The Spurious Correlation of Ratios that Have Common Variables: A Monte Carlo Examination of Pearson's Formula

Abstract

Pearson (1897) investigated correlations of ratios of bone measurements and found that although the correlations among the original measures were low, the correlations among ratios with common measures were about .5. To understand this result, he developed an approximate equation for the correlations of ratios. In the present study, Monte Carlo methods were used to show that Pearson's equation is fairly accurate and that correlations among ratios with common elements (e.g., X/C, Y/C) are indeed at least partly spurious, as Pearson concluded. This finding should serve as a two-fold warning to those who might correlate ratios that have common elements: (a) Interpretation of the observed relationship between such ratios may at best be tenuous, and (b) the relationships among the elements themselves may call into question the usefulness of addressing a hypothesis that can be tested only with a correlation between ratios that share elements.