Covid tests

[Harryn01]

In the UK the chances of a false positive test are 0.8%. We recently did 2 lots of tests for 86 people. With 172 tests you would expect 1.376 false positives. We had 5 positive tests and some argue, frustratingly that all 5 are false as there was no repeat positives. Assuming the accuracy of government statistics, what are the odds of all 5 being false.

My answer is 1/30517578125 however I'm not sure of 1/125 to the power of 5 is correct

 

[Dr.Peterson]

There are a couple issues here. First, I checked and found a report that says your false positive rate may be between 0.8% and 4.0%. Second, this rate applies to your tests specifically if you assume no one is really positive; a proper estimate of the probability of your results would require estimating the rate of cases.

But the important thing is that your calculation doesn't take into account how many tests you did. You found the probability that ALL of 5 tests of healthy people would be positive, which of course is extremely unlikely. You need the binomial probability that at least 5 out of 172 tests of healthy people would be positive.

 

[Cubist]

You also need to take account of the probability of a false negative too, this may be different to the chance of a false positive. (That is, someone has Covid19 but the test comes back negative.) Also, does the test only detect if they currently have Covid 19, but won't detect someone who has already had it (I assume your two tests were performed at different times). There's a few things to think about here!

 

[Dr.Peterson]

If we assume a false positive rate of 0.8%, and suppose that no one tested actually has the disease, then the probability that at least 5 of 172 tests would be false positives is 1.29%, if I did that right. If the rate is 2%, the probability is 26.2%; if 4%, it's 82.1%.

 

[Cubist]

... if I did that right

FYI: I reproduced your figures.

I like the way you gave a fair representation of the OP's result by considering a simple and plausible scenario, "no one tested has the disease". I would have "over engineered" the calculation!

 

[Dr.Peterson]

We are essentially doing a hypothesis test, where the null hypothesis is that no one has the virus (so that all the positives are false).

It would be a much more interesting to find the actual probability that everyone is healthy, assuming specific rates of cases, false positives, and false negatives. I'll leave that for someone else.