|Sorption Isotherm Spreadsheet|
Microsoft Excel Spreadsheets for Fitting Sorption Data
Carl H. Bolster, USDA-ARS
The spreadsheets have been recently updated (
Introduction: Sorption models are commonly used for describing solute and metal sorption to soils. When fitting sorption models to sorption data, however, the user must be aware that certain statistical limitations exist with both linear and nonlinear versions of the models. Ongoing research at the Animal Waste Management Research Unit of the USDA-ARS addresses the effect of these statistical limitations on fitting phosphorus sorption data with various sorption models. Currently, the following studies have been completed:
1. Problems associated with using linearized versions of the Langmuir equation: Although linearized versions of the Langmuir equation are mathematically equivalent to the nonlinear equation, the transformation of data required for linearization can result in modifications of error structure, introduction of error into the independent variable, and alteration of the weight placed on each data point. This can often times lead to differences in fitted parameter values between linear and nonlinear versions of the Langmuir model. In addition, our research has shown that the use of linearized Langmuir equations needlessly limits the ability to accurately model sorption data with more sophisticated sorption models (C.H. Bolster and G.M. Hornberger. 2007. On the Use of Linearized Langmuir Equations. Soil Science Society of America Journal 71:1796-1806 see also the erratum). Because the use of linearized Langmuir equations is largely due to the ease of using linear regression, we make available an easy-to-use Microsoft Excel spreadsheet capable of fitting nonlinear sorption equations to isotherm data. The spreadsheet generates best-fit parameters, parameter uncertainties, parameter correlations, and goodness-of-fit measures. The accuracy of the spreadsheet has been thoroughly tested by comparing model fits and parameter estimates with a more sophisticated software package (SAS 9.2).To download the spreadsheet click here and save to desktop. NOTE – due to our findings that weighting by the inverse of the variance in the dependent variable can lead to poorer parameter estimates and uncertainties than with unweighted least squares regression (see below), the weighting option has been removed from the spreadsheet (3/20/2010). If weighting of data is needed, please contact me at email@example.com.
2. Addressing the errors-in-variables problem associated with fitting the Langmuir model to sorption data: The Langmuir model is often fit to sorption data using least squares regression. An important assumption of least squares regression is that the predictor variable is error free. In the case of sorption data, however, this assumption is not valid because the equilibrium concentration - treated as the predictor variable in the Langmuir equation - is the variable usually measured; therefore the potential for parameter bias exists when obtaining parameter estimates through least squares regression. Although not commonly used, alternative regression methods do exist which either explicitly account for error in the predictor variable or significantly reduce the error in the predictor variable by modifying the Langmuir equation so that the true independent variable, initial concentration, is used as the predictor variable. In this study, the differences in fitted parameters and model fits between three different regression methods are explored by fitting P sorption data collected on 26 different soil samples. For a majority of soils tested in this study, the differences in model fits between the three regression methods were not statistically significant. Statistical differences, however, were observed in over a third of the soils suggesting that under some conditions errors in the predictor variable may be large enough to produce biased parameter estimates (C.H. Bolster 2008. Revisiting a Statistical Shortcoming When Fitting the Langmuir Model to Sorption Data. Journal of Environmental Quality 37:1986-1992). A spreadsheet with a modified version of the Langmuir equation which uses the initial concentration, rather than the equilibrium concentration, as the independent variable can be downloaded by clicking here and saving to your desktop. Please note, however, that the use of this equation can give the false impression that the Langmuir model is fitting the data adequately. After fitting, the data and model fits should be plotted as S vs C for inspection.
3. Properly weighting sorption data for least-squares analysis: In this study we examined the role of proper weighting in the least-squares (LS) analysis of phosphorus sorption data when both the dependent (y) and independent (x) variables contain heteroscedastic errors. We compared parameter estimates and uncertainties obtained with unweighted LS (ULS) regression with those obtained using two different weighted LS (
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Disclaimer: Although I have tested these spreadsheets against the output from SAS I cannot guarantee that the coding is error free. If you find what appears to be an error please contact me immediately. Please regularly check the Web site for updates and/or corrections.