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. • Proposes a novel fiducial inference method for the widely used Rocke-Lorenzato calibration model (Rocke and Lorenzato, 1995), which has been cited over 259 times and is foundational in interlaboratory analytical chemistry. A key innovation lies in successfully extending fiducial inference to a nonlinear random-effects framework, where random effects enter the model in a non-linear fashion—an analytically challenging setting not adequately handled by traditional methods. This extension enables robust interval estimation in complex hierarchical calibration settings with heteroscedastic measurement errors. • Highlights the superiority of the fiducial approach over frequentist methods for in- terval estimation, especially in small-sample scenarios. While frequentist approaches typically rely on large-sample normal approximations and standard errors—often failing to capture the true uncertainty—fiducial inference yields intervals that better reflect the underlying sampling distribution, as shown in works by Mathew, Krishnamoorthy, and Hannig. In contrast to computationally expensive bootstrap methods and Zhao et al. (2021)’s standardized LRT approach (which involves two layers of approximation and still underperforms for some settings), our method achieves accurate, assumption-light inference. Figure 1 in the manuscript visualizes this advantage by comparing fiducial, bootstrap, and Wald densities for a simulated example. • Demonstrates through extensive simulations (Section 5.2, Tables 1, 2, 3, 4 in the manuscript) that the proposed fiducial method consistently achieves superior coverage probabilities and comparable expected interval widths compared to Zhao’s modified likelihood ratio tests (LRT), bootstrap-based methods, and conventional Wald intervals. Unlike Zhao’s method, which exhibits limitations in coverage, and computationally intensive bootstrap methods, the fiducial approach excels by robustly representing uncertainty while maintaining computational simplicity. • In addition to interval estimation, we evaluate point estimation via the fiducial mode, which shows comparable performance to maximum likelihood estimates (MLEs) (Section 5.2, Table 5 in the manuscript). This finding reinforces that the fiducial approach not only surpasses existing methods in uncertainty quantification but also delivers reliable and efficient point estimates, further solidifying its utility as a comprehensive alternative to traditional statistical methods in nonlinear calibration models. • Highlights significant computational efficiency advantages: while bootstrap-based approaches and Zhao’s LRT require extensive computational resources, especially problematic when estimating multiple unknown concentrations, the fiducial approach markedly reduces computational effort (Discussed in Section 5.3). Its innovative strategy allows pivot quantities to be efficiently reused across multiple unknown concentrations, making rigorous interval estimation practical even with limited computational resources. • Emphasizes practical relevance by enabling laboratories to effectively ”borrow strength” from multiple interlaboratory measurements. This enhancement directly supports more reliable and accurate environmental pollutant quantification, facilitating regulatory compliance and public health protection (Gibbons and Coleman (1994); Rocke and Lorenzato (1995)). Its efficiency and minimal assumptions enhance the accessibility of advanced statistical calibration methods for interdisciplinary researchers and practitioners.
Sahu et al. (Thu,) studied this question.