Bivariate Data and Calibration of Experimental Systems

  • Surendra P. VermaEmail author


In this chapter, statistical methodology for bivariate data and calibration of experimental systems is developed. Experimental systems include analytical instruments useful in geochemistry. From the freely available BiDASys software, we can apply both the conventional ordinary least-squares linear regression (OLR) and the new uncertainty weighted least-squares linear regression (UWLR) models. BiDASys was used for achieving and comparing the OLR and UWLR models for the calibration of a high-performance liquid chromatography equipment for the determination of rare-earth elements. Equations are provided for both regressions. The advantages of the UWLR model over the OLR are clearly documented. This is followed by linearity tests, which are useful for deciding whether a linear or a curvilinear fit is more appropriate. ANOVA for the evaluation of fitting is finally presented and exemplified from citations of literature on new precise and accurate critical values for the F and t tests.


  1. Andaverde, J., Verma, S. P., & Santoyo, E. (2005). Uncertainty estimates of static formation temperatures in boreholes and evaluation of regression models. Geophysical Journal International, 160, 1112–1122.CrossRefGoogle Scholar
  2. Asuero, A. G., & González, G. (2007). Fitting straight lines with replicated observations by linear regression. III weighting data. Critical Reviews in Analytical Chemistry, 37, 143–172.CrossRefGoogle Scholar
  3. Baumann, K. (1997). Regression and calibration for analytical separation techniques. Part II: Validation, weighted and robust regression. Process Control and Quality, 10, 75–112.Google Scholar
  4. Bevington, P. R. (1969). Data reduction and error analysis for the physical sciences. New York: Mc-Graw Hill Book Company.Google Scholar
  5. Bevington, P. R., & Robinson, D. K. (2003). Data reduction and error analysis for the physical sciences. Boston: McGraw Hill.Google Scholar
  6. Cruz-Huicochea, R., & Verma, S. P. (2013). New critical values for F and their use in the ANOVA and Fisher’s F tests for evaluating geochemical reference material granite G-2 (U.S.A.) and igneous rocks from the Eastern Alkaline Province (Mexico). Journal of Iberian Geology, 39, 13–30.CrossRefGoogle Scholar
  7. Draper, N. R., & Smith, H. (1998). Applied regression analysis (3rd ed.). New York: John Wiley & Sons.Google Scholar
  8. Guevara, M., Verma, S. P., Velasco-Tapia, F., Lozano-Santa Cruz, R., & Girón, P. (2005). Comparison of linear regression models for quantitative geochemical analysis: an example using x-ray fluorescence spectrometry. Geostandards and Geoanalytical Research, 29, 271–284.CrossRefGoogle Scholar
  9. Hinich, M. J., & Talwar, P. P. (1975). A simple method for robust regression. Journal of the American Statistical Association, 70, 113–119.CrossRefGoogle Scholar
  10. Kataoka, Y. (1989). Standardless x-ray fluorescence spectrometry (Fundamental Parameter Method using Sensitivity Library). The Rigaku Journal, 7, 33–40.Google Scholar
  11. Mahon, K. L. (1996). The New “York” regression: application of an improved statistical method to geochemistry. International Geology Review, 38, 293–303.CrossRefGoogle Scholar
  12. Mashima, H. (2016). XRF analyses of major and trace elements in silicate rocks calibrated with synthetic standard samples. Natural Resource Environment and Humans, 6, 39–50.Google Scholar
  13. Meier, P. C., & Zünd, R. E. (1992). Statistical methods in analytical chemistry. New York: John Wiley & Sons Inc.Google Scholar
  14. Mendenhall, W., & Sincich, T. L. (1996). Second Course in Statistics, A: Regression Analysis. Upper Saddle River, New Jersey: Prentice Hall.Google Scholar
  15. Miller, J. M. (1991). Basic statistical methods for analytical chemistry. Part 2. Calibration and regression methods. A review. Analyst, 116, 3–14.CrossRefGoogle Scholar
  16. Miller, J. N., & Miller, J. C. (2005). Statistics and chemometrics for analytical chemistry (5th ed.). Essex CM20 2JE, England: Pearson Prentice Hall.Google Scholar
  17. Miller, J. N., & Miller, J. C. (2010). Statistics and chemometrics for analytical chemistry (6th ed.). Essex CM20 2JE, England: Pearson Prentice Hall.Google Scholar
  18. Mocak, J., Bond, A. M., Mitchell, S., & Scollary, G. (1997). A statistical overview of standard (IUPAC and ACS) and new procedures for determining the limits of detection and quantification: application to voltametric and stripping techniques. Pure & Applied Chemistry, 69, 297–328.CrossRefGoogle Scholar
  19. Otto, M. (1999). Chemometrics. Statistics and computer application in analytical chemistry. Weinheim: Wiley-VCH.Google Scholar
  20. Pearson, K. (1897). Mathematical contribution to the theory of evolution. - on a form of spurious correlation which may arise when indices are used in the measurement of organs. Proceedings of the Royal Society of London, 60, 489–502.CrossRefGoogle Scholar
  21. Potts, P. J. (1987). A handbook of silicate rock analysis. Glasgow: Blackie.CrossRefGoogle Scholar
  22. Rollinson, H. R. (1993). Using geochemical data: evaluation, presentation, interpretation. Essex: Longman Scientific Technical.Google Scholar
  23. Rosales Rivera, M. (2018). Desarrollo de herramientas estadísticas computacionales con nuevos valores críticos generados por simulación computacional. In: Instituto de Investigación en Ciencias Básicas y Aplicadas, Centro de Investigación en Ciencias, pp. 105. Cuernavaca, Morelos, Mexico: Universidad Autónoma del Estado de Morelos.Google Scholar
  24. Rosales-Rivera, M., Díaz-González, L., & Verma, S. P. (2018). A new online computer program (BiDASys) for ordinary and uncertainty weighted least-squares linear regressions: case studies from food chemistry. Revista Mexicana de Ingeniería Química, 17, 507–522.CrossRefGoogle Scholar
  25. Rosales-Rivera, M., Díaz-González, L., & Verma, S. P. (2019). Evaluation of nine USGS reference materials for quality control through Univariate Data Analysis System, UDASys3. Arabian Journal of Geosciences, 12(2), 40. Scholar
  26. Rousseeuw, P. J., & Leroy, A. M. (1987). Robust regression and outlier detection. New York: John Wiley & Sons.CrossRefGoogle Scholar
  27. Ryan, T. P. (1997). Modern regression analysis. New York: Wiley.Google Scholar
  28. Santoyo, E., & Verma, S. P. (2003). Determination of lanthanides in synthetic standards by reversed-phase high-performance liquid chromatography with the aid of a weighted least-squares regression model: estimation of method sensitivities and detection limits. Journal of Chromatography A, 997, 171–182.CrossRefGoogle Scholar
  29. Sayago, A., & Asuero, A. G. (2004). Fitting straight lines with replicated observations by linear regression: Part II. testing for homogeneity of variances. Critical Reviews in Analytical Chemistry, 34, 133–146.CrossRefGoogle Scholar
  30. Sayago, A., Boccio, M., & Asuero, A. G. (2004). Fitting straight lines with replicated observations by linear regression: the least squares postulates. Critical Reviews in Analytical Chemistry, 34, 39–50.CrossRefGoogle Scholar
  31. Taylor, J. K. (1990). Statistical techniques for data analysis. Michigan, USA: Lewis Publishers Inc.Google Scholar
  32. Tellinghuisen, J. (2007). Weighted least-squares in calibration: what difference does it make? Analyst, 132, 536–543.CrossRefGoogle Scholar
  33. Verma, S. P. (1991). Usefulness of liquid chromatography for determination of thirteen rare-earth elements in rocks and minerals. Lanthanide and Actinide Research, 3, 237–257.Google Scholar
  34. Verma, S. P. (2005). Estadística básica para el manejo de datos experimentales: aplicación en la Geoquímica (Geoquimiometría). México, D.F.: UNAM.Google Scholar
  35. Verma, S. P. (2012). Geochemometrics. Revista Mexicana de Ciencias Geológicas, 29, 276–298.Google Scholar
  36. Verma, S. P. (2016). Análisis estadístico de datos composicionales. CDMX: Universidad Nacional Autónoma de México.Google Scholar
  37. Verma, S. P., & Cruz-Huicochea, R. (2013). Alternative approach for precise and accurate Student´s t critical values and application in geosciences. Journal of Iberian Geology, 39, 31–56.Google Scholar
  38. Verma, S. P., & Santoyo, E. (2005). Is odd-even effect reflected in detection limits? Accreditation and Quality Assurance, 10, 144–148.CrossRefGoogle Scholar
  39. Verma, S. P., Verma, S. K., Rivera-Gómez, M. A., Torres-Sánchez, D., Díaz-González, L., Amezcua-Valdez, A., Rivera-Escoto, B. A., Rosales-Rivera, M., Armstrong-Altrin, J. S., López-Loera, H., Velasco-Tapia, F. & Pandarinath, K. (2018). Statistically coherent calibration of X-ray fluorescence spectrometry for major elements in rocks and minerals. Journal of Spectroscopy, 2018, Article ID 5837214, 13p, Scholar
  40. Verma, S. P., Rosales-Rivera, M., Rivera-Gómez, M. A. & Verma, S. K. (2019). Comparison of matrix-effect corrections for ordinary and uncertainty weighted linear regressions and determination of major element mean concentrations and total uncertainties of 62 international geochemical reference materials from wavelength-dispersive X-ray fluorescence spectrometry. In: Colloquium Spectroscopicum Internationale XLI (CSI XLI) and I Latin-American Meeting on Laser Induced Breakdown Spectroscopy (I LAMLIBS). Mexico City.Google Scholar
  41. Zorn, M. E., Gibbons, R. D., & Sonzogni, W. C. (1997). Weighted least-squares approach to calculating limits of detection and quantification by modeling variability as a function of concentration. Analytical Chemistry, 69, 3069–3075.CrossRefGoogle Scholar

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© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Instituto de Energías RenovablesUniversidad Nacional Autónoma de MéxicoTemixcoMexico

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