Doctor Glyndwr wrote:
A very timely and topical reminder of Bayesian mathematics:
Suppose you have a coronavirus antibody test that can tell you if you have had covid-19 or not. It has a 95% accuracy rate; so:
1) if you have had covid-19 there's a 5% chance the test says negative and
2) if you haven't had covid-19 there's a 5% chance the test says positive.
You get tested and it's positive. What is the probability you have had covid-19?
I can answer this now. The short answer is, you can't know, there's not enough data. But the longer answer is: a lot less than it sounds like.
Think of administering this test to a million people chosen at random. Now let's make an assumption: let's assume 10% of the population has genuinely had covid-19. (This is almost certainly a significant over-estimate at the moment.)
So you have 100,000 people who have had it, and a 95% test success rate. That's 95k who get a positive test and have had it. And it's 5k who have had it, but the test comes back negative (a false negative result).
But you also have 900,000 people who haven't had it, and another 95% test success rate. So you get 45k people whose test is positive but they haven't had it (a false positive result.)
So we have a pool of 140k people who get positive results, with 95k who have had it and 45k who haven't. So if your 95% accurate test comes back positive, and 10% of the population have been infected, then
there's only a 68% chance you've actually had covid-19.
Now, this all turns on the 10% infection figure I put in earlier, yes. But even if you make it 30% - which isn't possible to reach for a very long time without hopelessly overwhelming the hospitals - then the point stands: the false positive rate means your test is deceptive a lot more often than it sounds like it is.
Quote:
How dangerous is it - to you and to society - for your doctor to now say "you can resume your normal daily life"?
I unintentionally muddied the waters with this bit. I should have written "even assuming that you get 100% immunity to coronavirus" - ie the danger is just from the risk of false diagnosis. My intention is to frame this in terms of the "I have had it and am now immune" passport idea that's floating around.