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For the lateral flow testing kits (same swab method, but different test to PCR), the BMJ found that the sensitivity (True positive rate) dropped from 73% with healthcare staff to 58% with members of the public.
The evaluation found that the test performed best when used by laboratory scientists when the sensitivity was 79% (156/197 positive: 79.2% (95% confidence interval 72.8% to 84.6%)).
Sensitivity dropped to 73% when used by trained healthcare staff (92/126 positive: 73.0% (64.3% to 80.5%)) and to 58% with self-trained members of the public (214/372 positive: 57.5% (52.3% to 62.6%)).
https://www.bmj.com/content/371/bmj.m4469
PCR test have a much higher sensitivity to start with but presumably there will be a similar drop off due to a relatively poorer swabbing technique by the self-trained public.
Wonder how much impact not having medical staff doing the tests has on fake positives or negatives?