Comparison of linear and non-linear equations for univariate calibration

Univariate data accumulated for the purpose of calibration of chromatographic and spectroscopic methods often exhibit slight but definite curvature. In this paper the performance of a non-linear calibration equation with the capacity to account empirically for the curvature, y=a+bxm, (m≠1) is compared with the commonly used linear equation, y=a+bx, as well as the quadratic equation, y=a+bx+cx2. All equations were applied to high quality HPLC calibration data using unweighted least squares. Parameter estimates and their standard errors were calculated for each equation. Standard errors and 95% prediction intervals in analyte concentrations were estimated with the aid of the fitted equations and their respective covariance matrices. Results indicate that the non-linear and quadratic equations each provide a better fit than the linear equation to the data considered here, as judged by the Akaikes information criterion (AIC), the adjusted coefficient of multiple determination, the magnitude and scatter of residuals, standard errors in estimated analyte concentrations and lack of fit analysis of variance (ANOVA). While the difference between the equations y=a+bx+cx2 and y=a+bxm as judged by the same criteria is more marginal, this work suggests that the non-linear calibration equation should be considered when a curve is required to be fitted to low noise calibration data which exhibit slight curvature.