How to test for COVID-19 efficiently

“Group testing” protocols tailored to particularities of the COVID-19 pandemic promise more-informative test results.

In the absence of a vaccine, a valuable measure for controlling the spread of COVID-19 is large-scale testing. The limited availability of test kits, however, means that testing has to be done as efficiently as possible.

The most efficient testing protocol is group testing, in which test samples from multiple subjects are tested together. If the test is perfectly reliable, then a negative test for a group clears all of its members at once. Clever group selection enables the protocol to zero in on infected patients with fewer tests than would be required to test each patient individually.

Group testing is a well-studied problem, but particular aspects of COVID testing — among them the relatively low infection rate among the test population, the false-positive rate of the tests, and practical limits on the number of samples that can be pooled in a single group — mean that generic test strategies dictated by existing theory are suboptimal.

My colleagues and I have written a paper that presents optimal strategies for COVID testing in several different circumstances. The paper is currently under submission for publication, but we have posted it to arXiv in the hope that our ideas can help stimulate further advances in COVID test design.

The key to group testing is that a given test sample is tested in several different groups, each of which combines it with a different assortment of samples. By cross-referencing the results of all the group tests, it’s possible to predict with high probability the correct result for any given sample.

Graphic illustratration of groupings
The intuition behind the researchers’ non-adaptive-testing algorithm. Circles represent individual patients; each grouping assigns patients to different groups. A 1 in the Tests column indicates that the group tested positive, a 0 that it tested negative. Cross-referencing results across groupings identifies infected individuals more efficiently than individual testing would.
Credit: Stacy Reilly

In this respect, the problem exactly reproduces the classical problem of error-correcting codes in information theory. Each parity bit in an error-correcting code encodes information about several message bits, and by iteratively cross-referencing message bits and parity bits, it’s possible to determine whether errors have crept into either.

Accordingly, we treat the problem of deciding how to pool test samples as a coding problem, and the problem of interpreting the test results as a decoding problem, and we use the information-theoretic concept of information gain to evaluate test protocols.

Adaptive testing

Group testing comes in two varieties: adaptive and non-adaptive. In the adaptive setting, tests (or groups of tests) are conducted in sequence, and the outcomes of one round of testing inform the group selection for the next round. In non-adaptive testing, groups are selected without any prior information about group outcomes.

In our paper, we consider adaptive testing involving relatively small numbers of patients — less than 30. We also consider non-adaptive testing for much larger numbers — say, thousands. In both settings, using the tools of information theory, we factor in prior knowledge about the probability of infection — some patients’ risk is higher than others’ — and the false-positive and false-negative rates of the tests.

Even with small numbers of patients, given the uncertainty of the test results and the mixture of prior infection probabilities, calculating the optimal composition of the test groups is an intractably complex problem. We show that in the COVID-19 setting, evolutionary strategies offer the best approximation of the optimal composition.

With evolutionary strategies, test groups are assembled at random, and the likely information gain is computed (given the prior probability of a positive test for each patient). Then some of the group compositions are randomly varied and tested again. Variations that lead to greater information are explored further; those that don’t are abandoned.

This procedure will produce the best approximation of the optimal group composition, but it could take a while: there’s no theoretical guarantee about how quickly evolutionary strategies will converge on a solution. As an alternative in the context of adaptive testing with small numbers of patients, we also consider a greedy group composition strategy. 

With the greedy strategy, we first assemble the group that, in itself, maximizes the information gain for one round of testing. Then we select the group that maximizes the information gain in the next round, and so on. In our paper, we show that this approach is very likely to arrive at a close approximation of the ideal group composition, with tighter guarantees on the convergence rate than evolutionary strategies offer.

Non-adaptive testing

For large-scale, non-adaptive tests, the conventional approach is to use Bloom filter pooling. The Bloom filter is a mechanism designed to track data passing through a network in a streaming, online context. 

The Bloom filter uses several different hash functions to hash each data item it sees to several different locations in an array of fixed size. Later, if any location corresponding to a given data item is empty, the filter can guarantee that that item hasn’t been seen. False positives, however, are possible.

Group testing has appropriated this design, using the multiple hash functions to assign a single patient’s sample to multiple locations and grouping samples that hash to the same location. But no matter how good the hash functions are, the distribution across groups may not be entirely even. If the groups average, say, 20 members each, some might have 18, others 22, and so on. That compromises the accuracy of the ensuing predictions of infection.

The Bloom filter design assumes that the number of data items seen in the streaming, network setting is unpredictable and open ended. But in the group-testing context, we know exactly how many patient samples we’re distributing across groups. So we can exactly control the number of samples assigned to each group.

If we have no prior probabilities of infection rates, an even distribution is optimal. If we do have priors, then we can distribute samples accordingly: maximizing information gain might require that we reduce the sizes of groups containing high-probability samples and increase the sizes of groups containing low-probability samples.

Similarly, because the Bloom filter was designed for the streaming, networked setting, the algorithm for determining whether an item has been seen must be highly efficient; the trade-off is that it doesn’t minimize the risk of a false positive. 

In the context of group testing, we can afford a more involved but accurate decoding algorithm. In our paper, we show that a message-passing algorithm, of a type commonly used to decode error-correcting codes, is much more effective than the standard Bloom filter decoding algorithm.

Related content

US, WA, Seattle
Amazon.com strives to be Earth's most customer-centric company where customers can shop in our stores to find and discover anything they want to buy. We hire the world's brightest minds, offering them a fast paced, technologically sophisticated and friendly work environment. Economists at Amazon partner closely with senior management, business stakeholders, scientist and engineers, and economist leadership to solve key business problems ranging from Amazon Web Services, Kindle, Prime, inventory planning, international retail, third party merchants, search, pricing, labor and employment planning, effective benefits (health, retirement, etc.) and beyond. Amazon Economists build econometric models using our world class data systems and apply approaches from a variety of skillsets – applied macro/time series, applied micro, econometric theory, empirical IO, empirical health, labor, public economics and related fields are all highly valued skillsets at Amazon. You will work in a fast moving environment to solve business problems as a member of either a cross-functional team embedded within a business unit or a central science and economics organization. You will be expected to develop techniques that apply econometrics to large data sets, address quantitative problems, and contribute to the design of automated systems around the company. We are open to hiring candidates to work out of one of the following locations: Arlington, VA, USA | Bellevue, WA, USA | Boston, MA, USA | Los Angeles, CA, USA | New York, NY, USA | San Francisco, CA, USA | Seattle, WA, USA | Sunnyvale, CA, USA
US, WA, Seattle
Amazon.com strives to be Earth's most customer-centric company where customers can shop in our stores to find and discover anything they want to buy. We hire the world's brightest minds, offering them a fast paced, technologically sophisticated and friendly work environment. Economists at Amazon partner closely with senior management, business stakeholders, scientist and engineers, and economist leadership to solve key business problems ranging from Amazon Web Services, Kindle, Prime, inventory planning, international retail, third party merchants, search, pricing, labor and employment planning, effective benefits (health, retirement, etc.) and beyond. Amazon Economists build econometric models using our world class data systems and apply approaches from a variety of skillsets – applied macro/time series, applied micro, econometric theory, empirical IO, empirical health, labor, public economics and related fields are all highly valued skillsets at Amazon. You will work in a fast moving environment to solve business problems as a member of either a cross-functional team embedded within a business unit or a central science and economics organization. You will be expected to develop techniques that apply econometrics to large data sets, address quantitative problems, and contribute to the design of automated systems around the company. We are open to hiring candidates to work out of one of the following locations: Arlington, VA, USA | Bellevue, WA, USA | Boston, MA, USA | Los Angeles, CA, USA | New York, NY, USA | San Francisco, CA, USA | Seattle, WA, USA | Sunnyvale, CA, USA
US, WA, Seattle
Amazon.com strives to be Earth's most customer-centric company where customers can shop in our stores to find and discover anything they want to buy. We hire the world's brightest minds, offering them a fast paced, technologically sophisticated and friendly work environment. Economists at Amazon partner closely with senior management, business stakeholders, scientist and engineers, and economist leadership to solve key business problems ranging from Amazon Web Services, Kindle, Prime, inventory planning, international retail, third party merchants, search, pricing, labor and employment planning, effective benefits (health, retirement, etc.) and beyond. Amazon Economists build econometric models using our world class data systems and apply approaches from a variety of skillsets – applied macro/time series, applied micro, econometric theory, empirical IO, empirical health, labor, public economics and related fields are all highly valued skillsets at Amazon. You will work in a fast moving environment to solve business problems as a member of either a cross-functional team embedded within a business unit or a central science and economics organization. You will be expected to develop techniques that apply econometrics to large data sets, address quantitative problems, and contribute to the design of automated systems around the company. We are open to hiring candidates to work out of one of the following locations: Arlington, VA, USA | Bellevue, WA, USA | Boston, MA, USA | Los Angeles, CA, USA | New York, NY, USA | San Francisco, CA, USA | Seattle, WA, USA | Sunnyvale, CA, USA
LU, Luxembourg
The Decision, Science and Technology (DST) team part of the global Reliability Maintenance Engineering (RME) is looking for a Senior Operations Research Scientist interested in solving challenging optimization problems in the maintenance space. Our mission is to leverage the use of data, science, and technology to improve the efficiency of RME maintenance activities, reduce costs, increase safety and promote sustainability while creating frictionless customer experiences. As a Senior OR Scientist in DST you will be focused on leading the design and development of innovative approaches and solutions by leading technical work supporting RME’s Predictive Maintenance (PdM) and Spare Parts (SP) programs. You will connect with world leaders in your field and you will be tackling customer's natural language challenges by carrying out a systematic review of existing solutions. The appropriate choice of methods and their deployment into effective tools will be the key for the success in this role. The successful candidate will be a self-starter comfortable with ambiguity, with strong attention to detail and outstanding ability in balancing technical leadership with strong business judgment to make the right decisions about model and method choices. Key job responsibilities • Provide technical expertise to support team strategies that will take EU RME towards World Class predictive maintenance practices and processes, driving better equipment up-time and lower repair costs with optimized spare parts inventory and placement • Implement an advanced maintenance framework utilizing Machine Learning technologies to drive equipment performance leading to reduced unplanned downtime • Provide technical expertise to support the development of long-term spares management strategies that will ensure spares availability at an optimal level for local sites and reduce the cost of spares A day in the life As a Senior OR Scientist in DST you will be focused on leading the design and development of innovative approaches and solutions by leading technical work supporting RME’s Predictive Maintenance (PdM) and Spare Parts (SP) programs. You will connect with world leaders in your field and you will be tackling customer's natural language challenges by carrying out a systematic review of existing solutions. The appropriate choice of methods and their deployment into effective tools will be the key for the success in this role. About the team Our mission is to leverage the use of data, science, and technology to improve the efficiency of RME maintenance activities, reduce costs, increase safety and promote sustainability while creating frictionless customer experiences. We are open to hiring candidates to work out of one of the following locations: Luxembourg, LUX
US, WA, Seattle
Amazon.com strives to be Earth's most customer-centric company where customers can shop in our stores to find and discover anything they want to buy. We hire the world's brightest minds, offering them a fast paced, technologically sophisticated and friendly work environment. Economists at Amazon partner closely with senior management, business stakeholders, scientist and engineers, and economist leadership to solve key business problems ranging from Amazon Web Services, Kindle, Prime, inventory planning, international retail, third party merchants, search, pricing, labor and employment planning, effective benefits (health, retirement, etc.) and beyond. Amazon Economists build econometric models using our world class data systems and apply approaches from a variety of skillsets – applied macro/time series, applied micro, econometric theory, empirical IO, empirical health, labor, public economics and related fields are all highly valued skillsets at Amazon. You will work in a fast moving environment to solve business problems as a member of either a cross-functional team embedded within a business unit or a central science and economics organization. You will be expected to develop techniques that apply econometrics to large data sets, address quantitative problems, and contribute to the design of automated systems around the company. We are open to hiring candidates to work out of one of the following locations: Arlington, VA, USA | Bellevue, WA, USA | Boston, MA, USA | Los Angeles, CA, USA | New York, NY, USA | San Francisco, CA, USA | Seattle, WA, USA | Sunnyvale, CA, USA
US, WA, Seattle
Amazon.com strives to be Earth's most customer-centric company where customers can shop in our stores to find and discover anything they want to buy. We hire the world's brightest minds, offering them a fast paced, technologically sophisticated and friendly work environment. Economists in the Forecasting, Macroeconomics & Finance field document, interpret and forecast Amazon business dynamics. This track is well suited for economists adept at combining cutting edge times-series statistical methods with strong economic analysis and intuition. This track could be a good fit for candidates with research experience in: macroeconometrics and/or empirical macroeconomics; international macroeconomics; time-series econometrics; forecasting; financial econometrics and/or empirical finance; and the use of micro and panel data to improve and validate traditional aggregate models. Economists at Amazon are expected to work directly with our senior management and scientists from other fields on key business problems faced across Amazon, including retail, cloud computing, third party merchants, search, Kindle, streaming video, and operations. The Forecasting, Macroeconomics & Finance field utilizes methods at the frontier of economics to develop formal models to understand the past and the present, predict the future, and identify relevant risks and opportunities. For example, we analyze the internal and external drivers of growth and profitability and how these drivers interact with the customer experience in the short, medium and long-term. We build econometric models of dynamic systems, using our world class data tools, formalizing problems using rigorous science to solve business issues and further delight customers. We are open to hiring candidates to work out of one of the following locations: Arlington, VA, USA | Bellevue, WA, USA | Boston, MA, USA | Los Angeles, CA, USA | New York, NY, USA | San Francisco, CA, USA | Seattle, WA, USA | Sunnyvale, CA, USA
US, WA, Seattle
Amazon.com strives to be Earth's most customer-centric company where customers can shop in our stores to find and discover anything they want to buy. We hire the world's brightest minds, offering them a fast paced, technologically sophisticated and friendly work environment. Economists in the Forecasting, Macroeconomics & Finance field document, interpret and forecast Amazon business dynamics. This track is well suited for economists adept at combining cutting edge times-series statistical methods with strong economic analysis and intuition. This track could be a good fit for candidates with research experience in: macroeconometrics and/or empirical macroeconomics; international macroeconomics; time-series econometrics; forecasting; financial econometrics and/or empirical finance; and the use of micro and panel data to improve and validate traditional aggregate models. Economists at Amazon are expected to work directly with our senior management and scientists from other fields on key business problems faced across Amazon, including retail, cloud computing, third party merchants, search, Kindle, streaming video, and operations. The Forecasting, Macroeconomics & Finance field utilizes methods at the frontier of economics to develop formal models to understand the past and the present, predict the future, and identify relevant risks and opportunities. For example, we analyze the internal and external drivers of growth and profitability and how these drivers interact with the customer experience in the short, medium and long-term. We build econometric models of dynamic systems, using our world class data tools, formalizing problems using rigorous science to solve business issues and further delight customers. We are open to hiring candidates to work out of one of the following locations: Arlington, VA, USA | Bellevue, WA, USA | Boston, MA, USA | Los Angeles, CA, USA | New York, NY, USA | San Francisco, CA, USA | Seattle, WA, USA | Sunnyvale, CA, USA
US, WA, Seattle
Economists in the Forecasting, Macroeconomics & Finance field document, interpret and forecast Amazon business dynamics. This track is well suited for economists adept at combining cutting edge times-series statistical methods with strong economic analysis and intuition. This track could be a good fit for candidates with research experience in: macroeconometrics and/or empirical macroeconomics; international macroeconomics; time-series econometrics; forecasting; financial econometrics and/or empirical finance; and the use of micro and panel data to improve and validate traditional aggregate models. Economists at Amazon are expected to work directly with our senior management and scientists from other fields on key business problems faced across Amazon, including retail, cloud computing, third party merchants, search, Kindle, streaming video, and operations. The Forecasting, Macroeconomics & Finance field utilizes methods at the frontier of economics to develop formal models to understand the past and the present, predict the future, and identify relevant risks and opportunities. For example, we analyze the internal and external drivers of growth and profitability and how these drivers interact with the customer experience in the short, medium and long-term. We build econometric models of dynamic systems, using our world class data tools, formalizing problems using rigorous science to solve business issues and further delight customers. We are open to hiring candidates to work out of one of the following locations: Arlington, VA, USA | Bellevue, WA, USA | Boston, MA, USA | Los Angeles, CA, USA | New York, NY, USA | San Francisco, CA, USA | Seattle, WA, USA | Sunnyvale, CA, USA
US, WA, Seattle
Amazon.com strives to be Earth's most customer-centric company where customers can shop in our stores to find and discover anything they want to buy. We hire the world's brightest minds, offering them a fast paced, technologically sophisticated and friendly work environment. Economists at Amazon partner closely with senior management, business stakeholders, scientist and engineers, and economist leadership to solve key business problems ranging from Amazon Web Services, Kindle, Prime, inventory planning, international retail, third party merchants, search, pricing, labor and employment planning, effective benefits (health, retirement, etc.) and beyond. Amazon Economists build econometric models using our world class data systems and apply approaches from a variety of skillsets – applied macro/time series, applied micro, econometric theory, empirical IO, empirical health, labor, public economics and related fields are all highly valued skillsets at Amazon. You will work in a fast moving environment to solve business problems as a member of either a cross-functional team embedded within a business unit or a central science and economics organization. You will be expected to develop techniques that apply econometrics to large data sets, address quantitative problems, and contribute to the design of automated systems around the company. We are open to hiring candidates to work out of one of the following locations: Arlington, VA, USA | Bellevue, WA, USA | Boston, MA, USA | Los Angeles, CA, USA | New York, NY, USA | San Francisco, CA, USA | Seattle, WA, USA | Sunnyvale, CA, USA
US, WA, Seattle
Amazon.com strives to be Earth's most customer-centric company where customers can shop in our stores to find and discover anything they want to buy. We hire the world's brightest minds, offering them a fast paced, technologically sophisticated and friendly work environment. Economists at Amazon partner closely with senior management, business stakeholders, scientist and engineers, and economist leadership to solve key business problems ranging from Amazon Web Services, Kindle, Prime, inventory planning, international retail, third party merchants, search, pricing, labor and employment planning, effective benefits (health, retirement, etc.) and beyond. Amazon Economists build econometric models using our world class data systems and apply approaches from a variety of skillsets – applied macro/time series, applied micro, econometric theory, empirical IO, empirical health, labor, public economics and related fields are all highly valued skillsets at Amazon. You will work in a fast moving environment to solve business problems as a member of either a cross-functional team embedded within a business unit or a central science and economics organization. You will be expected to develop techniques that apply econometrics to large data sets, address quantitative problems, and contribute to the design of automated systems around the company. We are open to hiring candidates to work out of one of the following locations: Arlington, VA, USA | Bellevue, WA, USA | Boston, MA, USA | Los Angeles, CA, USA | New York, NY, USA | San Francisco, CA, USA | Seattle, WA, USA | Sunnyvale, CA, USA