Computing Confidence Intervals Research Articles

Confidence Intervals for Piecewise Normal Distributions and Stochastic Variational Inequalities

In this paper, we first show how to obtain easy-to-compute confidence intervals for the center of a piecewise normal distribution given a sample from this distribution (assuming that the center belongs to a known affine set parallel to the common lineality space of all cones defining the piecewise normal distribution) by using certain skewed projectors on that space. We then extend this method to an asymptotic setting. Next, we apply this method to compute confidence intervals for the true solution of a stochastic variational inequality given a solution to a sample average approximation (SAA) problem for the general situation in which the asymptotic distribution of SAA solutions is piecewise normal. For stochastic complementarity problems, we obtain asymptotic normality of certain estimators of the true solution when the asymptotic distribution of the SAA solutions is piecewise normal. Funding: The research reported in this paper was supported the National Science Foundation [Grant DMS-1814894].

A parametric bootstrap approach for computing confidence intervals for genetic correlations with application to genetically determined protein-protein networks

Genetic correlation refers to the correlation between genetic determinants of a pair of traits. When using individual-level data, it is typically estimated based on a bivariate model specification where the correlation between the two variables is identifiable and can be estimated from a covariance model that incorporates the genetic relationship between individuals, e.g., using a pre-specified kinship matrix. Inference relying on asymptotic normality of the genetic correlation parameter estimates may be inaccurate when the sample size is low, when the genetic correlation is close to the boundary of the parameter space, and when the heritability of at least one of the traits is low. We address this problem by developing a parametric bootstrap procedure to construct confidence intervals for genetic correlation estimates. The procedure simulates paired traits under a range of heritability and genetic correlation parameters, and it uses the population structure encapsulated by the kinship matrix. Heritabilities and genetic correlations are estimated using the close-form, method of moment, Haseman-Elston regression estimators. The proposed parametric bootstrap procedure is especially useful when genetic correlations are computed on pairs of thousands of traits measured on the same exact set of individuals. We demonstrate the parametric bootstrap approach on a proteomics dataset from the Jackson Heart Study.

Estimation of the incubation time distribution in the singly and doubly interval censored model

We analyze nonparametric estimators for the distribution function of the incubation time in the singly and doubly interval censoring model. The classical approach is to use parametric families like Weibull, log‐normal or gamma distributions in the estimation procedure. We propose nonparametric estimates for functions of the observations, which stay closer to the data than the classical parametric methods. We also give explicit limit distributions for discrete versions of the models and apply this to compute confidence intervals. The methods complement the analysis of the continuous model in Groeneboom (2021, 2023). R scripts for computation of the estimates are provided in Groeneboom (2020).

Classical and Bayesian Inference of Unit Gompertz Distribution Based on Progressively Type II Censored Data

In this article, two frequentist approaches and a Bayesian approach employing progressive Type II censored data are used to estimate parameters of a unit Gompertz distribution. In frequentist approach, besides conventional maximum likelihood estimation, maximum product of spacing method is proposed for parameter estimation as an alternative approach to common maximum likelihood method. Both Newton-Raphson and stochastic expectation minimization algorithms are used for computing the MLEs, while Bayes estimates are obtained using both the product of spacing function and the likelihood function. Additionally, the highest posterior density (HPD) credible intervals are compared with the approximate confidence intervals (CIs) for the parameters of the model that were derived using both traditional approaches. Moreover, percentile bootstrap technique is utilized to compute confidence intervals. Numerical comparisons are presented of the proposed estimators with respect to various criteria quantities using Monte Carlo simulations. Further, using different optimality criteria, an optimal censoring scheme has been suggested. Besides, one-sample and two-sample prediction problems based on observed sample and predictive intervals under Bayesian framework are discussed. Finally, to demonstrate the proposed methodology in a real-life scenario, maximum flood level data is considered to show the applicability of the proposed methods.

Assessing torrentiality in catchments of the tropical Andes: A morphometric approach

Torrential flows represent one of the most catastrophic natural hazards in the Andes, and Colombia is not an exception. We computed and analyzed ten morphometric indexes, classified by the catchment geometry and relief characteristics, for 149 torrential catchments across Colombian Andes three branches (Western, Central, and Eastern cordilleras). We divided the study area into six tectonic-stratigraphic regions, and employed statistical tests such as the Shapiro-Wilk and ANOVA One-Way Permutation tests to assess index distributions within each region. The ANOVA One-Way Permutation test results revealed significant differences in the distributions of form factor (p-value 0.048), mean gradient (p-value 0.009), relief ratio (p-value 0.003), Melton ratio (p-value 0.006), and hypsometric integral (p-value 0.000) indices among regions. Likewise, the pairwise comparison tests identified that the following regions are different between them for Melton ratio: Central Cordillera and Plateau Antioqueño (p-value 0.032), Garzón Massif and Plateau Antioqueño (p-value 0.015), Garzón Massif and Santander Massif (p-value 0.05), and Plateau Antioqueño and Western Cordillera (p-value 0.032); for relief ratio: Central Cordillera and Plateau Antioqueño (p-value 0.0219), Estern Cordillera and Plateau Antioqueño (p-value 0.0425), Garzón Massif and Plateau Antioqueño (p-value 0.0058), Plateau Antioqueño and Western Cordillera (p-value 0.053); for hypsometric integral: Central Cordillera and Western Cordillera (p-value 0.0004), Eastern Cordillera and Western Cordillera (p-value 0.0004), Garzón Massif and Western Cordillera (p-value 0.0004), Plateau Antioqueño and Western Cordillera (p-value 0.0161), and Santander Massif and Western Cordillera (p-value 0.002). Based on the findings we computed confidence intervals for the population median of the studied morphometric parameters according to the region employing a non-parametric bootstrapping method. The statistical analysis highlights the importance of the geological and geomorphological features in the morphometric indices analysis as part of the torrential flow susceptibility assessment.Applying our findings, we tested proposed thresholds in a case study of the Santander Massif (Eastern Cordillera of Colombia), focusing on the 2020 Piedecuesta Flow. In the study case, we demonstrated the relevance of identifying the feeder catchments where the morphometric indices would be calculated from, especially in areas where significant hydrologic changes occur. Volumetric concentration variations along the Piedecuesta Flow path highlight the need to prioritize the recognition of feeder catchments as an imperative part of the susceptibility assessment. Our approach enriches the knowledge base for identifying torrential-prone basins and has important implications for natural hazard management, aiming at mitigating the impact of torrential flows in the Andes.

Deep learning based normative models identify structures that become abnormal with Alzheimer’s disease

AbstractBackgroundImproving dementia diagnosis is a major global concern, given that up to 20% of subjects are misdiagnosed. Normative modelling, which quantifies how far individuals deviate from healthy controls (HC), may help to better situate subjects along the spectrum. To determine whether this is the case, we implemented a deep learning based normative model and tested it on structural neuroimaging data from subjects with HC, mild cognitive impairment (MCI), and Alzheimer’s disease (AD).MethodWe developed an adversarial autoencoder (AAE) (Figure 1) leveraging FreeSurfer features, namely cortical surface area of 68 cortical subregions (34 by hemisphere; Desikan‐Killiany atlas) and volumes of 33 neuroanatomical structures (Aseg atlas). Our model not only returns individual “deviation” scores—divergence from HC—but also identifies regions of interest that contribute to said score, facilitating interpretability. We retrieved T1‐weighted MRI scans (n = 204) from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (HC/MCI/AD 105/29/60; Table 1). We trained our AAE on 71 HC samples (n = 71) and tested it on the rest (n = 133). We used bootstrapping to compute confidence intervals (CI) for each point estimate and examined regions where scores differed for the considered groups [1].ResultDeviation scores were lowest for HC (mean 0.74 [95%‐CI 0.63–0.88]), highest for AD (mean 1.21 [95%‐CI 1.03–1.50]), and intermediate for MCI (mean 1.04 [95%‐CI 0.88–1.27]), implying subjects indeed fell along the AD spectrum (Figure 2A). Said scores also enabled classifying subjects into HC, MCI, and AD with a mean area under the curve (AUC) of 75% (Figure 2B‐C; HC vs. MCI: 0.75 [95%‐CI 0.62–0.85]; HC vs. AD: 0.76 [95%‐CI 0.64–0.82]). According to the AAE, the hippocampus, parahippocampal cortex, ventricles, and inferior parietal lobe contributed to such differences (Figure 2D‐E).ConclusionNormative models can identify structures that become abnormal as the brain undergoes ageing and AD, in line with the literature. Further testing on a larger and more heterogeneous sample will reveal the potential of this tool and its potential deployment in clinics.

Cost-effectiveness analysis of PCI in focal and diffuse coronary artery disease

Abstract Aims The pattern of CAD predicts angina relief after PCI. Our aim was to evaluate the cost-effectiveness of PCI in diffuse coronary artery disease (CAD) to deferral from PCI in these patients. Methods This is a sub-analysis of the TARGET-FFR randomized clinical trial (NCT03259815). The pullback pressure gradient (PPG) was calculated to differentiate focal from diffuse CAD. Healthcare resource use associated with the index hospitalization and with follow-up outpatient visits, medications, was recorded prospectively. We did not discount costs because of the limited follow-up period. Costs of procedures and follow-up consultations were derived from Healthcare Resource Group reference costs and drug costs from the National Health Service (NHS). Quality-adjusted life-years (QALYs) were derived from health-related quality of life and survival during the 3-month time horizon of the trial. Quality-of-life indexes (utilities) were evaluated at baseline and 3 months with the European Quality of Life–5 Dimensions (EQ-5D) instrument with UK weights scaled from 0 (death) to 1 (perfect health). The overall QALYs for each patient were estimated as the area under the curve determined by the utility values at baseline and 3 months. We computed confidence intervals for differences in costs and QALYs and in the incremental cost-effectiveness rate (ICER) using the bootstrap technique with the percentile method with 10 000 replications. A Markov model was used to simulate deferral of diffuse disease from PCI and not using PPG to select treatment. Probabilistic sensitivity analysis was undertaken to test the robustness of results to parameter uncertainty. Results Patients who underwent PCI for focal CAD had higher EQ-5D index values at follow-up (0.89 ± 0.18 vs 0.76 ± 0.28, p=0.004) and higher QALYs after 3 months follow-up (0.19 ± 0.06 vs 0.21 ± 0.04, p=0.022). Mean cumulative costs were £1574.3 ± 230.5 and significantly higher in patients treated for diffuse CAD as compared to patients treated for focal CAD (£1697 ± 293.4 vs £1393 ± 264.3; p<0.001). The ICER was £7746/QALY (95% CI 5024 – 11007). PCI in focal CAD saved £1522.2/QALY when compared to PCI in diffuse CAD (£7009.2 ± 1919.4 /QALY vs. £8531.4±2753.4 /QALY, p< 0.001). Deferring 30% of the diffuse CAD patients from PCI to optimized medical therapy would save £673.8/QALY (£7072.2 ± 1730.147 /QALY vs. £7746 ± 1943.5/QALY, p< 0.001). Conclusion PCI is more cost-effective in focal than diffuse CAD. Treatment of diffuse CAD results in worst QALYs and costlier procedures. Every 5% of the diffuse CAD patients that were deferred from PCI saved aproximately £100/QALY. Further trials are needed to determine if the cost-effectiveness of PCI can be improved by deferring patients with diffuse CAD.Quality of life in focal and diffuse CADCost-effectiveness focal vs diffuse CAD

Robust variance estimation in small meta-analysis with the standardized mean difference.

Conventional random-effects models in meta-analysis rely on large sample approximations instead of exact small sample results. While random-effects methods produce efficient estimates and confidence intervals for the summary effect have correct coverage when the number of studies is sufficiently large, we demonstrate that conventional methods result in confidence intervals that are not wide enough when the number of studies is small, depending on the configuration of sample sizes across studies, the degree of true heterogeneity and number of studies. We introduce two alternative variance estimators with better small sample properties, investigate degrees of freedom adjustments for computing confidence intervals, and study their effectiveness via simulation studies.

An analytical approach to confidence interval estimation of river microplastic sampling

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.

Trends and predictors of incidence of hypospadias in a tertiary hospital in South Africa

BackgroundHypospadias is a penile congenital abnormality characterized by an ectopic urethral opening, ventral curvature and incomplete foreskin closure with a dorsal hooded prepuce. Management of the outcome demands high cost. The aim of this study was to determine the trends and predictors of hypospadias among babies born at Charlotte Maxeke Johannesburg Academic Hospital between 01 January 2014 and 31 December 2016.MethodsMothers’ files from the transfer unit of the labour ward were reviewed. A case–control study compared boys with hypospadias and those without born approximately at the same time. Data were retrospectively collected on data collection forms. EpiData version 3.1 and STATA version 15.1 were used for statistical analysis. The Mantel Haenszel method determined combined odds ratios for hypospadias outcome on boys whose mothers were exposed to certain conditions versus boys of unexposed mothers. Woolf’s method was used to compute confidence intervals of the odds ratios. Logistic regression analysis was used to assess the independent contributing maternal factors to the risk of hypospadias.ResultsRecords of 221 male babies were collected of which 73 were cases and 148 controls in the ratio of 1:2. Hypospadias was more frequent in boys born by Caesarean Section (p < 0.001), those with low birth weight (p < 0.001) and those small for gestational age (p < 0.002). Alcohol use (odds ratio 3.1), smoking (odds ratio 1.54), herbal use (odds ratio 2.05), medical history (odds ratio 2.8), multiple pregnancies (odds ratio 1.69) and maternal congenital abnormalities (odds ratio 3.03) indicated an increased risk of hypospadias. Surgical history (odds ratio 1.29), pre-natal vaccination (odds ratio 0.92), employment (odds ratio 0.85), and education (odds ratio 0.48) were not associated with the risk of hypospadias.ConclusionsMothers older than 36 years presented a stratifying effect on disease outcome. Our findings did not indicate major changes in trends of incidence of hypospadias. The years assessed did not have a significant effect on the number of cases.

Examining Schizotypal Personality Scales Within and Across Interpersonal Circumplex Surfaces.

Differing perspectives on the operationalization of schizotypal personality pathology (STPP) have led to numerous multidimensional assessment measures. The current study applied the interpersonal construct validation approach to self-report data from 856 undergraduate students to formally examine the interpersonal content, similarities, and differences in the subscales of four measures of STPP within and across two interpersonal circumplex surfaces using a bootstrapping methodology for computing confidence intervals around circumplex structural summary method parameters. Results suggested that negative-like expressions of STPP are prototypically and distinctively interpersonal constructs associated with cold and socially avoidant interpersonal problems and hypersensitivity to others' warmth and affection. Positive-like expressions of STPP as assessed by two out of four measures were prototypically and distinctively interpersonal constructs associated with vindictiveness. Across four measures, there was notable overlap in interpersonal correlates among related subscales, suggesting convergent validity. However, subscales containing social anxiety content were associated with more submissive (i.e., socially avoidant) interpersonal problems than subscales without social anxiety content.

Evaluating COVID-19 in Portugal: Bootstrap confidence interval

&lt;abstract&gt;&lt;p&gt;In this paper, we consider a compartmental model to fit the real data of confirmed active cases with COVID-19 in Portugal, from March 2, 2020 until September 10, 2021 in the Primary Care Cluster in Aveiro region, ACES BV, reported to the Public Health Unit. The model includes a deterministic component based on ordinary differential equations and a stochastic component based on bootstrap methods in regression. The main goal of this work is to take into account the variability underlying the data set and analyse the estimation accuracy of the model using a residual bootstrapped approach in order to compute confidence intervals for the prediction of COVID-19 confirmed active cases. All numerical simulations are performed in R environment (&lt;inline-formula id="math-09-02-136-M1"&gt;&lt;inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="math-09-02-136-M1.jpg"/&gt;&lt;/inline-formula&gt; version. 4.0.5). The proposed algorithm can be used, after a suitable adaptation, in other communicable diseases and outbreaks.&lt;/p&gt;&lt;/abstract&gt;

A detailed kinetic model for aromatics formation from small hydrocarbon and gasoline surrogate fuel combustion

This work develops a detailed chemical kinetic model for Polycyclic Aromatic Hydrocarbon (PAH) chemistry that builds on a chemical model for gasoline surrogates by Cai et al. [Proc. Combust. Inst. 37 (2019) 639–647]. Informed by ab-initio and experimental studies, the model development emphasizes the prediction of soot precursors starting from C3H4 isomers up to the size of acepyrene. The proposed model was validated against experimental measurements from 79 publications, including ignition delay times, laminar burning velocities, and speciation data for several fuels in various conditions. A validation against measured peak mole fractions from 31 counterflow flames demonstrates an average prediction error of aromatic species up to the size of acenaphthalene below a factor of three. Additionally, the effects of experimental uncertainty are quantitatively discussed by computing confidence intervals for the true model prediction error. The formation of two-ring aromatic species is often still considered poorly understood. Accordingly, special attention is given to the formation of indene and naphthalene. Reaction flux analyses based on rigorously defined species selection criteria reveal that indene formation pathways are initiated by reactions of phenyl and benzyl and described by the reaction kinetics from theoretical calculations. In contrast, naphthalene formation is dominated by more uncertain pathways relying on estimated rates that are initiated by reactions of benzene and fulvenallenyl. Five-member rings on the periphery of PAH species and soot particles play a crucial role in recent experimental and theoretical studies. Therefore, this work investigates acenaphthene and other C12H8 isomers. The most abundant C12H8 isomers are acenaphthene, 1-ethynylnaphthalene, and 2-ethynylnaphthalene, and their respective concentrations are predicted to be similar in many flames. Furthermore, reaction flux analyses show that their formation is dominated by HACA routes starting from naphthalene.

The failure of four bootstrap procedures for estimating confidence intervals for predicted-to-expected ratios for hospital profiling

BackgroundHealthcare provider profiling involves the comparison of outcomes between patients cared for by different healthcare providers. An important component of provider profiling is risk-adjustment so that providers that care for sicker patients are not unfairly penalized. One method for provider profiling entails using random effects logistic regression models to compute provider-specific predicted-to-expected ratios. These ratios compare the predicted number of deaths at a given provider given the case-mix of its patients with the expected number of deaths had those patients been treated at an average provider. Despite the utility of this metric in provider profiling, methods have not been described to estimate confidence intervals for these ratios. The objective of the current study was to evaluate the performance of four bootstrap procedures for estimating 95% confidence intervals for predicted-to-expected ratios.MethodsWe used Monte Carlo simulations to evaluate four bootstrap procedures: the naïve bootstrap, a within cluster-bootstrap, the parametric multilevel bootstrap, and a novel cluster-specific parametric bootstrap. The parameters of the data-generating process were informed by empirical analyses of patients hospitalized with acute myocardial infarction. Three factors were varied in the simulations: the number of subjects per cluster, the intraclass correlation coefficient for the binary outcome, and the prevalence of the outcome. We examined coverage rates of both normal-theory bootstrap confidence intervals and bootstrap percentile intervals.ResultsIn general, all four bootstrap procedures resulted in inaccurate estimates of the standard error of cluster-specific predicted-to-expected ratios. Similarly, all four bootstrap procedures resulted in 95% confidence intervals whose empirical coverage rates were different from the advertised rate. In many scenarios the empirical coverage rates were substantially lower than the advertised rate.ConclusionExisting bootstrap procedures should not be used to compute confidence intervals for predicted-to-expected ratios when conducting provider profiling.

Nonparametric methods for clustered data in pre-post intervention design

In pre-post factorial designs involving clustered units, parametric methods such as generalized linear mixed effects models are used to handle within subject correlations. However, the distributional and parametric model assumptions in these methods are not always satisfied, especially so with modern data sets, and are often difficult to verify in practice. Often times, the assumptions may not even be realistic when data are measured in a non-metric scale as commonly happens, for example, in Quality-of-Life outcomes. In this article, nonparametric effect-size measures for clustered data in factorial designs with pre-post measurements will be introduced. In our setup, the pre and post occasions may also be viewed as two treatment conditions. Arbitrary dependence among observations within a cluster and across treatment groups is allowed. The effect-size estimators along with their asymptotic properties for computing confidence intervals and performing hypotheses tests are investigated. The proposed methods handle within-cluster as well as between-cluster treatment assignments seamlessly. The methods are shown to be effective in finite samples using simulation studies and they have high powers, especially in situations with multiple forms of clustering. Applications are illustrated with data from a three-arm Randomized Trial of Indoor Wood Smoke reduction.

Confidence intervals for the random forest generalization error

We show that the byproducts of the standard training process of a random forest yield not only the well known and almost computationally free out-of-bag point estimate of the model generalization error, but also open a direct path to compute confidence intervals for the generalization error which avoids processes of data splitting and model retraining. Besides the low computational cost involved in their construction, these confidence intervals are shown through simulations to have good coverage and appropriate shrinking rate of their width in terms of the training sample size.

Examining the Degree of Changes in Korean Elementary Schools' Physical Activity Promotion System Grades amidst the COVID-19 Pandemic

The study aimed to provide basic data about the changes in the Korean elementary schools' Physical Activity Promotion System (PAPS) amidst the COVID-19 pandemic and to present future direction for school and physical education. Data were collected for 2019-2021 from the PAPS, conducted by the Korea Ministry of Education. Participants were fifth and sixth level students from 1,214 Korean elementary schools, with 124,871, 126,396, and 117,953 participants in 2019, 2020, and 2021, respectively. Five physical fitness variables were evaluated-cardiopulmonary endurance, flexibility, muscle strength, power, and obesity-and graded from 1 (highest physical fitness level) to 5 (lowest physical fitness level). Data were analyzed using repeated measures analysis of variance and Bonferroni for multiple comparisons for computing confidence intervals. The findings indicated significant changes in the ratios of grades 1, 2, 4, and 5 according to the year (P<0.05). Grades 1 and 5 decreased and increased in 2021 from 2019 and 2020, respectively (P<0.001). Grades 2 and 4 continuously decreased and increased from 2019 to 2020 (P=0.001, P<0.001, respectively) and further in 2021 (P<0.001), respectively. Moreover, significant changes were identified in the grade ratio by year according to sex (P<0.05). Male students reported higher grade 1 ratios than female students in 2019 (P=0.018) and 2021 (P<0.001). However, for all 3 years, female students reported higher grade 2 ratio (P<0.001), while male students reported higher grade 4 and 5 ratios (P<0.001). Lastly, male students reported higher grade 3 ratios than female students in 2019 (P=0.003), but this finding was reversed in 2021 (P=0.004). The pandemic decreased students' physical activity and weakened their health, and this reduced fitness was greater pronounced in male students. In the future, physical educators should plan and present solutions to tackle reduced physical activity post the pandemic.

Confidence intervals for discrete log-linear models when MLE does not exist

The aim of this paper is to provide a methodology and MATLAB programs to compute confidence intervals for the cell probability parameters in a high-dimensional discrete log-linear model when the maximum likelihood estimate of these parameters does not exist. To do so, we use the geometry of exponential families as well as recent methodology to identify the submodel for which the maximum likelihood estimate exists. We illustrate our results with both simulated and real world data.

Analyzing nested experimental designs-A user-friendly resampling method to determine experimental significance.

While hierarchical experimental designs are near-ubiquitous in neuroscience and biomedical research, researchers often do not take the structure of their datasets into account while performing statistical hypothesis tests. Resampling-based methods are a flexible strategy for performing these analyses but are difficult due to the lack of open-source software to automate test construction and execution. To address this, we present Hierarch, a Python package to perform hypothesis tests and compute confidence intervals on hierarchical experimental designs. Using a combination of permutation resampling and bootstrap aggregation, Hierarch can be used to perform hypothesis tests that maintain nominal Type I error rates and generate confidence intervals that maintain the nominal coverage probability without making distributional assumptions about the dataset of interest. Hierarch makes use of the Numba JIT compiler to reduce p-value computation times to under one second for typical datasets in biomedical research. Hierarch also enables researchers to construct user-defined resampling plans that take advantage of Hierarch's Numba-accelerated functions.

Greenhouse Gas Emission Reduction Potentials in Europe by Sector: A Bootstrap-Based Nonparametric Efficiency Analysis

The reduction of greenhouse gas emissions is the key action to limit global warming. An important source of greenhouse gas emissions and pollution is the inefficiency of production processes. We report results from a stochastic nonparametric efficiency analysis using directional distance functions to take account of undesirable outputs like greenhouse gases. With this approach, we are able to provide estimates of the potential emission reductions for 7 main sectors in 16 European countries. A specially adapted bootstrapping approach allows to implement a bias correction of the estimates and to compute confidence intervals. The results show that static efficiency improvements are a quantitatively important element of the emission reductions which are required to achieve the reduction targets of the European Union.