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Published:
Journal of Analytical Toxicology,
ISSN 0146-4760,
Volume 32, Number 5, June,
pp.329-338
Comparison
of Ordinary, Weighted, and Generalized Least-Squares Straight-Line
Calibrations for LC–MS–MS, GC–MS, HPLC, GC, and
Enzymatic Assay
Wayne C. Duer1, Paul J. Ogren2,
Alison Meetze3, Chester J. Kitchen4,
Ryan Von Lindern5, Dustin C. Yaworsky6, Christopher
Boden7, and
Jeffery A. Gayer8
1Hillsborough County Medical Examiner Department and Pathology
and Cell Biology Department, College of Medicine, University of
South Florida, 401 South Morgan Street, Tampa, Florida 33602;
2Chemistry
Department, Earlham College, Richmond, Indiana 47374;
3Florida
Department of Environmental Protection, 13051 North Telecom Parkway,
Temple Terrace, Florida 33637;
4Merck and Company, 770 Sumneytown
Pike, West Point, Pennsylvania 19486;
5The George Washington University
Law School, Washington, D.C., 20052;
6Waters Corporation, 34 Maple
Street, Milford, Massachusetts 01757;
7Medicolegal Investigations,
653 West 23rd Street, #246, Panama City, Florida 32405-3922; and
8Florida Department of Law Enforcement, Orlando Regional Crime
Laboratory, 500 West Robinson Street, Orlando, Florida 32801
The impact of experimental errors in one or both
variables on the use of linear least-squares was investigated
for method calibrations (response = intercept plus slope times
concentration, or equivalently, Y = a1 + a2X
) frequently used in analytical toxicology. In principle, the
most reliable calibrations should consider errors from all sources,
but consideration of concentration (X) uncertainties has not
been common due to complex fitting algorithm requirements. Data
were obtained for liquid chromatography–tandem mass spectrometry,
gas chromatography–mass
spectrometry, high-performance liquid chromatography, gas chromatography,
and enzymatic assay. The required experimental uncertainties
in response were obtained from replicate measurements. The required
experimental uncertainties in concentration were determined from
manufacturers’ furnished uncertainties in stock solutions
coupled with uncertainties imparted by dilution techniques. The
mathematical fitting techniques used in the investigation were
ordinary least-squares, weighted least-squares (WOLS), and generalized
least-squares (GLS). GLS best-fit results, obtained with an efficient
iteration algorithm implemented in a spreadsheet format, are
used with a modified WOLS-based formula to derive reliable uncertainties
in calculated concentrations. It was found that while the values
of the intercepts and slopes were not markedly different for
the different techniques, the derived uncertainties in parameters
were different. Such differences can significantly affect the
predicted uncertainties in concentrations derived from the use
of the different linear least-squares equations.
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