Evaluating the Suitability of Linear and Nonlinear Regression Approaches for the Langmuir Adsorption Model as Applied toward Biomass-Based Adsorbents: Testing Residuals and Assessing Model Validity

Regression analysis is a powerful tool in adsorption studies. Researchers often favor linear regression for its simplicity when fitting isotherm models, such as the Langmuir equation. Validating regression assumptions is crucial to ensure that the model accurately represents the data and allows appropriate inferences. This study provides a detailed examination of assumption checking in the context of adsorption studies while simultaneously evaluating the robustness of linear regression methods for fitting the Langmuir equation to isotherm data from 2,4-dichlorophenol (DCP) adsorption onto various biomass-based adsorbents and activated carbon. Different linearized Langmuir equations (Hanes-Woolf, Lineweaver-Burk, Eadie-Hofstee, and Scatchard) were compared to nonlinear regression, and each method was validated by rigorous residual checking. This included visual plots of residuals as well as statistical tests, including the Durbin-Watson test for autocorrelation (independence), the Shapiro-Wilk test for normality, and the White test for homoscedasticity. Key findings indicate that the Hanes-Woolf (type 1) and Lineweaver-Burk (type 2) linearizations were the best for most biomass adsorbents studied and that Eadie-Hofstee (type 3) and Scatchard (type 4) were generally invalid due to the negative parameters or assumption violations. For activated carbon, all linearization methods were unsuitable due to independence violations. In the case of nonlinear regression, there were no major assumption violations for all of the adsorbents. Symbolic regression identified the Langmuir equation only for activated carbon (AC). This study revealed shortcomings in relying solely on linearized Langmuir models. A proposed workflow recommends using nonlinear or weighted nonlinear regression, starting with Hanes-Woolf or Lineweaver-Burk linearization results as initial values for parameter estimation. If assumptions remain violated with nonlinear techniques, novel methods such as symbolic regression should be employed. This advanced regression technique can improve adsorption models' accuracy and predictive behavior without the stringent need for assumption checking. Symbolic regression can also aid in understanding mechanisms of novel adsorbents.