What’s the value of a confirmatory PCR test?

<span>Photograph: Stéphane Mahé/Reuters</span>
Photograph: Stéphane Mahé/Reuters

After a wave of cases in which a positive lateral flow device (LFD) test was followed by a negative PCR test, a private laboratory handling swab tests has been suspended.

But conflicting results are not a new problem. Back in June, when secondary school students with a positive LFD were retested with a PCR check, over one in eight came back negative. And even without laboratory problems, it is unclear why a negative PCR should trump a positive LFD.

Imagine a (rather strange) legal case with the prosecution alleging that you harbour the virus. In court, it is becoming common to quote a “likelihood ratio” provided by forensic evidence — the relative support for the prosecution versus the defence.

First, the positive LFD is presented by the prosecution. If the virus were present, a recent study estimates around an 80% chance of a positive LFD – higher if you were infectious. Alternatively, if the defence is correct, there is a less than one in 1,000 chance of a false positive LFD. The likelihood ratio is therefore at least 800 (0.8/0.001). As a comparison, the curvature of the spine found on the skeleton in a Leicester car park contributed an estimated likelihood ratio of 200 in favour of the remains being those of Richard III.

The defence retorts with the negative PCR test. If you were infected, the PCR test might miss it around one in 20 times. If there were no virus, then that test is almost certain to be negative. Here, the likelihood ratio is around one in 20.

Combining these two conflicting pieces of evidence gives an overall likelihood ratio of about 40 (800 divided by 20). In a court, that might be reported as “moderate evidence” in favour of you having an infection.

As viral prevalence changes, then the probability of infection following conflicting test results also changes. At the current infection rate in England of one in 60 people, and with labs working well, out of 100 people with a positive LFD followed by a negative PCR, around 40 would actually have the virus and be falsely reassured.

The negative PCR does not outweigh the positive LFD.

• 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