False positives and false negatives
29 November 2020
False positive and false negative rates are important to bear in mind in a mass testing scenario.
By way of example: you reside in an area with a 1 in 1,000 background probability of infection with blanket testing. You take a test, get a positive result. You learn the test has a 1% false positives, and 10% false negatives. What is the probability you are infected?
False negative rate (FNR)=0.1
False positive rate (FPR)=0.01
True positive rate (TPR)=1−FNR=0.9
Let’s assume 100 are infected from 100×1000=100000 tests, which accords with the 10001 probability.
100000−100=99900 are not infected.
True positives (TPs)=100×TPR=90
False positives (FPs)=99900×FPR=999
Probability is then true positives over all positive test results
TPs+FPsTPs=90+99990≈0.083
If we then re-test all the positives we can take c. 0.083 as our background probability, which means that a second positive test result will yield a much higher probability of infection.