There are two mathematically equivalent ways to describe the protection that a vaccine offers against such specific outcomes as severe disease:

• relative risk

• effectiveness

Specialists understand the quirks of these two measures, but the rest of us may not until someone calls attention to them. One of these quirks is that effectiveness activates a cognitive bias that misleads people about the size of any change in the protection offered by a vaccine that is in use.

In the current context, this means that statements about effectiveness will tend to minimize the significance of the fall in protection that triggered the decision to offer a third dose of the vaccines from Pfizer and Moderna.

Although there are debates about the true magnitude of the change, I will use simple, specific values to illustrate the bias about the size of any change that is implicit in comparisons of effectiveness. Suppose initially that $$r = 0.04$$ is the relative risk of severe disease for people who have been double vaccinated. This means that the probability that they get severe disease is ||0.04|| times the probability that someone who is not vaccinated gets severe disease; or equivalently, that people who are not vaccinated are ||25|| times more likely to get severe disease.

The effectiveness of the vaccine is defined as $$e = 1 -r$$

The convention is to state ||e|| as a percentage, so ||r = 0.04|| means that the vaccine is ||96\%|| effective against severe disease.

Compared to relative risk, effectiveness has the advantage that a higher number corresponds to a better outcome. Unfortunately, it has the disadvantage that it that it converts a large proportional change in relative risk into a small proportional change in effectiveness.

To illustrate this effect, suppose that some change – perhaps the emergence of a new variant of the virus – means that people who are double vaccinated face a relative probability of severe disease that is twice as big as before. Their relative risk doubles to:

$$ r^\prime = 0.08$$

With this change, the effectiveness of the vaccine falls from

$$e = 96\% \rightarrow e^\prime = 92\%.$$

After the change, someone could correctly say that “the vaccine continues to be highly effective against the endpoint of severe disease” or that “the vaccine continues to be more than ||90\%|| effective against severe disease.” These statements, while true, could easily be misinterpreted to mean that the change in the protection is insignificant.

Expressed in terms of relative risk, as a change from $$r = 0.04 \rightarrow r^\prime = 0.08,$$ a reader or listener is more likely to understand that a doubled risk is a nontrivial change; that the protection offered by the vaccine has diminished substantially. Everything else equal, twice as many people who are double vaccinated will suffer a breakthrough infection that ends in severe disease.