From R-squared to coefficient of model accuracy for assessing "goodness-of-fits"

Abstract. Modelers tend to focus more on advancing methods of statistical and mathematical modeling than developing novel techniques for comparing modeled results with observations or establishing metrics for model performance assessment. Perhaps solely the most extensively applied "goodness-of-fit" measure especially for assessing performance of regression models is the coefficient of determination R². Normally, high R² tends to be associated with an efficient model. Nevertheless, R² has been cited to have no importance in the classical model of regression. Even in its use in descriptive statistics, R² is known to have questionable justification. R² is inadequate in assessing model performance because it does not give any information on the model residuals. Furthermore, R-squared can be low for an effective model. Contrastingly, a very poor model fit can yield high R². Regressing X on Y yields R² which is the same as that if Y is regressed on X thereby invalidating its use as a coefficient of determination. Taking into account the drawbacks of using R², this paper introduces coefficient of model accuracy (CMA) the derivation of which comprises an analogy to the R². However, instead of simply squaring an ordinary Pearson's product-moment correlation coefficient to obtain R², CMA comprises the product of nonparametric sample correlation and model bias. Acceptability of the introduced method can be found demonstrated through comparison of results from simulations by hydrological models calibrated using CMA and other existing objective functions.