Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Bayesian analysis of homogeneity studies in the production of reference materials

  • 434 Accesses

  • 4 Citations

Abstract

For almost two decades, the batch homogeneity in the production of reference materials has been evaluated using analysis of variance (ANOVA) to determine the between-bottle standard deviation. This approach replaced at that time the use of the F-test in ANOVA to determine whether the ratio of the mean squares \({MS}_{\mathrm {between}}/{MS}_{\mathrm {within}}\) is statistically significant. Problems arise when \({MS}_{\mathrm {between}} < {MS}_{\mathrm {within}}\), because classical ANOVA provides a negative between-bottle variance, which is then often set to zero. By using a Bayesian hierarchical model, based on the same assumptions as traditional ANOVA, we show that even if \({MS}_{\mathrm {between}} < {MS}_{\mathrm {within}}\), there can be a relevant level of between-bottle inhomogeneity to account for. The Bayesian analysis produces a nonzero value for the between-bottle standard deviation, which dismisses the practice of setting this standard deviation to 0. At the same time, it dismisses the current guidance given in ISO Guide 35 under these circumstances. Finally, it is shown that traditional ANOVA, meta-analysis methods and Bayesian analysis give very similar answers as long as \({MS}_{\mathrm {between}} > {MS}_{\mathrm {within}}\), so there is no need to discourage using these methods in favour of a Bayesian analysis, provided that the repeatability of the measurement method used to conduct the between-bottle homogeneity study is sufficient to characterise the dispersion across the bottles (items) in a batch of a reference material.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

References

  1. 1.

    Pauwels J, Lamberty A, Schimmel H (1998) Homogeneity testing of reference materials. Accred Qual Assur 3(2):51–55. doi:10.1007/s007690050186

  2. 2.

    Pauwels J, van der Veen AMH, Lamberty A, Schimmel H (2000) Evaluation of uncertainty of reference materials. Accred Qual Assur 5(3):95–99. doi:10.1007/s007690050020

  3. 3.

    van der Veen AMH, Pauwels J (2000) Uncertainty calculations in the certification of reference materials. 1. Principles of analysis of variance. Accred Qual Assur 5(12):464–469. doi:10.1007/s007690000237

  4. 4.

    van der Veen AMH, Linsinger TP, Pauwels J (2001) Uncertainty calculations in the certification of reference materials. 2. Homogeneity study. Accred Qual Assur 6(1):26–30. doi:10.1007/s007690000238

  5. 5.

    ISO Guide 35 (2006) Reference materials—general and statistical principles for certification. International Organization for Standardization, Geneva

  6. 6.

    ISO Guide 34 (2009) General requirements for the competence of reference material producers. International Organization for Standardization, Geneva

  7. 7.

    ISO 17034 (2016) General requirements for the competence of reference material producers. International Organization for Standardization, Geneva

  8. 8.

    ISO 13528 (2015) Statistical methods for use in proficiency testing by interlaboratory comparison. International Organization for Standardization, Geneva

  9. 9.

    ISO/IEC 17043 (2010) Conformity assessment—general requirements for proficiency testing. International Organization for Standardization, Geneva

  10. 10.

    Ulrich JC, Sarkis JES, Hortellani MA (2015) Homogeneity study of candidate reference material in fish matrix. J Phys Conf Ser 575:012,040. doi:10.1088/1742-6596/575/1/012040

  11. 11.

    Linsinger TPJ, Pauwels J, van der Veen AMH, Schimmel H, Lamberty A (2001) Homogeneity and stability of reference materials. Accred Qual Assur 6(1):20–25. doi:10.1007/s007690000261

  12. 12.

    Gelman A, Carlin J, Stern H, Dunson D, Vehtari A, Rubin D (2013) Bayesian data analysis, 3rd edn. Chapman and Hall/CRC, Boca Raton

  13. 13.

    Ellison SLR (2015) Homogeneity studies and ISO Guide 35:2006. Accred Qual Assur 20(6):519–528. doi:10.1007/s00769-015-1162-z

  14. 14.

    Hoff PD (2009) A first course in Bayesian statistical methods. Springer, New York. doi:10.1007/978-0-387-92407-6

  15. 15.

    BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, OIML (2008) Guide to the expression of uncertainty in measurement, JCGM 100:2008, GUM 1995 with minor corrections. BIPM, Sèvres

  16. 16.

    Beelen R (2016) Preparation of a homogeneous set of PT materials. Technical report, VSL, Delft

  17. 17.

    DerSimonian R, Laird N (1986) Meta-analysis in clinical trials. Controlled Clin Trials 7(3):177–188. doi:10.1016/0197-2456(86)90046-2. http://www.sciencedirect.com/science/article/pii/0197245686900462

  18. 18.

    ISO 6143 (2001) Gas analysis—comparison methods for determining and checking the composition of calibration gas mixtures, 2nd edn. International Organization for Standardization, Geneva

  19. 19.

    ISO 6974-1 (2012) Natural gas—determination of composition with defined uncertainty by gas chromatography—part 1: guidelines for tailored analysis. International Organization for Standardization, Geneva

  20. 20.

    ISO 6974-2 (2012) Natural gas—determination of composition with defined uncertainty by gas chromatography—part 2: measuring-system characteristics and statistics for processing of data. International Organization for Standardization, Geneva

  21. 21.

    Gelman A (2006) Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper). Bayesian Anal 1:515–534. doi:10.1214/06-ba117a

  22. 22.

    DerSimonian R, Kacker R (2007) Random-effects model for meta-analysis of clinical trials: an update. Contemp Clin Trials 28(2):105–114. doi:10.1016/j.cct.2006.04.004

  23. 23.

    Kacker RN (2004) Combining information from interlaboratory evaluations using a random effects model. Metrologia 41(3):132–136. doi:10.1088/0026-1394/41/3/004

  24. 24.

    Rivier C, Désenfant M, Crozet M, Rigaux C, Roudil D, Tufféry B, Ruas A (2014) Use of an excess variance approach for the certification of reference materials by interlaboratory comparison. Accred Qual Assur 19(4):269–274. doi:10.1007/s00769-014-1066-3

  25. 25.

    Stan Developers Team (2016) Stan modeling language. User’s guide and reference manual. http://mc-stan.org/documentation/

  26. 26.

    Klauenberg K, Wübbeler G, Mickan B, Harris P, Elster C (2015) A tutorial on bayesian normal linear regression. Metrologia 52(6):878–892. doi:10.1088/0026-1394/52/6/878

  27. 27.

    O’Hagan A (2014) Eliciting and using expert knowledge in metrology. Metrologia 51(4):S237–S244. doi:10.1088/0026-1394/51/4/s237

  28. 28.

    R Core Team (2016) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/

  29. 29.

    Carpenter B, Gelman A, Hoffman M, Lee D, Goodrich B, Betancourt M, Brubaker MA, Guo J, Li P, Riddell A (2016) Stan: a probabilistic programming language. J Stat Softw (in press)

  30. 30.

    Klauenberg K, Elster C (2016) Markov chain monte carlo methods: an introductory example. Metrologia 53(1):S32–S39. doi:10.1088/0026-1394/53/1/s32

  31. 31.

    Bich W, Cox MG, Dybkaer R, Elster C, Estler WT, Hibbert B, Imai H, Kool W, Michotte C, Nielsen L, Pendrill L, Sidney S, van der Veen AMH, Wöger W (2012) Revision of the "Guide to the expression of Uncertainty in Measurement". Metrologia 49(6):702–705 http://stacks.iop.org/0026-1394/49/i=6/a=702

Download references

Acknowledgements

The work presented in this paper was funded by the Ministry of Economic Affairs of the Netherlands.

Author information

Correspondence to Adriaan M. H. van der Veen.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

van der Veen, A.M.H. Bayesian analysis of homogeneity studies in the production of reference materials. Accred Qual Assur 22, 307–319 (2017). https://doi.org/10.1007/s00769-017-1292-6

Download citation

Keywords

  • Measurement uncertainty
  • Bayesian analysis
  • Hierarchical model
  • Meta-analysis
  • Analysis of variance
  • Reference material
  • Homogeneity study