A Note on the Relationship between the Pearson Product-Moment and the Spearman Rank-Based Coefficients of Correlation
- 1 Department of CQMSE (Quantitative Methods-Statistics), Southern Illinois University, Carbondale, USA
Abstract
This note derives the relationship between the Pearson product-moment coefficient of correlation and the Spearman rank-based coefficient of correlation for the bivariate normal distribution. This new derivation shows the relationship between the two correlation coefficients through an infinite cosine series. A computationally efficient algorithm is also provided to estimate the relationship between the Pearson product-moment coefficient of correlation and the Spearman rank-based coefficient of correlation. The algorithm can be implemented with relative ease using current modern mathematical or statistical software programming languages e.g. R, SAS, Mathematica, Fortran, et al. The algorithm is also available from the author of this article.
- Rodgers, J.L. and Nicewander, W.A. (1988) Thirteen Ways to Look at the Correlation Coefficient. The American Statistician, 42, 59-66. https://doi.org/10.2307/2685263
- Stein, S.K. and Barcellos, A. (1992) Calculus and Analytic Geometry. 5th Edition, McGraw-Hill, Inc., New York.
- Pearson, K. (1907) Mathematical Contributions to the Theory of Evolution. XVI. On Further Methods of Determining Correlation. Drapers Company of Research Memoirs, Biometric Series, Cambridge University Press, Cambridge.
- Moran, P.A.P. (1948) Rank Correlation and Product-Moment Correlation. Biometrika, 35, 203-206. https://doi.org/10.1093/biomet/35.1-2.203
- Headrick, T.C. (2010) Statistical Simulation: Power Method Polynomials and Other Transformations. Chapman & Hall/CRC, Boca Raton.
- Höffding, W. (1948) A Class of Statistics with Asymptotically Normal Distributions. The Annals of Mathematical Statistics, 19, 293-325. https://doi.org/10.1214/aoms/1177730196
- Gibbs, J.W. (1899) Fourier Series. Nature, 59, 200, 606.