Bayes’ theorem for COVID-19 tests

Especially for PhDs in feminist basket weaving and medical doctors 😛

Do you want to know, what your chances are to actually be infected with SARS-COV-2 when your PCR test result is “positive”?

The PCR tests are those, when you feel like getting a brain puncture as some nurse sticks a very long thin swab-stick in your nose, in order to get a sample of your mucus. The actual test is done in a lab with pre-manufactured test kits.

In the first few months of the ‘pandemic’, these PCR tests were rather inaccurate. Some say many of the Chinese tests were already spoiled by the manufacturer. Who knows, but you don’t have to assume bad manufacturing practices if you look at the math.

1. mass testing

Politicians talk a lot about mass testing, thinking this would solve all the problems.’_theorem

At first you have to make an assumption about the prevalence of COVID-19 infections in the population. Let’s say for this example, that 1% of the population is currently infected with the virus.

The early, inaccurate tests had a sensitivity and specificity above 50%, but probably below 90%. Let’s take a sensitivity of 80% and a specificity of 60% for this example. That means, 80% of those who were actually infected, were also tested positive (= false negative rate of 20%). And it means that 60% of those who were not infected correctly tested negative (=false positive rate of 40%). Remember the papaya and goat tests in Tanzania?

For this scenario, the probability of you actually being infected with SARS-COV-2 after testing positive, calculated with Bayes’ theorem, is just 2%. Not a winner!

The tests have improved greatly in quality and accuracy. They claim sensitivity and specificity in the 99% range. So, I use for this example a sensitivity of 98.5% and a specificity of 99.5%. Based on this, your probability of actually being infected after testing positive is ONLY 66.5%. Based on these “highly accurate tests” your government may force you into quarantine, and let you out only after you test negative for 2 (or 3?) times.

2. testing of risk groups only

In the early ‘pandemic’ stages, tests were rare and expensive (they are still expensive, but not rare anymore). So they were used on people who were likely to be more at risk catching the virus, like doctors, nurses, etc. (and of course effing politicians).

Let’s say the prevalence of being infected in these groups was much higher than 1%, let’s make it 10%.

Using the early “papaya test” accuracy, we can calculate that your chance of having Covid if you tested positive was only 18%.

Using the new “super accurate” tests, your chance of actually having Covid after you test positive is 95.6%.

Now, what you hear in the media, from the testing companies, from alleged experts etc. is 99% accuracy or better!!!1eleven!!! You never hear about the roughly 66% probability. Wonder why? Ask your trusted PhD (not me) 😛

PS: I made the calculations on my smartphone calculator. If I messed up, correct me.

PS2: Oh well, I could have used this:

PS3: Oh, and if you don’t even know a single person who had a positive Covid test, like most of us, and therefore assume the prevalence is only 1 in 1,000 – the probability of the positive test being right is only 16.5%.