Optimal Risk-Based Group Testing

Abstract

Group testing (i.e., testing multiple subjects simultaneously with a single test) is essential for classifying a large population of subjects as positive or negative for a binary characteristic (e.g., presence of a disease). We study optimal group testing designs under subject-specific risk characteristics and imperfect tests, considering classification accuracy-, efficiency- and equity-based objectives, and characterize important structural properties of optimal testing designs. These properties allow us to model the testing design problems as partitioning problems, develop efficient algorithms, and derive insights on equity versus accuracy trade-off. One of our models reduces to a constrained shortest path problem, for a special case of which we develop a polynomial-time algorithm. We also show that determining an optimal risk-based Dorfman testing scheme that minimizes the expected number of tests is tractable, resolving an open conjecture. We demonstrate the value of optimal risk-based testing schemes with a case study of public health screening. This paper was accepted by Yinyu Ye, optimization.

Group Testing in a Pandemic: The Role of Frequent Testing, Correlated Risk, and Machine Learning

“…Given the complicated tradeoff between the benefits of increasing group size and the cost of follow-up testing for a positive result, a large literature has emerged on optimal group testing strategies. This literature uses a set of assumptions grounded in the common use cases to date: one-time testing of a set of samples, e.g., screening donated blood for infectious disease, 1 with independent risk of infection (Dorfman (1943); Sobel and Groll (1959); Hwang (1975); Du et al (2000); Saraniti (2006); Feng et al (2010); Li et al (2014); Aprahamian et al (2018Aprahamian et al ( , 2019).…”

Static Risk-Based Group Testing Schemes Under Imperfectly Observable Risk

“…McMahan et al (2012) extend the analysis in Hwang (1975) to the case of imperfect tests, but conjecture that the problem of determining riskbased Dorfman testing schemes that minimize the expected number of tests, under imperfect tests and perfectly observable subject risk, is intractable and they develop heuristics. More recently, focusing on dynamic testing schemes, Aprahamian et al (2019) demonstrate that the generalization of Hwang's model that takes into account imperfect tests, but with perfectly observable subject risk, is in fact tractable, resolving the conjecture in the literature; establish the equivalence of the dynamic testing design problem to a specific form of a network flow problem, namely, the constrained-shortest path problem; and develop exact algorithms to determine optimal dynamic risk-based Dorfman testing schemes.…”

“…More specifically, both Dorfman's original model and the majority of the subsequent research impose unrealistic assumptions, such as perfect tests, that is, there are no classification errors, a homogeneous population, that is, the risk of the binary characteristic is identical across subjects, and infinite testing batch sizes (e.g., Dorfman 1943, Sobel et al 1959, Sterrett 1957. Although several papers extend the analysis of Dorfman testing schemes to imperfect tests (e.g., Graff and Roeloffs 1972, Johnson et al 1991, Kim et al 2007, McMahan et al 2012, there is very limited work on Dorfman testing for a heterogeneous population, that is, with subject-specific risk, and the few papers that consider a heterogeneous population (e.g., Hwang 1975, McMahan et al 2012, Aprahamian et al 2019) mainly do so under restrictive assumptions, including that subject risk is perfectly observable, or they determine testing schemes heuristically. In particular, Hwang (1975) determines optimal risk-based Dorfman testing schemes for a heterogeneous population, but under the assumption that the test is perfect (hence, the objective is to minimize the number of tests) and the subject risk is perfectly observable.…”

“…Although the aforementioned studies have improved our understanding of optimal risk-based Dorfman testing for a heterogeneous population, they leave out other important aspects of the problem, such as implementability of the testing scheme and uncertainty in subject risk estimates. For example, both Aprahamian et al (2019) and Hwang (1975) assume that the decision-maker can construct an optimal dynamic testing scheme, customized for each batch of subjects, and subject risk values are deterministic and perfectly observable by the tester. Although the first assumption may be justified in certain settings, in other settings the decisionmaker may not have the flexibility to modify the testing scheme for every batch, as discussed above.…”

“…Although the first assumption may be justified in certain settings, in other settings the decisionmaker may not have the flexibility to modify the testing scheme for every batch, as discussed above. McMahan et al (2012) consider a static testing scheme (same group sizes and assignment policy are used for every batch), which partially resolves the implementability issue, but this is done by: (i) ordering the subjects in nondecreasing order of their risk and simply setting the risk of each subject to the expected risk of the corresponding order statistics and (ii) determining testing schemes that attempt to minimize the expected number of tests using heuristics, which are based on structures that are not guaranteed to exist in an optimal testing scheme (e.g., group sizes are nonincreasing for a risk-ordered batch; see Aprahamian et al 2019). A lack of properties and algorithms for optimal static Dorfman testing schemes in this setting is not surprising, because various functions of order statistics (for batch sizes that are in the hundreds) arise in an exact formulation, substantially complicating the analysis.…”