Nonparametric statistics. Part 3. Correlation coefficients
https://doi.org/10.21323/2414-438X-2023-8-3-237-251
Abstract
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.
Keywords
correlation,
association coefficient,
contingency coefficient,
rank correlation,
grading scales,
test of significance
Supporting agencies: The research was supported by state assignment of V. M. Gorbatov Federal Research Centre for Food Systems of RAS, scientific research No. FNEN-2019–0008.
About the Authors
M. A. Nikitina
V. M. Gorbatov Federal Research Center for Food Systems
Russian Federation
Russian Federation
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)
I. M. Chernukha
V. M. Gorbatov Federal Research Center for Food Systems
Russian Federation
Russian Federation
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)
References
1. Shmoilova, R.A., Minashkin, V.G., Sadjvnikova, N.A., Shuvalova, E.B. (2014). Theory of statistics. Moscow: Finance and Statistics, 2014. (In Russian)
2. Cuvier, G. (1805). Leçons d’anatomie comparée (volumes in-8). Paris: Baudouin. (In French)
3. Eliseeva, I.I., Yuzbashev, M.M. (2004). General theory of statistics. Moscow: Finance and Statistics, 2004. (In Russian)
4. 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.1888.0082
5. Pearson, 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.1895.0041
6. Kraemer, 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/10.1177/0962280206070650
7. Bavrina, A.P., Borisov, I.B. (2021). Modern rules of the application of correlation analysis. Medicinskij al’manah, 3(68), 70–79. (In Russian)
8. Grjibovski, А.M. (2008). Correlation analysis. Human Ecology, 9, 50–60. (In Russian)
9. Leach, C. (1979). Introduction to statistics: a nonparametric approach for the social sciences. Chichester: Wiley, 1979.
10. Noether, G.E. (1981). Why Kendall Tau? Teaching Statistics, 3(2), 41–43. https://doi.org/10.1111/j.1467-9639.1981.tb00422.x
11. Bonett, 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/BF02294183
12. Kendall, M.G. (1938). A new measure of rank correlation. Biometrika, 30(1–2), 81–93. https://doi.org/10.1093/biomet/30.1-2.81
13. Gubler, E.V., Genkin, A.A. (1973). Application of nonparametric statistical criteria in biomedical research. Leningrad: Medicine, 1973. (In Russian)
14. Rosenbaum, S. (1954). Tables for a nonparametric test of location. Annals of Mathematical Statistics, 25(1), 146–150. https://doi.org/10.1214/aoms/1177728854
15. Stepanov, V.G. (2019). Application of nonparametric statistical methods in agricultural biology and veterinary medicine research. St-Petersburg: Lan. 2019. (In Russian)
16. 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)
17. Klyachkin, V.N. (2021). Expert methods. Chapter in a book: Statistical methods in quality management: computer technologies. Moscow: Finance and Statistics, 2021. (In Russian)
18. 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)90109-9
19. Cramer, H. (1946). Variables and Distributions. Chapters in a book: Mathematical Methods of Statistics. Princeton: Princeton University Press, 1946.
20. 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/10.2307/23401263
21. 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.
22. 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.0177678
23. Cramer, H. (1946). The two-dimensional case. Chapter in a book: Mathematical Methods of Statistics. Princeton: Princeton University Press, 1946.
24. Fechner, G.T. (1897). Kollektivmasslehre. Leipzig: Verlag von Wilhelm Engelmann, 1897. (In German)
25. Lakin, G.F. (1990). Biometrics. Moscow: High School. 1990. (In Russian)
26. Chaddock, R.E. (1925). Principles and methods of statistics. Boston, New York, 1925.
27. Cohen, J. (1973). Statistical power analysis and research results. American Educational Research Journal, 10(3), 225–229. https://doi.org/10.2307/1161884
28. Cohen, J. (1988). Statistical power analysis for the behavioral sciences. Hillsdale. Mahwah, NJ: Lawrence Erlbaum Associates, 1988.
29. Cohen, J. (1992). A power primer. Psychological Bulletin, 112(1), 155–159. https://doi.org/10.1037/0033-2909.112.1.155
30. Divaris, 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.2012.0071
31. Lomax, R.G., Hahs-Vaughn, D.L. (2012). Statistical Concepts. A Second Course. New-York: Taylor and Francis Group, LLC, 2012.
32. 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_02
33. Berry, 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/10.1007/978-3-319-98926-6
34. de 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.2013.14.9.4965
35. Bourne, 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.2016.01.555558
36. Mukaka, M.M. (2012). Statistics Corner: A guide to appropriate use of correlation coefficient in medical research. Malawi Medical Journal, 24(3), 69–71.
37. 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.0000000000002864
38. Kotrlik, 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.2011.01132
39. Hinkle, D.E., Wiersma, W., Jurs, S.G. (1979). Applied Statistics for the Behavioral Sciences. Chicago: Rand McNally College Pub. Co., 1979.
40. Hinkle, D.E., Wiersma, W., Jurs, S.G. (2003). Applied Statistics for the Behavioral Sciences. 5th Ed. Boston: Houghton Mifflin, 2003.
41. 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.
42. Moore, D., Notz, W.I., Fligher, M.A. (2012). The Basic Practice of Statistics. Publisher: W. H. Freeman, 2012.
43. Rumsey, D.J. (2016). Statistics For Dummies. 2nd Ed. New York: For Dummies, 2016.
44. Evans, J.D. (1996). Straightforward statistics for the behavioral sciences. Pacific Grove, Calif.: Brooks/Cole Publ. Co: An International Thomson Publ. Co., 1996.
45. 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.
46. 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)
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.
48. Gerguri, D. (2018). Leader-staff relationships in Kosovo customs: leadership and its impact on customs effectiveness. Styles of Communication, 10(1), 108–124.
49. 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.
50. 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.
51. Dancey, C.P., Reidy, J. (2007). Statistics without Maths for Psychology. Harlow: Pearson Education Limited, 2007.
52. 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.2018.08.001
53. Koterov, 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
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For citations:
Nikitina M.A., Chernukha I.M. Nonparametric statistics. Part 3. Correlation coefficients. Theory and practice of meat processing. 2023;8(3):237-251. https://doi.org/10.21323/2414-438X-2023-8-3-237-251
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