The Many-to-Many Mapping Between the Concordance Correlation Coefficient and the Mean Square Error

We derive the mapping between two of the most pervasive utility functions, the mean square error (MSE) and the concordance correlation coefficient (CCC, ρc). Despite its drawbacks, MSE is one of the most popular performance metrics (and a loss function); along with lately ρc in many of the sequence prediction challenges. Despite the ever-growing simultaneous usage, e.g., inter-rater agreement, assay validation, a mapping between the two metrics is missing, till date. While minimisation of Lp norm of the errors or of its positive powers (e.g., MSE) is aimed at ρc maximisation, we reason the often-witnessed ineffectiveness of this popular loss function with graphical illustrations. The discovered formula uncovers not only the counterintuitive revelation that `MSE1<MSE2' does not imply `ρc1>ρc2', but also provides the precise range for the ρc metric for a given MSE. We discover the conditions for ρc optimisation for a given MSE; and as a logical next step, for a given set of errors. We generalise and discover the conditions for any given Lp norm, for an even p. We present newly discovered, albeit apparent, mathematical paradoxes. The study inspires and anticipates a growing use of ρc-inspired loss functions e.g., left|fracMSEσXYright|, replacing the traditional Lp-norm loss functions in multivariate regressions.