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Heterogeneity in agents' preferences is generally analysed through mixed logit models, which assume taste parameters are distributed in the population according to a certain mixing distribution. As a result, if the utility function is linear in attributes, the willingness to pay is the ratio of two random parameters and is itself random. This paper proposes a technique built on the Delta method, partly analytical and partly based on simulations, to obtain the sampling distribution of the willingness to pay, accounting for both heterogeneity and sampling error. The paper contributes to the literature by: (i) redressing some imprecisions in Bliemer and Rose (2013) that produce biased results; (ii) proposing a faster estimation process, compared to the Krinsky and Robb (1986, 1990) method that, relying on simulation only, proves computationally more demanding; (iii) comparing the performance of different methods using both synthetic and real data sets. The paper shows, via a Monte Carlo study, that the method we develop and the Krinsky and Robb one produce similar results, while outperforming that proposed by Bliemer and Rose. • Uncertainty in estimating Willingness To Pay under Mixed Logit Model arises from both heterogeneity and sampling error. • Delta method can be used, in combination with normal mixtures, to estimate both sources of uncertainty. • Both confidence and prediction intervals can be computed for Willingness To Pay by means of Delta method. • Delta method, being partly analytical, proves time-saving when compared to Krinsky and Robb method. [ABSTRACT FROM AUTHOR]

This paper proposes an operating performance index (OPI) based directly on the Six Sigma process quality index. We considered the following case: the number of customers arriving at a store follows a Poisson process of rate λ and the sequence of inter-arrival times is an exponential random variable with mean τ , where τ = 1/ λ. In operating performance assessments, using the statistical testing to evaluate the OPI can lower the risk of misjudgment caused by sampling error. A smaller sample size entails a larger sampling error, while a larger sample size entails a smaller sampling error. Thus, using critical value for statistical testing can result in inconsistencies for different sample size. We therefore referred to the one-tailed fuzzy method proposed by Buckley (2005) for the parameters of population distributions and developed a two-tailed fuzzy test for the OPI to assess operating performance. The advantage of this approach is that conventional measurement methods can be used during data collection, and then the confidence interval of the OPI can be used to construct a fuzzy membership function for fuzzy testing. This not only lowers the risk of misjudgment caused by sampling error, but also enhances testing accuracy. [ABSTRACT FROM AUTHOR]

Microplastics (MPs), plastic particles <5 mm in diameter, are emerging ubiquitous pollutants in natural environments, including freshwater ecosystems. As rivers facilitate efficient transport among landscapes, monitoring is crucial for elucidating the origin, dynamics, and fate of MPs. However, standardized methodologies for in situ sampling in freshwater environments remain undefined to date. Specifically, evaluating the sampling error of MP concentration estimates is crucial for comparing results among studies. This study proposes a novel method for computing confidence intervals (CIs) from a single estimate of numerical concentration (expressed in particles·m−3). MPs are expected to disperse according to purely random processes, such as turbulent diffusion, and to consequently exhibit a random distribution pattern that follows a Poisson point process. Accordingly, the present study introduced a framework based on the Poisson point process to compute CIs, which were validated using MP samples from two urban rivers in Chiba, Japan, obtained using a mesh with an opening size of 335 μm. Random number simulations revealed that the CIs were applicable when ≥10 MPs were present in a sample. Further, when ≥50 MPs were present in a sample, the sampling error (95% CI) was within ±30% of the concentration estimates. The proposed framework allows for the intercomparison of single river MP samples despite the lack of sample replicates. Further, the present study emphasizes that the volume of sampled river water is the only controllable parameter that can reduce the sampling error. [Display omitted] • Assuming Poisson process allows computation of CIs for single MP samplings. • Collecting ≥50 MPs allows 95% CI within ±30% of concentration estimate. • Sample water volume is the only controllable parameter to reduce sampling error. [ABSTRACT FROM AUTHOR]

The usefulness of biodiversity indicators strongly increases if accompanied by measures of uncertainty. In the case of indicators that combine population indices of species, however, the inclusion of the uncertainty of the species indices has shown to be hard to realize, usually due to imperfections in monitoring programmes. Missing values and time series of different lengths preclude the use of analytical approaches, whereas bootstrapping across sites requires the raw abundance data on the site level, which may not always be available. Sometimes bootstrapping across species rather than sites is opted for, but this approach ignores the uncertainty attached to species indices. We developed a method to account for sampling error of species indices in the calculation of multi-species indicators based on Monte Carlo simulation of annual species indices. The construction of confidence intervals enables various trend assessments, like testing for linear or smooth trends, testing for changes between two time points, testing the significance of a suspected change-point and testing for differences between two multi-species indicators. Here, we compare our method with conventional methods and illustrate the benefits of our approach using Dutch breeding bird indicators. [ABSTRACT FROM AUTHOR]

• TCATA data explored and visualized using PCA. • Bootstrap procedures used to investigate uncertainty in scores. • Uncertainty in scores visualized in different ways. • Index proposed to evaluate discriminability of scores. • New graphical features proposed for PCA biplots. Temporal check-all-that-apply (TCATA) data can be summarized and explored using principal component analysis (PCA). Here we analyze TCATA data on Syrah wines obtained from a trained sensory panel. We evaluate new and existing methods to explore the uncertainty in the PCA scores. To do so, we use the bootstrap procedure to obtain many virtual panels from the real panel's data. Virtual-panel PCA scores are obtained using two methods. The first method, called the partial bootstrap (PB), obtains virtual-panel scores from regression. The second method, called the truncated total bootstrap (TTB), applies PCA to the virtual-panel results to obtain scores, which are truncated and superimposed on the real-panel scores by Procrustes rotation. We use the virtual scores from each method to investigate uncertainty in the real-panel PCA scores visually and numerically. To understand the uncertainty of the scores, we obtain confidence ellipses (CEs) and their areas, as well as confidence intervals (CIs) and their widths. Next, to determine whether PCA scores for different samples are well separated, we propose a procedure for approximating the standard errors of sample differences and correcting for multiple comparisons. We propose a discriminability index, and show that it can enhance the interpretability of PCA results. We incorporate graphical features into our PCA biplots to visualize discriminability. We did not find a large difference between the PB and TTB methods for understanding the uncertainty and discriminability in PCA scores. Although the TCATA data that we analyzed have a special structure, the methodological approaches presented here can be readily adapted to other applications of PCA. [ABSTRACT FROM AUTHOR]

• A novel PCAC to measure products with multi-quality characteristic. • Pareto optimality to achieve the performance optimization for CM selection. • A feedback mechanism to advise ineligible CMs on quality issues. • A rigid coupling to validate the proposed methodology. Outsourcing is a crucial strategy of the mechanical manufacturing industry. When outsourcing, it is necessary for companies to carefully inspect the performance of contract manufacturers (CMs) to ensure high-quality products and satisfactory services. This study presents an integrated methodology for the quality performance assessment and optimization of products with multiple quality characteristics for CM selection in the mechanical manufacturing industry. First, we present a novel process capability analysis chart (PCAC) using the 100(1– α)% upper confidence limits of C pmh , C puh , and C plh to gauge the process capabilities of products with multiple quality characteristics and reduce the influence of sampling errors. Second, we apply Pareto optimality to reexamine eligible CMs under the objectives of maximizing performance in individual quality characteristics and minimizing variance in overall performance. A feedback mechanism is included to advise ineligible CMs on means of improvement. Finally, we present a case study to demonstrate the feasibility and effectiveness of the proposed methodology. [ABSTRACT FROM AUTHOR]

High fidelity computational fluid dynamics simulations of ship airwakes are often performed with time accurate, turbulence resolving methods, such as large eddy simulation or detached eddy simulation. In our review of the literature we find that the finite sample error in the results of these simulations is rarely expressed in terms of uncertainty in the statistics. This makes it difficult to determine if any disagreement with experiment or other simulations can be attributed to finite sample time, grid resolution, or modelling approximations. In this paper, turbulence resolving simulations of the airwake over the SFS1 geometry are used to demonstrate a method for quantifying the uncertainty due to finite sample error with an estimated confidence interval. Based on this quantification, a stopping criterion is then formulated so that the simulation time can be set by a predetermined requirement for the confidence in the statistics. [Display omitted] • A comprehensive review of the literature on CFD simulations of ship airwakes is presented. • The dependence of velocity spectra on grid resolution in ship airwake simulations is studied. • A method for estimating the confidence intervals for flow statistics from ship airwake simulations is formulated. • A stopping criterion based on estimated confidence interval widths is proposed. • A concise set of recommendations for grid resolutions, time steps, and simulation lengths for ship airwake CFD is made. [ABSTRACT FROM AUTHOR]

• Measured mineral grades generally confirmed geochemical static test results. • Mineralogy did not explain pH > 7 in humidity cell tests; liberation data were needed. • Mineralogy/texture data enabled sampling error and confidence interval assessment. • Sampling error evaluation for coarse material is key for accurate ARD assessment. • Lack of mineralogy and texture data greatly limits ARD test result interpretation. The value of quantitative mineralogy and texture is often not fully appreciated within minerals applications, and particularly so in the analysis of acid rock drainage (ARD). This is partly due to the absence of a clear understanding as to why quantitative textural tools such as QEMSCAN are valuable for ARD assessment. Additionally, most ARD prediction work is performed on a coarse particle scale, for which statistical representativeness of quantitative textural and mineralogical data becomes difficult to achieve. The current study provides an application of mineralogical and textural data (Fe-sulfide liberation, association, grain size distribution) to geochemical characterisation and humidity cell tests and applies these data to the estimation of the fundamental sampling error (FSE) and mineralogical error through binomial distribution and confidence intervals for liberation data. For the ARD characterisation tests, the mineralogy was key in interpreting the classification of the samples as potentially acid forming, non-acid forming or uncertain. For ARD prediction tests, the large degree of Fe-sulfide encapsulation in intermediate weathering, slow weathering and inert minerals aided in understanding the circumneutral behaviour of potentially acid forming and uncertain samples after 60 weeks of the humidity cell testing. The particle size distribution, in conjunction with texture assessment, showed that ±50% of the Fe-sulfides were unliberated, leading to the expectation that reactivity in the humidity cells would occur predominantly as a result of mineral reaction in finer fractions. The evaluation of uncertainties suggested that "one size fits all" tools such as Gy's safety line or the binominal distribution approximation may not be adequate in all cases, particularly for coarse material, such as that of the parent samples or humidity cell tests. In these situations, material-specific approaches for error consideration should be taken, which in this study were the calculation of the FSE for each sample, and the plotting of size-by-size confidence intervals for the Fe-sulfide liberation data. [ABSTRACT FROM AUTHOR]

Evidence is accumulating about the impacts of plastics on marine life. The prevalence of plastics in seabird nests has been used as an indicator of levels of this pollutant in the ocean. However, the lack of a framework for defining sample sizes and errors associated with estimating the prevalence of plastic in nests prevents researchers from optimising time and reducing impacts of fieldwork. We present a method to determine the confidence intervals for the prevalence of debris in seabird nests and provide, for the first time, information on the prevalence of these items in nests of the Hartlaub's gull Larus hartlaubii , the African penguin Spheniscus demersus , the great white pelican Pelecanus onocrotalus , and the white-breasted cormorant Phalacrocorax lucidus in South Africa. The method, based on observations and resampling simulations and tested here for nests of 12 seabird species from 15 locations worldwide, allows for straightforward hypothesis testing. Appropriate sample sizes can be defined by combining this method with a Bayesian approach. We show that precise estimates of prevalence of debris in nests can be obtained by sampling around 250 nests. Smaller sample sizes can be useful for obtaining rough estimates. For the Hartlaub's gull, the African penguin, the great white pelican, and the white-breasted cormorant, debris were present in 0.75%, 3.00%, 6.41%, and 25.62% of the respective nests. Our approach will help researchers to determine errors associated with the prevalence of debris recorded in seabird nests and to optimise time and costs spent collecting data. It can also be applied to estimate confidence intervals and define sample sizes for assessing prevalence of plastic ingestion by any organism. Image 1 • A method to define errors and sample size for the prevalence of debris in nests. • We use data from 12 seabird species from 15 locations worldwide. • Sampling 250 nests provide precise estimates of prevalence of debris. • Our method can be used also for data on prevalence of debris ingestion. • The method is freely available as an R package. A method for defining confidence intervals and sample size for estimating the prevalence of plastic debris in seabird nests. [ABSTRACT FROM AUTHOR]