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\text{False negative rate (FNR)} = 0.1
False positive rate (FPR)=0.01\text{False positive rate (FPR)} = 0.01
True positive rate (TPR)=1FNR=0.9\text{True positive rate (TPR)} = 1-\text{FNR} = 0.9

Let’s assume 100100 are infected from 100×1000=100000100 \times 1000 = 100000 tests, which accords with the 11000\frac{1}{1000} probability.

100000100=99900100000-100=99900 are not infected.

True positives (TPs)=100×TPR=90\text{True positives (TPs)} = 100 \times \text{TPR} = 90
False positives (FPs)=99900×FPR=999\text{False positives (FPs)} = 99900 \times \text{FPR} = 999

Probability is then true positives over all positive test results
TPsTPs+FPs=9090+9990.083\frac{\text{TPs}}{\text{TPs}+\text{FPs}} = \frac{90}{90+999} \approx 0.083

If we then re-test all the positives we can take c. 0.083\text{c. } 0.083 as our background probability, which means that a second positive test result will yield a much higher probability of infection.