A friend asked about the significance of her positive Covid tests. She had suggestive symptoms and an antibody test was positive. What did this mean? False negatives may occur if you test too soon, before antibodies have developed or the test might have been faulty. Counterintuitively, the significance to her of a positive test depends on the local prevalence of infection. False positives are rare but if the infection prevalence is very low then there may be more false positives than true positives (this would be more likely if she had no suggestive symptoms). If the infection prevalence is high then there would be more true positives than false positives. She had had two positive antibody tests (separated by weeks and by provider of the test) and so she almost certainly had had Covid. Whether she was less likely to suffer from another bout of Covid is a separate question. Failure to take note of prevalence rates when screening asymptomatic patients for a disease (women under 40 with mammograms for breast cancer for example) can lead to disproportionate over investigation of women who were at low risk.
This is standard Bayesian thinking about probabilities. There are prior probabilities (hopefully well-informed guesses) that can be (repeatedly) modified by fresh data that influence prior probabilities to produce more accurate probabilities, the posteriors (the updated priors). Bayes theorem can be used to manage probabilities mathematically to produce better assessments that the latest posterior is more likely to be correct or very nearly correct
Similar problems arise when assessing rare occurrences. A lady had given birth twice and her two babies unexpectedly died. Professor Sir Roy Meadow (an expert in cot deaths but not statistics) gave an opinion that it must have been double murder because the chance of two sudden cot deaths was so remote (the chances of one cot death was about 1 in 8,543, so the chance of two cot deaths would be 8,543 squared = about 73 million to 1 against). But none of those involved with trial realised that the chance of one child murder were much less than 1 in 8,543, even allowing for gross under identification of murders, so the chances of two child murders were hugely less than 1 in 75 million. His use of such statistics in rare events was incompetent. At the time I knew about this evidence but presumed that someone would point out the fallacy. But they didn’t and the lady went to jail. I now have similarly worries about our not having enough staff to do the number of Covid tests required. I worry that those responsible have they heard of pooling of specimens? Say for example 5% of samples will test positive. One hundred patients. Pool 20 groups each of 5 patients’ samples and test the pooled samples 20 TESTS. At a maximum five pooled groups could test positive. Then go back and test each of five in the five positive groups TWENTY-FIVE TESTS. A maximum total of 45 TESTS, the workload being more than halved.
I also worry that attempts to eradicate Covid from the population, especially by too vigorous lockdowns, will be economically disastrous and the national health will suffer – at some stage the national economy will equate with the national health. It might be best to shield those most at risk and to allow those at minimal risk to get on with life and keep the nation’s economy in a reasonable state to the benefit of all. Infection would be widespread but mortality rates might be lower than otherwise. Incidentally the purpose of lockdowns is to smooth the numbers of cases over time so that society does not have to cope with abrupt spikes that might overwhelm health services and the stability of society.
Another example. I had a patient on anti-HIV medication with a painful ankle. The House Officer had requested an HLA-B27 test “to exclude Ankylosing spondylitis,“ a very remote possibility. This was a waste of money because, about 85 percent of people with ankylosing spondylitis will be HLA-B27 positive (of 1000 people 1-8 will have ankylosing spondylitis and will test positive) but, because about 8 percent of normal Caucasians will be HLA-B27 positive anyway, 8 will test positive spondylitis. So the test result was irrelevant. If the patient had low back pain the prior probability of ankylosing spondylitis would have been higher as would the posterior after the HLA-B27 results were known. The patient had gout – I was kind and told the House Officer it could have been Spondylosing ankylitis.