Evaluating\nthe Suitability of Linear and Nonlinear\nRegression Approaches for the Langmuir Adsorption Model as Applied\ntoward Biomass-Based Adsorbents: Testing Residuals and Assessing Model\nValidity

Regression analysis is a powerful tool in adsorption\nstudies. Researchers\noften favor linear regression for its simplicity when fitting isotherm\nmodels, such as the Langmuir equation. Validating regression assumptions\nis crucial to ensure that the model accurately represents the data\nand allows appropriate inferences. This study provides a detailed\nexamination of assumption checking in the context of adsorption studies\nwhile simultaneously evaluating the robustness of linear regression\nmethods for fitting the Langmuir equation to isotherm data from 2,4-dichlorophenol\n(DCP) adsorption onto various biomass-based adsorbents and activated\ncarbon. Different linearized Langmuir equations (Hanes–Woolf,\nLineweaver–Burk, Eadie–Hofstee, and Scatchard) were\ncompared to nonlinear regression, and each method was validated by\nrigorous residual checking. This included visual plots of residuals\nas well as statistical tests, including the Durbin–Watson test\nfor autocorrelation (independence), the Shapiro–Wilk test for\nnormality, and the White test for homoscedasticity. Key findings indicate\nthat the Hanes–Woolf (type 1) and Lineweaver–Burk (type\n2) linearizations were the best for most biomass adsorbents studied\nand that Eadie–Hofstee (type 3) and Scatchard (type 4) were\ngenerally invalid due to the negative parameters or assumption violations.\nFor activated carbon, all linearization methods were unsuitable due\nto independence violations. In the case of nonlinear regression, there\nwere no major assumption violations for all of the adsorbents. Symbolic\nregression identified the Langmuir equation only for activated carbon\n(AC). This study revealed shortcomings in relying solely on linearized\nLangmuir models. A proposed workflow recommends using nonlinear or\nweighted nonlinear regression, starting with Hanes–Woolf or\nLineweaver–Burk linearization results as initial values for\nparameter estimation. If assumptions remain violated with nonlinear\ntechniques, novel methods such as symbolic regression should be employed.\nThis advanced regression technique can improve adsorption models’\naccuracy and predictive behavior without the stringent need for assumption\nchecking. Symbolic regression can also aid in understanding mechanisms\nof novel adsorbents.