Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Measurement Science Review

The Journal of Institute of Measurement Science of Slovak Academy of Sciences

6 Issues per year

IMPACT FACTOR 2016: 1.344

CiteScore 2016: 1.88

SCImago Journal Rank (SJR) 2016: 0.495
Source Normalized Impact per Paper (SNIP) 2016: 1.419

Open Access
See all formats and pricing
More options …
Volume 14, Issue 2


Linear Mixed Models: Gum and Beyond

Barbora Arendacká / Angelika Täubner / Sascha Eichstädt / Thomas Bruns / Clemens Elster
Published Online: 2014-05-08 | DOI: https://doi.org/10.2478/msr-2014-0009


In Annex H.5, the Guide to the Evaluation of Uncertainty in Measurement (GUM) [1] recognizes the necessity to analyze certain types of experiments by applying random effects ANOVA models. These belong to the more general family of linear mixed models that we focus on in the current paper. Extending the short introduction provided by the GUM, our aim is to show that the more general, linear mixed models cover a wider range of situations occurring in practice and can be beneficial when employed in data analysis of long-term repeated experiments. Namely, we point out their potential as an aid in establishing an uncertainty budget and as means for gaining more insight into the measurement process. We also comment on computational issues and to make the explanations less abstract, we illustrate all the concepts with the help of a measurement campaign conducted in order to challenge the uncertainty budget in calibration of accelerometers.

Keywords: Linear mixed models; uncertainty; GUM; ANOVA; random effects


  • [1] BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, OIML. (2008). Guide to the expression of uncertainty in measurement (GUM 1995 with minor corrections). JCGM 100:2008. http://www.bipm.org/utils/common/documents/jcgm/JCGM_100_2008_E.pdf.Google Scholar

  • [2] ISO. (2005). Measurement uncertainty for metrological applications - repeated measurements and nested experiments. ISO/TS 21749:2005.Google Scholar

  • [3] Sakurai, H., Ehara, K. (2011). Evaluation of uncertainties in femtoampere current measurement for the number concentration standard of aerosol nanoparticles. Measurement Sience and Technology, 22, 024009.Google Scholar

  • [4] Lee, J., Yang, J., Yang, S., Kwak, J. (2007). Uncertainty analysis and ANOVA for the measurement reliability estimation of altitude engine test. Journal of Mechanical Science and Technology, 21 (4), 664-671.Web of ScienceGoogle Scholar

  • [5] Wang, D.Y., Lin, K.-H., Lo Huang, M.-N. (2002). Variability studies on EMI data for electronic, telecommunication and information technology equipment. IEEE Transactions on Electromagnetic Compatibility, 44 (2), 385-393.CrossrefGoogle Scholar

  • [6] Toman, B. (2006). Linear statistical models in the presence of systematic effects requiring a Type B evaluation of uncertainty. Metrologia, 43 (1), 27-33.CrossrefGoogle Scholar

  • [7] von Martens, H.-J., Link, A., Schlaak, H.-J., T¨aubner, A., Wabinski, W., G¨obel, U. (2004). Recent advances in vibration and shock measurements and calibrations using laser interferometry. In Sixth International Conference on Vibration Measurements by Laser Techniques: Advances and Applications. SPIE, Vol. 5503, 1-19.Google Scholar

  • [8] ISO. (1999). Methods for the calibration of vibration and shock transducers - Part 11: Primary vibration calibration by laser interferometry. ISO 16063-11:1999.Google Scholar

  • [9] Jackett, R.J., Barham, R.G. (2013). Phase sensitivity uncertainty in microphone pressure reciprocity calibration. Metrologia, 50 (2), 170-179.Web of ScienceCrossrefGoogle Scholar

  • [10] Pinheiro, J.C., Bates, D.M. (2000). Mixed-effects Models in S and S-PLUS. Springer.Google Scholar

  • [11] Searle, S.R., Casella, G., McCulloch, C.E. (1992). Variance Components. John Wiley & Sons.Google Scholar

  • [12] Burdick, R.K., Graybill, F.A. (1992). Confidence Intervals on Variance Components. Marcel Dekker.Google Scholar

  • [13] West, B.T., Welch, K.B., Gałecki, A.T. (2007). Linear Mixed Models: A Practical Guide Using Statistical Software. Chapman and Hall/CRC.Google Scholar

  • [14] Witkovský, V. (2012). Estimation, testing, and prediction regions of the fixed and random effects by solving the Henderson’s mixed model equations. Measurement Science Review, 12 (6), 234-248.Web of ScienceGoogle Scholar

  • [15] Witkovský, V. (2000). mixed.m - Matlab algorithm for solving Henderson’s mixed model equations. http://www.mathworks.com/matlabcentral/fileexchange/200-mixed.Google Scholar

  • [16] Gelman, A., Hill, J. (2009). Data Analysis Using Regression and Multilevel /Hierarchical Models. Cambridge University Press.Google Scholar

  • [17] W¨ubbeler, G., Mickan, B., Elster, C. (2013). Bayesian analysis of sonic nozzle calibration data. In: FLOMEKO 2013: 16th International Flow Measurement Conference, 24-26 September 2013. CEESI.Google Scholar

About the article

Received: 2013-12-16

Accepted: 2014-03-28

Published Online: 2014-05-08

Published in Print: 2014-04-01

Citation Information: Measurement Science Review, Volume 14, Issue 2, Pages 52–61, ISSN (Online) 1335-8871, DOI: https://doi.org/10.2478/msr-2014-0009.

Export Citation

© by Barbora Arendacká. This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

N. Garg, K. Soni, A. Kumar, and T. K. Saxena
MAPAN, 2015, Volume 30, Number 2, Page 91

Comments (0)

Please log in or register to comment.
Log in