Friday, March 27, 2020

Group testing

Christian Gollier and Olivier Gossner pass on a beautiful and simple idea: Group testing. It's also known as test pooling.)

To stop the virus, we need testing. But we don't have enough tests. As a result, a trillion dollars a month stands to go down the toilet, unemployment is skyrocketing, and a big financial crisis looms. What to do?

Test groups. Group testing works particularly well given that so far, the percentage of infected people is low.

Get a group of 32 people, and they each spit in a bucket. Test the bucket. (Metaphorically. Actually, the samples are swabs, and we mix parts of the samples.) If it's negative, everyone in the group is clean and they can go back to work.

If not, split the samples into two groups of 16, and test again. Again, if a group of 16 is negative, they're all clear. Keep going 8, 4, 2, 1. (You don't get new samples, of course. You take the original samples and split them apart, and test them again.)

If nobody has it, you find out in 1 test, not 32. If 1 out of the 32 has it, you find him or her with 12 tests not 32.

Often the goal of testing is not to find one particular person. And if tests take a day to come back, repeated testing is impractical. But with two rounds of testing you can at least very quickly find groups of 32 and 16 who are all clear, and isolate the smaller groups.

There are distinct reasons to test. If you have a very sick patient and you need to find out what he or she has, you need to test that person. But now testing has moved to public health questions. We want to find and certify the vast majority who do not have it. We want to find out what fraction of a neighborhood has it. And so on. For these purposes, group testing makes sense.

This idea strikes me as particularly good because of the spatial concentration of a virus. With one test we can find out if a city of 10,000 has any infected people. With one test, we can find out if a zip code is free of virus.

Update:  This seems like an especially useful idea to get business going again. Every morning, test the group sample of everyone at a business, plant, or building, say even groups of 100. As long as they are all clear the business stays open. To show a business does not have a virus, you only need to test the group.

Update 2: In response to comments. For the purpose of slowing a virus, it doesn't have to be perfect. Paul Romer's simulations are good on this. Just lower the probabilities and you lower transmission rates.


An Israeli team does group tests in practice

Larry Kotlikoff thinks through the practicalities of group testing, a way to cut the costs of testing by orders of magnitude.


  1. Would this actually work?

  2. But the test is a nasopharyngeal swab. Is spit a sensitive alternative?

  3. You've overlooked an important question: testing for what?

    The current testing scheme looks for the actual virus (meaning current infection), with a fairly well-documented (and large) rate of false positives, and a less well-documented (but, I suspect, also large) rate of false negatives. Actual virus testing is important to isolate carriers and administer cures. Since we don't have a cure, however, isolating carriers is all that's left and it's way too late for that. By some estimates it was too late for that in February.

    Testing for the virus now is like looking for hay in a haystack. It's also a good way to waste a finite supply of testing materials.

    That's why the drive-through testing didn't really take off, and why the administration (with Dr. Birx as the mild-mannered public face) is pushing a new testing scheme: the antibody assays. Testing for antibodies is testing for current infection AND past infection. That provides us with much more useful data. In particular, it reveals the infection fatality rate (instead of the current estimates and models, that even the people who created them are now admitting are wildly overstated).

    As Dr. Birx said yesterday, "the predictions of the models don’t match the reality on the ground in either China, South Korea, or Italy." Put differently, not nearly as many people are dying as the models said would die (by, it seems, many orders of magnitude).

    1. False positives seems low cost compared to false negatives in the big scheme of things. "Who mislaid the loss function?"

      And let us never, never forget that the reason these cheap tests have to be rationed is on account of our miraculous CDC/FDA.

      Under the circumstances, group testing sounds right.

  4. It's a old idea, but an excellent one. It was developed by Robert Dorfman in the 1940s to test for syphilis - each person gave two blood samples, one of which was pooled and the other was kept in reserve. The Wasserman Test was applied to the pooled sample. If the test was negative, a number of people would be cleared with a single test. If not, each of the second blood samples was tested. The Wikipedia article ( is excellent. It is used today for managing communications among IoT devices.

  5. Seems like a promising idea, but at the moment the bottleneck for mass testing is not test throughput or reagents but administration. No way that nasopharyngeal swabbing can be done properly at home -- it makes essentially everyone gag (and a non-trivial number vomit) and no one is going to do that. Same with taking a blood draw for a serological test.

    If there is a genetic test doable with saliva and a serological test doable with a finger prick, then this could work.

  6. 1) The gains are much smaller if you actually want to retest people after a positive result. Not 18,4 people to work after a test with 50 persons and 2% prevalence, but just 4,5. See the simulations here:

    2) Does it matter if false positives/negatives are high? I think high false positives roughly means more prevalence, so that would negate some of the benefits of group testing. How about false negatives?

  7. 1) The gains are much smaller if you actually want to retest people after a positive result. Not 18,4 people to work after a test with 50 persons and 2% prevalence, but just 4,5. See the simulations here:
    2) What about false positives and false negatives? If false positives are high you have higher prevalence, which negates some of the benefits of group testing. How should we think about false negatives in group testing?

  8. Don't do just simple group testing but rank people 1 to 10 (group of 10). Make 1st person spit once into the bucket, 2nd one spits twice... 10th one ten spits. We have 55 spits. Test it and count antibodies. If it's lowest then person ranked 1 is infected. 5 times, it's the fifth person, 10 times you guessed it. This has been used in environmental sampling. Now if two are infected then identifying is a problem but making groups of 10 by randomizing will reduce when virality is rare ...


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