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We address the problem of linear comparative calibration, a special case of linear calibration where both variables are measured with errors, and the analysis of the uncertainty of the measurement results obtained with the calibrated instrument. The concept is explained in detail using the calibration experiment of the pressure transducer and the subsequent analysis of the measurement uncertainties. In this context, the calibration and the measurements with the calibrated instrument are performed according to ISO Technical Specification 28037:2010 (here referred to as ISO linear calibration), based on the approximate linear calibration model and the application of the law of propagation of uncertainty (LPU) in this approximate model. Alternatively, estimates of the calibration line parameters, their standard uncertainties, the coverage intervals and the associated probability distributions are obtained using the Monte Carlo method (MCM) based on the law of propagation of distributions (LPD). Here we also obtain the probability distributions and the coverage interval for the quantities measured with the calibrated instrument. Furthermore, motivated by the model structure of this particular example, we conducted a simulation study that presents the empirical coverage probabilities of the ISO and MCM coverage intervals and investigates the influence of the sample size, i.e. the number of calibration points in the measurement range, and the different combinations of measurement uncertainties. The study generally confirms the good properties and validity of the ISO technical specification within the considered (limited) framework of experimental designs motivated by real-world application, with small uncertainties in relation to the measurement range. We also point out the potential weaknesses of this method that require increased user attention and emphasise the need for further research in this area.

Suggested is a non-parametric method and algorithm for estimating the probability distribution of a stochastic sum of independent identically distributed continuous random variables, based on combining and numerically inverting the associated empirical characteristic function (CF) derived from the observed data. This is motivated by classical problems in financial risk management, actuarial science, and hydrological modelling. This approach can be naturally generalized to more complex semi-parametric modelling and estimating approaches, e.g., by incorporating the generalized Pareto distribution fit for modelling heavy tails of the considered continuous random variables, or by considering the weighted mixture of the parametric CFs (used to incorporate the expert knowledge) and the empirical CFs (used to incorporate the knowledge based on the observed or historical data). The suggested numerical approach is based on combination of the Gil-Pelaez inversion formulae for deriving the probability distribution (PDF and CDF) from the associated CF and the trapezoidal quadrature rule used for the required numerical integration. The presented non-parametric estimation method is related to the bootstrap estimation approach, and thus, it shares similar properties. Applicability of the proposed estimation procedure is illustrated by estimating the aggregate loss distribution from the well-known Danish fire losses data.

Omnia in mensura et numero et pondere disposuisti is a famous Latin phrase from Solomon’s Book of Wisdom, dated to the mid first century BC, meaning that all things were ordered in measure, number, and weight. Naturally, the wisdom is appearing in its relation to man. The Wisdom of Solomon is understood as the perfection of knowledge of the righteous as a gift from God showing itself in action. Consequently, a natural and obvious conjecture is that measurement science is the science of sciences. In fact, it is a basis of all experimental and theoretical research activities. Each measuring process assumes an object of measurement. Some science disciplines, such as quantum physics, are still incomprehensible despite complex mathematical interpretations. No phenomenon is a real phenomenon unless it is observable in space and time, that is, unless it is a subject to measurement. The science of measurement is an indispensable ingredient in all scientific fields. Mathematical foundations and interpretation of the measurement science were accepted and further developed in most of the scientific fields, including physics, cosmology, geology, environment, quantum mechanics, statistics, and metrology. In this year, 2020, Measurement Science Review celebrates its 20th anniversary and we are using this special opportunity to highlight the importance of measurement science and to express our faith that the journal will continue to be an excellent place for exchanging bright ideas in the field of measurement science. As an illustration and motivation for usage and further development of mathematical methods in measurement science, we briefly present the simple least squares method, frequently used for measurement evaluation, and its possible modification. The modified least squares estimation method was applied and experimentally tested for magnetic field homogeneity adjustment.

Application of the exact statistical inference frequently leads to non-standard probability distributions of the considered estimators or test statistics. The exact distributions of many estimators and test statistics can be specified by their characteristic functions, as is the case for the null distribution of the Bartlett's test statistic. However, analytical inversion of the characteristic function, if possible, frequently leads to complicated expressions for computing the distribution function and the corresponding quantiles. An efficient alternative is the well-known method based on numerical inversion of the characteristic functions, which is, however, ignored in popular statistical software packages. In this paper, we present the explicit characteristic function of the corrected Bartlett's test statistic together with the computationally fast and efficient implementation of the approach based on numerical inversion of this characteristic function, suggested for evaluating the exact null distribution used for testing homogeneity of variances in several normal populations, with possibly unequal sample sizes.

Gaseous mixtures of ethanol in nitrogen are prepared by the gravimetric method according to ISO 6142. The liquid component, ethanol, is added into the cylinder by the gas tight syringe. The exact amount of added liquid is determined by weighing the syringes. The exact amount of ethanol is calculated before starting filling the cylinder in accordance with the concentration of the final mixture. This calculated amount of liquid is filled into the syringe. The same type and volume of the empty syringe is used as a reference syringe. Before filling, the full syringe is weighed vs. the reference syringe. After filling, the empty syringe is weighed vs. the reference syringe. This procedure and the determination of the uncertainty of the weighing have a considerable influence on the certified value of the final mixture. Here, based on this simple experiment, we present the approach for uncertainty analysis used at the Slovak Institute of Metrology and compare it with different approaches, which includes the GUM uncertainty framework, the Monte Carlo method, and the characteristic function approach.

We investigated how the attenuation of pain with cognitive interventions affects brain connectivity using neuroimaging and a whole brain novel analysis approach. While receiving tonic cold pain, 20 healthy participants performed three different pain attenuation strategies during simultaneous collection of functional imaging data at seven tesla. Participants were asked to rate their pain after each trial. We related the trial-by-trial variability of the attenuation performance to the trial-by-trial functional connectivity strength change of brain data. Across all conditions, we found that a higher performance of pain attenuation was predominantly associated with higher functional connectivity. Of note, we observed an association between low pain and high connectivity for regions that belong to brain regions long associated with pain processing, the insular and cingulate cortices. For one of the cognitive strategies (safe place), the performance of pain attenuation was explained by diffusion tensor imaging metrics of increased white matter integrity.

The effectiveness of cognitive strategies to attenuate pain has been reported in various behavioural studies, however the underlying neuronal mechanisms are only now beginning to be understood. Using a 7 T fMRI, we investigated three different pain attenuation strategies in 20 healthy subjects via: (a) non-imaginal distraction by counting backwards in steps of seven; (b) imaginal distraction by imagining a safe place; and (c) reinterpretation of the pain valence (reappraisal). Although we found considerable variability in the performances, all strategies exhibited a significant relief of pain compared to an unmodulated pain condition. Our finding argues against a subject's potential predisposition for a certain attenuation approach, as some of the subjects performed well on all attenuation tasks yet others performed low on all attenuation tasks. We further investigated the variability of performance within-subjects and explored the cortical regions that contribute to successful single attempts of pain attenuation at trial level. For each of the three tasks, we found a different pattern of brain activity that reflects the performance of pain attenuation. The more successful trials are related to reduced activity of different parts of the insular cortex. Behavioural data suggest that distraction is the preferable cognitive strategy to modulate pain perception. For three different cognitive strategies we revealed brain regions that are suggested to reliably modulate the perception of pain. The findings could be of utmost benefit for future attempts to integrate neuroscientific techniques into the treatment of pain. Further studies are necessary to investigate whether the present results are transferable to patients as an essential part of the multimodal therapy for chronic pain. These patients may also benefit from additional neurofeedback techniques by combining the strategies with the cortical feedback in order to modulate pain-related brain activity.

We consider a problem of constructing the exact and/or approximate coverage intervals for the common mean of several independent distributions. In a metrological context, this problem is closely related to evaluation of the interlaboratory comparison experiments, and in particular, to determination of the reference value (estimate) of a measurand and its uncertainty, or alternatively, to determination of the coverage interval for a measurand at a given level of confidence, based on such comparison data. We present a brief overview of some specific statistical models, methods, and algorithms useful for determination of the common mean and its uncertainty, or alternatively, the proper interval estimator. We illustrate their applicability by a simple simulation study and also by example of interlaboratory comparisons for temperature. In particular, we shall consider methods based on (i) the heteroscedastic common mean fixed effect model, assuming negligible laboratory biases, (ii) the heteroscedastic common mean random effects model with common (unknown) distribution of the laboratory biases, and (iii) the heteroscedastic common mean random effects model with possibly different (known) distributions of the laboratory biases. Finally, we consider a method, recently suggested by Singh et al., for determination of the interval estimator for a common mean based on combining information from independent sources through confidence distributions.

We introduce a class of logarithmic Lambert W random variables for a specific family of distributions. In particular, we characterize the log-Lambert W random variables for chi-squared distributions which naturally appear in the likelihood based inference of normal random variables.

Analyzed is the comparative calibration problem in the case when linear relationship is assumed between two considered measuring devices. The method deriving the approximate confidence region for the unknown parameters of the calibration line based on estimation of the calibration line via replicated errors-in-variables model is given inWimmer et al. (2004). Essential point in this approach is the use of the F-approximation of the distribution of the F-statistic suggested by Kenward and Roger (1997). Moreover, in Wimmer and Witkovský (2005) such a calibration procedure is conducted which enables proper interval estimates of several unknown values based on the next independent measurement results obtained by the less precise measuring device. The present contribution is a continuation of work presented in Wimmer et al. (2004) and Wimmer and Witkovský (2005). Derived is the Scheffé-type confidence region for the (whole) calibration line. This enables to construct the interval estimators for the multiple-use calibration case.

Linear models with variance-covariance components are used in a wide variety of applications. In most situations it is possible to partition the response vector into a set of independent subvectors, such as in longitudinal models where the response is observed repeatedly on a set of sampling units (see, e.g., Laird & Ware 1982). Often the objective of inference is either a test of linear hypotheses about the mean or both, the mean and the variance components. Confidence intervals for parameters of interest can be constructed as an alter- native to a test. These questions have kept many statisticians busy for several decades. Even under the assumption that the response can be modeled by a multivariate normal distribution, it is not clear what test to recommend except in a few settings such as balanced or orthogonal designs. Here we investigate statistical properties, such as accuracy of p-values and powers of exact (Crainiceanu & Ruppert 2004) tests and compare with properties of approximate asymptotic tests. Simultaneous exact confidence regions for variance components and mean parameters are constructed as well.

We propose an implementation of the straight-line fit algorithm to evaluate data obtained as part of measurements of line structures, e.g. a grating line, which allows us to determine, at a conventional threshold value, the coordinates of the edges, the position and the width of its photometric profile as well as the related standard uncertainty contribution. These are derived for the (1 ? ?)100% confidence intervals with ? [0, 1]. The implementation is demonstrated using data obtained by the PTB Nanometer Comparator (Fluegge and K?ning 2001 Proc. SPIE 4401 275?83). The investigation revealed that large parts of the uncertainty of the position and the width of the line are due to the choice of the fit interval and the uncertainty of the determination of the minimum and maximum level of the photoelectric signal of the line.

We consider the comparative calibration problem in the case when linear relationship is assumed between two considered measuring devices with possibly different units and precisions. The first method for obtaining the approximate confidence region for unknown parameters of the calibration line applies the maximum likelihood estimators of the unknown parameters. The second method is based on estimation of the calibration line via replicated errors-in-variables model. Essential point in this approach is approximation of the small sample distribution of the Wald-type test statistic. This enables to construct the interval estimators for the multiple-use calibration case.

We consider the problem of making statistical inference about the mean of a normal distribution based on a random sample of quantized (digitized) observations. This problem arises, for example, in a measurement process with errors drawn from a normal distribution and with a measurement device or process with a known resolution, such as the resolution of an analog-to-digital converter or another digital instrument. In this paper we investigate the effect of quantization on subsequent statistical inference about the true mean. If the standard deviation of the measurement error is large with respect to the resolution of the indicating measurement device, the effect of quantization (digitization) diminishes and standard statistical inference is still valid. Hence, in this paper we consider situations where the standard deviation of the measurement error is relatively small. By Monte Carlo simulations we compare small sample properties of the interval estimators of the mean based on standard approach (i.e. by ignoring the fact that the measurements have been quantized) with some recently suggested methods, including the interval estimators based on maximum likelihood approach and the fiducial approach. The paper extends the original study by Hannig et al. (2007).

We present an exact likelihood ratio (LR) based test for testing the simple null hypothesis on all parameters of the linear regression model with normally distributed errors. In particular, we consider simultaneous test for the regression parameters (beta) and the error standard deviation (sigma). The critical values of the LR test are presented for small sample sizes and small number of explanatory variables with standard significance level, alpha = 0.05.

In this paper, we deal with the comparative calibration problem, i.e. with the situation when one instrument or measurement technique is calibrated against another, each of which is subject to the measurement error. We propose an approximate, small sample, calibration confidence interval of the unknown true value of the measured substance in units of the more precise instrument, given measurement in units of the less precise instrument. Here we deal with the simplest case—single-use linear univariate calibration, i.e. the case in which we assume linear relationship between the two measurement techniques (instruments), and, further, that the calibration procedure is conducted in order to obtain one value for an unknown, reported together with an interval estimate. The method for deriving the approximate confidence interval is based on estimation of the calibration line via the replicated errors-in-variables model. The model is locally linearized and the Wald-type F-statistic is constructed. An essential po...

Although humans are generally capable of distinguishing single events of pain or touch, recent research suggested that both modalities activate a network of similar brain regions. By contrast, less attention has been paid to which processes uniquely contribute to each modality. The present study investigated the neuronal oscillations that enable a subject to process pain and touch as well as to evaluate the intensity of both modalities by means of Electroencephalography. Nineteen healthy subjects were asked to rate the intensity of each stimulus at single trial level. By computing Linear mixed effects models (LME) encoding of both modalities was explored by relating stimulus intensities to brain responses. While the intensity of single touch trials is encoded only by theta activity, pain perception is encoded by theta, alpha and gamma activity. Beta activity in the tactile domain shows an on/off like characteristic in response to touch which was not observed in the pain domain. Our results enhance recent findings pointing to the contribution of different neuronal oscillations to the processing of nociceptive and tactile stimuli.

The Invariant Quadratic Estimators, the Maximum Likelihood Estimator (MLE) and Restricted Maximum Likelihood Estimator (REML) of variances in an orthogonal Finite Discrete Spectrum Linear Regression Model (FDSLRM) are derived and the problems of unbiasedness and consistency of these estimators are investigated.

The studied topic is motivated by the problem of interlaboratory comparisons. This paper focuses on the confidence interval estimation of the between group variance in the unbalanced heteroscedastic one-way random effects model. Several interval estimators are proposed and compared by means of the simulation study. The most recommended (safest) is the confidence interval based on Bonferroni's inequality.

The inversion formula for evaluation of the distribution of a linear combination of independent t and F random variables, respectively, is suggested. The method is applied to computing the exact confidence intervals for the common mean of several normal populations. This method is compared with the known approximate methods.

We introduce a new method for rounding of measured values and their uncertainties. We define a new concept called `e-properly rounding' and describe exactly the probability properties of the rounded measured values and their `e-proper confidence intervals'. We propose general rules for proper rounding of the measurement results under normality assumptions. Finally we examine the probabilistic properties of the standard methods of rounding given in the Guide to the Expression of Uncertainty in Measurement of the ISO (1995).

We present a brief overview of the methods for making statistical inference (testing statistical hypotheses, construction of confi dence and/or prediction intervals and regions) about linear functions of the fix ed effects and/or about the fixed and random effects simultaneously, in conventional simple linear mixed model. The presented approach is based on solutions from the Henderson’s mixed model equations.

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 exercis e on a stationary bicycle). Determination of statistical detection lim its 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 det ection (LOD), respectively, based on a theoretical model of the measurement process is outlined.

Background This study was performed to clarify variations in breath isoprene concentrations with age, gender, body mass index (BMI) and total serum cholesterol. Our cohort consisted of 205 adult volunteers of different smoking background without health complaints. Total cholesterol in blood serum was measured in 79 of these volunteers. Methods Mixed expiratory exhaled breath was sampled using Tedlar bags. Concentrations of isoprene were then determined using proton transfer reaction-mass spectrometry. Results Isoprene concentrations ranged from 5.8 to 274.9 ppb, with an overall geometric mean (GM) of 99.3 ppb. There was no statistically significant difference in mean isoprene in breath between males and females (GM 105.4 and 95.5 ppb, respectively). Ageing led to a decrease in concentration in men, with an estimated slope of the regression line for log-transformed isoprene concentrations of -0.0049, but did not influence isoprene levels in women. We did not observe any significant correlation between isoprene breath content and cholesterol level in blood, even after adjusting for the possible influence of age. Similarly, no correlation was found between isoprene levels and BMI. Conclusions Isoprene concentrations in exhaled breath showed gender-specific correlations with respect to age. Further investigations are necessary to clarify the relation between isoprene concentrations in exhaled breath and cholesterol levels and synthesis rates in blood.

The collection contains more than 60 original papers and reflects current research topics in linguistics and text analysis. Most of the papers present recent results of empirical quantitative investigations; others focus on methodological issues, whereas some of them are of a more theoretical, systems-theoretical/semiotic character. Finally, a number of contributions form typical integrative deductive-inductive studies. The volume is a valuable source of information about the current state-of-the-art in quantitative linguistic research, presented by renowned representatives of the field.

Although the Heydemann correction is widely used to demodulate the phase of quadrature homodyne interferometer and encoder signals, the related measurement uncertainty is considered only in a few publications. The statistical uncertainty of the phase, which is the contribution of the high-frequency noise in the input signals, due to the use of a simplified fitting approach in the Heydemann correction, cannot be determined exactly. Therefore, neglecting the correlation between the input signals and the fit parameters, an approximation is provided and verified using Monte Carlo simulations and an iterative fitting method that allows exact determination of the statistical uncertainty of the phase. For the high-quality signals encountered in dimensional metrology applications, only minor differences of a few percent in the uncertainty estimates are observed. Furthermore, for cases in which a number of points used in the fit is sufficiently large, a new, simple, analytic expression for the statistical uncertainty of the phase is derived. It represents a practical limit of optical quadrature displacement interferometry, which already has been reached experimentally. To achieve an expanded measurement uncertainty of 10 pm a signal-to-noise greater than 104 has to be realized experimentally.