meatTheory and practice of meat processingТеория и практика переработки мяса2414-438X2414-441XФГБНУ «Федеральный научный центр пищевых систем им. В.М. Горбатова» РАН10.21323/2414-438X-2023-8-3-237-251meat-286Research ArticleСтатьиNonparametric statistics. Part 3. Correlation coefficientshttps://orcid.org/0000-0002-8313-4105NikitinaM. A.

Marina A. Nikitina, Doctor of Technical Sciences, Docent, Leading Scientific Worker, the Head of the Direction of Information Technologies of the Center of Economic and Analytical Research and Information Technologies

26, Talalikhina, 109316, Moscow

Tel: +7–495–676–95–11 (297)

m.nikitina@fncps.ruhttps://orcid.org/0000-0003-4298-0927ChernukhaI. M.

Irina M. Chernukha, Doctor of Technical Sciences, Professor, Academician of the Russian Academy of Sciences, Head of the Department for Coordination of Initiative and International Projects

26, Talalikhina, 109316, Moscow

Tel: +7–495–676–95–11 (109)

imcher@inbox.ruV. M. Gorbatov Federal Research Center for Food SystemsRussian Federation20231010202383237251Copyright © Nikitina M.A., Chernukha I.M., 20232023Nikitina M.A., Chernukha I.M.Nikitina M.A., Chernukha I.M.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.meatjournal.ru/jour/article/view/286

A measure of correlation or strength of association between random variables is the correlation coefficient. In scientific research, correlation analysis is most often carried out using various correlation coefficients without explaining why this particular coefficient was chosen and what the resulting value of this coefficient means. The article discusses Spearman correlation coefficient, Kendall correlation coefficient, phi (Yule) correlation coefficient, Cramér’s correlation coefficient, Matthews correlation coefficient, Fechner correlation coefficient, Tschuprow correlation coefficient, rank-biserial correlation coefficient, point-biserial correlation coefficient, as well as association coefficient and contingency coefficient. The criteria for applying each of the coefficients are given. It is shown how to establish the significance (insignificance) of the resulting correlation coefficient. The scales in which the correlated variables should be located for the coefficients under consideration are presented. Spearman rank correlation coefficient and other nonparametric indicators are independent of the distribution law, and that is why they are very useful. They make it possible to measure the contingency between such attributes that cannot be directly measured, but can be expressed by points or other conventional units that allow ranking the sample. The benefit of rank correlation coefficient also lies in the fact that it allows to quickly assess the relationship between attributes regardless of the distribution law. Examples are given and step-by-step application of each coefficient is described. When analyzing scientific research and evaluating the results obtained, the strength of association is most commonly assessed by the correlation coefficient. In this regard, a number of scales are given (Chaddock scale, Cohen scale, Rosenthal scale, Hinkle scale, Evans scale) grading the strength of association for correlation coefficient, both widely recognized and not so well known.

correlationassociation coefficientcontingency coefficientrank correlationgrading scalestest of significanceThe research was supported by state assignment of V. M. Gorbatov Federal Research Centre for Food Systems of RAS, scientific research No. FNEN-2019–0008.ReferencesShmoilova, R.A., Minashkin, V.G., Sadjvnikova, N.A., Shuvalova, E.B. (2014). Theory of statistics. Moscow: Finance and Statistics, 2014. (In Russian)Shmoilova, R.A., Minashkin, V.G., Sadjvnikova, N.A., Shuvalova, E.B. (2014). Theory of statistics. Moscow: Finance and Statistics, 2014. (In Russian)Cuvier, G. (1805). Leçons d’anatomie comparée (volumes in-8). Paris: Baudouin. (In French)Cuvier, G. (1805). Leçons d’anatomie comparée (volumes in-8). Paris: Baudouin. (In French)Eliseeva, I.I., Yuzbashev, M.M. (2004). General theory of statistics. Moscow: Finance and Statistics, 2004. (In Russian)Eliseeva, I.I., Yuzbashev, M.M. (2004). General theory of statistics. Moscow: Finance and Statistics, 2004. (In Russian)Galton, F. (1888). Co-relations and their measurement, chiefly from Anthropometric Data. Proceedings of the Royal Society of London, 45, 135–145. https://doi.org/10.1098/rspl.1888.0082Galton, F. (1888). Co-relations and their measurement, chiefly from Anthropometric Data. Proceedings of the Royal Society of London, 45, 135–145. https://doi.org/10.1098/rspl.1888.0082Pearson, K. (1895). Notes on regression and inheritance in the case of Two Parents. Proceedings of the Royal Society of London, 58, 240–242. https://doi.org/10.1098/rspl.1895.0041Pearson, K. (1895). Notes on regression and inheritance in the case of Two Parents. Proceedings of the Royal Society of London, 58, 240–242. https://doi.org/10.1098/rspl.1895.0041Kraemer, H.C. (2006). Correlation coefficients in medical research: from product moment correlation to the odds ratio. Statistical Methods in Medical Research, 15(6), 525–544. https://doi.org/10.1177/0962280206070650Kraemer, H.C. (2006). Correlation coefficients in medical research: from product moment correlation to the odds ratio. Statistical Methods in Medical Research, 15(6), 525–544. https://doi.org/10.1177/0962280206070650Bavrina, A.P., Borisov, I.B. (2021). Modern rules of the application of correlation analysis. Medicinskij al’manah, 3(68), 70–79. (In Russian)Bavrina, A.P., Borisov, I.B. (2021). Modern rules of the application of correlation analysis. Medicinskij al’manah, 3(68), 70–79. (In Russian)Grjibovski, А.M. (2008). Correlation analysis. Human Ecology, 9, 50–60. (In Russian)Grjibovski, А.M. (2008). Correlation analysis. Human Ecology, 9, 50–60. (In Russian)Leach, C. (1979). Introduction to statistics: a nonparametric approach for the social sciences. Chichester: Wiley, 1979.Leach, C. (1979). Introduction to statistics: a nonparametric approach for the social sciences. Chichester: Wiley, 1979.Noether, G.E. (1981). Why Kendall Tau? Teaching Statistics, 3(2), 41–43. https://doi.org/10.1111/j.1467-9639.1981.tb00422.xNoether, G.E. (1981). Why Kendall Tau? Teaching Statistics, 3(2), 41–43. https://doi.org/10.1111/j.1467-9639.1981.tb00422.xBonett, D.G., Wright, T.A. (2000). Sample size requirements for estimating Pearson, Kendall and Spearman correlations. Psychometrica, 65, 23–28. https://doi.org/10.1007/BF02294183Bonett, D.G., Wright, T.A. (2000). Sample size requirements for estimating Pearson, Kendall and Spearman correlations. Psychometrica, 65, 23–28. https://doi.org/10.1007/BF02294183Kendall, M.G. (1938). A new measure of rank correlation. Biometrika, 30(1–2), 81–93. https://doi.org/10.1093/biomet/30.1-2.81Kendall, M.G. (1938). A new measure of rank correlation. Biometrika, 30(1–2), 81–93. https://doi.org/10.1093/biomet/30.1-2.81Gubler, E.V., Genkin, A.A. (1973). Application of nonparametric statistical criteria in biomedical research. Leningrad: Medicine, 1973. (In Russian)Gubler, E.V., Genkin, A.A. (1973). Application of nonparametric statistical criteria in biomedical research. Leningrad: Medicine, 1973. (In Russian)Rosenbaum, S. (1954). Tables for a nonparametric test of location. Annals of Mathematical Statistics, 25(1), 146–150. https://doi.org/10.1214/aoms/1177728854Rosenbaum, S. (1954). Tables for a nonparametric test of location. Annals of Mathematical Statistics, 25(1), 146–150. https://doi.org/10.1214/aoms/1177728854Stepanov, V.G. (2019). Application of nonparametric statistical methods in agricultural biology and veterinary medicine research. St-Petersburg: Lan. 2019. (In Russian)Stepanov, V.G. (2019). Application of nonparametric statistical methods in agricultural biology and veterinary medicine research. St-Petersburg: Lan. 2019. (In Russian)Edelbaeva, N.A., Lebedinskaya, O.G., Kovanova, E.S., Tenetova, E.P., Timofeev, A.G. (2019). Fundamentals of nonparametric statistics. Moscow: YUNITY-DANA. 2019. (In Russian)Edelbaeva, N.A., Lebedinskaya, O.G., Kovanova, E.S., Tenetova, E.P., Timofeev, A.G. (2019). Fundamentals of nonparametric statistics. Moscow: YUNITY-DANA. 2019. (In Russian)Klyachkin, V.N. (2021). Expert methods. Chapter in a book: Statistical methods in quality management: computer technologies. Moscow: Finance and Statistics, 2021. (In Russian)Klyachkin, V.N. (2021). Expert methods. Chapter in a book: Statistical methods in quality management: computer technologies. Moscow: Finance and Statistics, 2021. (In Russian)Matthews, B.W. (1975). Comparison of the predicted and observed secondary structure of T4 phage lysozyme. Biochimica et Biophysica Acta (BBA) — Protein Structure, 405(2), 442–451. https://doi.org/10.1016/0005-2795(75)90109-9Matthews, B.W. (1975). Comparison of the predicted and observed secondary structure of T4 phage lysozyme. Biochimica et Biophysica Acta (BBA) — Protein Structure, 405(2), 442–451. https://doi.org/10.1016/0005-2795(75)90109-9Cramer, H. (1946). Variables and Distributions. Chapters in a book: Mathematical Methods of Statistics. Princeton: Princeton University Press, 1946.Cramer, H. (1946). Variables and Distributions. Chapters in a book: Mathematical Methods of Statistics. Princeton: Princeton University Press, 1946.Yule, G.U. (1912). On the methods of measuring association between two attributes. Journal of the Royal Statistical Society, 75(6), 579–652. https://doi.org/10.2307/23401263Yule, G.U. (1912). On the methods of measuring association between two attributes. Journal of the Royal Statistical Society, 75(6), 579–652. https://doi.org/10.2307/23401263Aaron, B., Kromrey, J.D., Ferron, J.M. (1998, 2–4 November). Equating r-based and d-based effect-size indices: Problems with a commonly recommended formula. Paper presented at the annual meeting of the Florida Educational Research Association. Orlando, FL. Retrieved from https://files.eric.ed.gov/fulltext/ED433353.pdf Accessed June 20, 2023.Aaron, B., Kromrey, J.D., Ferron, J.M. (1998, 2–4 November). Equating r-based and d-based effect-size indices: Problems with a commonly recommended formula. Paper presented at the annual meeting of the Florida Educational Research Association. Orlando, FL. Retrieved from https://files.eric.ed.gov/fulltext/ED433353.pdf Accessed June 20, 2023.Boughorbel, S., Jarray, F., El-Anbari, M. (2017). Optimal classifier for imbalanced data using Matthews coefficient metric. PLOS ONE, 12(6), Article e0177678. https://doi.org/10.1371/journal.pone.0177678Boughorbel, S., Jarray, F., El-Anbari, M. (2017). Optimal classifier for imbalanced data using Matthews coefficient metric. PLOS ONE, 12(6), Article e0177678. https://doi.org/10.1371/journal.pone.0177678Cramer, H. (1946). The two-dimensional case. Chapter in a book: Mathematical Methods of Statistics. Princeton: Princeton University Press, 1946.Cramer, H. (1946). The two-dimensional case. Chapter in a book: Mathematical Methods of Statistics. Princeton: Princeton University Press, 1946.Fechner, G.T. (1897). Kollektivmasslehre. Leipzig: Verlag von Wilhelm Engelmann, 1897. (In German)Fechner, G.T. (1897). Kollektivmasslehre. Leipzig: Verlag von Wilhelm Engelmann, 1897. (In German)Lakin, G.F. (1990). Biometrics. Moscow: High School. 1990. (In Russian)Lakin, G.F. (1990). Biometrics. Moscow: High School. 1990. (In Russian)Chaddock, R.E. (1925). Principles and methods of statistics. Boston, New York, 1925.Chaddock, R.E. (1925). Principles and methods of statistics. Boston, New York, 1925.Cohen, J. (1973). Statistical power analysis and research results. American Educational Research Journal, 10(3), 225–229. https://doi.org/10.2307/1161884Cohen, J. (1973). Statistical power analysis and research results. American Educational Research Journal, 10(3), 225–229. https://doi.org/10.2307/1161884Cohen, J. (1988). Statistical power analysis for the behavioral sciences. Hillsdale. Mahwah, NJ: Lawrence Erlbaum Associates, 1988.Cohen, J. (1988). Statistical power analysis for the behavioral sciences. Hillsdale. Mahwah, NJ: Lawrence Erlbaum Associates, 1988.Cohen, J. (1992). A power primer. Psychological Bulletin, 112(1), 155–159. https://doi.org/10.1037/0033-2909.112.1.155Cohen, J. (1992). A power primer. Psychological Bulletin, 112(1), 155–159. https://doi.org/10.1037/0033-2909.112.1.155Divaris, K., Vann, W.F., Baker, A.D., Lee, J.Y. (2012). Examining the accuracy of caregivers’ assessments of young children’s oral health status. The Journal of the American Dental Association, 143(11), 1237–1247. https://doi.org/10.14219/jada.archive.2012.0071Divaris, K., Vann, W.F., Baker, A.D., Lee, J.Y. (2012). Examining the accuracy of caregivers’ assessments of young children’s oral health status. The Journal of the American Dental Association, 143(11), 1237–1247. https://doi.org/10.14219/jada.archive.2012.0071Lomax, R.G., Hahs-Vaughn, D.L. (2012). Statistical Concepts. A Second Course. New-York: Taylor and Francis Group, LLC, 2012.Lomax, R.G., Hahs-Vaughn, D.L. (2012). Statistical Concepts. A Second Course. New-York: Taylor and Francis Group, LLC, 2012.Rosenthal, J.A. (1996). Qualitative descriptors of strength of association and effect size. Journal of Social Service Research, 21(4), 37–59. https://doi.org/10.1300/j079v21n04_02Rosenthal, J.A. (1996). Qualitative descriptors of strength of association and effect size. Journal of Social Service Research, 21(4), 37–59. https://doi.org/10.1300/j079v21n04_02Berry, K.J., Johnston, J.E., Mielke, P.W. (2018). The Measurement of Association. A Permutation Statistical Approach. Cham: Springer Nature Switzerland AG, 2018. https://doi.org/10.1007/978-3-319-98926-6Berry, K.J., Johnston, J.E., Mielke, P.W. (2018). The Measurement of Association. A Permutation Statistical Approach. Cham: Springer Nature Switzerland AG, 2018. https://doi.org/10.1007/978-3-319-98926-6de Menezes, R.F., Bergmann, A., Thuler, L.C.S. (2013). Alcohol consumption and risk of cancer: a systematic literature review. Asian Pacific Journal of Cancer Prevention, 14(9), 4965–4972. https://doi.org/10.7314/apjcp.2013.14.9.4965de Menezes, R.F., Bergmann, A., Thuler, L.C.S. (2013). Alcohol consumption and risk of cancer: a systematic literature review. Asian Pacific Journal of Cancer Prevention, 14(9), 4965–4972. https://doi.org/10.7314/apjcp.2013.14.9.4965Bourne, P.A., Hudson-Davis, A. (2016). Psychiatric induced births in Jamaica: homicide and death effects on pregnancy. Psychology and Behavioral Science International Journal, 1(2), Article 555558. https://doi.org/10.19080/PBSIJ.2016.01.555558Bourne, P.A., Hudson-Davis, A. (2016). Psychiatric induced births in Jamaica: homicide and death effects on pregnancy. Psychology and Behavioral Science International Journal, 1(2), Article 555558. https://doi.org/10.19080/PBSIJ.2016.01.555558Mukaka, M.M. (2012). Statistics Corner: A guide to appropriate use of correlation coefficient in medical research. Malawi Medical Journal, 24(3), 69–71.Mukaka, M.M. (2012). Statistics Corner: A guide to appropriate use of correlation coefficient in medical research. Malawi Medical Journal, 24(3), 69–71.Schober, P., Boer, C., Schwarte, L.A. (2018). Correlation coefficients: appropriate use and interpretation. Anesthesia and Analgesia, 126(5), 1763–1768. https://doi.org/10.1213/ANE.0000000000002864Schober, P., Boer, C., Schwarte, L.A. (2018). Correlation coefficients: appropriate use and interpretation. Anesthesia and Analgesia, 126(5), 1763–1768. https://doi.org/10.1213/ANE.0000000000002864Kotrlik, J.W., Williams, H.A., Jabor, M.K. (2011). Reporting and interpreting effect size in quantitative agricultural education research. Journal of Agricultural Education, 52(1), 132–142. https://doi.org/10.5032/jae.2011.01132Kotrlik, J.W., Williams, H.A., Jabor, M.K. (2011). Reporting and interpreting effect size in quantitative agricultural education research. Journal of Agricultural Education, 52(1), 132–142. https://doi.org/10.5032/jae.2011.01132Hinkle, D.E., Wiersma, W., Jurs, S.G. (1979). Applied Statistics for the Behavioral Sciences. Chicago: Rand McNally College Pub. Co., 1979.Hinkle, D.E., Wiersma, W., Jurs, S.G. (1979). Applied Statistics for the Behavioral Sciences. Chicago: Rand McNally College Pub. Co., 1979.Hinkle, D.E., Wiersma, W., Jurs, S.G. (2003). Applied Statistics for the Behavioral Sciences. 5th Ed. Boston: Houghton Mifflin, 2003.Hinkle, D.E., Wiersma, W., Jurs, S.G. (2003). Applied Statistics for the Behavioral Sciences. 5th Ed. Boston: Houghton Mifflin, 2003.Correlation Coefficients. Applied Statistics. Lesson 5. (2005). Andrews University (Michigan). Retrieved from https://www.andrews.edu/~calkins/math/edrm611/edrm05.htm. Accessed June 20, 2023.Correlation Coefficients. Applied Statistics. Lesson 5. (2005). Andrews University (Michigan). Retrieved from https://www.andrews.edu/~calkins/math/edrm611/edrm05.htm. Accessed June 20, 2023.Moore, D., Notz, W.I., Fligher, M.A. (2012). The Basic Practice of Statistics. Publisher: W. H. Freeman, 2012.Moore, D., Notz, W.I., Fligher, M.A. (2012). The Basic Practice of Statistics. Publisher: W. H. Freeman, 2012.Rumsey, D.J. (2016). Statistics For Dummies. 2nd Ed. New York: For Dummies, 2016.Rumsey, D.J. (2016). Statistics For Dummies. 2nd Ed. New York: For Dummies, 2016.Evans, J.D. (1996). Straightforward statistics for the behavioral sciences. Pacific Grove, Calif.: Brooks/Cole Publ. Co: An International Thomson Publ. Co., 1996.Evans, J.D. (1996). Straightforward statistics for the behavioral sciences. Pacific Grove, Calif.: Brooks/Cole Publ. Co: An International Thomson Publ. Co., 1996.Neill, J. (2018). Survey research and design in psychology. Lecture 4. Retrieved from https://upload.wikimedia.org/wikiversity/en/f/fd/SRDP_Lecture04Handout_Correlation_6slidesperpage.pdf. Accessed June 20, 2023.Neill, J. (2018). Survey research and design in psychology. Lecture 4. Retrieved from https://upload.wikimedia.org/wikiversity/en/f/fd/SRDP_Lecture04Handout_Correlation_6slidesperpage.pdf. Accessed June 20, 2023.Yavna, D.V., Kupriyanov, I.V., Bunyaeva, M.V. (2016). Sensory and perceptual processes: a tutorial. Rostov-on-Don: Publishing House of the Southern Federal University. 140. (In Russian)Yavna, D.V., Kupriyanov, I.V., Bunyaeva, M.V. (2016). Sensory and perceptual processes: a tutorial. Rostov-on-Don: Publishing House of the Southern Federal University. 140. (In Russian)Chakkapark, J, Vinitwatanakun, W. (2017). The relationship between division heads’ leadership styles and teacher satisfaction at Siam Commercial College of Technology. Scholar: Hum Sciences, 9(1), 36–47.Chakkapark, J, Vinitwatanakun, W. (2017). The relationship between division heads’ leadership styles and teacher satisfaction at Siam Commercial College of Technology. Scholar: Hum Sciences, 9(1), 36–47.Gerguri, D. (2018). Leader-staff relationships in Kosovo customs: leadership and its impact on customs effectiveness. Styles of Communication, 10(1), 108–124.Gerguri, D. (2018). Leader-staff relationships in Kosovo customs: leadership and its impact on customs effectiveness. Styles of Communication, 10(1), 108–124.Miletic, M., Vukusic, M., Mausa, G., Galinac, T. (2017, 11–13 September). Relationship between design and defects for software in evolution. Proceedings of the Workshop of Software Quality, Analysis, Monitoring, Improvement, and Applications. Belgrade, Serbia, 2017.Miletic, M., Vukusic, M., Mausa, G., Galinac, T. (2017, 11–13 September). Relationship between design and defects for software in evolution. Proceedings of the Workshop of Software Quality, Analysis, Monitoring, Improvement, and Applications. Belgrade, Serbia, 2017.Pearson’s correlation. (2011). Site Statstutor. Statistics support for students. UK. Retrieved from www.statstutor.ac.uk/resources/uploaded/pearsons.pdf. Accessed June 20, 2023.Pearson’s correlation. (2011). Site Statstutor. Statistics support for students. UK. Retrieved from www.statstutor.ac.uk/resources/uploaded/pearsons.pdf. Accessed June 20, 2023.Dancey, C.P., Reidy, J. (2007). Statistics without Maths for Psychology. Harlow: Pearson Education Limited, 2007.Dancey, C.P., Reidy, J. (2007). Statistics without Maths for Psychology. Harlow: Pearson Education Limited, 2007.Akoglu, H. (2018). User’s guide to correlation coefficients. Turkish Journal of Emergency Medicine, 18(3), 91–93. https://doi.org/10.1016/j.tjem.2018.08.001Akoglu, H. (2018). User’s guide to correlation coefficients. Turkish Journal of Emergency Medicine, 18(3), 91–93. https://doi.org/10.1016/j.tjem.2018.08.001Koterov, A.N., Ushenkova, L.N., Zubenkova, E.S., Kalinina, M.V., Biryukov, A.P., Lastochkina, E.M. et al. (2019). Strength of association. Report 2. Graduations of correlation size. Medical Radiology and Radiation Safety, 64(6), 12–24. https://doi.org/10.12737/1024-6177-2019-64-6-12-24Koterov, A.N., Ushenkova, L.N., Zubenkova, E.S., Kalinina, M.V., Biryukov, A.P., Lastochkina, E.M. et al. (2019). Strength of association. Report 2. Graduations of correlation size. Medical Radiology and Radiation Safety, 64(6), 12–24. https://doi.org/10.12737/1024-6177-2019-64-6-12-24

The authors declare that there are no conflicts of interest present.