The Limit of Detection in Generalized Least-Squares Calibrations: An Example Using Alprazolam Liquid Chromatography-Tandem Mass Spectrometry Data

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Limit of detection (LOD) values provide useful indicators for the suitability of an analytical method for samples with low analyte levels. An LOD value can also be used to estimate the false positive probability (p(x >or= LOD)) of a result for a sample with no analyte present, as well as the false negative probability (p(x <or= 0)) for a sample analyte level at the LOD. In straight line least-squares calibrations, the LOD of a calculated concentration (LOD(x)) results from uncertainties in low-level signal (y) measurements and in the calibration intercept (a(1)) and slope (a(2)) parameters. In generalized least-squares (GLS) calibrations, uncertainties in both the concentration (x(i)) and signal (y(i)) calibration data contribute to the fit parameter uncertainties. We define LOD(X) as 3 sigma(x = 0), where the calculated standard deviation, sigma(x = 0), includes all of these uncertainty contributions. Our GLS results can be understood in terms of small nonlinear distortions of the weighted ordinary least-squares (WOLS) problem. Because these distortions lead to skewed distributions of calculated x values, we also obtain exact GLS results for the asymmetric values. Differences between WOLS and GLS approaches are smallest when calibration uncertainties in x(i) are small, as in our alprazolam example. A second example, using synthetic data from earlier GLS work, shows greater differences between sigma(x = 0) and sigma(+/-)(x). Monte Carlo calculations for this example show that GLS-derived false positive and negative probabilities can differ by factors of two or more from normal distribution predictions, even when sigma(+)(x = 0) is used in place of sigma(x = 0) in the LOD definition. From a user standpoint, spreadsheet or other computational implementations of the GLS approach are as easy to use as those for a WOLS treatment, and GLS results will be the same as WOLS results when x uncertainties are ignored. Therefore, because GLS is a more complete and correct approach, it should be used as the standard method for weighted least-squares calibrations.