A Decomposition Of The Pearson's Correlation Coefficient Into One- And Two-Group Measures

We reformulate and decompose the Pearson and Spearman correlation coefficients into two components. We recommend the first component for detecting linear or monotonic relationships and the second for recognizing patterns of two parallel lines, providing robust versions to outliers. Thus, we reveal the unknown aspect of the Pearson coefficient that identifies a two-group association other than a group, producing Type I or III errors. Finally, we apply members of the proposed family of coefficients with permutation tests to simulated and real data in economics. Novel coefficients identify two-parallel line correlation, hidden, mixed, influenced, or weaker but significant relationships not recognized by standard coefficients