False Negatives and False Positives

Yesterday, I posted some comments on Twitter regarding the press focus on lateral flow test (LFT) false positives on the eve of the return of most students to school in England. It seems that I should probably have written a short blog post about this rather than squeezing it into a few tweets, given the number of questions I’ve had about the post since then. This is my attempt.

The press seem to be focusing on the number of false positives we are likely to see on return to school and the unnecessarily lost learning this may cause. My view is that given the current high case numbers and slow rate of decline, this is not the primary issue we should be worried about. Primarily, we should be worried about the false negative rate of these tests. My concern is that the small number of true positives caught by these tests may have less impact on reducing the rate of infection than behavioural relaxation induced by these tests has on increasing the rate of infection. Time will tell, of course.

Let me explain some of the data that has been released, the conclusions I draw from it, and why false positive rate is important for these conclusions regarding false negative rates.

For any given secondary-age pupil, if given both a LFT and a PCR test, excluding void tests there are four possible outcomes: LFT+ & PCR+ (LFT positive & PCR positive), LFT+ & PCR-, LFT- & PCR+ and LFT- & PCR-. We can imagine these in a table of probabilities, as below.

LFT+LFT-Total
PCR+??0.34% to 1.45%
PCR-???
Total~0.19%?100%

Here the total LFT+ figure of 0.19% comes from the SchoolsWeek article for secondary school students based on Test and Trace data for late February, while the total PCR+ figure is the confidence interval provided in the REACT-1 9b data from, which says “In the latter half of round 9 (9b), prevalence varied from 0.21% (0.14%, 0.31%) in those aged 65 and over to 0.71% (0.34%, 1.45%) in those aged 13 to 17 years.” Note that REACT-1 9b ran over almost the same period as the Test and Trace data on which the SchoolsWeek article is based.

What I think we all really want to know is what is the probability that a lateral flow test would give me a negative result when a PCR test would give me a positive result? We cannot get this information directly from the table, but we can start to fill in some of the question marks. Clearly the total LFT- probability will be 100% – 0.19% = 99.81%, and the total PCR- probability will be 98.55% to 99.66%. What about the four remaining question marks?

Let’s consider the best case specificity of these tests: that all the 0.19% of LFT+ detected were true positives, i.e. the tests were producing no false positives at all. In that case, the table would look something like this:

LFT+LFT-Total
PCR+~0.19%~0.15% to ~1.26%0.34% to 1.45%
PCR-0.00%98.55% to 99.66%98.55% to 99.66%
Total~0.19%~99.81%100%

Under these circumstances, we can see from the table that the LFTs are picking up 0.19% out of 0.34% to 1.45%, so we can estimate that the most they’re picking up is 0.19/0.34 = 56% of the true positive cases.

However, this assumed no false positives at all, which is a highly unrealistic assumption. What if we consider a more realistic assumption on false positives? The well-cited Oxford study gives a confidence interval for self-trained operatives of 0.24% to 0.60% false positives. Note that the lower end of this confidence interval would suggest that we should see at least 0.24% x 98.55% = ~0.24% of positive LFTs just from false positives alone. This is a higher value than the LFT positive rate we saw over this period of 0.19% (as noted by Deeks here). So this means it’s also entirely feasible that none of the LFT+ results were true positives, i.e. the results table could look more like this:

LFT+LFT-Total
PCR+0%0.34% to 1.45%0.34% to 1.45%
PCR-~0.19%~98.36% to ~99.47%98.55% to 99.66%
Total~0.19%~99.81%100%

Now this time round, we can see that the tests are picking up 0% out of 0.34% to 1.45%, so we can estimate that they’re picking up 0% of the true positive cases (i.e. 100% false negatives).

This is why I think we do need to have a conversation about false positives. Not because of a few days of missed school, as reported in the press, but because hiding behind these numbers may be a more significant issue of a much higher false negative rate than we thought, leading to higher infections in schools as people relax after receiving negative lateral flow tests.

Perhaps most importantly, I think the Government needs to commit to following the recommendations of the Royal Statistical Society, which would enable us to get to the bottom of exactly what is going on here with false positive and false negative rates.

(Note that I have assumed throughout this post that LFTs are being used as a ‘quick and easy’ substitute for PCRs, so that the ideal LFT outcome is to mirror a PCR outcome. I am aware that there are those who do not think this is the case, and I am not a medical expert so will not pass further comment on this issue.)

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