Short Communication Utility of the Coefficient of Determination (r) in Assessing the Accuracy of Interspecies Allometric Predictions: Illumination or Illusion?

The appropriateness of relying on the coefficient of determination (r) as a statistical metric for judging the predictability of human clearance (CL) based on interspecies animal data was assessed. An explicit mathematical expression was derived for r as a function of species body weight and the corresponding measured value of CL. The derived mathematical function demonstrated that r is numerically large in most instances. Simulations using random CL generated from a common combination of species of mouse, rat, and monkey resulted in an r of 0.75 as the minimum, and 0.95 and 0.98 at 50th and 75th percentiles, respectively, given that total CL values increase with increasing species body weight. Analysis of literature data also indicated that the prediction accuracy of human CL was not correlated with values of r. Therefore, it is concluded that r is a limited statistical measure when assessing allometric scaling for the purpose of predicting human CL. Allometric scaling has been widely used in predicting human pharmacokinetic (PK) parameters, although the allometric approach is empirical and numerous examples of substantial prediction errors have been observed (Boxenbaum 1982; Mahmood and Balian, 1996; Nagilla and Ward, 2004; Tang and Mayersohn, 2006). The allometric relationship for PK parameters across animal species and the confidence in extrapolation of this relationship to humans are often assessed with use of the coefficient of determination (r). The latter is obtained from linear regression of log-transformed animal body weights and the corresponding measured values of (log) PK parameters. High r values (ca. greater than 0.90) have been cited for most of the allometric relationships reported in the literature (Mahmood and Balian, 1996; Hu and Hayton, 2001). By definition, r is the fraction of the total squared error explained by the model. It is generally recognized that r is not a good statistical measure for nonlinear models. For example, overparameterized models could easily lead to high r values, whereas such models usually have little predictive value. It has also been long recognized that the log-log transformation of the allometric power function (P a W) would minimize deviations from the regression line (Smith, 1984). Therefore, it is reasonable to speculate that r may not offer a good measure for examining the predictive quality of the allometric relationship. We report here an explicit mathematical function of r derived to quantitatively assess the appropriateness of using r as a statistical measure in allometric scaling. Literature data were also evaluated to assess the relationship between r and the prediction performance by allometric scaling. Materials and Methods Theory. Expression of predicted PK parameters among species. The function relating predicted PK parameters (P̂) in humans or animal species to animal body weights (Wi, i 1 to n, where n is the number of animal species) and observed animal PK parameters (Pi) has been described previously (Tang and Mayersohn, 2005). The following highlights the major mathematical functions needed in the subsequent derivations.