A relationship between orthogonal regression and the coefficient of determination under rotation of data sets

Abstract

In this article, we explore patterns of R 2 across data sets to reveal a fundamental equivalence. If any set of bivariate data is rotated, the R 2 value changes and defines a curve as a function of the rotation angle θ. We show that for any data set whose rotated R 2 curve has the same maximum value, the entire curve is identical up to a shift in the rotation angle. We also find that a recently introduced measure of linearity, Q 2, provides a definitive relation between these data sets, with equal Q 2 values resulting in identically rotated R 2 curves. Finally, we also show that we can consider R 2 as the combination of two components, the linearity of the data and a rotation angle. While rotating data or axes for non spatial data loses the specific meaning of the variables, these results may provide additional understanding of the relationships within the data and can reveal similarities between data sets, the role of sensitivity in correlation, and the nature of correlation itself.