Last Monday, schools in England returned, with secondary school students having “lateral flow” tests (LFTs). These device kits operate like pregnancy tests – a liquid sample flows along a pad. Instead of expecting a baby, positive tests mean students and families are expected to isolate.

All tests are imperfect, but the statistics of diagnostic tests are notoriously confusing.

The false-negative rate is the proportion of infected people who get a negative test: the most recent estimate for people without symptoms, published last Wednesday by NHS Test and Trace and Public Health England, is a LFT false-negative rate of around 50% relative to a PCR (polymerase chain reaction) test. The LFT picks up only around half of infected people without symptoms, so even with a negative test you could still be infectious: rapid tests are not green lights.

On the other hand, the false-positive rate is the share of uninfected people who get a positive result, recently estimated in LFTs to be at most 0.1% and possibly as low as 0.03% (3 in 10,000). This sounds excellent. But there are serious statistical issues with diagnostic screening in populations when the disease is rare and it’s best not to trust intuition.

Crucially, this does not mean that if you get a positive test, there is only 0.03% chance that the test is wrong. Far from it. In fact, very far from it.

The latest data from Test and Trace says that, in secondary schools, five in 10,000 LFTs came out positive. But we have seen that we would expect at least three false positives in every 10,000 tests of uninfected people. So out of those five positive tests, at least three will be false positives. Whilst lateral flow tests act as a dragnet, this explains the need for PCR confirmation.

If you pick what looks like a needle from a haystack, then your confidence that it really is a needle depends on how many are in the haystack in the first place. If you know there’s just one in there, then you should be very cautious when making a claim that you’ve found it.

David Spiegelhalter is chair of the Winton Centre for Risk and Evidence Communication at Cambridge. Anthony Masters is statistical ambassador for the Royal Statistical Society



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