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A negative binomial approximation in group testing

Part of: Distribution theory Communication, information Limit theorems

Published online by Cambridge University Press:  28 October 2022

Letian Yu ,
Fraser Daly  and
Oliver Johnson Open the ORCID record for Oliver Johnson [Opens in a new window]
Letian Yu
Affiliation:
Department of System Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin NT, Hong Kong. E-mail: letian.yu@link.cuhk.edu.hk
Fraser Daly
Affiliation:
Department of Actuarial Mathematics and Statistics, Heriot–Watt University, Edinburgh EH14 4AS, UK. E-mail: f.daly@hw.ac.uk
Oliver Johnson
Affiliation:
School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol BS8 1UG, UK. E-mail: o.johnson@bristol.ac.uk
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Abstract

We consider the problem of group testing (pooled testing), first introduced by Dorfman. For nonadaptive testing strategies, we refer to a nondefective item as “intruding” if it only appears in positive tests. Such items cause misclassification errors in the well-known COMP algorithm and can make other algorithms produce an error. It is therefore of interest to understand the distribution of the number of intruding items. We show that, under Bernoulli matrix designs, this distribution is well approximated in a variety of senses by a negative binomial distribution, allowing us to understand the performance of the two-stage conservative group testing algorithm of Aldridge.

Keywords

Group testingPooled testingStein–Chen method

MSC classification

Primary: 62E17: Approximations to distributions (nonasymptotic)
Secondary: 60F05: Central limit and other weak theorems 94A20: Sampling theory

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 37 , Issue 4 , October 2023 , pp. 973 - 996
Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

Aldridge, M.P. (2019). Individual testing is optimal for nonadaptive group testing in the linear regime. IEEE Transactions on Information Theory 65(4): 2058–2061.CrossRefGoogle Scholar
Aldridge, M.P. (2020). Conservative two-stage group testing. arXiv:2005.06617.Google Scholar
Aldridge, M.P. & Ellis, D. (2022). Pooled testing and its applications in the COVID-19 pandemic. In M. del Carmen Boado-Penas, J. Eisenberg, & S. Sahin (eds), Pandemics: Insurance and social protection. Cham: Springer, pp. 217–249.CrossRefGoogle Scholar
Aldridge, M.P., Baldassini, L., & Johnson, O.T. (2014). Group testing algorithms: Bounds and simulations. IEEE Transactions on Information Theory 60(6): 3671–3687.CrossRefGoogle Scholar
Aldridge, M.P., Johnson, O.T., & Scarlett, J.M. (2019). Group testing: An information theory perspective. Foundations and Trends in Information Theory 15(3–4): 196–392.CrossRefGoogle Scholar
Barbour, A.D., Holst, L., & Janson, S. (1992). Poisson approximation. Oxford: Clarendon Press.Google Scholar
Barbour, A.D., Gan, H.L., & Xia, A. (2015). Stein factors for negative binomial approximation in Wasserstein distance. Bernoulli 21(2): 1002–1013.CrossRefGoogle Scholar
Brown, T.C. & Phillips, M.J. (1999). Negative binomial approximation with Stein's method. Methodology and Computing in Applied Probability 1(4): 407–421.CrossRefGoogle Scholar
Chan, C.L., Che, P.H., Jaggi, S., & Saligrama, V. (2011). Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms. In Proceedings of the 49th Annual Allerton Conference on Communication, Control, and Computing, September, pp. 1832–1839.CrossRefGoogle Scholar
Coja-Oghlan, A., Gebhard, O., Hahn-Klimroth, M., & Loick, P. (2020). Optimal group testing. Proceedings of 33rd Conference on Learning Theory (COLT’20), pp. 1374–1388.Google Scholar
Cormode, G. & Muthukrishnan, S. (2005). What's hot and what's not: Tracking most frequent items dynamically. ACM Transactions on Database Systems (TODS) 30(1): 249–278.CrossRefGoogle Scholar
Denuit, M., Dhaene, J., & Ribas, C. (2001). Does positive dependence between individual risks increase stop-loss premiums? Insurance: Mathematics and Economics 28: 305–308.Google Scholar
Denuit, M., Lefèvre, C., & Utev, S. (2002). Measuring the impact of dependence between claims occurrences. Insurance: Mathematics and Economics 30: 1–19.Google Scholar
Dorfman, R. (1943). The detection of defective members of large populations. The Annals of Mathematical Statistics 14(4): 436–440.Google Scholar
Du, D. & Hwang, F. (1993). Combinatorial group testing and its applications. Series on Applied Mathematics. Singapore: World Scientific.CrossRefGoogle Scholar
Erlich, Y., Gilbert, A., Ngo, H., Rudra, A., Thierry-Mieg, N., Wootters, M., Zielinski, D., & Zuk, O. (2015). Biological screens from linear codes: Theory and tools. bioRxiv, p. 035352.CrossRefGoogle Scholar
Esary, J.D., Proschan, F., & Walkup, D.W. (1967). Association of random variables, with applications. The Annals of Mathematical Statistics 38: 1466–1474.CrossRefGoogle Scholar
Fortuin, C.M., Kasteleyn, P.W., & Ginibre, J. (1971). Correlation inequalities on some partially ordered sets. Communications in Mathematical Physics 22(2): 89–103.CrossRefGoogle Scholar
Gaunt, R.E., Pickett, A.M., & Reinert, G. (2017). Chi-square approximation by Stein's method with application to Pearson's statistic. The Annals of Applied Probability 27(2): 720–756.CrossRefGoogle Scholar
Harremoës, P., Johnson, O.T., & Kontoyiannis, I. (2010). Thinning, entropy and the law of thin numbers. IEEE Transactions on Information Theory 56(9): 4228–4244.CrossRefGoogle Scholar
Hong, E.S. & Ladner, R.E. (2002). Group testing for image compression. IEEE Transactions on Image Processing 11(8): 901–911.CrossRefGoogle ScholarPubMed
Hwang, F.K. (1972). A method for detecting all defective members in a population by group testing. Journal of the American Statistical Association 67(339): 605–608.CrossRefGoogle Scholar
Johnson, O.T., Aldridge, M.P., & Scarlett, J. (2019). Performance of group testing algorithms with near-constant tests-per-item. IEEE Transactions on Information Theory 65(2): 707–723.CrossRefGoogle Scholar
Kautz, W.H. & Singleton, R.C. (1964). Nonrandom binary superimposed codes. IEEE Transactions on Information Theory 10(4): 363–377.CrossRefGoogle Scholar
Luk, H.M. (1994). Stein's method for the gamma distribution and related statistical applications. PhD thesis, University of Southern California.Google Scholar
Mutesa, L., Ndishimye, P., Butera, Y., Souopgui, J., Uwineza, A., Rutayisire, R., Musoni, E., Rujeni, N., Nyatanyi, T., Ntagwabira, E., Semakula, M., Musanabaganwa, C., Nyamwasa, D., Ndashimye, M., Ujeneza, E., Mwikarago, I., Muvunyi, C., Mazarati, J., Nsanzimana, S., Turok, N., & Ndifon, W. (2021). A strategy for finding people infected with SARS-CoV-2: Optimizing pooled testing at low prevalence. Nature 589: 276–280. doi:10.1038/s41586-020-2885-5CrossRefGoogle ScholarPubMed
Polyanskiy, Y., Poor, H.V., & Verdú, S. (2010). Channel coding rate in the finite blocklength regime. IEEE Transactions on Information Theory 56(5): 2307–2359.CrossRefGoogle Scholar
Ross, N. (2011). Fundamentals of Stein's method. Probability Surveys 8: 210–293.CrossRefGoogle Scholar
Ross, N. (2013). Power laws in preferential attachment graphs and Stein's method for the negative binomial distribution. Advances in Applied Probability 45(3): 876–893.CrossRefGoogle Scholar
Shevtsova, I. (2011). On the absolute constants in the Berry-Esseen type inequalities for identically distributed summands. arXiv:1111.6554.Google Scholar
Weba, M. (1999). Bounds for the total variation distance between the binomial and the Poisson distribution in case of medium-sized success probabilities. Journal of Applied Probability 36(1): 97–104.CrossRefGoogle Scholar
Wolf, J.K. (1985). Born again group testing: Multiaccess communications. IEEE Transactions on Information Theory 31(2): 185–191.Google Scholar
Yelin, I., Aharony, N., Tamar, E.S., Argoetti, A., Messer, E., Berenbaum, D., Shafran, E., Kuzli, A., Gandali, N., Shkedi, O., Hashimshony, T., Mandel-Gutfreund, Y., Halberthal, M., Geffen, Y., Szwarcwort-Cohen, M., & Kishony, R. (2020). Evaluation of COVID-19 RT-qPCR test in multi sample pools. Clinical Infectious Diseases 71(16): 2073–2078.CrossRefGoogle ScholarPubMed