It seems that in this #Covid-19 crisis we are sometimes forgetting some basic aspects about interpreting test results, in particular with regard to issuing 'immunity passports'. Immunity passports are a tempting wayto loosen lockdowns and move towards a sense of normality.

The idea is that people with antibodies are protected against infection, so could be allowed to leave the lockdown earlier than others. But how reliable would an immunity passport be? It depends, of course, on how likely it is that a test gives the correct answer.

The sensitivity of a test is the proportion of positive individuals that are correctly identified as positive; its specificity is the proportion of negative individuals correctly identified as negative. The test by Cellex, for example, has 95% sensitivity and specificity.

It correctly identifies 95 out of 100 positive samples and 95 out of 100 negative ones. That seems rather good, right? Well, there's more to it. Let's take as an example a recent study from Germany, in which 14% of the tested people tested positive for antibodies.

What does that tell us about the true prevalence (which I'll call P)? Well, the measured percentage of positives (so 14%) is the sum of the correct positive tests (so 95% of the true positives, 0.95 P) and the falsely positive tests (5% of the true negatives, 0.05(100-P)).

Thus, 14 = 0.95P + 0.05(100-P). This tells us that the expected frequency of true positives (P) is 10%. Although the test gives false results only for 5% of the antibody-carriers and 5% of the people without antibodies, the number of carriers is overestimated by a factor 1.4.

Nevertheless, with the corrections I just showred we can estimate the true frequency of cases in a population. So we can use the test to tell us how rapidly the disease is spreading and whether our efforts at control are having an impact.

But how reliable is the test for an individual? Well, out of the 14% of the tests that were positive, 0.95*10% (so 9.5%) have antibodies, so only 68% of the people who tested positive are indeed positive.

In other words, even if you test positive, you are half as likely to never have been infected as to have been infected! The test therefore gives a false and risky sense of safety. A third of the people who rtes positive are, in fact, not protected and therefore risk infection;

a third of the people with positive tests could become infected and further spread the virus. We cannot trust the tests to give us reliable information about who is protected from infection (unless the prevalence is quite a bit higher than 10%).

Mass-testing is necessary to understand how the virus is spreading. But we shouldn't misuse the tests by assuming that we can identify immune individuals and issue immune passports.