Government policy on testing is worryingly misleading
There’s been a bit of a Covid outbreak in our neighbourhood, including our eldest child (he’s fine). All the cases were first detected with rapid, at-home “lateral flow” tests, or LFTs.
Here’s my reasoning. This is just Bayes’ theorem: remember that getting a positive result on a 99% accurate test does not mean that there’s a 99% chance you have the disease.
Let’s say about 1% of people who do an LFT have Covid. It’ll actually be more than that — about one person in 85 in England has the disease, more in the other three nations, and if you’re doing an LFT you may have reason to think you have it — but let’s go with it.
Imagine you test one million people. Of those one million people, 10,000 will have the disease, and 990,000 will not.
According to the Government’s own figures, LFTs only give a false positive result at most one time in every 1,000. There’s some argument about how often they give false negatives, but this Cochrane review suggests (with caveats) about one time in three.
So you test your million people. Of the 10,000 who have the disease, LFTs correctly tell you that 6,700 do, but wrongly tell you that 3,300 don’t. Of the 990,000 who don’t have the disease, they correctly tell you that 989,010 don’t. But they wrongly tell you that 990 do. That means that you get a total of 7,690 positive results. But now you go and “confirm” them with PCR tests.
What we need to know now is how often PCRs give false negatives. According to this paper, it’s about one time in three; according to this one, it’s one time in 10. The Office for National Statistics’ infection survey relies on this paper, which says one time in 20. (False positives are so rare that we can ignore them.)
If we take the ONS figure of 5%, then of our 6,700 people who got a true LFT positive, your PCR will wrongly tell 335 of them that they’re disease-free. If we take the largest figure of 30%, it’ll wrongly reassure 2,010 of them.
Given that it will also correctly identify the 990 false positives, that means that — depending on how accurate we think PCR testing is — between 25% and 67% of everyone given the all-clear by a PCR after a positive LFT actually have the disease.
The problem may be even worse than that. I was using conservative estimates of Covid prevalence among the people tested — and among schoolchildren, the prevalence is higher still, around 4% of kids in secondary and 2.6% of primary. (Possibly because we keep telling people to send their kids back to school despite a positive LFT.) For children, the risk of a false negative on PCR after a true positive will be even higher.
There is, I think, a tendency to assume that PCRs outrank LFTs. But that’s not how it works: both have some chance of being right and some chance of being wrong, in either direction. A negative PCR definitely does not mean you are safe to go back into the office or school. It may be government policy, but you should be extremely careful about ignoring a positive LFT, no matter what the “confirmation” test says. Perhaps the risk is worth it — in our mainly vaccinated population, Covid is less deadly than it used to be — but let’s not pretend that it isn’t there.