedagogical purposes only. In this case we find that increased practice of safe behaviors in general reduces the likelihood of dying a COVID-19-associated death by a factor of 16; after which, we must come to the ridiculous (in the author's opinion) conclusion that the vaccine has a negative effect increasing the chance of mortality by a factor of 1.4. Note 1: No attempt is made here to disaggregate data by age, general health, occupation or any other factor other than the practicing of safe behaviors.Analysis Define the following sets: LetS_{v} be the set of fully vaccinated people, S_{n} = ~S_{v} be the set of not fully vaccinated people (n for natural), S_{d} be the set of people who die a COVID-19-associated death, S_{s} be the set of people who practice safe behaviors in general, and S_{r} = ~S_{s} be the set of people who do not practice safe behaviors in general (r for risky). Further, LetP_{x,y}= |Sx ∩ Sy|/|Sx| — the proportion of people in Sx who are also in Sy, P_{v,d} = P_{v,s}*P_{s,d} + P_{v,r}*P_{r,d} — the proportion of fully vaccinated people who die a COVID-19-associated death, P_{n,d} = P_{n,s}*P_{s,d} + P_{n,r}*P_{r,d} — the proportion of not fully vaccinated people who die a COVID-19-associated death, b = P_{n,d}/P_{v,d} — the portion of the reduction in mortality attributable to safe behaviors in general practiced by the vaccinated (b for behaviors), e = 11.3/b — the portion of the reduction in mortality attributable to the efficacy of vaccines (e for efficacy). Example 1: Let's assume thatP_{v,s} = 0.75, P_{n,s} = 0.05, P_{s,d} = 0.005, and P_{r,d} = 0.05. Therefore:P_{v,d} = 0.75 x 0.005 + 0.25 x 0.05 = 0.01625 (1,625 of 100,000 fully vaccinated people are expected to die a death associated with COVID-19), P_{n,d} =.05 x 0.005 + 0.95 x 0.05 = 0.04775 (4,775 of 100,000 not fully vaccinated people are expected to die a death associated with COVID-19), b=2.9 (reduction in mortality attributable to safe behaviors in general practiced by the vaccinated), and e=3.9 (reduction in mortality attributable to the efficacy of vaccines). Example 2: Let's assume thatP_{v,s} = 0.95 (increased from 0.75 in Example 1), P_{n,s} = 0.05 (same as Example 1), P_{s,d} = 0.005 (same as Example 1), and P_{r,d} = 0.05 (same as Example 1). Therefore:P_{v,d} = 0.95 x 0.005 + 0.05 x 0.05 = 0.00725 (725 of 100,000 fully vaccinated people are expected to die a death associated with COVID-19), P_{n,d} =.05 x 0.005 + 0.95 x 0.05 = 0.04775 (4,775 of 100,000 not fully vaccinated people are expected to die a death associated with COVID-19), b=6.6 (reduction in mortality attributable to safe behaviors in general practiced by the vaccinated), and e=1.7 (reduction in mortality attributable to the efficacy of vaccines). Example 3: Let's assume thatP_{v,s} = 0.95 (same as Example 2), P_{n,s} = 0.05 (same as Example 2), P_{s,d} = 0.0005 (decreased by a factor of 10 from 0.005 in Example 2), and P_{r,d} = 0.05 (same as Example 2). Therefore:P_{v,d} = 0.95 x 0.0005 + 0.05 x 0.05 = 0.002975 (297.5 of 100,000 fully vaccinated people are expected to die a death associated with COVID-19), P_{n,d} =.05 x 0.0005 + 0.95 x 0.05 = 0.047525 (47,752.5 of 100,000 not fully vaccinated people are expected to die a death associated with COVID-19), b=16.0 (reduction in mortality attributable to safe behaviors in general practiced by the vaccinated), and e=0.7 (1/e=1.4 is the increase in mortality attributable to the negative effect of vaccines). Conclusion So, we see that the efficacy of vaccines is strongly dependent on correlations with the practice of safe behaviors in general which could vary widely. It is hoped that future analyses of the efficacy of vaccines will consider this important aspect of vaccine efficacy. The author would conjecture that the true factor for the reduction of deaths due to vaccine efficacy at this time in the United States would be somewhere in the neighborhood of 3.9, which was found in Example 1 above. |