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Anthony Browne MP is a member of the Treasury Select Committee and former CEO of the British Bankers’ Association.

The usual reaction – not just among those affected, but headteachers and senior politicians – is to bang your head against the wall in incredulity, and shout it is “nuts”. The Times reported one parent denouncing it as “illogical and psychologically cruel.”

The BBC reported it was ruining the return to school, and said the government couldn’t explain it. Matt Hancock, the Health Secretary, and Gavin Williamson, his counterpart at Education, have both been robustly challenged on it.

The source of all the anguish is the Government policy that if a child tests positive for Covid on the less-accurate lateral flow device (LFD) test at school, but subsequently tests negative by the more accurate laboratory PCR test, then the more accurate second test does NOT over-ride the less accurate first one: the child and their close contacts at school still need to self-isolate for ten days.

It happened at a school I know this week, where 18 A Level students missed their mocks because one student tested positive on the LFD test on Monday despite subsequently being cleared by the PCR test on Tuesday.

I was bombarded by apoplectic parents, and went into battle. Dredging up my maths degree, I created an algorithm for the problem and last night locked horns with the Department of Health mathematicians, plugging in all the real world data.

The headline is that with the virus at its current prevalence (0.5 per cent of people have it nationwide) then the proportion of people who test positive on the first LFD test and subsequently test negative on the PCR test but are actually infected is astonishingly high: 30 per cent. In other words, nearly one third of pupils with a negative
result from the second PCR test after a positive LFD test are actually infected – and that is a big enough risk to justify them being required to isolate.

However, as the prevalence of the virus falls, then that risk goes down rapidly as well. When the prevalence of the virus is down to 0.1 per cent (i.e. one in a thousand people have it), then the proportion who get a positive LFD result then a negative PCR result who are actually infected will be eight per cent – i.e. more than 90 per cent won’t be. So as the virus becomes rarer, we can rely more on the PCR result, and the Government policy will change.

Now, I can almost hear you thinking “this makes absolutely no sense”. The PCR test is much more accurate that the LFD tests. End of. Rely on it. But this is the world of false negatives (a result saying someone doesn’t have a virus when they do) and false positives (saying someone does have the virus when they don’t), and all I can say is that this is a time when good old common sense is a very false friend.

The mathematics is deeply counterintuitive, to the extent that many people even when they do understand the maths cannot bring themselves to believe it. To take a much simpler example than the two tests, imagine a world where 0.1 per cent of people have a virus, and a test which is 99.9 per cent specific (i.e. produces 0.1 per cent false positives), then what proportion of people who get a positive result are actually infected?

It is just 50 per cent. A test is 99.9 per cent accurate but only half of people tested positive are actually infected? Nuts! But true.

In our real world example, it is true that the PCR test is far more accurate than the LFD test, both in producing far fewer false negatives and false positives (it has higher sensitivity and specificity, in the jargon). But – and this is critical – the PCR test is nearly 200 times more likely to produce a false negative result (i.e. tell an infected person they are not infected) than the LFD test is to produce a false positive.

If the PCR test did not produce any false negatives, or if the LFD didn’t produce any false positives, then in either case we wouldn’t have this problem, but they do, and we have.

So crunching through the numbers (and frankly I am amazed you are still reading this), using current real world data from the NHS and ONS, if you test one million people with the LFD then 2,804 will get a positive test result (and 89 per cent of those will actually be infected). If they all go on to get the laboratory PCR test, then 2,375 will be confirmed as positive (i.e. they definitely have it), but 429 will come back negative.

But because about five per cent of PCR results are false negatives, of those 429 in fact 130 will actually be infected and only 299 be true negatives. That is, 30 per cent of those with the second negative PCR test will actually be infected. With millions of kids tested every week, that means there are many hundreds being confounded – and forced to isolate – by this paradox.

You might point out that if the first LFD test is done at home, then the Government allows the second PCR negative result to override it. But LFDs done at home are far less accurate, and full of so many uncertainties (have the identities been mixed up?) that it can he justified relying on the laboratory test alone.

It is mind-boggling, it is not common sense, and it is definitely infuriating. But this policy of not allowing a second PCR negative test to override a school done LFD positive test is based on sound statistics. We are definitely being led by the science, and I am glad we are. As the virus gets rarer, the statistics will change, and the policy can then revert to being common sense. Amen.

For the mathematicians among you, who want to work it out yourselves, here is the data: For the LFD, the sensitivity is 50.1 per cent (i.e. 49.9 per cent false negatives) and specificity is 99.97 per cent (i.e. 0.03 per cent false positives). For the PCR, the sensitivity is 94.8 per cent (i.e. 5.2 per cent false negatives) and specificity is 100 per cent (i.e. if the PCR test says you have the virus, you do, no ifs or buts).