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 real-world variability in measurement processes. Traditional large-sample interval estimation methods are inadequate for small samples, leading to the use of an alternative approach, namely the fiducial approach. The calibration curve that accurately captures the heteroscedastic nature of the data, results in more reliable estimates across diverse laboratory conditions. It turns out that the fiducial approach, when used to construct a confidence interval for an unknown concentration, produces a slightly wider width while achieving the desired coverage probability. 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.

翻译：本文通过采用随机效应校准曲线模型及其推广形式来解决分析化学中的校准挑战，以捕捉分析物浓度的变异性。该模型的提出源于分析化学中的特定问题，即在低浓度下测量误差保持恒定，但随着浓度升高而按比例增加。为此，该模型允许校准曲线的参数（这些参数将仪器响应与真实浓度关联起来）在不同实验室间变化，从而反映测量过程中实际存在的变异性。传统的大样本区间估计方法对于小样本而言并不适用，因此我们采用了一种替代方法，即基准方法。能够准确捕捉数据异方差性的校准曲线，可在不同实验室条件下产生更可靠的估计。结果表明，当使用基准方法构建未知浓度的置信区间时，区间宽度会略宽，但同时能达到期望的覆盖概率。所考虑的应用包括分析物存在的判定以及未知真实分析物浓度的区间估计。本文通过模拟数据和真实实验室间数据（包括蒸馏水中铜和镉的测定实例）对所提方法进行了验证。