Date of Award:

5-2024

Document Type:

Thesis

Degree Name:

Master of Science (MS)

Department:

Mathematics and Statistics

Committee Chair(s)

Stephen J. Walsh

Committee

Stephen J. Walsh

Committee

Brennan Bean

Committee

Yan Sun

Abstract

In metrology, the science of measurements, straight line calibration models are frequently employed. These models help understand the instrumental response to an analyte, whose chemical constituents are unknown, and predict the analyte’s concentration in a sample. Techniques such as ordinary least squares and generalized least squares are commonly used to fit these calibration curves. However, these methods may yield biased estimates of slope and intercept when the calibrant, substance used to calibrate an analytical procedure with known chemical constituents (x-values), carries uncertainty. To address this, Ripley and Thompson (1987) proposed functional relationship estimation by maximum likelihood (FREML), which considers uncertainties in both x and y variables, providing unbiased estimates of slope and intercept. This project aims to develop a robust mechanism for calculating uncertainty in final measurement quantities, integrating international standard guidelines such as GUM and parametric bootstrapping to handle uncertainties in both x and y calibration points. Initial validation follows Ripley and Thompson’s assumptions of known population standard deviations. Subsequently, the model is extended to accommodate cases where calibration input uncertainties are estimated, incorporating their degrees-of-freedom and propagating these uncertainties into parameter estimates.

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