An evaluation of R2 as an inadequate measure for nonlinear models in pharmacological and biochemical research: a Monte Carlo approach

Abstract

Background: It is long known within the mathematical literature that the coefficient of determination R(2) is an inadequate measure for the goodness of fit in nonlinear models. Nevertheless, it is still frequently used within pharmacological and biochemical literature for the analysis and interpretation of nonlinear fitting to data.

Results: The intensive simulation approach undermines previous observations and emphasizes the extremely low performance of R(2) as a basis for model validity and performance when applied to pharmacological/biochemical nonlinear data. In fact, with the 'true' model having up to 500 times more strength of evidence based on Akaike weights, this was only reflected in the third to fifth decimal place of R(2). In addition, even the bias-corrected R(2)(adj) exhibited an extreme bias to higher parametrized models. The bias-corrected AICc and also BIC performed significantly better in this respect.

Conclusion: Researchers and reviewers should be aware that R(2) is inappropriate when used for demonstrating the performance or validity of a certain nonlinear model. It should ideally be removed from scientific literature dealing with nonlinear model fitting or at least be supplemented with other methods such as AIC or BIC or used in context to other models in question.

Figure 1

Graph illustrating the noise model used for the simulations. 2000 simulations of random gaussian noise with mean = 0 and s.d. = 0.4 were added to the fitted values of a three-parameter log-logistic model (L3) fit to real-time quantitative PCR data. This resulted in the point cloud (black dots) of response values and the band of red lines reflecting the fitted curves of all simulations with the same model L3 applied.

Figure 2

Performance of the nine different sigmoidal models on the fitted values from a three-parameter log-logistic model. (A) The nine different sigmoidal models were fit by nonlinear least-squares to the fitted data from the L3 model. (B) Residual plot depicting the performance of each of the nine models in respect to fitting to the data from the L3 model. As expected, model L3 (light green, reference curve) has zero residual value as having been fit to the data obtained from the same model. Several other models also fit the data exceptionally well (L5, L4, baro5) and are not visible due to being overlayed by the L3 curve. Descriptions for the abbreviated models can be found under Formulas 1-9.

Figure 3

Analysis of adjusted R² and corrected AIC of nine different sigmoidal models on fitted data from a three-parameter log-logistic model (L3). Three different magnitudes of homoscedastic gaussian noise (low: 0.02; medium: 0.1; high: 0.4) were added to the fitted data (2000 simulations), each of the nine sigmoidal model fit by nonlinear least-squares and the two measures collected for each simulation. Finally, the measures were averaged and displayed as point graphs. Upper panel: AICc, lower panel: R²_adj. Descriptions for the abbreviated models can be found under Formulas 1-9. Coefficients of variation for all simulations were below 5% and hence omitted. More detailed data for the measures can be found in Table 1.

Figure 4

Analysis of model selection bias between for different measures of goodness-of-fit. Three different magnitudes of homoscedastic gaussian noise (0.02%; 0.1%; 0.4%) were added to the fitted data of model L3. The nine different sigmoidal models were fit and the different measures for goodness-of-fit collected at each iteration. The best model was selected for each measure and displayed for all iterations as a coloured selection heatmap. Light green reflects the 'true' model L3.