Bayes’s rule and Covid test accuracy
https://mashable.com/article/is-covid-test-accurate
I recently read a news about the accuracy of COVID-19 test. Regardless of which method is used in the covid test, the accuracy would not be 100%, the false rate is even higher in the at home antigen test, which is faster, cheaper, but not that accurate. The false positive results, which may caused by the low percent of specificity, would be very annoying. According to the data given in this news by a researcher in UCLA health, the specificity rate of antigen test is around 96%, which is lower than that of PCR test, which can be as high as 99 percent. It means, patients who tested with antigen methods is much more likely to get a false positive result. The data for sensitive rate of these two methods are about 60% for antigen test and 98% for PCR.
Relate to what we learned in Network class, we can calculate the probability of really been infected with COVID when the test result is positive respectively to compare these two method. Let’s first assume that 3% of people in the population who are tested have covid-19, to calculate the conditional probability, we will apply the Bayes’s method.
For antigen test,
Pr (C | TP) = Pr (TP | C) Pr (C) / ( Pr (TP | C) Pr (C) + Pr (TP | C-bar).Pr (C-bar) )
= 0.96 * 0.03 / (0.96 * 0.03) + (0.04 * 0.97) = 0.43
For PCR test,
Pr (C | TP) = Pr (TP | C) Pr (C) / ( Pr (TP | C) Pr (C) + Pr (TP | C-bar).Pr (C-bar) )
=0.99 * 0.03 / (0.99 * 0.03) + (0.01 * 0.97) = 0.75
From the calculation, we can define that the average false positive rate of PCR test is much lower than the antigen test, which means PCR test could tell a much more accurate information when the result is positive. Thus, if the patient has enough time to wait and be indifferent about the cost, PCR test is a better choice which could give a greater average accuracy for the result.