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Pool testing with dilution effects and heterogeneous priors

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Abstract

The Dorfman pooled testing scheme is a process in which individual specimens (e.g., blood, urine, swabs, etc.) are pooled and tested together; if the merged sample tests positive for infection, then each specimen from the pool is tested individually. Through this procedure, laboratories can reduce the expected number of tests required to screen the population, as individual tests are only carried out when the pooled test detects an infection. Several different partitions of the population can be used to form the pools. In this study, we analyze the performance of ordered partitions, those in which subjects with similar probability of infection are pooled together. We derive sufficient conditions under which ordered partitions outperform other types of partitions in terms of minimizing the expected number of tests, the expected number of false negatives, and the expected number of false positive classifications. These sufficient conditions can be easily verified in practical applications once the dilution effect has been estimated. We also propose a measure of equity and present conditions under which this measure is maximized by ordered partitions.

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Notes

  1. Bateman et al. [9] estimate that the probability of detecting COVID-19 from an infected subject is 6% lower when his sample is diluted with the sample of 4 healthy subjects. The percentage reduction in the precision of the test is 8% when the infected sample is further diluted with 9 non-infected samples, and 18% when the infected sample is diluted with 49 non-infected specimens.

  2. See Section 9 for details.

  3. Details provided in the online Appendix.

  4. Hrayer et al. [5] had previously shown that, when all the pools have the same size \(k\ge 2\), and the following condition holds

    $$\begin{aligned} I\frac{\partial ^2 h(I,k)}{\partial I^2}+2 \frac{\partial h(I,k)}{\partial I}\ge 0\quad \forall I\in [0,k], \end{aligned}$$

    then grouping subjects according to an ordered partition minimizes the expected number of false negatives. In the online Appendix we show that this condition implies that hypothesis 1 holds, but the converse is not necessarily true.

  5. The number of different partitions from a set S of 20 individuals is approximately \(5.17e+13\).

  6. Hrayer et al. [4] shows that the number of different ordered partitions of S is \(2^{|S|-1}\).

  7. Indeed, consider \(u=(u_1,u_2,u_3,u_4)=(1,1,2,4)\) and \(u=(\tilde{u_1},\tilde{u_2},\tilde{u_3},\tilde{u_4})=(36/37,36/37,3,3)\). Then clearly \(\min u_i>\min \tilde{u_i}\) and \(\sum u_i>\sum \tilde{u_i}\). However \(\sum \frac{u_i^{1-\alpha }}{1-\alpha }<\sum \frac{\tilde{u_i}^{1-\alpha }}{1-\alpha }\) for \(\alpha =2\).

  8. For simplification, this measure does not incorporate health-related reduction in quality of life, nor costs associated with transmissions that a true positive would have averted.

  9. https://wonder.cdc.gov/std-race-age.html

  10. In part, many end up not being screened because they exhibit no symptoms: approximately 75% of women and 50% of infected men exhibit no symptoms [14].

  11. Having a large number of subjects being tested helps to smooth out the graphs for instances in which n is not a multiple of k.

  12. For simplification, our approach does not incorporate costs associated with transmissions that a true positive would have averted.

  13. If the dilution effect is too strong, pool testing may be deemed too imprecise to be considered as a good alternative to individual testing.

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Acknowledgements

I am grateful to Dr. Shane Allwright for authorizing the usage of her dataset regarding the prevalence of HBV among Irish prisons’ inmates. I would also like to thank all of those who provided invaluable feedback to this research: especial thanks goes to Luis Martins Abreu, Michael Kuhlman, Joshua M. Tebbs, Paulo Saraiva and two anonymous referees.

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  1. Business School, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Macul, Santiago, Chile

    Gustavo Quinderé Saraiva

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Saraiva, G.Q. Pool testing with dilution effects and heterogeneous priors. Health Care Manag Sci 26, 651–672 (2023). https://doi.org/10.1007/s10729-023-09650-7

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