Fiducial inference for random-effects calibration models: Advancing reliable quantification in environmental analytical chemistry

Abstract

This article addresses calibration challenges in analytical chemistry by employing a random-effects calibration curve model and its generalizations to capture variability in analyte concentrations. The model is motivated by specific issues in analytical chemistry, where measurement errors remain constant at low concentrations but increase proportionally as concentrations rise. To account for this, the model permits the parameters of the calibration curve, which relate instrument responses to true concentrations, to vary across different laboratories, thereby reflecting the potential variability in measurement processes. The calibration curve that accurately captures the heteroscedastic nature of the data results in more reliable estimates across diverse laboratory conditions. Noting that traditional large-sample interval estimation methods are inadequate for small samples, an alternative approach, namely the fiducial approach, is explored in this work. It turns out that the fiducial approach, when used to construct a confidence interval for an unknown concentration, outperforms all other available approaches in terms of maintaining the coverage probabilities. Applications considered include the determination of the presence of an analyte and the interval estimation of an unknown true analyte concentration. The proposed method is demonstrated for both simulated and real interlaboratory data, including examples involving copper and cadmium in distilled water.