Robust estimates of the theoretical standard deviation to be used in interlaboratory precision experiments

Interlaboratory experiments often contain results that strongly deviate from other results obtained in the same laboratory under repeatability conditions, or laboratory means that strongly deviate from other laboratory means. In ISO 5725-2 [1] and IUPAC [2], the basic standards for the organisation and analysis of interlaboratory experiments for the determination of precision of a measurement method, outlier tests are performed in order to detect such situations and to finally decide whether these values are retained in the analysis or discarded as outliers. This outlier treatment, which always has a subjective aspect, becomes unnecessary by using robust estimates of the repeatability and reproducibility standard deviation. This paper proposes to use Rousseeuw’s Q _n as an extremely robust and efficient estimate of the standard deviation. Two examples show the performance of the new method.     

A simple method of constructing a confidence interval for the mean probability of detection in collaborative studies of binary measurement methods

We deal with collaborative studies where each of k laboratories performs n repeated binary measurements (measurement result x = 0: “not detected”; measurement result x = 1: “detected”), and present a simple method of constructing a confidence interval for the mean probability of detection of the laboratories. This method is based on an approximation of the distribution of the number y of detections among n independent measurements of a randomly chosen laboratory by a binomial distribution. The confidence interval is not only much easier to calculate but also more accurate than the profile likelihood interval presented by Uhlig et al.     

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Ohne Zusammenfassung     

Note on the bimodal distribution of PT results

The practitioner’s report "Treatment of bimodality in proficiency test of pH in bioethanol matrix" by Sarmanho et al. deals with a proficiency test (PT) for the measurement of pH in bioethanol. One group of 19 participating laboratories used an electrode that contained saturated LiCl and the other one KCl. The results of the LiCl group are biased, and those of the KCl group are unbiased as compared with the assigned reference value \(y_\mathrm{{RM}}\), and hence, the distribution of the laboratory means is bimodal. In order to deal with this situation, Sarmanho et al. present and carry out two different methods of fitting the bimodal distribution of the laboratory means: a fit by a mixture of two normal distributions and a Gaussian kernel estimation. Since this rather advanced statistical analysis does not help to solve the practical problem, some recomendations of ISO 17043 are discussed that might be useful for the coordinator of the PT and as well for the participating laboratories.     

Rules for stating when a limiting value is exceeded

The paper describes rules for stating whether a measurement result indicates that the value of the measurand, e.g. the concentration of a substance in the blood, is in conformity or not in conformity with given specifications. Examples of the intended applications are control of doping in sports, alcohol level in drivers’ blood, trace metals levels in workers’ blood, the proportion of gold in alloys, certain ingredients in food products and pesticides in drinking water. When limiting value(s) have been established a permissible measurand value is said to be in conformity and a non-permissible value it is said to be in non-conformity. One of the following three statements is asserted: Statement A: The value of the measurand is beyond any reasonable doubt in conformity, Statement B: The value of the measurand is beyond any reasonable doubt in non-conformity, Statement C: Neither conformity nor non-conformity could be demonstrated. The test can be performed as either a one-stage or a two-stage procedure. A two-stage procedure is recommended as especially appropriate when a cheap and fast screening method can be used in the first stage. For normally distributed measurements, a freeware computer program, conform1e.exe, is available for the calculations of the probabilities for Statements A, B and C.     

The determination of precision of qualitative measurement methods by interlaboratory experiments

In analogy to quantitative measurements, the precision of qualitative measurement methods can be characterised by the repeatability standard deviation and the reproducibility standard deviation of the sensitivity. In order to determine these standard deviations, the ISO 5725-2 approach is used. A simulation study and an example demonstrate the advantages of the method.     

Das Zeitverhalten von einfachen offenen exponentiellen Warteschlangensystemen mit unendlich vielen Warteplätzen

Zusammenfassung Die zeitabhängige (instationäre) Lösung für die Zustandswahrscheinlichkeiten und für einige Kenngrößen von Warteschlangensystemen mit einer Bedienungsstation, unendlich vielen Warteplätzen, exponentiellem Zu- und Abgang und beliebigem Anfangszustand wird bestimmt. Die ZustandswahrscheinlichkeitenP _v (), d. h. die Wahrscheinlichkeiten für Einheiten im System zur Zeit, ergeben sich als Integrale, in denen modifizierteSessel-Funktionen 1. Art auftreten. Der ErwartungswertL () und die VarianzV() der Zahl von Einheiten im System lassen sich als Integrale darstellen, in denen nur die ZustandswahrscheinlichkeitP ₀() auftritt.Für<1 und erreichen die Systeme einen stationären Zustand (für den die Lösung bekannt ist); für1 und giltP _v ()0 für alle, L(),V().Ist>1, dann wachsenL() undV() für große linear mit; ihre Asymptoten werden berechnet. Ist=1, dann wachsenL() und die Standardabweichung() für große mit ; einfache Näherungsformeln werden gefunden.

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Summary The time dependent solution is determined for the state probabilities and for some characteristic values of queuing systems with a single server, an infinite number of waiting places, exponentially distributed inter-arrival and service times, and any initial state. The state probabilitiesP _v (), i.e. the probabilities for units in the system at time, are given in the form of integrals in which modifiedBessel functions of the first kind occur. Integrating the state probalityP ₀() over leads to the meanL() and the varianceV() of the number of units in the system.For<1 and the systems tend to a steady state (for which the solution is known); for1 and we haveP _v ()0 for all, L(),V().If>1 asymptotic expansions for large are found givingL() andV() proportional to. If=1 simple approximate formulas for large are obtained givingL() and the standard deviation() proportional to .

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Vorgel. v.:J. Nitsche.     

Note on the correction of negative measured values if the measurand is nonnegative

Often, the concentration of a substance or the activity of radionuclides in a sample is measured indirectly as the difference between signal and noise, i.e. the difference between the measured value obtained at the sample and that obtained at a sample not containing the substance or the radionuclides (blank sample). The difference can be negative, especially if the concentration or the activity is low. Since a negative measurement result for a nonnegative measurand does not make sense, measured values must be corrected to nonnegative measurement results. Weise et al. and Korun/Maver Modec use a Bayesian approach in order to obtain a nonnegative estimate of the nonnegative measurand. Their estimate is the mean of the posterior distribution. Korun and Zorko modify this estimate slightly and obtain a less biased estimate. In this paper, it is proposed to use the mode of the posterior distribution instead of the mean. The resulting estimate is equal to the measured value if it is nonnegative and 0 if it is negative. This estimate has three advantages: it is extremely easy to calculate, it is less biased than the posterior mean proposed in the other papers, and it does not use the known standard deviation, i.e. it can also be used if the standard deviation is unknown.     

Note on the correction of negative measured values if the measurand is positive or 0 with known probability

Accreditation and Quality Assurance - Often the amount of a substance or the activity of radionuclides in a sample is measured indirectly as the difference between signal and noise, i.e. the...