Some people think of Rapid Antigen Tests (RATs) as a coin flip and question their value. This is based on a misunderstanding of Bayes’ rule. If we think of the PPA and NPA as the sensitivity and specificity of the tests, we can arrive at the posterior given any prior as a function of the false-negative rate. With a coin toss, the probability of either result is independent of having COVID, but a RAT’s results are not, even if the sensitivity is 0.50. In other words, if we hold sensitivity at 0.50, the posterior Pr(Have COVID | Positive RAT) increases as a function of the specificity. Flipping a coin should not update your beliefs about having COVID, but getting a positive RAT should make you more confident you have COVID, as long as the specificity is greater than 0.50, even if the sensitivity is 0.50, as a natural consequence of Bayes’ rule. If you know others around you have positive diagnoses or you have a positive PCR or you have symptoms, your prior should go up. If we have a low prior of 10% and we use the PPA and NPA from Allan-Blitz & Klausner (2021), we can arrive at the posterior probability of having COVID from an Abbott BinaxNOW COVID-19 antigen card.

PosteriorProbability = function(Prior, PPA, NPA){
  PPA * Prior / (PPA * Prior + NPA * (1 - Prior))}

Prior = 0.10

# Asymptomatic

PosteriorProbability(Prior, 0.438, 1 - 0.988)
## [1] 0.8021978
# Symptomatic

PosteriorProbability(Prior, 0.519, 1 - 0.985)
## [1] 0.793578

If you are symptomatic or even asymptomatic and you have a slight inclination that you might have COVID - say, 10% just because we’re in a pandemic and anyone could have it, a 10% conditional chance - then a positive COVID-19 antigen test result gives you a considerable probability of having COVID. This does not account for uncertainty or unreliability in the PCR results, which would downplay the utility of RATs. Note that this can also be used to explain why tests for rare diseases (i.e., diseases with low base rates) are more likely to deliver false rather than true positives: just lower the prior!

People who see published PPAs of 50% or less should think with Bayes’ rule in mind. Once they do, 50% is a very different type of coin toss.

References

Allan-Blitz, L.-T., & Klausner, J. D. (2021). A Real-World Comparison of SARS-CoV-2 Rapid Antigen Testing versus PCR Testing in Florida. Journal of Clinical Microbiology, 59(10), e0110721. https://doi.org/10.1128/JCM.01107-21