Non-linear regression analysis with errors in both variables: estimation of co-operative binding parameters

Abstract

Four different parameter estimation criteria, the geometric mean functional relationship (GMFR), the maximum likelihood (ML), the perpendicular least-squares (PLS) and the non-linear weighted least squares (WLS), were used to fit a model to the observed data when both regression variables were subject to error. Performances of these criteria were evaluated by fitting the co-operative drug-protein binding Hill model on simulated data containing errors in both variables. Six types of data were simulated with known variances. Comparison of the criteria was done by evaluating the bias, the relative standard deviation (S.D.) and the root-mean-squared error (RMSE), between estimated and true parameter values. Results show that (1) for data with correlated errors, all criteria perform poorly; in particular, the GMFR and ML criteria. For data with uncorrelated errors, all criteria perform equally well with regard to the RMSE. (2) Use of GMFR and ML lead to lower values for S.D. but higher biases compared with WLS and PLS. (3) WLS performs less well when equal dispersion is applied to the two observed variables.

Revisiting a Statistical Shortcoming when Fitting the Langmuir Model to Sorption Data

“…Previous studies have reported biased parameter estimates when using Model I regression with errors‐in‐variables models due to the fact that an inherent assumption in vertical least squares regression is that the predictor variable is error free (de Brauwere et al, 2005; Valsami et al, 2000). Because the predictor variable, C e , in the original Langmuir equation is not error free, the potential exists that Langmuir sorption constants obtained by fitting Eq.…”

“…An additional limitation to using least squares regression for fitting the original Langmuir equation is that S is calculated from measured concentrations of C e ; therefore, S and C e are not measured independently, and any measurement error in C e leads to a negatively correlated error in S Thus, an overestimate of C e leads to an erroneously low calculated value of S , with the resultant data point being shifted downward and to the right on the isotherm plot. Valsami et al (2000) reported that neither Model I nor Model II regression could provide good parameter estimates when errors in the response and predictor variables were correlated. In this study, however, fitting the original Langmuir equation using Model II regression yielded nearly identical parameter estimates and model fits as the modified Langmuir equation, an equation in which the response and predictor variables are measured independently, and thus the errors are uncorrelated.…”

“…When both model variables (i.e., the response and predictor) are subject to experimental error (often referred to as an errors‐in‐variables model), Model I regression has been shown to generate biased parameter estimates (de Brauwere et al, 2005; Seber and Wild, 2003; Valsami et al, 2000). Model II regression, on the other hand, accounts for errors in the predictor and response variables and is therefore more appropriate for fitting errors‐in‐variables models to data (de Brauwere et al, 2005; Ebert and Russell, 1994; Ko et al, 1997; Seber and Wild, 2003; Valsami et al, 2000). Schulthess and Dey (1996) used a form of Model II regression known as normal nonlinear least squares (NNLS) regression for fitting the Langmuir model to sorption data.…”

“…An additional statistical concern when fitting the Langmuir model to sorption data is that the predictor and response variables are not measured independently, and therefore the errors in the response and predictor variables are correlated. Valsami et al (2000) found that Model I and Model II regression provided unsatisfactory parameter estimates for the Hill protein‐binding model (an equation similar to the Langmuir model) when errors in the response and predictor variables were correlated. The authors recommended the use of more sophisticated methods, such as the method of scoring, when errors associated with the two variables are correlated.…”

Weighting Formulas for the Least-Squares Analysis of Binding Phenomena Data

“…Some workers have handled these problems through more sophisticated LS methods that allow for uncertainty in both x and y. 7,18,19 However, in many cases y and x are obtained from the same measurement, making them fully correlated. In some such cases it has been possible to reexpress the functional relations in terms of a truly independent variable to satisfy the assumptions of the standard linear and nonlinear LS methods.…”