Environmental and Ecological Statistics

, Volume 22, Issue 1, pp 161–177 | Cite as

Combining and comparing multiple serial dilution assays of particles in solution: application to brucellosis in elk of the Greater Yellowstone Ecosystem

  • Jarrett J. Barber
  • Pritam Gupta
  • William Edwards
  • Kiona Ogle
  • Lance A. Waller


The concentration detection threshold (CDT) is the concentration of particles in solution beyond which a (serial dilution) assay detects particle presence. By our account, CDTs typically are not estimated but are fixed at some value. Setting a CDT to zero (\(d=0\)) implies perfect detection, a common assumption, and setting \(d>0\) gives results that are “denominated” in units of \(d\), i.e., are relative to the choice of \(d\). Using multiple, different serial dilution assays, each with its own CDT, we choose a “reference assay,” to which we assign a fixed CDT value, to obtain relative estimates of the remaining assays’ CDTs and the underlying particle concentration. We present the CDTs as a novel way to account for or to compare different serial dilution assays, “sensitivities”. We apply our methodology to data from four assays of the presence of bacterial (B. abortus) antibodies in the serum of elk in the Greater Yellowstone Ecosystem, where transmission of brucellosis—the disease ensuing from infection—to commercial livestock is managed by the Wyoming Game and Fish Department to avoid the primary symptom of abnormal fetal abortion. Results agree qualitatively with the more traditional notion of sensitivity as the true positive rate.


Brucellosis CDT Concentration detection threshold  Particle concentration Sensitivity Serial dilution assay 



The authors wish to thank Jessica Jennings-Gaines and Hally Killionof for compiling the data. The first two authors received partial support from U.S. Department of Agriculture Cooperative State Research, Education, and Extension Service (CSREES) Grant USDACSRE45232BA.


  1. Anscombe FJ (1948) The transformation of poisson, binomial and negative-binomial data. Biometrika 35(3/4):246–254.
  2. Block J, Chavance M (1998) A mixed model for repeated dilution assays. Biometrics 54(2):482–492.
  3. Brooks S, Gelman A (1998) General methods for monitoring convergence of iterative simulations. J Comput Graph Stat 7(4):434–455Google Scholar
  4. Chase GR, Hoel DG (1975) Serial dilutions: error effects and optimal designs. Biometrika 62(2):329–334.
  5. Cheville NF, McCullough DR, Paulson R (1998) Brucellosis in the Greater Yellowstone Area. National Academy PressGoogle Scholar
  6. Cochran WG (1950) Estimation of bacterial densities by means of the “most probable number”. Biometrics 6(2):105–116.
  7. Cornfield J (1954) Measurement and comparison of toxicites: the quantal response. In: Kempthorne O, Bancroft TA, Gowen JW, Lush JL (eds) Statistics and mathematics in biology. Iowa State College Press, AmesGoogle Scholar
  8. Cornfield J, Mantel M (1977) A discussion of “Estimation of safe doses in carcinogenic experiments” by Hartley and Sielken. Biometrics 33(1):21–24Google Scholar
  9. Cross PC, Edwards WH, Scurlock BM, Maichak EJ, Rogerson JD (2007) Effects of management and climate on elk brucellosis in the Greater Yellowstone Ecosystem. Ecol Appl 17:957–964CrossRefPubMedGoogle Scholar
  10. Davidian M, Giltinan DM (1995) Nonlinear models for repeated measurement data. Chapman & Hall/CRC, Boca RatonGoogle Scholar
  11. Dellaportas P, Stephens DA (1995) Bayesian analysis of errors-in-variables regression models. Biometrics 51(3):1085–1095.
  12. Elliott P, Wakefield J, Best N, Briggs D (2000) Spatial epidemiology. Oxford University Press, New York. iSBN 0-19-262941-7Google Scholar
  13. Finney DJ (1978) Statistical method in biological assays. MacMillan, New YorkGoogle Scholar
  14. Fisher R (1922) On the mathematical foundations of theoretical statistics. Philos Trans R Soc A 222:309–368CrossRefGoogle Scholar
  15. Gelman A, Chew GL, Shnaidman M (2004) Bayesian analysis of serial dilution assays. Biometrics 60(2):407–417.
  16. Giltinan D, Davidian M (1994) Assays for recombinant proteins: a problem in non-linear calibration. Stat Med 13:1165–1179CrossRefPubMedGoogle Scholar
  17. Hamilton MA, Rinaldi MG (1988) Descriptive statistical analyses of serial dilution data. Stat Med 7:535–544CrossRefPubMedGoogle Scholar
  18. Lee MLT, Whitmore GA (1999) Statistical inference for serial dilution assay data. Biometrics 55(4):1215–1220CrossRefPubMedGoogle Scholar
  19. McCrady MH (1915) The numerical interpretation of fermentation-tube results. J Infect Dis 17(1):183–212.
  20. McCullagh P, Nelder J (1989) Generalized linear models, 2nd edn. Monographs on statistics and applied probability, vol 37. Chapman & Hall, LondonGoogle Scholar
  21. Mehrabi Y, Matthews JNS (1998) Implementable Bayesian designs for limiting dilution assays. Biometrics 54(4):1398–1406.
  22. Morton JK, Thorne ET, Thomas GM (1981) Brucellosis in elk III: serologic evaluation. J Wildl Dis 17(1):23–31CrossRefPubMedGoogle Scholar
  23. Nelder J, Mead R (1965) A simplex algorithm for function minimization. Comput J 7:308–313CrossRefGoogle Scholar
  24. Plummer M (2003) JAGS: a program for analysis of Bayesian graphical models using Gibbs sampling. In: Proceedings of the 3rd international workshop on distributed statistical computing (DSC 2003)Google Scholar
  25. R Core Team (2012) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria., ISBN 3-900051-07-0
  26. Ragan VE (2002) The Animal and Plant Health Inspection Service (APHIS) brucellosis eradication program in the united states. Veterinary Microbiol 90(1–4):11–18. doi: 10.1016/S0378-1135(02)00240-7.
  27. Ridout M (2005) Serial dilution assay. Encycl Biostat 4079–4084. doi: 10.1002/0470011815.b2a06022
  28. Roffe TJ, Jones LC, Coffin K, Drew ML, Sweeney SJ, Hagius SD, Elzer PH, Davis D (2004) Efficacy of single calfhood vaccination of elk with Brucella abortus strain 19. J Wildl Manag 68:830–836CrossRefGoogle Scholar
  29. Smith BL (2001) Winter feeding of elk in western North America. J Wildl Manag 65:173–190CrossRefGoogle Scholar
  30. Thorne ET (2001) Brucellosis. In: Williams ES, Barker IK (eds) Infectious diseases of wild mammals. Iowa State Press, Ames, pp 372–395CrossRefGoogle Scholar
  31. Thorne E, Morton J, Blunt F, Dawson H (1978) Brucellosis in elk ii. Clinical effects and means of transmission as determined through artificial infections. J Wildl Dis 14:280–291CrossRefPubMedGoogle Scholar
  32. Thorne ET, Smith SG, Aune K, Hunter D, Roffe TJ (1997) Brucellosis: the disease in elk. In: Thorne ET, Boyce MS, Nicolletti P, Kreeger TJ (eds) Brucellosis, bison, elk, and cattle in the Greater Yellowstone Area: defining the problem, exploring solution. Wyoming Game and Fish Department, Cheyenne, pp 33–44Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Jarrett J. Barber
    • 1
  • Pritam Gupta
    • 2
  • William Edwards
    • 3
  • Kiona Ogle
    • 4
  • Lance A. Waller
    • 5
  1. 1.School of Mathematical and Statistical SciencesArizona State UniversityTempeUSA
  2. 2.Novartis Healthcare Pvt. Ltd.HyderabadIndia
  3. 3.Wyoming Game and Fish DepartmentLaramieUSA
  4. 4.School of Life SciencesArizona State UniversityTempeUSA
  5. 5.Department of Biostatistics and Bioinformatics, Rollins School of Public HealthEmory UniversityAtlantaUSA

Personalised recommendations