In general, tests are good at saying that someone is positive for COVID, but really bad in saying that someone is negative for COVID.
This goes for both RT-PCR tests and antigen kits. So if you test positive, you’re probably really infected. If you test negative, don’t be complacent. Ideally, continue quarantine. During the surge now, it’s probably a false negative especially if you have symptoms. False positives are extremely rare.
In epidemiologic terms, these tests have a very high Positive Predictive Value (PPV) and a very low Negative Predictive Value (NPV), high Positive Likelihood Ratio (PLR), low Negative Likelihood Ratio (NLR), owing to its low sensitivity (Sn) but high specificity (Sp).
Given a population size N with It and St being the number of infected and healthy patients respectively. Let p be the prevalence of a disease, say COVID-19.
The prevalence is the number of infected per population at a given time. Therefore:
p = It / N
It = p * N
St = (1-p) * N
Let Sn and Sp be the Sensitivity and Specificity of a given test respectively. Sensitivity is the chance of a test actually producing a positive result on a person infected with the disease. Specificity is the chance of a test actually producing a negative result on a healthy person. These two accuracy parameters are inherent to the test.
We then construct a confusion matrix shown in the attached picture.
Consequently, calculating the Positive Predictive Value (PPV):
PPV = Sn * It / [Sn * It + St * (1-Sp)]
Substituting It = p * N and St = (1-p)* N, you have:
PPV = (Sn * p * N) / [Sn * p * N + (1-p) * N * (1-Sp)]
Cancelling out N,
PPV = (p * Sn) / [p * Sn + (1-p) * (1-Sp)]
Then, a graph for a supposedly very ideal gold-standard RT-PCR test with at least 90% sensitivity and 99% specificity and a less ideal Antigen test 80% sensitivity and 97% specificity are made. (These values are estimated based on literature; not necessarily the exact values in our local setting.)
Likewise, calculating the Negative Predictive Value (NPV):
NPV = Sp * St / [It * (1-Sn) + Sp * St]
Substituting It = p * N and St = (1-p) * N, you have:
NPV = [Sp * (1-p) * N] / [p * N * (1-Sn) + Sp * (1-p) * N]
Cancelling out N,
NPV = [(1-p) * Sp] / [p * (1-Sn) + (1-p) * Sp]
On the other hand,
PLR = Sn / (1-Sp)
NLR = (1-Sn) / Sp
Here are ggplot2-generated graphs in R using the formulas above.
As you can see here, the more cases of COVID-19 there are, the less reliable a negative result is.
