https://slate.com/technology/2022/01/rapid-testing-covid-math-false-negatives-sensitivity.html
But they cut my draft down dramatically. I wanted to add a discussion of what is called “positive predictive value” (PPV) i.e. the answer to the question “if you test positive, do you have the disease?”
If a disease is relatively rare (alas not Omicron) even a positive result with a very specific test can be very misleading and unfortunately can confuse your doctor (https://www.nejm.org/doi/pdf/10.1056/NEJM197811022991808).
Here’s an example of what can go wrong. Suppose you have a test that is 99% specific. If you read my Slate article you now know this means it only has 1% false positives. That is a pretty good test in the real world. But suppose the disease is also really rare, say only 1 in 1,000 people have it. Then it turns out that a positive test, even though the test is pretty darn good, isn’t telling you as much as you think! Why? Well, suppose you test 1,000 people. Since we are assuming the disease prevalence is only 1 in a 1000, you had only one person with the disease in your group of a 1,000. Now our test is 99% specific so there are 1% false positives. So among our 1,000 people, you will have about 10 false positives (.01*1000=10 although technically it is .01*9,999 =9.99). The 10 false positives dwarf the 1 true positive and the odds of you not having the disease are 10 to 1
(A clear treatment of PPV in the context of Covid here: https://www.someweekendreading.blog/weekend-editrix-exposed/)
In your example of testing a 1-in-1000 frequency disease (0.1% true positive rate) with a 1-in-100 (1%) FPR is a good one:
(1) True, if you test 1000 people you get 1 true positive and 10 false positives, for 11 total positives. The 10 false positives dwarf the single true positive.
(2) But, if you’re in the pool of people who test positive, you’re now 1/11 = 9.1% chance of having the disease, vs 0.1% in the general population. So the test has enriched the pool of sick people over the general population by about 100-fold, which is really good information in the real world.
This is why when you get a positive test, your doc orders another test of a different kind, to winnow out the false positives. Get a couple positives in a row from different kinds of tests, and your doc can be pretty sure you’ve got the disease.
Our present problem is people look at test like oracles, which Give The Answer immediately.
Even the oracles were confusing (https://en.wikipedia.org/wiki/Ibis_redibis_nunquam_per_bella_peribis).