Reversed inverse regression for the univariate linear calibration and its statistical properties derived using a new methodology

Abstract

Abstract. Since simple linear regression theory was established at the beginning of the 1900s, it has been used in a variety of fields. Unfortunately, it cannot be used directly for calibration. In practical calibrations, the observed measurements (the inputs) are subject to errors, and hence they vary, thus violating the assumption that the inputs are fixed. Therefore, in the case of calibration, the regression line fitted using the method of least squares is not consistent with the statistical properties of simple linear regression as already established based on this assumption. To resolve this problem, "classical regression" and "inverse regression" have been proposed. However, they do not completely resolve the problem. As a fundamental solution, we introduce "reversed inverse regression" along with a new methodology for deriving its statistical properties. In this study, the statistical properties of this regression are derived using the "error propagation rule" and the "method of simultaneous error equations" and are compared with those of the existing regression approaches. The accuracy of the statistical properties thus derived is investigated in a simulation study. We conclude that the newly proposed regression and methodology constitute the complete regression approach for univariate linear calibrations.Keywords: bias / classical regression / error propagation / mean-data-point-based variance / populationregression-line-based variance / reversed inverse regression / simultaneous error equations / Taylor approximation

Misuse of Beer–Lambert Law and other calibration curves

“…The authors of [14] also arrived at the same conclusions through some simulation studies (see also references therein in the same vein), although other authors criticize that recommendation. For example, in the recent paper [15], the authors introduced a new methodology, the ‘reverse inverse regression’ to address the same problem, assuming that the inputs (concentration values) vary according to Gaussian distributions, which allow them to derive some statistical properties, and criticize the inverse regression approach based on the treatment of the inputs (absorbance values) as determined with small error. But they compare their method against classical calibration and inverse regression using a simulation study, and have to recognize the best behaviour of the latter in the sense of minimizing the variance of the prediction interval.In brief, leaving aside assumptions that could, or not, be accomplished (that in case to be fulfilled allow to deduce some statistical properties for the linear regression model), if we are interested in prediction, the best approach nonetheless seems to be inverse regression.…”