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Statistical properties of limits of detection and their quotients

Daniel J Montville, University of Massachusetts Amherst


The instrumental limit of detection (LOD) lies at the heart of modern analytical chemistry, yet there is still much debate as to its role and reliability as a figure of merit. There are a number of possible ways to define the LOD, and in the present work four such sample test statistics were defined and their probability density functions (PDFs) were derived. The PDFs were easily evaluated via numerical integration and were used to obtain expectation values, population precisions, and confidence intervals. Monte Carlo simulation methods were used to prepare normalized histograms of one million LOD variates each, for homoscedastic linear calibration curve systems, and these were found to be in excellent agreement with the numerically obtained PDFs and associated statistics. With the realization that LODs are random variates governed by their PDFs it becomes evident that the quotient of two LODs is also a random variate governed by its own distribution. One case of a limit of detection definition was taken and the theoretical PDF for the quotient of two LODs was derived. Monte Carlo simulations were again used to generate normalized histograms of one million quotient of LOD variates, and were also in excellent agreement with the numerically evaluated theoretical PDF. In addition to the simulation and theoretical approaches, a real chemical measurement system was designed in an effort to collect large data sets that could be used to obtain experimental LODs. The data collected on the measurement system was carefully analyzed and, after a randomization algorithm, was deemed appropriate to generate a normalized histograms of quotients of LOD (in some cases up to 40,000 points). It was found that the distribution of values obtained from the real system almost perfectly matched that of the derived theory and Monte Carlo simulation providing a sound argument for the conclusions we have drawn. The software used in all aspects of the work is available with full commented source code.

Subject Area

Analytical chemistry

Recommended Citation

Montville, Daniel J, "Statistical properties of limits of detection and their quotients" (2007). Doctoral Dissertations Available from Proquest. AAI3282737.