The Correlation Coefficient: An Overview

TLDR

Correlation measures association while regression predicts outcomes, and although they share mathematical expressions, they differ in interpretation and application, especially when the predictor variable is random. The study aims to elucidate the interrelations among the coefficient of determination, multiple correlation coefficient, covariance, correlation coefficient, and coefficient of alienation for two variables, and to recommend using Fisher’s z transformation over raw r values. The authors derive and illustrate the mathematical connections among these statistics for two related variables, and discuss how r can be used to infer correlation or test linearity. The paper presents graphical illustrations and real‑world chemical examples to demonstrate the discussed statistical relationships.

Abstract

Correlation and regression are different, but not mutually exclusive, techniques. Roughly, regression is used for prediction (which does not extrapolate beyond the data used in the analysis) whereas correlation is used to determine the degree of association. There situations in which the x variable is not fixed or readily chosen by the experimenter, but instead is a random covariate to the y variable. This paper shows the relationships between the coefficient of determination, the multiple correlation coefficient, the covariance, the correlation coefficient and the coefficient of alienation, for the case of two related variables x and y. It discusses the uses of the correlation coefficient r, either as a way to infer correlation, or to test linearity. A number of graphical examples are provided as well as examples of actual chemical applications. The paper recommends the use of z Fisher transformation instead of r values because r is not normally distributed but z is (at least in approximation). For either correlation or for regression models, the same expressions are valid, although they differ significantly in meaning.