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date 11.Apr.2021

■ Understanding Bayesian conditional probabilities for Covid-19 testing

How many of you have heard of a friend or relative testing positive for Covid-19 only to be found clean upon retesting? When you test healthy populations with unreliable rapid tests, that's what happens. In statistics this is called a false positive test, and for diseases affecting a small percentage of the population, it is the norm rather than the exception.

When you test positive for Covid-19 and the test accuracy is, say 90%, then you are 90% certain to have the bug, right? Wrong answer! According to Bayes theorem the probability you are ill is only 15%. The majority of people get this figure completely wrong, but it is undeniable scientifically. As rapid tests are liable to flag "positive" people that are not ill (even by a small percent), you don't know whether a positive result is real or one of the test's error cases, especially if only 1% of the population have the bug on average. The math is very simple:

bayes formula

According to Bayes formula, the test accuracy (sensitivity) is diluted by the prior probability of the illness, so the chances you have covid after a single positive test are still low. To be certain you must have a second test, preferably independent of the first one. If the second one comes also positive then you are 76% likely to end up in intensive care <g>

Mathematically the probability increase is justified because your prior probability P(illness) after the first positive test is 15%, not 1% as before

So if the rapid test is not very reliable, and the illness is not widespread, you need at least two tests to get a statistically significant indicator of the test result.

Armed with our theoretical background, let's apply it to the covid rapid test reliability data. We need numbers for P(illness) and P(accuracy) for the Bayes formula above; immediately we stumble into several obstacles:

Given the lack of reliable statistical data, all I can do is investigate a range of test accuracies and infection levels, and see what's the worst case scenario. The table below shows the real (Bayesian) probability of covid infection for a person with one positive test, for a range of accuracies from 80-99%, and for infection levels of 1:1000 till 1:10.

For simplicity, I equate false positive rate to (100-accuracy), which is not necessarily true but this is just a paper exercise.

Infected  Rapid test sensitivity (accuracy)
1:10 30.77% 50.0% 91.67%
1:100 3.88% 8.33% 50.0%
1:1000 0.40% 0.89% 9.02%
Table 1. Probability of covid infection after one positive rapid test for various cases

Taking a middle estimation (1 in 100 people infected and 90% accuracy), a single test translates to only 8% chance of real covid infection. So counting single test results as "daily covid infection statistics" is grossly exaggerating the real extent of the disease. Likewise quarantining people on the evidence of a single test is unwarranted. Health authorities are aware of the problem but book-sellers-turned-politicians won't take heed.

The case against mass covid-19 testing

Another indicator of the desperation of our "decision makers" is the consideration of mass covid testing for every man, wife and their dogs. The idea of testing 7 billion of mostly healthy people on a weekly basis would be absurd, even if the available tests were reliable; counting in their sorry state exposed above, mass testing is totally ridiculous. It reminds me of people that compensate their ignorance making knots by tying 10 messy knots one on top of the other.

Perhaps they have mistaken real time control with statistical sampling. Real time control of the spread of a disease is infeasible, it would require people being tested continuously 24 hours a day.

If you read basic quality control, or elementary statistical sampling, you understand that you don't need to test everyone to assess a situation. You don't ask everybody's intention to vote in a poll, you choose a small sample. You don't need to taste all pasta in a boiling pot to see if it is cooked or not, just the one spaghetti piece will do.

Lately we hear that they are thinking of introducing mandatory weekly tests to all elementary school kids. A classroom is the definition of a biased sample for disease, as kids spend 6-7 hours each day in close contact. If one catches something, the rest will also show it quickly. You don't need to test everyone, and definitely not with an inaccurate, painful, nose test kit. Teachers are being tested weekly, and that's probably enough for a statistical indicator of the entire class.

Unless somebody's making money selling test kits, then it would all make sense!
Let's hope that with mass vaccinations under way (good luck with beta testing :) our Experts will calm down and propose reasonable, feasible measures.

You can fool all the people some of the time, and some of the people all the time, but you cannot fool all the people all the time.
Abraham Lincoln

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