Featured image: A healthcare worker collects a swab sample of a man for COVID-19 test from the swab collection booth during the nationwide lockdown in the wake of coronavirus pandemic, in Vijayawada, Saturday, April 11, 2020. Photo: PTI
On March 16, the Director General of the World Health Organisation sent out a “… simple message for all countries: test, test, test”.
Many developing countries have not quite followed the dictum, mainly due to the unavailability of sufficient test kits. Over time, this has caused medical professionals and scientists to urge their governments to substantially expand testing. Recognis
ing the limited availability of test kits, several policy-makers and academics have recently encouraged the adoption of “group testing” for the novel coronavirus.
The logic of a group RT-PCR test[footnote]reverse transcription polymerase chain reaction[/footnote] is as follows:
Suppose we want to test ten people. We collect their swab samples and create a group-sample by mixing the individual samples, and first run a single test on the group-sample.
If the test comes out negative, we inform the ten people that they are disease-free. If the test is positive, then we subsequently test the ten samples to individually identify the infected.
Of course, if our initial estimate of the population infection rate is quite low, then we expect that we will need no more than a single test-kit to test the ten people. That is the selling point of a group test, given kit scarcity.
A basic problem with this story is that it implicitly assumes the RT-PCR test to be quite accurate in detecting infection. Many accounts by doctors and health professionals suggest that that might not be the case.
If fact, while the test is considered to be very specific (giving very few “false positives”), its effective sensitivity might not be very high.
The innate sensitivity of the test, coupled with the fact that it requires sufficient skill to be able to collect existing virus in a nasal swab – the novel coronavirus has been called “lethal but shy” – can mean that for up to 30% of truly infected cases, the test will give a “false negative” result.
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Unfortunately, if this is really the case (and there is evidence for it), it calls into question the advice of expanding testing in general, and group testing in particular.
Let’s consider a simple example.
Suspend disbelief, and imagine a country with 20 citizens and in possession of a single test-kit. We will study two different scenarios – one in which the initial infected population proportion is 10% (so that initially, we expect two infected patients in the populace), and the other in which the proportion is 20% (so we expect four infections initially).
Possessing only one test-kit, the head-of-state might just throw it away, and exhort to the citizens: “Dear countrymen, we have no test-kits. So let us behave as if each of us have a very high chance of infecting our neighbours! Let us practice personal hygiene and social distancing as if our lives depended on it!”
Following this advice, the citizens might indeed behave more cautiously, resulting in a countrywide disease reproduction number R which, while positive, is quite a bit lower than the “R0” of the novel coronavirus (about 2.5). So, if the country shuns testing, it will expect to begin its COVID journey with two spreaders spreading at R if the infection proportion is 10%, and with four spreaders spreading at R if the infection proportion is 20%.
Now suppose a friendly foreign leader calls up and says: “If you need it really badly, we can get you ten more kits!” Then the head-of-state faces a choice:
(a) she can immediately request for those ten kits and randomly select eleven citizens and separately test them, or
(b) she can use her single kit to group-test ten randomly-chosen citizens, and then plead for the ten extra kits only if the group test is positive.
The following table summarises the expected outcomes for the country from following the different testing strategies.
To understand the case of separate testing, consider asking: when initial infection proportion is 20%, what should we expect if eleven people are separately tested and the untested are urged to act with caution?
Then, approximately two untested spreaders will spread at R; there will be about 42% chance of one more spreader spreading at R0 (this being a single infected person falsely identified); and there will be about 9% chance of two more spreaders spreading at R0 (both being infected and falsely identified).
Is it obvious that separate testing of a sample population, using a test that is not very sensitive, is preferable to not testing at all?
Not quite; the basic trade-off is that we lower the number of cautious spreaders at the cost of increasing the chance of there being misinformed super-spreaders (and at the cost of having to acquire additional kits).
It is, however, quite obvious that except for reducing the probability for needing to acquire extra kits (note that the contingency of needed extra kits is not eliminated, just lowered), the group testing scheme does not unambiguously outperform the separate testing scheme; rather, it can be significantly worse.
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That is entirely due to the fact that sequential testing simply exacerbates the probability of false negatives. If the group-sample is determined to be wrongly negative 30% of the time, that falsely releases the entire tested group into the unsuspecting populace at one stoke – that is why the probability of there being multiple super-spreaders is necessarily greater under group- testing than under separate testing.
That has a significant social cost, so much so that disease transmission might be lower with no testing than with group-testing, and then the (probabilistic) cost-savings from not having to acquire additional kits become irrelevant.
The idea of group-testing is an ingenious response to a genuine scarcity. However, it is important to be aware of the potentially large social cost of this strategy when the underlying test lacks sufficient sensitivity. If a more accurate diagnostic test for COVID-19 is soon discovered, but is initially available in limited quantities, it might indeed be a very good candidate for group-testing.
Arijit Sen is a Professor of Economics at the Indian Institute of Management Calcutta. Views are personal.