For many materials or specimen, their resistance to stress is an important (safety) characteristic. The target value is the event stress, i.e., the value of the stress at which the (adverse) event occurs. Depending on the test procedure and the test items, the event stress cannot always be measured directly but must be determined by binary sensitivity tests. A method for the evaluation of binary sensitivity tests based on the principle of Maximum Likelihood is presented and compared to other methods (traditionally) applied. Sensitivity testing of explosives and fatigue testing of metallic materials are used as real examples to illustrate application of the method and to compare actual results. The method presented delivers not only the estimate of the stress at any specified probability of the event but also the respective 95 % confidence interval. Application of the method is facilitated because it does not require the data to be obtained in a staircase procedure, can process any kind of input data (e.g., as measured or logarithmic), and does not require the stress levels to be equidistant. The method is easily applicable for any laboratory because it is implemented into an Excel file, which is made freely available at https://www.statistics-wilrich.info.

Zusammenfassung Emil J. Gumbel ist der Namensgeber der jährlichen Gumbel-Vorlesung auf der Statistischen Woche. Leider ist der Namensgeber dieser Vorlesung nur noch wenigen Teilnehmern und Vortragenden bekannt. Dieser Artikel möchten diesem Defizit abhelfen. Denn Gumbel war nicht nur der Namensgeber irgendeiner statistischen Verteilung. Der Aufsatz beschreibt den Lebensweg Gumbels vom Weltkriegsteilnehmer zu einem weithin bekannten Pazifisten, der die politischen Morde in der frühen Weimarer Republik mit statistischen Schlußweisen analysierte. Seine Zahlenwerke legten die Defizite der Rechtssprechung in der Weimarer Republik offen. Die Richter kompromittierten sich mit ihren Urteilen. Seine Statistik-Professur an der Universität Heidelberg verlor er nach Angriffen der NS-Studenten und einer deutsch-nationalistischen Universitätsleitung noch vor Beginn der NS-Herrschaft. Wir schildern Gumbels Emigration erst nach Frankreich und von da in die USA und seine vergeblichen Versuche, wieder an deutschen Universitäten aufgenommen zu werden. Im zweiten Teil skizzieren wir Gumbels wissenschaftliches Werk im Bereich der Statistik der Extremwerte. Gumbel schrieb nicht nur den Klassiker dieses Statistik-Bereichs. Er baute auch vielen Ingenieuren einen fachlichen Zugang mit leicht berechenbaren und gut interpretierbaren Diagrammen. Schließlich schildert einer der Autoren, der Gumbel noch persönlich kannte, seine Erinnerungen an den freundlichen und interessierten Gastprofessor aus den USA.

Background: We deal with interlaboratory experiments (collaborative studies) in which k participating laboratories, selected randomly from a population of laboratories, use samples from one and the same material or matrix. They perform binary microbiological measurements for which the measurement results are either "0" (target microorganisms not detected) or "1" (target microorganisms detected). The performance of such a measurement method is described by its probability of detection (POD) function, i.e., the POD as a function of the contamination of the sample (CFU per gram or CFU per milliliter), or by the level of detection (LODp), i.e., the contamination level of the sample that is detected (measurement result "1") with a specified probability p. Objective: We derive an approximate statistical analysis that is simple enough to be implemented in a spreadsheet application. Methods: Under the assumption of a Poisson distribution of the number of CFU in the samples, we estimate the mean POD function of the laboratories and the SD of the laboratory effect based on a complementary log-log model, a special case of the Generalized Linear Model in the special situation in which the contamination level is known by means other than the POD. The estimates are obtained by maximization of the Laplace approximation of the likelihood function. By simulation, a bias correction factor for the estimate of the SD is obtained. With the estimated POD function, LODs can be estimated. The model can also be used to evaluate the relative LOD of an alternative method with repect to a reference method. Results: The EXCEL program PODLOD-interlab_ver1.xls for this method of statistical analysis can be downloaded from http://www.wiwiss.fu-berlin.de/fachbereich/vwl/iso/ehemalige/wilrich. Highlights: A simple approximate statistical method for the estimation of the POD and LOD is derived. The method also allows the estimation of the RLOD of an alternative method with respect to reference method. The method is implemented in an EXCEL program that can be downloaded from http://www.wiwiss.fu-berlin.de/fachbereich/vwl/iso/ehemalige/wilrich.

The lifetime (time to failure) of a product is modeled as Weibull distributed (with unknown parameters); in this case the logarithms of the lifetimes are Gumbel distributed. Lots of items shall be accepted if their fraction p of nonconforming items (items the lifetime of which is smaller than a lower specification limit t L ) is not larger than a specified acceptable quality limit. The acceptance decision is based on the r ≤ n observed lifetimes of a sample of size n which is put under test until a defined censoring time t C is reached (Type I censoring). A lot is accepted if r = 0 or if the test statistic \(y = \hat {\mu } - k\hat {\sigma }\) is not smaller than the logarithm of the specification limit, \(x_L = \log (t_L)\), where k is an acceptance factor and \(\hat {\mu }\) and \(\hat {\sigma }\) are the Maximum Likelihood estimates of the parameters of the Gumbel distribution. The parameters of the sampling plan (acceptance factor k, sample size n and censoring time t C ) are derived so that lots with p ≤ p1 shall be accepted with probability not smaller than 1 − α. On the other hand, lots with fractions nonconforming larger than a specified value p2 shall be accepted with probability not larger than β. n and t C are not obtained separately but as a function that relates the sample size n to the censoring time t C . Of course, n decreases if the censoring time t C is increased. For t C  →∞ the smallest sample size, i.e. that of the uncensored sample, is obtained. Unfortunately, the parameters of the sampling plan do not only depend on the two specified points of the OC, P1(p1, 1 − α) and P2(p2, β), but directly on the parameters τ and δ of the underlying Weibull distribution or equivalently, on the parameters \(\mu = \log (\tau )\) and σ = 1/δ of the corresponding Gumbel distribution. Since these parameters are unknown we assume that the hazard rate of the underlying Weibull distribution is nondecreasing (δ ≥ 1). For the design of the sampling plan we use the limiting case δ = 1 or σ = 1/δ = 1. A simulation study shows that the OC of the sampling plan is almost independent of σ if the censoring time t C is not smaller than the specification limit t L .

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 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. We deal with the situation in which it is known that the measurand is 0 with a probability \(p_0\) that is a priori known, and that the standard deviation \(\sigma\) of the measurement is known. For this case Korun, Vodenik and Zorko extend an earlier paper by Korun and Zorko and derive the mean of the posterior distribution as a Bayesian estimator of the measurand. We offer an estimator that is based on the posterior probability \(\hat{p}_0\) of the measurand being 0. If \(\hat{p}_0 > 1 - \hat{p}_0\) it is 0 and otherwise the mode of the posterior distribution. This estimate is easier to calculate and less biased than that of Korun et al.

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.

In analogy to quantitative measurement methods the precision of binary measurement methods used in a population of laboratories can be characterised by the repeatability standard deviation and the reproducibility standard deviation of the probability of detection, POD. In order to estimate these standard deviations an interlaboratory experiment with k laboratories, each performing n repeated binary measurements at identical samples, is carried out according to ISO 5725-2 and analysed with a one-way analysis of variance. The variance estimates are, e.g., used for a test of equal POD of all laboratories and for the determination of a 90 %-expectation tolerance interval for the PODs of the laboratories.

We deal with sampling inspection by variables, i.e. acceptance sampling procedures wherein the acceptability of a lot is statistically established from the measurement results of a specified continuous variable X obtained at the items in a sample from the lot. An item is qualified as nonconforming if its measured quality characteristic x is larger than a defined upper specification limit U. We accept the lot if (x) over bar + k sigma <= U or (x) over bar + ks <= U in the case of known or unknown lot standard deviation sigma, respectively, where (x) over bar and s are mean and standard deviation of a sample of size n drawn at random from the lot and k is an acceptance constant given as a parameter of the sampling plan.In some cases the acceptance procedure is extended by an additional limit U* = U + Delta with Delta is an element of R that must not be exceeded by any of the measurements x(1), x(2,) ... , x(n), i.e. for acceptance of the lot the largest measurement result x((n)) = max(x(1), x(2), ... ,x(n)) must be less or equal to U*, x((n)) <= U* = U + Delta. Of course, with this additional requirement for acceptance the probability of acceptance of the lot is smaller than without it for each fraction p of nonconforming items in the lot.Such extended sampling plans are, e.g., used for the evaluation of bacterial contamination in foods, the amount of active ingredient used in formulating drug products and the strength of concrete.The OC function of these extended sampling plans for inspection by variables is derived and the advantages/disadvantages in comparison with unextended sampling plans are discussed. It turns out that especially in the case of "known" sigma plans the extended sampling plans protect against a true standard deviation that is larger than the value being used in the acceptance criterion (x) over bar + k sigma <= U.

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.

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.

Bioelution assays are fast, simple alternatives to in vivo testing. In this study, the intra- and inter-laboratory variability in bioaccessibility data generated by bioelution tests were evaluated in synthetic fluids relevant to oral, inhalation, and dermal exposure. Using one defined protocol, five laboratories measured metal release from cobalt oxide, cobalt powder, copper concentrate, Inconel alloy, leaded brass alloy, and nickel sulfate hexahydrate. Standard deviations of repeatability (sr) and reproducibility (sR) were used to evaluate the intra- and inter-laboratory variability, respectively. Examination of the sR:sr ratios demonstrated that, while gastric and lysosomal fluids had reasonably good reproducibility, other fluids did not show as good concordance between laboratories. Relative standard deviation (RSD) analysis showed more favorable reproducibility outcomes for some data sets; overall results varied more between- than within-laboratories. RSD analysis of sr showed good within-laboratory variability for all conditions except some metals in interstitial fluid. In general, these findings indicate that absolute bioaccessibility results in some biological fluids may vary between different laboratories. However, for most applications, measures of relative bioaccessibility are needed, diminishing the requirement for high inter-laboratory reproducibility in absolute metal releases. The inter-laboratory exercise suggests that the degrees of freedom within the protocol need to be addressed.

According to ISO 5725-2 (1994), measurement results obtained in an interlaboratory experiment are inspected for consistency by plotting Mandel’s h and k statistics and for outliers by application of the Grubbs test and the Cochran test. Critical values of these statistics for significance levels α=5% and α=1% and for some numbers p of laboratories and n of repeated measurements in the laboratories are supplied in ISO 5725-2 without reference to methods for their calculation. In this paper, exact formulae for the critical values of Mandel’s h and k and approximate formulae for the critical values of the Single Grubbs test, the Double Grubbs test and the Cochran test are derived.

Performance of qualitative microbiological measurement methods where the results are either "O" (microorganism not detected) or "1" (microorganism detected) is described by their probability of detection (POD) function, i.e., the POD as a function of the level of contamination of the sample, expressed as CFU/g or CFU/mL, or by the level of detection (LODp), i.e., the contamination of the sample that is detected (measurement result "1") with a specified probability p. When it is impossible to obtain samples of known contamination, estimation of the POD and LOD is impossible. However, it may not be the LOD of the method that is of interest, but its LOD with respect to the LOD of a reference method. Hence, an intralaboratory experiment is performed with a reference method, R, and an alternative method, A, at different levels of unknown contamination. A complementary loglog model is used to statistically estimate the relative LOD (RLOD) of A with respect to R that is equal for all chosen values p of the POD. An intralaboratory experiment for the detection of Listeria monocytogenes in fish and eggs illustrates the method. In a simulation study, the bias of the estimate of the RLOD was investigated. This bias is due to the small number of repeated measurements in intralaboratory studies; the relative bias increases with increasing true values of the RLOD from 0 for true RLOD = 1 to about 20% for true RLOD = 3. If the number of CFUs in the test portions does not follow a Poisson distribution, but instead follows a negative binomial distribution, e.g., because of overdispersion, the bias of the estimate of the RLOD decreases. An EXCEL program RLOD_ver1. xlsm for this method of statistical analysis can be downloaded from http://www.wiwiss.fu-berlin.de/ instituteliso/mitarbeiterlwilrichlindex.html.

We select an appropriate sampling plan from a sampling inspection system for inspection by variables, i.e. ISO 3951, and base the lot acceptance decision and the adaptation of the sampling plan directly on the a posteriori distribution of the fraction of nonconforming items in the lots and especially the a posteriori estimate of the probability of the fraction of nonconforming items in the lot being larger than the acceptance quality limit AQL. We do not assume a prior distribution of the fraction nonconforming in the lots because the production process does not directly generate fractions of nonconforming items but items with a quantitative characteristic assumed to be normally distributed with parameters μ (lot mean) and σ2 (within-lot variance) varying in time, respectively from lot to lot. The process curve, i.e. the two-dimensional distribution of the lot means and the within-lots variances is assumed to be Normal-scaled-inverse-chi-squared. In our hierarchical Bayes model we estimate the parameters of the process curve directly by exponentially weighted means and variances of the sample averages and the sample variances of the already inspected lots. Switching between tightened, normal and reduced inspection turns out to be more straightforward than with the switching rules of ISO 3951. Furthermore, if the process curve of the variance is stable, it is possible to switch from sampling plans with unknown variance to sampling plans with known variance. We apply the Bartlett test for switching to σ-plans and a CUSUM-s 2-chart for switching back to s-plans.

We deal with Three-class sampling plans for the evaluation of bacterial contamination. A Two-class sampling plan for inspection by attributes is defined by the triple (n,c,m); n items (test samples) are drawn randomly from the inspection lot and analysed; if the measured log-concentration x of a test sample is larger than the specification limit m, the test sample is nonconforming; if the number y of nonconforming test samples is larger than the acceptance constant c, the lot is rejected. A Three-class sampling plan is defined by (n,c,m,M) with an additional specification limit M > m; the lot is also rejected if at least one of the n measured log-concentrations is larger than M. A Three-class sampling plan protects better against unacceptable lots than the underlying Two-class sampling plan. The same effect could also be achieved by using a twoclass sampling plan with a somewhat smaller specification limit m(star), but this statement is only true if the distributional assumptions hold (lognormal distribution of the concentration with known standard deviation). However, if these assumptions are violated the three-class sampling plan reacts to larger fractions of totally nonconforming test samples ( x > M) for identical fractions p of nonconforming test samples ( x > in) with smaller probabilities of acceptance. In order to gain more efficiency we propose to use three-class sampling plans for inspection by variables.

Recent foodborne crises have demonstrated the importance of monitoring food safety. In terms of microbiological criteria, food safety requires the reliable detection of pathogens such as Listeria monocytogenes along the food chain by appropriate analytical methods. However, indications exist that accompanying Listeria innocua strains suppress the growth of L. monocytogenes during selective enrichment, which may cause reduced or even inhibited detection. To study these effects, the limit of detection of L. monocytogenes was investigated in the presence of L. innocua using the International Organization for Standardization standard method ISO 11290-1 and the VIDAS LDUO system, an automated method based on enzyme-linked fluorescence technology. The challenge was to provide low initial Listeria concentrations at sufficient precision to quantify the influence on the probability of detection of L. monocytogenes. The application of reference materials appropriate for quantitative test methods and a standardized dilution procedure were necessary to ensure accurate CFU levels of defined proportions of mixtures of both Listeria species. During selective enrichment, overgrowth of L. monocytogenes by L. innocua could be confirmed, leading to high rates of false-negative results. Moreover, with both methods, a significant decrease in the detectability of L. monocytogenes could be quantified at ratios of 2:1 at very low concentrations representative of natural contamination levels often found in foods and environments. It is concluded that there is a need to improve existing procedures with respect to selective enrichment, as well as the detection techniques.

A large number of papers exists that deal with Bayesian sampling plans. Hald defines Bayesian sampling plans as "plans obtained by minimizing average costs, consisting of inspection, acceptance and rejection costs". In order to obtain such a plan one starts with an a priori distribution of the fraction of nonconforming items in the lots, i.e. an assumption about the process curve, and calculates the sampling plan that minimizes the Bayesian risk or cost (if cost parameters are given). However, once these plans have been obtained they are applied in the classical manner just by making acceptance/rejection decisions for the inspected lots. For a Bayesian, the calculation of the a posteriori distribution of the fraction nonconforming in the lot is the essential step of the Bayesian analysis because for him the complete information combining prior knowledge and sample information is incorporated in the a posteriori distribution. Hence, in this paper the lot acceptance decision is directly based on the a posteriori distribution of the fraction nonconforming in the lot and especially the a posteriori estimate of the probability of the fraction of nonconforming items in the lot being larger than the acceptance quality limit pAQL. This Bayesian method is applied to sampling by attributes based on a beta-binomial model.

Aims:The purpose of this work was to derive a simple Excel spreadsheet and a set of standard tables of most probable number (MPN) values that can be applied by users of International Standard Methods to obtain the same output values for MPN, SD of the MPN, 95% confidence limits and test validity. With respect to the latter, it is considered that the Blodgett concept of 'rarity' is more valuable than the frequently used approach of improbability (vide de Man).Methods and Results:The paper describes the statistical procedures used in the work and the reasons for introducing a new set of conceptual and practical approaches to the determination of MPNs and their parameters. Examples of MPNs derived using these procedures are provided. The Excel spreadsheet can be downloaded from http://www.wiwiss.fu-berlin.de/institute/iso/mitarbeiter/wilrich/index.html.Conclusions:The application of the revised approach to the determination of MPN parameters permits those who wish to use tabulated values, and those who require access to a simple spreadsheet to determine values for nonstandard test protocols, to obtain the same output values for any specific set of multiple test results. The concept of 'rarity' is a more easily understood parameter to describe test result combinations that are not statistically valid. Provision of the SD of the log MPN value permits derivation of uncertainty parameters that have not previously been possible.Significance and Impact of the Study:A consistent approach for the derivation of MPNs and their parameters is essential for coherence between International Standard Methods. It is intended that future microbiology standard methods will be based on the procedures described in this paper.

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.

Qualitative microbiological measurement methods in which the measurement results are either 0 (microorganism not detected) or 1 (microorganism detected) are discussed. The performance of such a measurement method is described by its probability of detection as a function of the contamination (CFU/g or CFU/mL) of the test material, or by the LOD(p), i.e., the contamination that is detected (measurement result 1) with a specified probability p. A complementary log-log model was used to statistically estimate these performance characteristics. An intralaboratory experiment for the detection of Listeria monocytogenes in various food matrixes illustrates the method. The estimate of LOD50% is compared with the Spearman-Kaerber method.