The data are derived from the studies used to gain approval of the RNA vaccines. Groups were either given the vaccine or a placebo and the number who developed Serious Adverse Events (SAEs) were counted. The data was normalized to 10,000 for ease of comparisons. To explain the Pfizer results:
"“The Pfizer trial exhibited a 36 % higher risk of serious adverse events in the vaccine group; risk difference 18.0 per 10,000 vaccinated (95 % CI 1.2 to 34.9); risk ratio 1.36 (95 % CI 1.02 to 1.83)."
With the Pfizer vaccine more developed SAEs compared to the placebo group. As in any study there is variation in the results, which when quantified showed that the number of additional SAEs could range from 1.2 to 34.9 with 95% statistical confidence, with 18 the most probable. The 36% higher risk put in numbers means that 50 out of 10,000 from the placebo group showed SAEs compared to 68 out 10,000 for the Pfizer vaccine—(68-50)/50 = 0.36. The risk ratio is Pfizer/placebo (68/50) = 1.36. Again, when taking into account observed variability, this ratio ranges from 1.02 to 1.82 at 95% confidence. The same goes for the Moderna data.
Note that the statement: " Combined, there was a 16 % higher risk of serious adverse events in mRNA vaccine recipients" does NOT mean 16% of vaccinated get SAEs. It means there is a 16% higher risk compared to those vaccinated with no RNA. The frequency of SAEs was relatively low, <100 per 10,000.
Also worth noting one conclusion of these authors who made a quick and superficial risk assessment just to provide some context to the data (AESI means adverse events of special interest which simply means bad stuff that from their biology are likely to be a direct result of the specific vaccine):
In the Moderna trial, the excess risk of serious AESIs (15.1 per 10,000 participants) was higher than the risk reduction for COVID-19 hospitalization relative to the placebo group (6.4 per 10,000 participants). In the Pfizer trial, the excess risk of serious AESIs (10.1 per 10,000) was higher than the risk reduction for COVID-19 hospitalization relative to the placebo group (2.3 per 10,000 participants)….[this data is summarized in the Discussion as:] Rational policy formation should consider potential harms alongside potential benefits. To illustrate this need in the present context, we conducted a simple harm-benefit comparison using the trial data comparing excess risk of serious AESI against reductions in COVID-19 hospitalization. We found excess risk of serious AESIs to exceed the reduction in COVID-19 hospitalizations in both Pfizer and Moderna trials.". https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9428332/
So they cautioned that for most healthy individuals using hospitalization as the metric, the data shows a possible greater risk from the vaccines than from Covid at the time of the analysis.
I want to emphasize that this is only one study of what will likely be many that are analyzing the data coming out of the pandemic. No doubt there will be some that disagree. Just keep in mind that the development and deployment of the Covid RNA vaccines were unprecedented in their speed so there is much less data on secondary effects than typical for vaccines. I am not saying the vaccines are dangerous, I’ve had three jabs of Moderna. But it may be prudent to consider the possible risks of future boosters to a Covid disease that appears to be becoming increasingly more mild.