Univariate calibration by reversed regression of heteroscedastic data: a case study

Abstract

In a study of calibration with HPLC data for acetaldehyde-DNPH, we have collected replicate data (5–11 points each) for 33 samples spanning the range 0.0004–3 µg of detected analyte. Over most of this range, the data uncertainty is proportional to the signal, implying that weighted least squares is required to obtain the calibration function, since minimum-variance estimation requires weights inversely proportional to the data variance. When a variance function derived from an analysis of the replicate statistics is used to assign weights, w_i = 1/σ_i², the resulting values of χ² for the calibration fit are too large by a factor of 400. This implies that the method error is dominated by sample preparation rather than measurement uncertainty, and it means that in the calibration fit, the peak area should be taken as the independent variable and the amount as the dependent. In this reversed regression, the generalized LS method (GLS) is used to estimate the total method variance function from the residuals. The resulting method variance function resembles the instrumental variance, in containing constant and proportional error terms. The calibration data demand at least a cubic polynomial for adequate representation, but other response functions are statistically equivalent, with the result that this model uncertainty is comparable to the directly computed statistical uncertainty of the calibration function. In these computations, emphasis is placed on the virtues of χ² as a statistical figure of merit over the widely used R.