A Maximum Likelihood Estimator for the Prevalence Rate Using Pooled Sample Tests

  • João Paulo MartinsEmail author
  • Rui Santos
  • Miguel Felgueiras
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 136)


Since Dorfman’s seminal work, research on methodologies involving pooled sample tests has increased significantly. Moreover, the use of pooled samples refers not only to the classification problem (identifying all the infected individuals in a population), but also refers to the problem of estimating the prevalence rate \(p\), as Sobel and Elashoff stated. The use of compound tests is not restricted to hierarchical algorithms where the most common example is Dorfman’s two-stage procedure. Matrix schemes such as the square array algorithm or multidimensional matrices schemes in certain cases outperform Dorfman’s procedure. Maximum likelihood estimates are quite difficult to compute when a procedure does not classify all individuals. This paper presents two innovative methods to compute maximum likelihood estimates in both type of procedures.


Compound tests Maximum likelihood estimator Prevalence rate 



Research partially sponsored by national funds through the Fundação Nacional para a Ciência e Tecnologia, Portugal — FCT under the project (PEst-OE/MAT/UI0006/2014).


  1. 1.
    Bilder, C.R., Zang, B., Schaarschmidt, F., Tebbs, J.M.: binGroup: a package for group testing. R J. 2, 56–60 (2010)Google Scholar
  2. 2.
    Chen, C.L., Swallow, W.H.: Using group testing to estimate a proportion, and to test the binomial model. Biometrics 46, 1035–1046 (1990)CrossRefGoogle Scholar
  3. 3.
    Dorfman, R.: The detection of defective members in large populations. Ann. Math. Stat. 14, 436–440 (1943)CrossRefGoogle Scholar
  4. 4.
    Finucan, H.M.: The blood testing problem. Appl. Stat. 13, 43–50 (1964)CrossRefGoogle Scholar
  5. 5.
    Garner, F.C., Stapanian, M.A., Yfantis, E.A., Williams, L.R.: Probability estimation with sample compositing techniques. J. Off. Stat. 5, 365–374 (1989)Google Scholar
  6. 6.
    Hung, M., Swallow, W.H.: Robustness of group testing in the estimation of proportions. Biometrics 55, 231–237 (1999)CrossRefzbMATHGoogle Scholar
  7. 7.
    Hwang, F.K.: Group testing with a dilution effect. Biometrika 63, 671–673 (1976)CrossRefzbMATHGoogle Scholar
  8. 8.
    Kim, H., Hudgens, M., Dreyfuss, J., Westreich, D., Pilcher, D.: Comparison of group testing algorithms for case identification in the presence of testing errors. Biometrics 63, 1152–1163 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Kim, H., Hudgens, M.: Three-dimensional array-based group testing algorithms. Biometrics 65, 903–910 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Lancaster, V.A., Keller-McNulty, S.: A review of composite sampling methods. J. Am. Stat. Assoc. 93, 1216–1230 (1998)CrossRefGoogle Scholar
  11. 11.
    Liu, S.C., Chiang, K.S., Lin, C.H., Chung, W.C., Lin, S.H., Yang, L.C.: Cost analysis in choosing group size when group testing for Potato virus Y in the presence of classification errors. Ann. Appl. Biol. 159, 491–502 (2011)CrossRefGoogle Scholar
  12. 12.
    Loyer, M.W.: Bad probability, good statistics, and group testing for binomial estimation. Am. Stat. 37, 57–59 (1983)Google Scholar
  13. 13.
    Martins, J.P., Felgueiras, M., Santos, R.: Meta-analysis techniques applied in prevalence rate estimation. Discuss. Math. Probab. Stat. 33, 79–97 (2013)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Santos, R., Pestana, D., Martins, J.P.: Extensions of Dorfman’s theory. In: Oliveira, P.E., Graa, M., Henriques, C., Vichi, M. (eds.) Studies in Theoretical and Applied Statistics, Recent Developments in Modeling and Applications in Statistics, pp. 179–189. Springer, New York (2013)Google Scholar
  15. 15.
    Sobel, M., Elashoff, R.M.: Group testing with a new goal, estimation. Biometrika 62, 181–193 (1975)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Sterret, A.: On the detection of defective members of large populations. Ann. Math. Stat. 28, 1033–1036 (1957)CrossRefGoogle Scholar
  17. 17.
    Wein, L.M., Zenios, S.A.: Pooled testing for HIV screening: capturing the dilution effect. Oper. Res. 44, 543–569 (1996)CrossRefzbMATHGoogle Scholar
  18. 18.
    Zenios, S., Wein, L.: Pooled testing for HIV prevalence estimation exploiting the dilution effect. Stat. Med. 17, 1447–1467 (1998)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • João Paulo Martins
    • 1
    • 2
    Email author
  • Rui Santos
    • 1
    • 2
  • Miguel Felgueiras
    • 1
    • 2
  1. 1.School of Technology and ManagementPolytechnic Institute of LeiriaLeiriaPortugal
  2. 2.CEAUL – Center of Statistics and Applications of the University of LisbonUniversity of LisbonLisbonPortugal

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