A new look at the correlation coefficient: Correlation as the difference-sum ratio of SSEs

Adriano Pareto

Communications in Statistics - Theory and Methods, 2023, vol. 52, issue 9, 2852-2859

Abstract: An interpretation of the Pearson’s correlation coefficient is presented that describes the coefficient as the difference-sum ratio between the sum of squared errors for the geometric mean regression line and the sum of squared errors for the line passing through the mean point, with a slope equal to that of the geometric mean regression line, but opposite sign. In this perspective, the Pearson’s r can be defined in terms of the amount of squared deviation from a ‘best-fitting line’ through a bivariate distribution, compared with the amount of squared deviation from a ‘counter-line’ representing the exact opposite theory of association of the two variables. Moreover, it is shown that even the geometric mean regression, similarly to ordinary least squares and orthogonal regression, can be obtained by minimizing the sum of squares of some kind of distances. An illustrative example is also provided.

Date: 2023
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DOI: 10.1080/03610926.2021.1961153

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