Decomposing the Pearson Correlation

Abstract

The Pearson correlation coefficient r, a scale free measure in [-1, 1], indicates the strength and direction of a linear relationship between two variables. When data are partitioned into two groups, r can be decomposed into a between-group component which is the product of two effect sizes (ESp) and a within-group component (WGc) which is the sum of within-group correlations weighted by a coefficient that is inversely related to ESp magnitude. Both zero and nonzero correlations can arise in multiple ways. A zero r could emerge from either (i) zero ESp and zero WGc, or (ii) ESp and WGc of similar magnitude but in opposed direction. A nonzero r could emerge from either (i) zero ESp and nonzero WGc, or (ii) zero WGc and nonzero WGc, or (iii) ESp and WGc in the same direction, or (iv) ESp and WGc that differ in magnitude and direction. Bipartitions that are theoretically interpretable include (i) natural zero points of variables (e.g., semantic differentials, IATs) and (ii) distinct data subsamples (e.g., males vs. females, Task orders). This decomposition delineates the respective contributions of concordance in group means and quantitative linearity to the observed magnitude and direction of the Pearson correlation.

Keywords: correlation, decomposition, bipartition, Pearson, effect size

JEL Classification: C10