from John Dee’s Almanac
I’ve been wondering if there’s a simple way of expressing some of my recent ranting and came up with this idea.
Whilst it is pretty obvious that the more people we test the more likely we are to find positive cases, what is not so obvious is the same situation applies to re-testing of the same people. The more people re-test themselves the more likely they are to obtain a positive result at some point. This is because the test, whether PCR or LFD, isn’t 100% perfect and will produce both false negatives and false positives as well as true negative and true positive results. This all boils down to probability theory and the fact that if we roll a die enough times we are going to get that six. We can get twelve people to each roll that die once or we can get six people to roll it twice, four people to roll it three times, three people to roll it four times, two people to roll it six times or one person to roll it twelve times. In each instance we are giving ourselves 12 chances of obtaining a six, and if our die is fair and true then we would expect two sixes to appear.
In the attached slide I’ve got Alan and Betty experiencing three different scenarios, with Alan testing positive on his second test. Since there are only two people involved the case rate is always 1 /2 = 50% but the detection rate declines as we offer Betty more test opportunities. The tricky issue is deciding which of these two rates offers the more accurate estimate of disease prevalence. If testing was 100% reliable then we could simply assume Alan has the virus but Betty has not and go for 50%; since testing is not 100% reliable then we have to take into account increasing risk of false results as more and more tests are undertaken. The detection rate may thus become a better proxy for disease prevalence under certain circumstances. We may attempt to solve the puzzle by testing a random sample of the population once and once only.
In the final example on the right-hand side six different people would be given one test and we’d see how these results compare with the re-testing of Alan and Betty.
The Shadows 