#### Preamble

This note started out as a reminder to myself about the definition of relative
risk and vaccine efficacy, and morphed into a perusal of the FDA briefs on the
Pfizer, Moderna, and J&J vaccines (links to all 3 briefs are at the bottom
of the article).

It's really worth looking at the actual numbers of COVID cases among people in
the studies -- they are surprisingly low. In some cases, they are so low that
they make inference about vaccine efficacy hard.

This is my first close look at the outcome of a clinical study. You have to
make a lot of semi-arbitrary decisions, it seems, in order to design a
clinical study. Even something as simple as a difference of 5 years in your
cutoff for the 'older' age group can have an effect on inference. The 3 teams
made all sorts of different decisions that make it hard to compare their
outcomes head-to-head.

Above all, while writing this note, I wished many times that I could have
gotten my hands on the actual data. I guess the current age of copious open
data has spoiled me.

Disclaimer: I do not have medical training, and nothing written here should be taken as medical advice.

#### Definition of efficacy

Vaccine efficacy is defined as:

$$1-\text{relative risk} = 1-\frac{\text{Prob(outcome|treatment)}}{\text{Prob(outcome|no treatment)}}.$$

If the experiment has roughly equal treatment and control groups (as all the vaccine clinical trials did), then the probabilities can be replaced by counts:

$$1-\text{relative risk} \approx 1-\frac{\text{Count(outcome|treatment)}}{\text{Count(outcome|no treatment)}}.$$

So 95% effectiveness means that

$$\frac{\text{Count(outcome|treatment)}}{\text{Count(outcome|no treatment)}}\approx 1 - 0.95 = \frac{1}{20};$$

that is, for every 1 event in the vaccinated group, there were 20 in the unvaccinated group.

$$1-\text{relative risk} = 1-\frac{\text{Prob(outcome|treatment)}}{\text{Prob(outcome|no treatment)}}.$$

If the experiment has roughly equal treatment and control groups (as all the vaccine clinical trials did), then the probabilities can be replaced by counts:

$$1-\text{relative risk} \approx 1-\frac{\text{Count(outcome|treatment)}}{\text{Count(outcome|no treatment)}}.$$

So 95% effectiveness means that

$$\frac{\text{Count(outcome|treatment)}}{\text{Count(outcome|no treatment)}}\approx 1 - 0.95 = \frac{1}{20};$$

that is, for every 1 event in the vaccinated group, there were 20 in the unvaccinated group.

####
What was the measured event (aka Primary Endpoint) used to measure vaccine
efficacy?

**TL;DR**: Patients needed to have more symptoms in order to satisfy the J&J or Moderna primary endpoints than to satisfy the Pfizer primary endpoint. All confirmed cases in all 3 clinical trials required positive PCR tests.

For Moderna: First Occurrence of confirmed COVID-19 (as defined by an adjudicated committee using a formal protocol) starting 14 Days after the Second Dose. Confirmed COVID-19 is defined on page 13 of the FDA brief, and requires at least 2 moderate COVID symptoms (i.e., fever, sore throat, cough, loss of taste or smell) or at least 1 severe respiratory symptom, as well as a positive PCR test.

Moderna primary endpoint results. |

For Pfizer: Confirmed COVID-19 beginning 7 days after the second dose. Confirmed cases had at least one symptom from the usual list of COVID symptoms, and a positive PCR test for COVID within 4 days of the symptom.

Pfizer primary endpoint results. |

for J&J: 'Molecularly confirmed' (by a PCR test) moderate-to-severe/critical COVID infection, measured at least 14 and at least 28 days post-vaccination. They also studied the rates of severe/critical COVID, which required signs of at least one of severe respiratory illness, organ failure, respiratory failure, shock, ICU admission, or death. Definitions of the COVID illness levels are on page 15 of the FDA brief, and are similar to the Moderna definition of Confirmed COVID-19.

#### Thoughts about the results

Moderna and Pfizer both reported very high efficacies of about 95%. These
were point estimates, i.e., single values summarizing the measured
efficacy.

But the confidence interval (CI) is the thing to look at for each result, not the point estimate. The CI gives you information about not only the point estimate for efficacy, but about the certainty of the efficacy measurement. The CI for efficacy always contains its point estimate, but the wider the CI, the less confidence you can have in the point estimate.

But the confidence interval (CI) is the thing to look at for each result, not the point estimate. The CI gives you information about not only the point estimate for efficacy, but about the certainty of the efficacy measurement. The CI for efficacy always contains its point estimate, but the wider the CI, the less confidence you can have in the point estimate.

__Moderna__
The vaccine was tested with roughly equal control and vaccine arms. There
were about 21,600 participants in each arm.

The 95% CI for people aged 18-65 is (90.6%, 97.9%), which is very high.

The point estimate of efficacy for people aged 65 and up was a bit lower, at 86.4%. The 95% confidence interval was (61.4%, 95.5%). The reason the confidence interval is wider is that only about 7000 people over 65 were enrolled in the clinical trial, and there were only 33 covid cases among that group (as opposed to 163 in the younger group). This caused the CI to be wider, reflecting increased uncertainty as to the true efficacy of the vaccine.

If the cutoff for the older age group were lower, there would have been more cases in that group, and more confidence in the result. It would have been nice to have access to the raw clinical trial data.

The 95% CI for people aged 18-65 is (90.6%, 97.9%), which is very high.

The point estimate of efficacy for people aged 65 and up was a bit lower, at 86.4%. The 95% confidence interval was (61.4%, 95.5%). The reason the confidence interval is wider is that only about 7000 people over 65 were enrolled in the clinical trial, and there were only 33 covid cases among that group (as opposed to 163 in the younger group). This caused the CI to be wider, reflecting increased uncertainty as to the true efficacy of the vaccine.

If the cutoff for the older age group were lower, there would have been more cases in that group, and more confidence in the result. It would have been nice to have access to the raw clinical trial data.

__Pfizer__

The vaccine was tested with roughly equal control and vaccine arms. There
were about 18200 people in each arm.

The division along age lines in this table occurs at age 55 years, rather than 65 years. This made the age groups a bit more balanced and resulted in more cases in the 55+ age group. Thus the 95% CI for the older age group is narrower than Moderna's, at (80.6%, 98.8%). The results for the younger group are even better.

The division along age lines in this table occurs at age 55 years, rather than 65 years. This made the age groups a bit more balanced and resulted in more cases in the 55+ age group. Thus the 95% CI for the older age group is narrower than Moderna's, at (80.6%, 98.8%). The results for the younger group are even better.

**Johnson & Johnson**
J&J had two endpoints, one corresponding to moderate illness, and one
to severe and critical illness. J&J has emphasized the efficacy of
their vaccine against their endpoint of severe or critical COVID-19, so
that's where I focused my attention.

The J&J study had some issues in its design that make it hard to draw conclusions. Because severe COVID is rarer, there were fewer cases of it in the final analysis, which means increased uncertainty for the conclusions. They also ran studies across several countries with wildly different base rates of covid, and with different dominant COVID

-19 strains. This makes me think nervously about aggregation confounding (Simpson's paradox) when all the results are thrown into one bucket. Again, access to the raw data would have been nice.

J&J's point estimate of 85% efficacy in the US against severe covid, which you hear about all the time, is of questionable value, because the 95% CI was (-9,% 99.7%)! That's because there were only 8 severe COVID cases in the US arm of the trial -- 7 in the placebo group and one in the vaccine group. That's not enough to base any conclusions on. The same problem with a low total case count was found in Brazil.

Probably the best estimate of J&J efficacy against severe covid came from the South African arm of the study, where the number of severe cases was largest (26 severe cases in both arms of the study after 28 days post-vaccination -- 22 in the placebo group and 4 in the vaccinated group). The point estimate there was 81.7%, and the 95% CI was (46.2%, 95.4%). Remember that the tough South African COVID variant was spreading during this study, so that's pretty good news as to J&J's efficacy against that variant.

If you throw all the people in those 3 locations into one bucket, you get this table describing the aggregate result for severe covid:

The J&J study had some issues in its design that make it hard to draw conclusions. Because severe COVID is rarer, there were fewer cases of it in the final analysis, which means increased uncertainty for the conclusions. They also ran studies across several countries with wildly different base rates of covid, and with different dominant COVID

-19 strains. This makes me think nervously about aggregation confounding (Simpson's paradox) when all the results are thrown into one bucket. Again, access to the raw data would have been nice.

J&J's point estimate of 85% efficacy in the US against severe covid, which you hear about all the time, is of questionable value, because the 95% CI was (-9,% 99.7%)! That's because there were only 8 severe COVID cases in the US arm of the trial -- 7 in the placebo group and one in the vaccine group. That's not enough to base any conclusions on. The same problem with a low total case count was found in Brazil.

Probably the best estimate of J&J efficacy against severe covid came from the South African arm of the study, where the number of severe cases was largest (26 severe cases in both arms of the study after 28 days post-vaccination -- 22 in the placebo group and 4 in the vaccinated group). The point estimate there was 81.7%, and the 95% CI was (46.2%, 95.4%). Remember that the tough South African COVID variant was spreading during this study, so that's pretty good news as to J&J's efficacy against that variant.

If you throw all the people in those 3 locations into one bucket, you get this table describing the aggregate result for severe covid:

I have two thoughts about this; one is that I'm suspicious of aggregation
effects, due to the fact that the studies in the 3 countries were so
different. The second is that the evidence for the effectiveness of
J&J's vaccine is significantly stronger for onset 28 days
post-vaccination than for 14 days post-vaccination; the jump in efficacy
against severe COVID in the younger age group is more than 10 percentage
points.

So, although I've read that you can consider yourself officially "J&J-immunized" after 14 days post-vaccination -- I intend to wait another 2 weeks after that, till the 28-day mark, before really relaxing the rules.

So, although I've read that you can consider yourself officially "J&J-immunized" after 14 days post-vaccination -- I intend to wait another 2 weeks after that, till the 28-day mark, before really relaxing the rules.

**References**

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