###### Abstract
Moreover, the linearization of isotherms with three, four, and more parameters is not possible, as two or more unknown components limit simple linearization [26]. Nonlinear regression analysis provides a rigorous method of defining adsorption parameters using isotherm equations in their original form. It involves a stochastic distribution used to minimize the errors between experimental and simulated isotherm data [27]. Hence, a trial-and-error approach can be a solution used to minimize or maximize the objective function. However, this approach can be time-consuming and may not always guarantee the optimal solution. With the development of computer algorithms, this process has become faster, more reliable, and more efficient, allowing for quicker optimization of the objective function [28]. The statistical error functions are more reliable objective functions for the development and comparison of suitable nonlinear regression methods. To analyze the errors, a number of mathematically rigorous equations are used [28; 29]. The sum of squared errors (SSE), the sum of absolute errors (SAE), the average relative error (ARE), the hybrid fractional error function (HYBRID), and Marquardt's percent standard deviation (MPSD) are some of the most commonly cited regression functions used to search for best-fitting relationships in a solid-liquid adsorption system [30; 31; 32; 33; 34]. These regression functions yield different sets of residuals arising from different mechanisms [27]. To compare the residuals, numerous studies have reported on the standard normalizing process using the sum of normalized errors (SNE), as stated by [PERSON] et al. [35; 36; 37; 38]. In comparison, some other studies documented the selection of the optimal isotherm model based on the value of the highest \(R^{2}\) and the lowest error function [4; 39; 40; 41]. From a statistical perspective, although this proof of concept shows some quantitative relation among the error values, it cannot correlate with the variance of the residuals. [PERSON] et al. explained the accuracy of error analysis based on the coefficient of non-determination (CND) to identify the best-fit isotherm model [42]. The CND is a useful metric for measuring the proportion of variance in the dependent variable that is explained by the independent variables. Still, it does not provide information about the variability of the residuals around the fitted line. To fully improve the accuracy and reliability of model predictions, the combination of other statistical measures, such as the SNE, root mean square error (RMSE), average percentage error (APE), or mean absolute error (MAE), can be particularly useful for complex datasets, where feature selection is crucial for model performance. In the present study, the equilibrium data of MB sorption on AG at four different test temperatures was computed using five linear and nonlinear adsorption isotherms, namely, Langmuir, Freundlich, Redlich-Peterson (R-P), Sips, and Toth. The underlying errors between the experimental and simulated isotherms were compared using five error functions, namely, the SSE, ARE, HYBRID, MPSD, and EABS. This offered a wide range of derived isotherm constants, making it challenging to determine the so-called optimal or the best-fitting relationship. Therefore, two different statistical methods, the SNE and CND, were used. The combination of these methods provided a more objective approach to quantifying the error and deviation between the observed and predicted data in different interpretations so as to compare the accuracy of the error analysis. Thus far, no studies combining ther SNE and CND to determine the best-fitting relationship with a detailed error analysis have been reported. Therefore, this study aims to fill this gap by proposing a novel approach that combines the SNE and CND methods to determine the best-fitting relationship with a comprehensive error analysis of dye sorption isotherms on various materials.