# Testing for SARS-CoV-2

Thursday 7th May 2020

Without definitive numbers, it is hard to understand the facts and figures about testing for SARS-CoV-2. Here is my take on the issue.

There are various estimates of the percentage of the population who actually have SARS-CoV-2, but 2% is roughly in the ballpark. So, for every 1000 randomly selected people, about 20 actually have the virus. If I am one of the 1000, then the probability that I have the virus is about 1 in 50.

Any test to determine whether someone has the virus or not needs to have high *sensitivity*, that is, the proportion of people who have the virus and who test positive (there will be some people who have the virus but who test negative). If the test for SARS-CoV-2 is comparable to, say, the mammogram commonly used in the UK, that figure is about 90%. We also need the test to have high *specificity*, that is, the proportion of people who do not have the virus and who produce a negative result (there will be some people who do not have the virus but who do test positive, the so-called, false positives). The figure for the mammogram is about 93%.

So, the big question is: if I take the SARS-CoV-2 test and it comes back positive, what is the probability that I do actually have the virus? Before reading on, write down your estimate: high (over 70%), medium (30%–70%), or low (less than 30%)? No, really, write down high, medium or low.

This is clearly a job for *Conditional Probability Man*! But probability, let alone conditional probability, is a closed book for many people, so let’s try a different approach.

Imagine a random sample of 1000 people are all given the test. We know that about 2% of them have the virus, so that is about 20 people. The test will correctly identify about 18 of these, as it is 90% sensitive. We also know that 980 of the sample do not have the virus and the test will correctly identify 93% of them – that is about 911 people, so 69 people who do not have the virus will test positive.

Now we know that the total number of positive results from the sample is \(18 + 69 = 87\), but only 20 of them are genuine cases. Therefore, when my result comes back positive, I can see the probability that I actually have the virus is now roughly ¼. Higher than before I took the test, but nowhere near as high as most people predict.