A Note on the Relationship between the Pearson Product-Moment and the Spearman Rank-Based Coefficients of Correlation

Show more

1. Introduction

The Pearson product-moment coefficient of correlation can be interpreted as the cosine of the angle between variable vectors in n dimensional space (e.g. [1] and [ [2] , p. 702]). Pearson [3] showed that the relationship of turning Spearman rank-based correlation coefficients () for the bivariate normal distribution into Pearson product-moment correlations (), which was contrived based on the so-called correlation of grades, for large samples to be:

(1)

For finite (small) samples, Moran [4] derived the relationship between the Pearson and Spearman coefficients of correlation for the bivariate normal distribution, which also appears in Headrick [ [5] p. 114], to be:

. (2)

Taking the limit as in Equation (2) will reduce Equation (2) to Equation (1). We would also note that Höffding [6] demonstrated that the Spearman rank correlation tends to normality for any given parent population.

2. Mathematical Development

In view of the above, this note derives the relationship between the Pearson product-moment correlation coefficient and the Spearman rank-based correlation coefficient for the bivariate normal distribution, in a different manner from either the Pearson [3] or the Moran [4] derivations, through the following infinite cosine series:

. (3)

Specifically, if we let, then

(4)

where it follows that for, that

(5)

Thus, from Equation (5) we have:

. (6)

The series associated with Equation (6) is uniformly convergent for all values of y and for. As such, integrating with respect to y, where yields:

(7)

Let x neither be zero nor a multiple of. As such, it necessarily follows that the series in Equation (3) is convergent. Hence, for; is positive, monotonic, decreasing, and bounded. Whence, the series

(8)

is, therefore, uniformly convergent for. Subsequently letting, noting again that x is neither zero nor a multiple of, it follows that Equation (3) can be expressed as

. (9)

3. Main Result and Conclusions

Setting in Equation (9), and through subsequent inverse exponentiation of Equation (9), yields the relationship (for large samples) between the Pearson product-moment correlation and the Spearman rank-based correlation coefficients as

(10)

for the bivariate normal distribution. In conclusion, the algorithm provided below in Equation (11), which has an oscillating effect of the Gibbs phenomenon [7] , to demonstrate the analytical derivation above is given as:

(11)

where, k is finite, and where Equation (11) converges to Equation (10) as. Finally, in terms of the error associated with Equation (11), it is straight-for- ward to see through real analysis, that and have a maximum absolute deviation when and hence Equation (10) would result in. As such, at this maximum point of deviation, given that in Equation (11), that the absolute error is less than when juxtaposed with Equation (10).

References

[1] 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

[2] Stein, S.K. and Barcellos, A. (1992) Calculus and Analytic Geometry. 5th Edition, McGraw-Hill, Inc., New York.

[3] 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.

[4] 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

[5] Headrick, T.C. (2010) Statistical Simulation: Power Method Polynomials and Other Transformations. Chapman & Hall/CRC, Boca Raton.

[6] 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

[7] Gibbs, J.W. (1899) Fourier Series. Nature, 59, 200, 606.