Suppose 100 individuals are to be tested. Let’s first arrange them in 20 groups of five each. Now, instead of testing swab of one suspect with a kit, swabs of all five individuals in a group are mixed, and then the mixed swab is tested. Such a test is applicable only if the test is such that all the five individuals can be declared “negative” if the outcome of testing the mixed swab is “negative”. On the other hand, if the test outcome is “positive”, at least one of the five individuals is Covid-19 “positive”. In that case, all five individuals are tested separately.
The percentage of “positive” in India out of the tested cases was 3.4 per cent as of April 1. There were 1,637 Covid-19 “positive” cases out of 47,951 tests. Thus, if we carry out the testing in groups of size five each, we may need one test or 1+5=6 tests for any group, depending on whether the combined test is “negative” or “positive”. Given that a person has a 3.4 per cent chance of being diagnosed “positive” (which means that the probability of “negative” diagnosis for an individual is 0.966), the probability that the additional five tests are needed for a particular group is the probability that at least one of them is “positive”, which is 1-0.9665 = 15.9 per cent. Following this procedure, on an average, less than 36 tests will be needed to screen 100 individuals in 20 groups of five each.
Using simple calculation, I find that the optimal group size is six (i.e., swabs of six suspects can be mixed and tested together to minimise the number of tests). And, by this approach, about 354 test kits are needed to test 1,000 individuals!
The procedure will, however, be less effective if the probability of a “positive” diagnosis is higher. If about 10 per cent cases are “positive”, an optimal strategy of testing a group of four individuals together would enable testing 100 individuals by about 60 kits. And, if we are able to test more individuals, the chance of a “positive” diagnosis would decrease in any case, making the “group test procedure” more effective.
This approach can be extended a bit more — maybe up to the second stage. A relatively larger group of individuals (15 or so) can be tested first. If the test result is “positive”, the samples can be divided into smaller groups, say of five each. And then a smaller “positive” group may be tested for each individual separately. The concerned physicians can construct the groups judiciously to make this procedure more effective.
Statistically, it’s alright. But, let me put a caveat: I maybe missing something from the medical perspective. This procedure is not applicable if the swab of some Covid-19 “negative” individuals mixed with a Covid-19 “positive” one results in a “negative” testing result — a possibility which I cannot comment on as a statistician.
Certainly, a race to develop antibody tests using a few drops of blood is going on in labs around the world, and are expected to be available soon. Such serological tests will provide quicker results and might become instrumental in the fight against the Covid-19 pandemic.
The writer is a professor of statistics at Indian Statistical Institute, Kolkata