Model Based Determination of Detection Limits for Proton Transfer Reaction Mass Spectrometer
Proton Transfer Reaction Mass Spectrometry (PTR-MS) is a chemical ionization mass spectrometric technique which allows to measure trace gases as, for example, in exhaled human breath. The quantification of compounds at low concentrations is desirable for medical diagnostics. Typically, an increase of measuring accuracy can be achieved if the duration of the measuring process is extended. For real time measurements the time windows for measurement are relatively short, in order to get a good time resolution (e.g. with breath-to-breath resolution during exercise on a stationary bicycle). Determination of statistical detection limits is typically based on calibration measurements, but this approach is limited, especially for very low concentrations. To overcome this problem, a calculation of limit of quantification (LOQ) and limit of detection (LOD), respectively, based on a theoretical model of the measurement process is outlined.
The paper deals with the comparative calibration model, i.e. with a situation when both variables are subject to errors. The calibration function is supposed to be a polynomial. From the statistical point of view, the model after linearization could be represented by the linear errors-in-variables (EIV) model. There are two different ways of using the Kenward and Roger’s type approximation to obtain the confidence region for calibration function coefficients. These two confidence regions are compared on a small simulation study. Calibration process and process of measuring with calibrated device are described under the assumption that the measuring errors are normally distributed.
Partial-sums discrete probability distributions occurred in description of many stochastic models. They were used also as a tool for creating new distributions, or as a link between known distributions. It is shown in this paper that every discrete distribution with only non-zero probabilities is a partial-sums distribution, and, moreover, that it has infinitely many parent distributions. The paper generalizes and unifies the concept of partial-sums distribution. Besides, it generalizes some risk models in insurance and revises some approaches to mathematical modelling in quantitative linguistics.