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False positives and the never-ending epidemic


This article was amended on October 7, 2020

TESTING is at the heart of the government’s response to Covid-19, but the current test for the virus is not 100 per cent reliable. It sometimes gives positive results for samples from people who haven’t got the illness. These are called false positives and the likelihood of such results happening is called the false positive rate (FPR).

Unfortunately, no one knows for sure what the false positive rate is for the Covid-19 test. However, previous research on a similar test for other viruses suggests that it’s somewhere 0.8 per cent and 4 per cent. In this article, I’m going to assume that it’s 1 per cent because it’s close to the low end of the range and makes the arithmetic easy.

As with all percentages, it’s important to know what it is 1 per cent of. In this case, we are talking about 1 per cent of the tests that should be negative, not 1 per cent of the positive results. So if we test a group of people who haven’t got the virus:

100 tests give 1 false positive

1,000 tests give 10 false positives

10,000 tests give 100 false positives

and so on.

As you can see, the more tests you do on people who are not infected, the more false positives you are likely to get. So the huge increase in testing since March is likely to have caused a huge increase in the number of false positives.

Why are false positives so important now?

Although the number of false positives is independent from the number of real positives, their effect is much greater when the vast majority of people haven’t got the virus.

Let’s imagine that we’ve done 1,000 tests in a virus hotspot where 900 people tested positive who genuinely had the disease.

The number of real positives = 900

The number of real negatives = 100

The number of false positives = 1 per cent of 100 = 1

So the total number of all positive tests is 901, which is so close to 900 that the false positives don’t have much effect on the overall number.

Now let’s imagine that we’ve done another 1,000 tests. This time we’re in an area with a lower incidence of the virus so only ten per cent of the people tested are really infected.

The number of real positives = 10

The number of real negatives = 990

The number of false positives = 1 per cent of 990 = 9.9, which is nine whole people

So the total number of all positive tests is 10 + 9 = 19 

This time the total number of positive tests is almost double the actual number of infections, which is very misleading. 

As you can see, the smaller the number of real positive results, the more the false positives distort the picture and suggest the situation is much worse than it really is. This is what is happening at the moment, when the number of people testing negative is far greater than the number testing positive and there is a real risk that distortion could be leading to bad decisions. 

To see how that might happen, let’s look at a recent real example, the figures for September 15. I haven’t picked that at random – it’s the day Chris Whitty and Patrick Vallance chose as the starting point for the non-prediction that resulted in their notorious graph.

On that day, the results of 207,718 tests were reported.

The total number of all positives (real and false) was 3,105.

So the total number of negative test results was 204,613.

(This excludes the false positives that should have been negative but it is close enough for our calculations.)

1 per cent of 204,613 is 2,046, so we would expect there to be 2,046 false positives.

So the number of real positives was 3,105 minus 2,046 = 1,059, roughly one-third of the 3,105 Whitty and Vallance used in their calculations. Any non-prediction based on that would have looked very different.

The never-ending epidemic

Health Secretary Matt Hancock has promised to provide 10million tests a day early next year. Let’s be optimistic and imagine that by then the number of real cases will have dropped to zero or be at negligible levels. In that case, virtually all the 10million people tested would not be infected.

1 per cent of 10,000,000 is 100,000 so he’s likely to find 100,000 false positives per day.

That’s 100,000 non-infected people on the first day who will have to self-isolate for two weeks.

On the second day, there will be another 100,000 so 200,000 in all.

On the third day, there will be another 100,000 so 300,000 in all.

Continuing like that, we can see that on the tenth day there would be one million people self-isolating who haven’t got the disease.

Of course, their contacts would have to self-isolate too. Let’s assume each person has four close contacts (which is a fairly conservative guess). That gives us 5million people self-isolating due to false positives just ten days after starting to test 10million people a day.

That adverse effect would still be there if the false positive rate is lower than the 1 per cent I have assumed. Even if it was only 0.1 per cent (which is far lower than predicted), there would still be half a million people self-isolating unnecessarily after ten days.

As long as we continue testing, there will always be false positives. That won’t change if we have an effective vaccine, and it won’t change if the virus disappears completely. So we will never reach a stage where the number of cases is zero. Unless the government takes this into account, the current crisis with its associated restrictions on our freedom will go on for ever.  

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Diana Kimpton
Diana Kimpton
Diana Kimpton is a maths graduate and an author. She specialises in writing about horses and numbers, but not usually at the same time.

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