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Fiducial Inference for Random-Effects Calibration Models: Advancing Reliable Quantification in Environmental Analytical Chemistry

Soumya Sahu, Thomas Mathew, Robert Gibbons, Dulal K. Bhaumik

TL;DR

This paper tackles uncertain interlaboratory calibration under heteroscedastic errors by formulating a nonlinear random-effects calibration model inspired by the Rocke–Lorenzato framework. It develops a fiducial inference procedure that constructs model-parameter and unknown-concentration distributions via structural equations, using a blockwise computation to achieve scalability. Through extensive simulations and real-data analyses (cadmium and copper in water), the fiducial method delivers accurate coverage for unknown concentrations with competitive or narrower intervals and substantially better computational efficiency than bootstrap or likelihood-based alternatives. The approach enables reliable estimation and decision-making in environmental monitoring by borrowing strength across laboratories while maintaining practical and interpretable uncertainty quantification.

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 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.

Fiducial Inference for Random-Effects Calibration Models: Advancing Reliable Quantification in Environmental Analytical Chemistry

TL;DR

This paper tackles uncertain interlaboratory calibration under heteroscedastic errors by formulating a nonlinear random-effects calibration model inspired by the Rocke–Lorenzato framework. It develops a fiducial inference procedure that constructs model-parameter and unknown-concentration distributions via structural equations, using a blockwise computation to achieve scalability. Through extensive simulations and real-data analyses (cadmium and copper in water), the fiducial method delivers accurate coverage for unknown concentrations with competitive or narrower intervals and substantially better computational efficiency than bootstrap or likelihood-based alternatives. The approach enables reliable estimation and decision-making in environmental monitoring by borrowing strength across laboratories while maintaining practical and interpretable uncertainty quantification.

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 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.

Paper Structure

This paper contains 21 sections, 9 equations, 1 figure, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivating Examples
  3. Choice of Model
  4. Fiducial Approach
  5. Choice of Structural Equations to Find Fiducial Pivotal Quantities and Non-Uniqueness.
  6. Fiducial Quantities for Model Parameters
  7. Computational motivation and initialization.
  8. Initial Fiducial Quantities for $\alpha_i$ and $\sigma_\epsilon$
  9. Fiducial Quantities for $\beta_i$
  10. Updated Fiducial Quantities for $\alpha_i$
  11. Fiducial Pivotal Quantity for $\sigma^2_\eta$ and Updated Fiducial Pivotal Quantity $\sigma^2_\epsilon$
  12. Numerical solutions: Existence and uniqueness.
  13. Fiducial Pivotal Quantity for an Unknown Concentration
  14. Fiducial Confidence Interval and Fiducial Point Estimate
  15. Simulation Studies
  16. ...and 6 more sections

Figures (1)

  • Figure 1: Density plots of the fiducial distribution, bootstrap distribution and Wald-Based normal distribution for an unknown concentration (true value is 50) for a sample from simulation setting 2.A where the MLE is 55.6. Vertical dashed lines shows 95% confidence intervals. The figure highlights the fact that only the fiducial interval includes the true value: a conclusion that agrees with the poor coverage probabilities of the other intervals, as reported in this section. Note that in the overlaid contour/density display, apparent colour changes at intersections can occur as a rendering artifact of overlapping densities and do not represent an additional method.