Comparison of interval estimation for extreme event proportions based on exact, approximate and Bayesian approaches

The coronavirus disease 2019 (COVID-19), caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), has become a global pandemic. No specific therapeutic agents or vaccines for COVID-19 are available, though several antiviral drugs, are under investigation as treatment agents for COVID-19. The use of convalescent plasma transfusion that contain neutralizing antibodies for COVID-19 has become the major focus. This requires mass screening of populations for these antibodies. While several countries started reporting population based antibody rate, its simple point estimate may be misinterpreted without proper estimation of standard error and confidence intervals. In this paper, we review the importance of antibody studies and present the 95% confidence intervals COVID-19 antibody rate for the Korean population using two recently performed antibody tests in Korea. Due to the sparsity of data, the estimation of confidence interval is a big challenge. Thus, we consider several confidence intervals using Asymptotic, Exact and Bayesian estimation methods. In this article, we found that the Wald method gives the narrowest interval among all Asymptotic methods whereas mid p-value gives the narrowest among all Exact methods and Jeffrey’s method gives the narrowest from Bayesian method. The most conservative 95% confidence interval estimation shows that as of 00:00 on September 15, 2020, at least 32,602 people were infected but not confirmed in Korea.

The subject of this research is distance and time of several city tour problems which known as traveling salesman problem (tsp). The goal is to find out the gaps of distance and time between two types of optimization methods in traveling salesman problem: exact and approximate. Exact method yields optimal solution but spends more time when the number of cities is increasing and approximate method yields near optimal solution even optimal but spends less time than exact methods. The task in this study is to identify and formulate each algorithm for each method, then to run each algorithm with the same input and to get the research output: total distance, and the last to compare both methods: advantage and limitation. Methods used are Brute Force (BF) and Branch and Bound (B&B) algorithms which are categorized as exact methods are compared with Artificial Bee Colony (ABC), Tabu Search (TS) and Simulated Annealing (SA) algorithms which are categorized as approximate methods or known as a heuristics method. These three approximate methods are chosen because they are effective algorithms, easy to implement and provide good solutions for combinatorial optimization problems. Exact and approximate algorithms are tested in several sizes of city tour problems: 6, 9, 10, 16, 17, 25, 42, and 58 cities. 17, 42 and 58 cities are derived from tsplib: a library of sample instances for tsp; and others are taken from big cities in Java (West, Central, East) island. All of the algorithms are run by MATLAB program. The results show that exact method is better in time performance for problem size less than 25 cities and both exact and approximate methods yield optimal solution. For problem sizes that have more than 25 cities, approximate method – Artificial Bee Colony (ABC) yields better time which is approximately 37% less than exact and deviates 0.0197% for distance from exact method. The conclusion is to apply exact method for problem size that is less than 25 cities and approximate method for problem size that is more than 25 cities. The gap of time will be increasing between two methods when sample size becomes larger.

The performance of statistical methods is often evaluated by means of simulation studies. In case of network meta-analysis of binary data, however, simulations are not currently available for many practically relevant settings. We perform a simulation study for sparse networks of trials under between-trial heterogeneity and including multi-arm trials. Results of the evaluation of two popular frequentist methods and a Bayesian approach using two different prior specifications are presented. Methods are evaluated using coverage, width of intervals, bias, and root mean squared error (RMSE). In addition, deviations from the theoretical surface under the cumulative rankings (SUCRAs) or P-scores of the treatments are evaluated. Under low heterogeneity and when a large number of trials informs the contrasts, all methods perform well with respect to the evaluated performance measures. Coverage is observed to be generally higher for the Bayesian than the frequentist methods. The width of credible intervals is larger than those of confidence intervals and is increasing when using a flatter prior for between-trial heterogeneity. Bias was generally small, but increased with heterogeneity, especially in netmeta. In some scenarios, the direction of bias differed between frequentist and Bayesian methods. The RMSE was comparable between methods but larger in indirectly than in directly estimated treatment effects. The deviation of the SUCRAs or P-scores from their theoretical values was mostly comparable over the methods but differed depending on the heterogeneity and the geometry of the investigated network. Multivariate meta-regression or Bayesian estimation using a half-normal prior scaled to 0.5 seems to be promising with respect to the evaluated performance measures in network meta-analysis of sparse networks.

The main aim of this work is to present results of the mechanical system's analysis based on the exact and approximate Galerkin's methods. The considered system is the flexural vibrating one‐dimension bending beam. The exact and approximate method were used to assign the dynamic flexibility of the considered system and results of this work were juxtaposed to verify the approximate method's accuracy. The correction coefficients were introduced into the approximate method to unify results of both methods. The aim of this work was to check accuracy of the approximate method and to verify if this method may be used to mechatronic system's analysis, where it is impossible to use the exact method. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

There are two major optimization methods: Exact and Approximate methods. A well known exact method, Branch and Bound algorithm (B&B) and approximate methods, Elimination-based Fruit Fly Optimization Algorithm (EFOA) and Artificial Atom Algorithm (A3) are used for solving the Traveling Salesman Problem (TSP). For 56 destinations, the results of total distance, processing time, and the deviation between exact and approximate method will be compared where the distance between two destinations is a Euclidean distance and this study shows that the distance of B&B is 270 , EFOA is 270 and A3 is 288.38 which deviates 6.81%. For time processing aspect, B&B needs 12.5 days to process, EFOA needs 36.59 seconds, A3 needs 35.34 seconds. But for 29 destinations, exact method is more powerful than approximate method.

Supply chain management and optimization in transportation logistics

With the ongoing rise of coronavirus disease 2019 (COVID-19) pandemic across the globe, interests in COVID-19 antibody testing, also known as a serology test has grown, as a way to measure how far the infection has spread in the population and to identify individuals who may be immune. Recently, many countries reported their population based antibody titer study results. South Korea recently reported their third antibody formation rate, where it divided the study between the general population and the young male youths in their early twenties. As previously stated, these simple point estimates may be misinterpreted without proper estimation of standard error and confidence intervals. In this article, we provide an updated 95% confidence intervals for COVID-19 antibody formation rate for the Korean population using asymptotic, exact and Bayesian statistical estimation methods. As before, we found that the Wald method gives the narrowest interval among all asymptotic methods whereas mid p-value gives the narrowest among all exact methods and Jeffrey’s method gives the narrowest from Bayesian method. The most conservative 95% confidence interval estimation shows that as of 00:00 November 23, 2020, at least 69,524 people were infected but not confirmed. It also shows that more positive cases were found among the young male in their twenties (0.22%), three times that of the general public (0.051%). This thereby calls for the quarantine authorities’ need to strengthen quarantine managements for the early twenties in order to find the hidden infected people in the population.

Meta-analysis combines multiple independent studies, which can increase power and provide better estimates. However, it is unclear how best to deal with studies with zero events; such studies are also known as double-zero-event studies (DZS). Several statistical methods have been proposed, but the agreement among different approaches has not been systematically assessed using real-world published systematic reviews. The agreement of five commonly used methods (i.e., the inverse-variance, Mantel-Haenszel, Peto, Bayesian, and exact methods) was assessed using the Cohen's κ coefficients using 368 meta-analyses with rare events selected from the Cochrane Database of Systematic Reviews. Three continuity corrections, including the correction of a constant 0.5, the treatment arm continuity correction (TACC), and the empirical (EMP) correction, were used to handle DZS when applying inverse-variance and Mantel-Haenszel methods. When the proportion of DZS studies was lower than 50% in a meta-analysis, different methods had moderately high agreement. However, when this proportion was increased to be over 50%, the agreement among the methods decreased to different extents. For the Bayesian, exact, and Peto methods and the inverse-variance and Mantel-Haenszel methods using the EMP correction, their agreement coefficients with the inverse-variance and Mantel-Haenszel methods using a constant 0.5 and TACC decreased from larger than 0.70 to smaller than 0.30. In contrast, the agreement coefficients only decreased slightly among the Bayesian, exact, and Peto methods and the inverse-variance and Mantel-Haenszel methods using the EMP correction. To utilize all available information and reduce research waste and avoid overestimating the effect, meta-analysts should incorporate DZS, rather than simply removing them. The Peto and other conventional methods with continuity correction should be avoided when the proportion of DZS is extremely high. The exact and Bayesian methods are highly recommended, except when none of the included studies have an event in one or both treatment arms.

The effect of selection on individual performance for a quantitative trait is studied theoretically for populations of finite size. The trait is assumed to be affected by environmental error and by segregation at a single locus. Exact formulae are derived to predict the change in gene frequency at this locus, initially by finding the probability distribution of the numbers of each genotype selected from a finite population of specified genotypic composition. Assuming that there is random mating and no natural selection the results are extended to describe repeated cycles of artificial selection for a monecious population. The formulae are evaluated numerically for the case of normally distributed environmental errors.Using numerical examples comparisons are made between the exact values for the predicted change in gene frequency with values obtained using approximate, but simpler, methods. Unless the gene has a large effect (α) on the quantitative trait, relative to the standard deviation of the environmental errors, the agreement between exact and approximate methods is satisfactory for most predictive purposes. The chance of fixation after repeated generations of selection is also evaluated using the exact method, and by means of a diffusion approximation and simple transition probability matrix methods. Except for very small values of population size (N) and large α the results from the diffusion equation agree closely with those from the exact method. Similar results are found from tests made of the prediction from the diffusion equation that the limit is only a function ofNα for a given intensity of selection and initial frequency, and that the rate of advance in gene frequency is proportional to 1/Nfor the same set of parameters.

It is generally agreed that a confidence interval (CI) is usually more informative than a point estimate or p-value, but we rarely encounter small proportions with CI in the pharmaco-epidemiological literature. When a CI is given it is sporadically reported, how it was calculated. This incorrectly suggests one single method to calculate CIs. To identify the method best suited for small proportions, seven approximate methods and the Clopper-Pearson Exact method to calculate CIs were compared. In a simulation study for 90-, 95- and 99%CIs, with sample size 1000 and proportions ranging from 0.001 to 0.01, were evaluated systematically. Main quality criteria were coverage and interval width. The methods are illustrated using data from pharmaco-epidemiology studies. Simulations showed that standard Wald methods have insufficient coverage probability regardless of how the desired coverage is perceived. Overall, the Exact method and the Score method with continuity correction (CC) performed best. Real life examples showed the methods to yield different results too. For CIs for small proportions (pi < or = 0.01), the use of the Exact method and the Score method with CC are advocated based on this study.

In this article, we point out some interesting relations between the exact test and the score test for a binomial proportion p. Based on the properties of the tests, we propose some approximate as well as exact methods of computing sample sizes required for the tests to attain a specified power. Sample sizes required for the tests are tabulated for various values of p to attain a power of 0.80 at level 0.05. We also propose approximate and exact methods of computing sample sizes needed to construct confidence intervals with a given precision. Using the proposed exact methods, sample sizes required to construct 95% confidence intervals with various precisions are tabulated for p = .05(.05).5. The approximate methods for computing sample sizes for score confidence intervals are very satisfactory and the results coincide with those of the exact methods for many cases.

Determination of the rotational diffusion tensor of macromolecules in solution from nmr relaxation data with a combination of exact and approximate methods--application to the determination of interdomain orientation in multidomain proteins.

BackgroundEffective coverage research is increasing rapidly in global health and development, as researchers use a range of measures and combine data sources to adjust coverage for the quality of services received. However, most estimates of effective coverage that combine data sources are reported only as point estimates, which may be due to the challenge of calculating the variance for a composite measure. In this paper, we evaluate three methods to quantify the uncertainty in the estimation of effective coverage.MethodsWe conducted a simulation study to evaluate the performance of the exact, delta, and parametric bootstrap methods for constructing confidence intervals around point estimates that are calculated from combined data on coverage and quality. We assessed performance by computing the number of nominally 95% confidence intervals that contain the truth for a range of coverage and quality values and data source sample sizes. To illustrate these approaches, we applied the delta and exact methods to estimates of adjusted coverage of antenatal care (ANC) in Senegal. We used household survey data for coverage and health facility assessments for readiness to provide services.ResultsWith small sample sizes, when the true effective coverage value was close to the boundaries 0 or 1, the exact and parametric bootstrap methods resulted in substantial over or undercoverage and, for the exact method, a high proportion of invalid confidence intervals, while the delta method yielded modest overcoverage. The proportion of confidence intervals containing the truth in all three methods approached the intended 95% with larger sample sizes and as the true effective coverage value moved away from the 0 or 1 boundary. Confidence intervals for adjusted ANC in Senegal were largely overlapping across the delta and exact methods, although at the sub-national level, the exact method produced invalid confidence intervals for estimates near 0 or 1. We provide the code to implement these methods.ConclusionsThe uncertainty around an effective coverage estimate can be characterized; this should become standard practice if effective coverage estimates are to become part of national and global health monitoring. The delta method approach outperformed the other methods in this study; we recommend its use for appropriate inference from effective coverage estimates that combine data sources, particularly when either sample size is small. When used for estimates created from facility type or regional strata, these methods require assumptions of independence that must be considered in each example.

Effective coverage research is increasing rapidly in global health and development, as researchers use a range of measures and combine data sources to adjust coverage for the quality of services received. However, most estimates of effective coverage that combine data sources are reported only as point estimates, which may be due to the challenge of calculating the variance for a composite measure. In this paper, we evaluate three methods to quantify the uncertainty in the estimation of effective coverage. We conducted a simulation study to evaluate the performance of the exact, delta, and parametric bootstrap methods for constructing confidence intervals around point estimates that are calculated from combined data on coverage and quality. We assessed performance by computing the number of nominally 95% confidence intervals that contain the truth for a range of coverage and quality values and data source sample sizes. To illustrate these approaches, we applied the delta and exact methods to estimates of adjusted coverage of antenatal care (ANC) in Senegal. We used household survey data for coverage and health facility assessments for readiness to provide services. With small sample sizes, when the true effective coverage value was close to the boundaries 0 or 1, the exact and parametric bootstrap methods resulted in substantial over or undercoverage and, for the exact method, a high proportion of invalid confidence intervals, while the delta method yielded modest overcoverage. The proportion of confidence intervals containing the truth in all three methods approached the intended 95% with larger sample sizes and as the true effective coverage value moved away from the 0 or 1 boundary. Confidence intervals for adjusted ANC in Senegal were largely overlapping across the delta and exact methods, although at the sub-national level, the exact method produced invalid confidence intervals for estimates near 0 or 1. We provide the code to implement these methods. The uncertainty around an effective coverage estimate can be characterized; this should become standard practice if effective coverage estimates are to become part of national and global health monitoring. The delta method approach outperformed the other methods in this study; we recommend its use for appropriate inference from effective coverage estimates that combine data sources, particularly when either sample size is small. When used for estimates created from facility type or regional strata, these methods require assumptions of independence that must be considered in each example.

In this paper the analysis of torsionally vibrating subsystem of complex mechanical and mechatronic systems by using the the exact and approximate methods were the main purposes to solve the task of assignment of frequency‐modal analysis and characteristics of mechatronic system. The characteristic of the elementary subsystems using the exact and approximate methods has been determined according to accepted frequencies and the correction coefficient. The frequencies were chosen from the spectrum in which the synthesis of complex systems will be conducted. It is very important that the difference of flexibility values in the spectrum was minimal. The coefficient of the correction has been determined according to the flexibility values of chosen points and it is equal to quotient of flexibility calculated using the exact method across the flexibility delivered by using the approximate method. The coefficient of the correction is the zero‐dimensional quantity. After determination of the correction coefficient the medium value which has been afterwards considered in correlation of dynamic characteristics has been calculated. (© 2015 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)