There has been much confusing news coverage about antibody tests for SARS-CoV-2. This post will discuss antibody tests in language that hopefully will clarify the issues surrounding antibody testing.

False Positive Tests

The news media have recently reported that antibody tests have a high false positive rate. The implication is that the tests are unreliable. Although many of the initial antibody tests that were sold were in fact unreliable, the FDA has withdrawn approval for almost all of those. The tests that are left perform very well, yet the false positive rate is still high. What is going on?

It turns out that no matter how good a test is, there will still be a high false positive rate in populations that have a low proportion of people who have had COVID-19.

Every medical test of any kind has two characteristics, called sensitivity and specificity. A test is very sensitive if it has a very low chance of missing the condition being tested for (low false negative rate) A test is very specific if it has a very low chance of being falsely positive (low false positive rate). There is no test that has 100% sensitivity and 100% specificity. Usually the higher the sensitivity, the lower the specificity and vice versa.

The sensitivity of the best SARS-COV-2 antibody test is 90% (Out of 100 people who have had COVID-19, 10 will have a false negative test). The specificity of the best test is 99% (Out of 100 positive tests, 1 will be a false positive)

From these numbers, it sounds like the false positive rate would be just 1%, but the chance of a positive test being false is also affected by the proportion of people likely to have the disease in the population being tested. This is called the pre-test probability. If the pre-test probability is low, that is if the condition you are testing for is not very common in the population of people being tested, then the chance of any positive test being false positive is high.

A reasonable assumption in most populations is that the number of people that have had COVID-19 is about 1 person out of 100. This number will be higher in places like New York City, and lower in places like Montana, but this is a reasonable number to use for the pre-test probability for most of the country.

The chance that a positive test is a false positive can actually be calculated using a formula called Baye’s Theorem.

Here is a description of the theorem in words: Probability of positive test being false positive equals the Sensitivity of the test (.90) times the pre-test probability (.01) divided by the total probability of a positive test (.0198). (The total probability of a positive test is equal to the probability of a false positive plus the probability of a true positive.)

If we plug these numbers into the equation we get:

(.9 x .01)/.0198 = .45

This means that even with a very good test there is more than a 50/50 chance that a positive test is a false positive.

The higher the proportion of a population that has had COVID-19 (whether they know it or not) the lower the chance that a positive test is a false positive.

For example, using the same test in a population where 40% of people have had COVID-19, the Bayes theorem gives a much different answer:

(.9 x .4)/.370 = .97

This time, 97% of positive tests are true positives and only 3% false positives

Thus, even with a very good test we must be very careful about interpreting positive results because of the dependence on the pre-test probability. You can see why, even with a very good test, it would not be a good idea to issue back-to-work permits based on positive antibody tests.

False Negatives

The false negative rate also depends upon the pre-test probability (the proportion of people in the population who have had COVID-19). Unlike the false positive rate, the lower the pre-test probability, the lower the false negative rate.

Here is the formula: False negative rate equals 1-Sensitivity times the pre-test probability. Plugging in the numbers we get:

False negative rate = (1-.9) x .01 = .1 x .01 = .001

There is only one in one thousand chance of getting a false negative antibody test if 1% of the population has had COVID-19

Lets see what happens if 40 % of people in a population have had COVID-19.

False negative rate = (1-.9) x .4 = .1 x .4 = .04

In this case the chance of false negative test is four in one hundred. This is forty times higher than if the pre-test probability is 1 in 100.

Bottom Line

The chance of false positive SARS-COV-2 antibody tests is too high for them to be used by indviduals or by employers to figure out who is immune. The only use for them at present is for health departments to figure out the approximate number of people in a population who have had COVID=19.