Fiducial Generalized Confidence Intervals

Comparison of Analytical Procedures in Method Transfer and Bridging Experiments

Comparison of two analytical procedures is the primary objective of a method transfer or when replacing an old procedure with a new one in a single lab. Guidance for comparing two analytical procedures is provided in USP <1010> based on separate tests for accuracy and precision. Determination of criteria is somewhat problematic for these comparisons because of the interdependence of accuracy and precision. In this paper, a total error approach is proposed that requires a single criterion based on an allowable out-of-specification (OOS) rate at the receiving lab. This approach overcomes the difficulty of allocating acceptance criteria between precision and bias. Computations can be performed with any simulation software. Numerical examples are provided for four experimental designs that are typical in a method transfer study. Finally, recommendations are provided to help the user set criteria that provide an acceptable probability of passing for practical sample sizes.

Simultaneous confidence intervals for quantile differences of several heterogeneous normal populations: With application to vitamin D supplement treatment on colorectal cancer patients

In many applications, comparing the q-quantiles of several normal populations are more advantageous than comparing their means. In this paper, we consider the problem of constructing simultaneous confidence intervals (SCIs) for quantile differences of several heterogeneous normal distributions. To the best of our knowledge, this problem remains unsolved. We propose a novel method for constructing a set of SCI. We propose two new sets of SCI by using the proposed technique and discuss two classic and two simulation-based SCIs. We show that the proposed classic SCIs are conservative for all population parameters configuration. We also show that the simulation-based SCIs have correct coverage probability asymptotically. We then compare these six sets of SCI in terms of average volume and coverage probability via an extensive simulation study. Results show that one of the proposed classic SCI can be recommended. Finally, the proposed methods are illustrated by a real example that is a vitamin D study on colorectal cancer patients.

Confidence Interval Estimation for the Common Mean of Several Zero-Inflated Gamma Distributions

In this study, we propose estimates for the confidence interval for the common mean of several zero-inflated gamma (ZIG) distributions based on the fiducial generalized confidence interval (GCI) and Bayesian and highest posterior density (HPD) methods based on the Jeffreys rule or uniform prior. Their performances in terms of their coverage probabilities and expected lengths are compared via a Monte Carlo simulation study. For almost all of the scenarios considered, the simulation results show that the fiducial GCI performed better than the Bayesian and HPD methods. Daily rainfall data from Chiang Mai Province, Thailand that contains several zero entries and follows a ZIG distribution is used to test the efficacies of the methods in real-world situations.

Estimation methods for the ratio of medians of three-parameter lognormal distributions containing zero values and their application to wind speed data from northern Thailand

Wind speed has an important impact on the formation and dispersion of fine particulate matter (PM), which can cause several health problems. During the transition from the winter to the summer season in northern Thailand, the wind speed has been low for longer than usual, which has resulted in fine PM accumulating in the air. Motivated by this, we have identified a need to investigate wind speed due to its effect on PM formation and dispersion and to raise awareness among the general public. The hourly windspeed can be approximated by using confidence intervals for the ratio of the medians of three-parameter lognormal distributions containing zero values. Thus, we constructed them by using fiducial, normal approximation, and Bayesian methods. By way of comparison, the performance measures for all ofthe proposed methods (the coverage percentage, lower and upper error probabilities (LEP and UEP,respectively), and expected length) were assessed via Monte Carlo simulation. The results of Monte Carlo simulation studies show that the Bayesian method provided coverage percentages close to the nominal confidence level and shorter intervals than the other methods. Importantly, it maintained a good balance between LEP and UEP even for large variation and percentage of zero-valued observations. To illustrate the efficacy of our proposed methods, we applied them to hourly wind speed data from northern Thailand.

Accurate interval estimation for the risk difference in an incomplete correlated 2 × 2 table: Calf immunity analysis

Interval estimation with accurate coverage for risk difference (RD) in a correlated 2 × 2 table with structural zero is a fundamental and important problem in biostatistics. The score test-based and Bayesian tail-based confidence intervals (CIs) have good coverage performance among the existing methods. However, as approximation approaches, they have coverage probabilities lower than the nominal confidence level for finite and moderate sample sizes. In this paper, we propose three new CIs for RD based on the fiducial, inferential model (IM) and modified IM (MIM) methods. The IM interval is proven to be valid. Moreover, simulation studies show that the CIs of fiducial and MIM methods can guarantee the preset coverage rate even for small sample sizes. More importantly, in terms of coverage probability and expected length, the MIM interval outperforms other intervals. Finally, a real example illustrates the application of the proposed methods.

Bayesian estimation for the mean of delta-gamma distributions with application to rainfall data in Thailand

Precipitation and flood forecasting are difficult due to rainfall variability. The mean of a delta-gamma distribution can be used to analyze rainfall data for predicting future rainfall, thereby reducing the risks of future disasters due to excessive or too little rainfall. In this study, we construct credible and highest posterior density (HPD) intervals for the mean and the difference between the means of delta-gamma distributions by using Bayesian methods based on Jeffrey's rule and uniform priors along with a confidence interval based on fiducial quantities. The results of a simulation study indicate that the Bayesian HPD interval based on Jeffrey's rule prior performed well in terms of coverage probability and provided the shortest expected length. Rainfall data from Chiang Mai province, Thailand, are also used to illustrate the efficacies of the proposed methods.

Confidence intervals for rainfall dispersions using the ratio of two coefficients of variation of lognormal distributions with excess zeros

Rainfall fluctuation is directly affected by the Earth’s climate change. It can be described using the coefficient of variation (CV). Similarly, the ratio of CVs can be used to compare the rainfall variation between two regions. The ratio of CVs has been widely used in statistical inference in a number of applications. Meanwhile, the confidence interval constructed with this statistic is also of interest. In this paper, confidence intervals for the ratio of two independent CVs of lognormal distributions with excess zeros using the fiducial generalized confidence interval (FGCI), Bayesian methods based on the left-invariant Jeffreys, Jeffreys rule, and uniform priors, and the Wald and Fieller log-likelihood methods are proposed. The results of a simulation study reveal that the highest posterior density (HPD) Bayesian using the Jeffreys rule prior method performed the best in terms of the coverage probability and the average length for almost all cases of small sample size and a large sample size together with a large variance and a small proportion of non-zero values. The performance of the statistic is demonstrated on two rainfall datasets from the central and southern regions in Thailand.

Bayesian computation for the common coefficient of variation of delta-lognormal distributions with application to common rainfall dispersion in Thailand

Rainfall fluctuation makes precipitation and flood prediction difficult. The coefficient of variation can be used to measure rainfall dispersion to produce information for predicting future rainfall, thereby mitigating future disasters. Rainfall data usually consist of positive and true zero values that correspond to a delta-lognormal distribution. Therefore, the coefficient of variation of delta-lognormal distribution is appropriate to measure the rainfall dispersion more than lognormal distribution. In particular, the measurement of the dispersion of precipitation from several areas can be determined by measuring the common coefficient of variation in the rainfall from those areas together. Herein, we compose confidence intervals for the common coefficient of variation of delta-lognormal distributions by employing the fiducial generalized confidence interval, equal-tailed Bayesian credible intervals incorporating the independent Jeffreys or uniform priors, and the method of variance estimates recovery. A combination of the coverage probabilities and expected lengths of the proposed methods obtained via a Monte Carlo simulation study were used to compare their performances. The results show that the equal-tailed Bayesian based on the independent Jeffreys prior was suitable. In addition, it can be used the equal-tailed Bayesian based on the uniform prior as an alternative. The efficacies of the proposed confidence intervals are demonstrated via applying them to analyze daily rainfall datasets from Nan, Thailand.

Confidence Intervals for Assessing Non-Inferiority with Assay Sensitivity in a Three-Arm Trial with Normally Distributed Endpoints

Various approaches including hypothesis test and confidence interval (CI) construction have been proposed to assess non-inferiority and assay sensitivity via a known fraction or pre-specified margin in three-arm trials with continuous or discrete endpoints. However, there is little work done on the construction of the non-inferiority margin from historical data and simultaneous generalized CIs (SGCIs) in a three-arm trial with the normally distributed endpoints. Based on the generalized fiducial method and the square-and-add method, we propose two simultaneous CIs for assessing non-inferiority and assay sensitivity in a three-arm trial. For comparison, we also consider the Wald-type Bonferroni simultaneous CI and parametric bootstrap simultaneous CI. An algorithm for evaluating the optimal sample size for attaining the pre-specified power is given. Simulation studies are conducted to investigate the performance of the proposed CIs in terms of their empirical coverage probabilities. An example taken from the mildly asthmatic study is illustrated using the proposed simultaneous CIs. Empirical results show that the proposed generalized fiducial method and the square-and-add method behave better than other two compared CIs.

Generalized fiducial methods for testing the homogeneity of a three-sample problem with a mixture structure

Recently, the likelihood ratio (LR) test was proposed to test the homogeneity of a three-sample model with a mixture structure. Because of the presence of the mixture structure, the null limiting distribution of the LR test has a complicated supremum form, which leads to challenges in determining p-values. In addition, the LR test cannot control type-I errors well under small to moderate sample size. In this paper, we propose seven generalized fiducial methods to test the homogeneity of the three-sample model. Via simulation studies, we find that our methods perform significantly better than the LR test method in controlling the type-I errors under small to moderate sample size, while they have comparable powers in most cases. A halibut data example is used to illustrate the proposed methods.

Generalized Confidence Intervals for Zero-Inflated Pareto Distribution

This paper considers interval estimations for the mean of Pareto distribution with excess zeros. Three approaches for interval estimation are proposed based on fiducial generalized pivotal quantities (FGPQs), respectively. Simulation studies are performed to assess the performance of the proposed methods, along with three measurements to determine comparisons with competing approaches. The advantages and disadvantages of each method are provided. The methods are illustrated using a real phone call dataset.

Simultaneous confidence intervals for all pairwise comparisons of the means of delta-lognormal distributions with application to rainfall data

Natural disasters such as flooding and landslides are important unexpected events during the rainy season in Thailand, and how to direct action to avoid their impacts is the motivation behind this study. The differences between the means of natural rainfall datasets in different areas can be estimated using simultaneous confidence intervals (SCIs) for pairwise comparisons of the means of delta-lognormal distributions. Our proposed methods are based on a parametric bootstrap (PB), a fiducial generalized confidence interval (FGCI), the method of variance estimates recovery (MOVER), and Bayesian credible intervals based on mixed (BCI-M) and uniform (BCI-U) priors. Their coverage probabilities, lower and upper error probabilities, and relative average lengths were used to evaluate and compare their SCI performances through Monte Carlo simulation. The results show that BCI-U and PB work well in different situations, even with large differences in variances σj2. All of the methods were applied to estimate pairwise differences between the means of natural rainfall data from five areas in Thailand during the rainy season to determine their abilities to predict occurrences of flooding and landslides.

Generalized fiducial inference on the mean of zero-inflated Poisson and Poisson hurdle models

Zero-inflated and hurdle models are widely applied to count data possessing excess zeros, where they can simultaneously model the process from how the zeros were generated and potentially help mitigate the effects of overdispersion relative to the assumed count distribution. Which model to use depends on how the zeros are generated: zero-inflated models add an additional probability mass on zero, while hurdle models are two-part models comprised of a degenerate distribution for the zeros and a zero-truncated distribution. Developing confidence intervals for such models is challenging since no closed-form function is available to calculate the mean. In this study, generalized fiducial inference is used to construct confidence intervals for the means of zero-inflated Poisson and Poisson hurdle models. The proposed methods are assessed by an intensive simulation study. An illustrative example demonstrates the inference methods.

Estimating the average daily rainfall in Thailand using confidence intervals for the common mean of several delta-lognormal distributions

The daily average natural rainfall amounts in the five regions of Thailand can be estimated using the confidence intervals for the common mean of several delta-lognormal distributions based on the fiducial generalized confidence interval (FGCI), large sample (LS), method of variance estimates recovery (MOVER), parametric bootstrap (PB), and highest posterior density intervals based on Jeffreys' rule (HPD-JR) and normal-gamma-beta (HPD-NGB) priors. Monte Carlo simulation was conducted to assess the performance in terms of the coverage probability and average length of the proposed methods. The numerical results indicate that MOVER and PB provided better performances than the other methods in a variety of situations, even when the sample case was large. The efficacies of the proposed methods were illustrated by applying them to real rainfall datasets from the five regions of Thailand.

Confidence intervals for the common coefficient of variation of rainfall in Thailand

The log-normal distribution is often used to analyze environmental data like daily rainfall amounts. The rainfall is of interest in Thailand because high variable climates can lead to periodic water stress and scarcity. The mean, standard deviation or coefficient of variation of the rainfall in the area is usually estimated. The climate moisture index is the ratio of plant water demand to precipitation. The climate moisture index should use the coefficient of variation instead of the standard deviation for comparison between areas with widely different means. The larger coefficient of variation indicates greater dispersion, whereas the lower coefficient of variation indicates the lower risk. The common coefficient of variation, is the weighted coefficients of variation based on k areas, presents the average daily rainfall. Therefore, the common coefficient of variation is used to describe overall water problems of k areas. In this paper, we propose four novel approaches for the confidence interval estimation of the common coefficient of variation of log-normal distributions based on the fiducial generalized confidence interval (FGCI), method of variance estimates recovery (MOVER), computational, and Bayesian approaches. A Monte Carlo simulation was used to evaluate the coverage probabilities and average lengths of the confidence intervals. In terms of coverage probability, the results show that the FGCI approach provided the best confidence interval estimates for most cases except for when the sample case was equal to six populations (k = 6) and the sample sizes were small (n _I  < 50), for which the MOVER confidence interval estimates were the best. The efficacies of the proposed approaches are illustrated with example using real-life daily rainfall datasets from regions of Thailand.

Comparing Medical Care Costs using Bayesian Credible Intervals for the Ratio of Means of Delta-Lognormal Distributions

When considering the medical care costs data with a high proportion of zero items of two inpatient groups, comparing them can be estimated using confidence intervals for the ratio of the means of two delta-lognormal distributions. The Bayesian credible interval-based uniform-beta prior (BCIh-UB) is proposed and compared with the generalized confidence interval (GCI), fiducial GCI (FGCI), the method of variance estimates recovery (MOVER), BCIh based on Jeffreys’ rule prior (BCIh-JR), and BCIh based on the normal-gamma prior (BCIh-NG). The coverage probability (CP), average length (AL), and lower and upper error rates were the performance measures applied for assessing the methods through a Monte Carlo simulation. A numerical evaluation showed that BCIh-UB had much better CP and AL than the others even with a large difference between the variances and with a high proportion of zero. Finally, to illustrate the efficacy of BCIh-UB, the methods were applied to two sets of medical care costs data.

Exact distributions of statistics for making inferences on mixed models under the default covariance structure

At this juncture when mixed models are heavily employed in applications ranging from clinical research to business analytics, the purpose of this article is to extend the exact distributional result of Wald (Ann. Math. Stat. 18: 586–589, 1947) to handle models involving a number of variance components.Due to the unavailability of exact distributional results for underlying statistics, currently available methods provide small group/sample inference only for balanced ANOVA models or simple regression models. The exact distributional results developed in this article should prove useful in making inferences by such methods as parametric bootstrap, fiducial, and generalized p-value approach, when there are a number of variance components to deal with.

The Bayesian confidence intervals for measuring the difference between dispersions of rainfall in Thailand

The coefficient of variation is often used to illustrate the variability of precipitation. Moreover, the difference of two independent coefficients of variation can describe the dissimilarity of rainfall from two areas or times. Several researches reported that the rainfall data has a delta-lognormal distribution. To estimate the dynamics of precipitation, confidence interval construction is another method of effectively statistical inference for the rainfall data. In this study, we propose confidence intervals for the difference of two independent coefficients of variation for two delta-lognormal distributions using the concept that include the fiducial generalized confidence interval, the Bayesian methods, and the standard bootstrap. The performance of the proposed methods was gauged in terms of the coverage probabilities and the expected lengths via Monte Carlo simulations. Simulation studies shown that the highest posterior density Bayesian using the Jeffreys’ Rule prior outperformed other methods in virtually cases except for the cases of large variance, for which the standard bootstrap was the best. The rainfall series from Songkhla, Thailand are used to illustrate the proposed confidence intervals.

Bayesian confidence intervals for the difference between variances of delta-lognormal distributions

Unnatural rainfall fluctuation can result in such severe natural phenomena as drought and floods. This variability not only occurs in areas with unusual natural features such as land formations and drainage but can also be due to human intervention. Since rainfall data often contain zero values, evaluating rainfall change is an important undertaking, which can be estimated via the confidence intervals for the difference between delta-lognormal variances using the highest posterior density-based reference (HPD-ref) and probability-matching (HPD-pm) priors. Simulation results indicate that HPD-pm performances were better than other methods in terms of coverage rates and relative average lengths for the difference in delta-lognormal variances, even with a large difference in variances. To illustrate the efficacy of our proposed methods, we applied them to daily rainfall data sets for the lower and upper regions of northern Thailand.

A Bayesian approach to construct confidence intervals for comparing the rainfall dispersion in Thailand

Natural disasters such as drought and flooding are the consequence of severe rainfall fluctuation, and rainfall amount data often contain both zero and positive observations, thus making them fit a delta-lognormal distribution. By way of comparison, rainfall dispersion may not be similar in enclosed regions if the topography and the drainage basin are different, so it can be evaluated by the ratio of variances. To estimate this, credible intervals using the highest posterior density based on the normal-gamma prior (HPD-NG) and the method of variance estimates recovery (MOVER) for the ratio of delta-lognormal variances are proposed. Monte Carlo simulation was used to assess the performance of the proposed methods in terms of coverage probability and relative average length. The results of the study reveal that HPD-NG performed very well and was able to meet the requirements in various situations, even with a large difference between the proportions of zeros. However, MOVER is the recommended method for equal small sample sizes. Natural rainfall datasets for the northern and northeastern regions of Thailand are used to illustrate the practical use of the proposed credible intervals.

Generalized prediction intervals for treatment effects in random-effects models

This article derives generalized prediction intervals for random effects in linear random-effects models. For balanced and unbalanced data in two-way layouts, models are considered with and without interaction. Coverage of the proposed generalized prediction intervals was estimated in a simulation study based on an agricultural field experiment. Generalized prediction intervals were compared with prediction intervals based on the restricted maximum likelihood (REML) procedure and the approximate methods of Satterthwaite and Kenward and Roger. The simulation study showed that coverage of generalized prediction intervals was closer to the nominal level 0.95 than coverage of prediction intervals based on the REML procedure.

Measuring the dispersion of rainfall using Bayesian confidence intervals for coefficient of variation of delta-lognormal distribution: a study from Thailand

Since rainfall data series often contain zero values and thus follow a delta-lognormal distribution, the coefficient of variation is often used to illustrate the dispersion of rainfall in a number of areas and so is an important tool in statistical inference for a rainfall data series. Therefore, the aim in this paper is to establish new confidence intervals for a single coefficient of variation for delta-lognormal distributions using Bayesian methods based on the independent Jeffreys', the Jeffreys' Rule, and the uniform priors compared with the fiducial generalized confidence interval. The Bayesian methods are constructed with either equitailed confidence intervals or the highest posterior density interval. The performance of the proposed confidence intervals was evaluated using coverage probabilities and expected lengths via Monte Carlo simulations. The results indicate that the Bayesian equitailed confidence interval based on the independent Jeffreys' prior outperformed the other methods. Rainfall data recorded in national parks in July 2015 and in precipitation stations in August 2018 in Nan province, Thailand are used to illustrate the efficacy of the proposed methods using a real-life dataset.

Equivalence testing using existing reference data: An example with genetically modified and conventional crops in animal feeding studies

An equivalence testing method is described to assess the safety of regulated products using relevant data obtained in historical studies with assumedly safe reference products. The method is illustrated using data from a series of animal feeding studies with genetically modified and reference maize varieties. Several criteria for quantifying equivalence are discussed, and study-corrected distribution-wise equivalence is selected as being appropriate for the example case study. An equivalence test is proposed based on a high probability of declaring equivalence in a simplified situation, where there is no between-group variation, where the historical and current studies have the same residual variance, and where the current study is assumed to have a sample size as set by a regulator. The method makes use of generalized fiducial inference methods to integrate uncertainties from both the historical and the current data.

Analytic posteriors for Pearson's correlation coefficient

Pearson's correlation is one of the most common measures of linear dependence. Recently, Bernardo (11th International Workshop on Objective Bayes Methodology, 2015) introduced a flexible class of priors to study this measure in a Bayesian setting. For this large class of priors, we show that the (marginal) posterior for Pearson's correlation coefficient and all of the posterior moments are analytic. Our results are available in the open‐source software package JASP.