The idea

When too few tests are available, or we want to determine the infective of large numbers of people in as short a time as possible, we need to do block testing based on pooling individual samples.

In pictures an example of the idea is this:

For groups of \(k=10\) people, when the proportion \(p\) of infected is 1 in 100, on average only 2 tests are needed to test each group of 10 people!

The group size \(k\) can be adjusted depending on what proportion \(p\) of the targetted population is infected. That is, it can be adapted to the population being targetted.

An Excel spreadsheet that calculates the average number of tests for different group sizes \(k\) and proportion infected \(p\) can be downloaded here.

Germany used this block testing in groups of 10 on its front line hospital workers … New York Times Article, April 4, 2020.

Much larger group sizes could be used for the public where the probability of infection is likely much lower.

The email to Premier Ford

What follows is the text of an email I sent to Premier Doug Ford, Health Minister Christine Elliott, and my local MPP Ms. Catherine Fife on Thursday, April 9, 2020.
A copy of this was also sent later the same day to Ontario’s Chief Medical Officer, Dr. David Williams, and Ontario’s Associate Chief Medical Officer, Dr. Barbara Yaffe.

The purpose of the letter to is ensure that our medical research team is aware of the well-known advantages of pooling samples in testing a population.

Minor adjustments have been made for clarification.

As of today (April, 11), there have been no responses from any of the persons contacted; this is understandable given the volume of email they must be receiving.

Dear Premier Ford (cc Minister Elliott and Ms. Fife),

My name is Wayne Oldford and I am a Professor of Statistics at the University of Waterloo.

I would be remiss if I did not ensure that what follows is well known within the government. Hence this email. It is lengthy but potentially important.
I have broken it into sections for ease of reading.

The punchline is that we need to seriously consider pooling samples from many people into a single test to speed up and dramatically increase our coverage of people tested in Ontario. I give figures and an attached spreadsheet to help.

There may be some challenge to combining samples and testing but I have heard that Germany is already doing this.

Hopefully, you have heard this all before and have already taken proper action and are working hard on the logistics. If so and we are following pooled testing, great!!!
(But then, I am surprised not to see the army called upon to be doing massive public testing in Ontario (perhaps this is in the works). That is one reason I am emailing you now.)

If the idea has not been proposed, please read on.

Premier Ford is right

Far too few tests are being done in Ontario today. More importantly, we do not know how many people would test positive or negative.

The number of people we know about and the number of tests conducted do not have to be the same.

Instead, test many people in a single test.

That is, bulk or pooled testing.

The problem: To test k people one at a time we use k tests.

  • If we can conduct 13000 tests per day, then we will only test 13,000 people per day. 3,000 per day, 3,000 people.
  • At the rate of 13,000 people per day (which we have NOT come close to achieving) it will take more than 3 years to test all Ontarions,
    • About 4 months to test only 10%
  • At 3000, it is much worse. We are talking about 16 months for 10%.

Test groups of people

To speed this up dramatically we could swab 10 or 100 people, combine the samples into one and test the combined sample.

Following that protocol, here are the average number of tests (rounded to whole numbers) that would be done to figure out the results for k people. Values of k change, and are given for different probabilities (p) that a person randomly drawn from that pool would test positive.

I have also attached an excel spreadsheet that will let you do your own calculations. (Click here to download the Excel spreadsheet.)

Some numbers

Picking out some examples from the table.

Row 2, column k = 1,000, means

  • we pool the samples from 1,000 people and
  • we think that amongst the group being tested, the probability of being tested positive is about 1 in 100,000
  • then only 11 tests are needed on average to determine the results for each group of 1,000 people

Row 3, column k = 50, means

  • we pool the samples from 50 people and
  • we think that amongst the group being tested, the probability of being tested positive is about 1 in 10,000
  • then a single test is needed on average to determine the results for 50 people
  • NOTE: the true average here is about 1.25; the results are rounded in the table

Last row, column k = 20, means

  • we pool the samples from 20 people and
  • we think that amongst the group being tested, the probability of being tested positive is high about 1 in 5
  • then 21 tests are needed on average to determine the results for 20 people
  • NOTE: there is no advantage to pooling
  • As you can see looking across the table this will eventually happen as k and p get large

Adaptive pooling strategies

This leads to different pooling strategies for different groups of people who have different probabilities (p) of testing positive (being infected)

  • front line health care workers,
  • patients in long term residences,
  • grocery store workers,
  • the general public.

There may also be a constraint on the maximum number of samples (k) which can reliably be combined to test. - This may need some testing in itself to determine

The major challenge will be logistics. We could do massive pooled testing of the public for example at drive throughs managed by army personnel.

Even if only pooled testing is done in the public we will have much better data at the aggregate level on which to base decisions.

The math behind the idea

Suppose the probability that a person would test positive with COVID-19 is \(p\) (\(p\) will be smaller the earlier this is done) The probability they test negative is \(1-p\).

If the swabs from \(k\) people are combined into one and tested, then for the combined sample to test negative all of the people in that sample would have to be negative. This only happens with probability \[ (1-p) \times (1-p) \times \cdots \times (1-p) ~~~~~~ \mbox{ that is }~ k ~\mbox{ products} \] or \[(1-p)^k\]

When that happens, there is no more testing of those people. They are clear. 1 test covers k people.

If at least one of the people would test positive, then the combined test would be positive. This is the alternative to all of them testing negative and so happens with probability
\[1 – Prob(\mbox{all test negative}) = 1 – (1-p)^k \] When that happens each INDIVIDUAL person is tested. That means k more tests so we have spent k + 1 tests to find out about k people.

The good news is that if p is small the probability of wasting that extra test is really small.

Following this procedure on average we would only need \[ (k +1) \times (1 – (1-p)^k) + 1 \times (1-p)^k \] Equals \[k + 1 – k(1-p)^k\] tests for a group of \(k\) people. That minus \(k(1-p)^k\) is what reduces the number of tests needed dramatically.

I hope this helps

Sincerely

Wayne Oldford

R.W. Oldford Professor of Statistics University of Waterloo

math.uwaterloo.ca/~rwoldfor

Postscript: the idea again in pictures

This was not part of the letter but explains the pooling method in pictures.

Suppose we have a group of 100 people of which only 1 in 100 are infected (and would test positive). The unknown infected person is in red.

Samples are taken from all 100 people.

If we were to individually test all 100 samples to learn who was infected and who was not we would have used up 100 tests.

Instead, we

The results of the combined sample would wither be negative or positive. Only one of the groups tested positive.

Take the one group of 10 people whose combined sample tested positive. Then test each of these individually.

We have now determined the test value of 100 people with only 20 tests (or about the average of 2 tests per group of 10).

That is, 5 times as many people tested compared to testing them individually.

More people can be tested with fewer tests.

The size of each group can be adjusted depending on the probability of an individual testing positive. For example, some workers might have a greater exposure to the virus and hence a higher probability of an individual worker testing positive. In this case the group size will be relatively small. In contrast, in the general public when the probability of testing positive is much smaller, the group size can be larger and so require fewer tests.

Germany used this blocked or pooled testing in groups of 10 … New York Times Article, April 4, 2020.

An even faster solution - binary search

A faster solution would have powers work in our favour but might be unacceptable for reasons of managerial execution or public intolerance. The idea would be to employ binary search for infected individuals.

This would require several swabs to be taken from each individual, recorded, and stored (viz. \(log_2(k)\) swabs compared with only 2 swabs for the proposal above).

Testing would proceed as follows:

  1. Choose \(k\) to be the largest number of people that can be sampled in the available time period and for which a reliable test can be made of their combined sample.

  2. Test the combined sample. It will be either positive or negative.

  3. If it tests negative, move on to the next group of \(k\) people (go back to step 1), otherwise continue to step 4.

  4. Split the group into two equal-sized groups; retrieve the samples for the persons in each group and, within group, combine their samples.

  5. For each group, repeat the process starting at step 2.

This would continue until all individual positive cases were identified. Alternatively, if resources/logistics do not permit, the process can be ended early and every individual in the group tested.

At the time of each split, the new groups would be determined randomly or, in the unusual case that high quality judgment is available, into two groups (of equal size) where one group is judged to contain those people most likely to test negative.