Parametric Bootstrap Research Articles

Confidence interval estimation for the difference and ratio of the means of two gamma distributions

Gamma distribution is widely used in applied fields due to its flexibility of accommodating right-skewed data. This article explores statistical methods for constructing confidence intervals for both the difference and ratio of two gamma means. We propose several methods based on the concepts of generalized inference, variance estimates recovery (MOVER) and parametric bootstrapping. The performances of proposed methods are evaluated and compared via a comprehensive simulation study. Simulation results show that the hybrid method that combines MOVER with parametric bootstrapping can provide confidence intervals with reasonable coverage probabilities and interval lengths even for parameter settings with small shape parameters. Finally, two real data examples from environmental and engineering studies are analyzed using the proposed methods.

Confidence Intervals Based on the Difference of Medians for Independent Log-Normal Distributions

In this paper, we study the inferences of the difference of medians for two independent log-normal distributions. These methods include traditional methods such as the parametric bootstrap approach, the normal approximation approach, the method of variance estimates recovery approach, and the generalized confidence interval approach. The simultaneous confidence intervals for the difference in the median for more than two independent log-normal distributions are also discussed. Our simulation studies evaluate the performances of the proposed confidence intervals in terms of coverage probabilities and average lengths. We find that the parametric bootstrap approach would be a suitable choice for smaller sample sizes for the two independent distributions and multiple independent distributions. However, the method of variance estimates recovery and normal approximation approaches are alternative competitors for constructing simultaneous confidence intervals, especially when the populations have large variance. We also include two practical applications demonstrating the use of the techniques on observed data, where one data set works for the PM2.5 mass concentrations in Bangkapi and Dindaeng in Thailand and the other data came from the study of nitrogen-bound bovine serum albumin produced by three groups of diabetic mice. Both applications show that the confidence intervals from the parametric bootstrap approach have the smallest length.

Rectangular multivariate normal prediction regions for setting reference regions in laboratory medicine

ABSTRACT Reference intervals are among the most widely used medical decision-making tools and are invaluable in the interpretation of laboratory results of patients. Moreover, when multiple biochemical analytes are measured on each patient, a multivariate reference region (MRR) is needed. Such regions are more desirable than separate univariate reference intervals since the latter disregard the cross-correlations among variables. Traditionally, assuming multivariate normality, MRRs have been constructed as ellipsoidal regions, which cannot detect componentwise extreme values. Consequently, MRRs are rarely used in actual practice. In order to address the above drawback of ellipsoidal reference regions, we propose a procedure to construct rectangular MRRs under multivariate normality. The rectangular MRR is computed using a prediction region criterion. However, since the population correlations are unknown, a parametric bootstrap approach is employed for computing the required prediction factor. Also addressed in this study is the computation of mixed reference intervals, which include both two-sided and one-sided prediction limits, simultaneously. Numerical results show that the parametric bootstrap procedure is quite accurate, with estimated coverage probabilities very close to the nominal level. Moreover, the expected volumes of the proposed rectangular regions are substantially smaller than the expected volumes obtained from Bonferroni simultaneous prediction intervals. We also explore the computation of covariate-dependent MRRs in a multivariate regression setting. Finally, we discuss real-life applications of the proposed methods, including the computation of reference ranges for the assessment of kidney function and for components of the insulin-like growth factor system in adults.

Likelihood ratio test for the parameters of exponentially modified Gaussian distribution

The exponentially modified Gaussian (EMG) distribution is adopted in many fields such as psychology, chromatographic peaks, and cell biology. In this article, we focus on the test for the parameters of the EMG distribution. The primary test statistic is the classical likelihood ratio. Although the EMG distribution is an asymmetric one, we proved that the logarithm of the likelihood ratio is asymptotically distributed as that is, the Wilks phenomenon. We employ Q–Q plots to illustrate the sample sizes needed for the asymptotic distribution as a good approximation to the null distribution of the loglikelihood ratio for different parameters. The available approximate distribution for the loglikelihood ratio is guaranteed by a relatively large sample size. Hence, when the sample size is small, the approximate distribution is inapplicable, since the EMG distribution is asymmetric. For the small sample size, we propose a parametric bootstrap algorithm, which is demonstrated to be successful by the simulation studies. An application of our method to a real data example shows that the test is satisfactory.

A note on statistical tests for homogeneities in multivariate extreme value models for block maxima

AbstractMathematical theory suggests to model annual or seasonal maxima by the generalized extreme value distribution. In environmental applications like hydrology, record lengths are typically small, whence respective parameter estimators typically exhibit a large variance. The variance may be decreased by pooling observations from different sites or variables, but this requires to check the validity of the inherent homogeneity assumption. The present paper provides an overview of (partly new) respective asymptotic significance tests. It is found that the tests' levels are often violated in typical finite‐sample situations, whence a parametric bootstrap approach based on max‐stable process models is proposed to obtain more accurate critical values. As a side product, we present an overview of asymptotic results on a variety of common estimators for GEV parameters in a multisample situation of varying record lengths.

Testing components of two-way interaction in multi-environment trials

Experiments with two factors are commonly analyzed using two-way analysis of variance, where testing significance of interaction is straightforward. However, using bilinear models, interaction can be analyzed further. The additive main effects and multiplicative interaction (AMMI) model uses singular value decomposition for partitioning interaction into multiplicative terms, such that the first terms typically account for a large portion of the sum of squares, whereas the last terms are of minor importance. This model is used extensively for analysis of genotype-by-environment interaction in multi-environment trials. A recurring question is how to determine the number of terms to retain in the model. If data is replicated, which is usually the case, the F R test can be used for this purpose. The simple parametric bootstrap method is another option, although this test was developed for unreplicated data. Since both of these tests of significance may be applied in cases with replication, researchers need advice on which of the methods to use. We discuss several statistical models and show that the two methods address different questions.

Parametric bootstrap procedures for three-factor ANOVA and multiple comparison procedures with unequal group variances

The issue of unmet equal variance assumption in multi-factor ANOVA has been addressed in the literature with several methods, and parametric bootstrap (PB) has been found in the one-way and two-way cases to outperform other methods. We extend previously developed PB procedures for one- and two-way ANOVA, and illustrate with a three-way ANOVA model with unequal group variances (heteANOVA model). We develop a framework for working with these models, analogous to usual multi-factor ANOVA procedures, where F-tests and Tukey’s simultaneous multiple comparison procedures are replaced by PB procedures. Using simulation, we compare these methods to F-tests for each step in model selection, as well as to Tukey’s test for multiple comparison procedures (MCP). The results of our simulations indicate that the PB methods outperform F-tests and Tukey’s test in terms of Type I error when data are unbalanced.

Confidence intervals for zero-inflated gamma distribution

The problems of constructing confidence intervals (CIs) for the mean of a gamma distribution containing zero values are considered. Three different methods for constructing such CIs based on Parametric Bootstrap (PB) method, fiducial inference and the method of variance of estimates recovery (MOVER) are developed. The performances of the proposed CIs are evaluated by Monte Carlo simulation and compared with a published method, asymptotic normal (AN) method. The simulation studies indicate that the proposed methods perform better than the existing method (AN) according to coverage probabilities (CPs), tail error rates, robustness and time comparison. All the methods are illustrated using a dataset for HIV risk behavior in adolescent girls and a dataset for the immunologic response to combination therapy for HIV children. Recommendations are made at the end of the articles.

Inferences of the Multicomponent Stress–Strength Reliability for Burr XII Distributions

Multicomponent stress–strength reliability (MSR) is explored for the system with Burr XII distributed components under Type-II censoring. When the distributions of strength and stress variables have Burr XII distributions with common or unequal inner shape parameters, the existence and uniqueness of the maximum likelihood estimators are investigated and established. The associated approximate confidence intervals are obtained by using the asymptotic normal distribution theory along with the delta method and parametric bootstrap procedure, respectively. Moreover, alternative generalized pivotal quantities-based point and confidence interval estimators are developed. Additionally, a likelihood ratio test is presented to diagnose the equivalence of both inner shape parameters or not. Conclusively, Monte Carlo simulations and real data analysis are conducted for illustration.

Estimation of Reliability Indices for Alpha Power Exponential Distribution Based on Progressively Censored Competing Risks Data

In reliability analysis and life testing studies, the experimenter is frequently interested in studying a specific risk factor in the presence of other factors. In this paper, the estimation of the unknown parameters, reliability and hazard functions of alpha power exponential distribution is considered based on progressively Type-II censored competing risks data. We assume that the latent cause of failures has independent alpha power exponential distributions with different scale and shape parameters. The maximum likelihood method is considered to estimate the model parameters as well as the reliability and hazard rate functions. The approximate and two parametric bootstrap confidence intervals of the different estimators are constructed. Moreover, the Bayesian estimation method of the unknown parameters, reliability and hazard rate functions are obtained based on the squared error loss function using independent gamma priors. To get the Bayesian estimates as well as the highest posterior credible intervals, the Markov Chain Monte Carlo procedure is implemented. A comprehensive simulation experiment is conducted to compare the performance of the proposed procedures. Finally, a real dataset for the relapse of multiple myeloma with transplant-related mortality is analyzed.

A framework of tolerance specification for freeform point clouds and capability analysis for reverse engineering processes

The combination of reverse engineering (RE) with additive manufacturing (AM) is widely used for solid freeform fabrication and overcomes limitations with the current capabilities of designers and CAD packages that are restricted to classical model shapes. The shape flexibility offered by RE typically leads to the generated CAD models being represented in point cloud formats or stereolithography files. One resulting challenge is the difficulty with tolerance specification for the RE-generated models, especially for freeform shapes. Geometric metrology for the AM-produced parts to determine their geometric conformation can also be involved. Furthermore, conventional process analyses cannot be performed directly after RE because of the complex geometric data structure. We propose to address all these issues with a new tolerance specification and process analysis framework based on volumetric data analysis. The tolerance zone constructed under our framework, and the corresponding geometric measurements, are consistent with the profile tolerance in geometric dimensioning and tolerancing standards. Reverse engineering's process capability is assessed under our methodology via a parametric bootstrap procedure in the size-and-shape space. The performance and utility of the proposed framework are validated via a process capability study on RE-generated CAD models of multiple, additively manufactured freeform objects.

Dietary Lysine Requirement of Women Over 60 Years Estimated Using Indicator Amino Acid Oxidation (IAAO) Technique, Is Not Different From Current Recommendations

ObjectivesThe goals of the current study were to 1) define the lysine requirement in female adults >60years using the indicator amino acid oxidation method (IAAO) method with L-[1–13C]phenylalanine as the indicator and to 2) compare the derived estimate to our previous estimates in young women done in both luteal and follicular phase of the mensural cycle. MethodsSixteen healthy women (>60 years) were randomly assigned to receive 3 to 7 lysine intakes from 10 to 80 mg. kg–1. d–1. Subjects were adapted to a standard liquid diet providing 1.0 g. kg–1.d–1protein and adequate energy, for 2 days, with indicator oxidation measurements on day 3. The rate of release of13CO2 from the oxidation of L-[1–13C]phenylalanine was measured in breath. A two-phase linear mixed effect model, and parametric bootstrap was used to determine the mean and 95% confidence interval (CI) of the lysine requirement. The overlap of the 95% CI between the older and younger women were used to compare the requirement estimates. The null hypothesis was accepted if the interval contained zero. ResultsThe mean and upper 95% CI of the lysine requirement of older women was 29 and 33 mg.kg–1.d–1respectively. This was not different from our previous estimate of 30.9 mg.kg–1.d–1for young women in the follicular phase of the mensural cycle. However, it was lower than our previous estimate of 37.7 mg.kg–1.d–1derived during the luteal phase of the mensural cycle. ConclusionsThis is the first study to report on the lysine requirement in older women >60 years. Although the estimate in older women is not different from current recommendations derived in young men, the results suggest that lysine requirement in young and older women may differ based on mensural cycle phase. Funding SourcesCIHR.

Indicator Amino Acid Oxidation Does Not Require a Lengthy Period of Adaptation When Estimating Amino Acid Requirements in Healthy Young Men

ObjectivesThe objective of the current study was to determine whether 3 or 7 days, compared to 8 hours of adaptation to the test level of threonine intake alters the threonine requirement in healthy young men.Methods11 men (19–35 years) were studied at 6 levels of threonine intake (5, 10, 15, 20, 25 and 35 mg.kg−1.d−1), in random order. Participants were first habituated to a diet containing intact milk protein (1.0 g.kg−1.d−1) and adequate calories for 2 days. Subjects then received an experimental diet containing the test threonine intakes for 7 days, with IAAO studies performed on days 1, 3, and 7. A total of 162 IAAO studies were completed; 54 for each length of adaptation. The rate of release of 13CO2 from the oxidation of L-[1–13C]phenylalanine (F13CO2) was measured. Phenylalanine flux and kinetics were determined from urinary enrichment of L-[1–13C]phenylalanine. The threonine requirement was determined by applying mixed effect change-point regression in R to the F13CO2 data. The 95% CI of each breakpoint was calculated using parametric bootstrap and the threonine requirement estimates measured on days 1, 3, and 7 were compared using ANOVA. The effect of threonine intake and days of adaptation on phenylalanine flux and kinetics were assessed using marginal linear mixed models and ANOVA.ResultsThe mean threonine requirement (upper, lower 95% CI) for days 1, 3, and 7 was 10.2 (5.4, 15), 10.6 (8, 13.3), and 12.1(9.5, 14.7) mg.kg−1.d−1 respectively, and were not statistically different from each other (p = 0.502). Phenylalanine flux was not affect by threonine intake, however flux significantly decreased with increasing length of adaptation (p = 0.000). The interaction between threonine intake and days of adaptation on phenylalanine balance was significant (p = 0.0015); decreased with increasing length of adaptation at intakes below 10 mg.kg−1.d−1 threonine, and increased with increasing length of adaptation above 15 mg.kg−1.d−1.ConclusionsWe have demonstrated that the short, 8 hour IAAO protocol is valid for the determination of amino acid requirements in humans. This length of adaptation is better positioned to study the amino acid needs of vulnerable populations such as infants, children, and older adults.Funding SourcesThis study was funded by The Canadian Institute of Health Research.

The Total Sulphur Amino Acid Requirement Is Higher in Older Men Than Women

ObjectivesThe primary aim of this study was to determine the total sulphur amino acid (TSAA) requirement of adults > 60 years, using the indicator amino acid oxidation (IAAO) method and to compare the requirement estimates between older males and females. The secondary aim was to compare the requirement estimates in older adults to the previously derived IAAO requirement in young males.MethodsFifteen healthy older adults (n = 7 male and n = 8 female participants), were randomized to receive 2 to 7 methionine intakes: 5, 10, 15, 19, 25, 35 and 40 mg. kg−1. d−1 without dietary cysteine. The TSAA requirement was determined using the IAAO method with L-[1–13C]phenylalanine as the indicator. The mean TSAA requirement was determined using two-phase linear mixed effect model, which identified a breakpoint in the F13CO2 in response to graded methionine intakes. The parametric bootstrap method was used to determine the 95% confidence interval (CI). To assess whether the breakpoints were different between older males and females or between older adults and young males the overlap in 95% CI was calculated. The null hypothesis was accepted if the interval contained zero.ResultsThe mean TSAA requirement for older males was 26.2 mg · kg−1· d−1 and 17.1 mg · kg−1· d−1 for older females. The TSAA requirement of older males is higher than older females (Difference in CI: 9.1 ± 8.85) and higher compared to young males based on our previous IAAO estimate of 12.6 mg · kg−1· d−1 (Difference in CI: 13.6 ± 5.98).ConclusionsTo our knowledge this is the first study to experimentally derive the TSAA requirement in older adults. The results can be used to assess whether the amino acid requirements of older adults can be met on a plant-based diet, especially given the current recommendations for increased plant protein consumption.Funding SourcesThis study was funded by CIHR. First author Alyssa Paoletti funded by Restracomp Scholarship provided by the Research Training Centre at SickKids Hospital.

Testing distributional assumption of unit-Lindley regression model

This paper proposes smooth goodness of fit test statistic and its components to test the distributional assumption of the unit-Lindley regression model, which is useful for describing data measured between zero and one. Orthonormal polynomials on the unit-Lindley distribution, score functions and Fisher's information matrix are provided for the smooth test. Deviance and Pearson's chi-square tests are also adapted to the unit-Lindley regression model. A parametric bootstrap simulation study is conducted to compare type I errors and powers of the tests under different scenarios. Empirical findings demonstrate that the first smooth component, deviance, and chi-square tests have undesirable behavior for the unit-Lindley regression model. A real data set is analyzed by using the developed tests to show the adequacy of the unit-Lindley regression model. Model selection criteria and residual analysis prove that the unit-Lindley regression model provides a better fit than the Beta and simplex regression models for the real data set.

Modified sequential bifurcation for simulation factor screening under skew-normal response model

Sequential bifurcation (SB) has been widely used for simulation factor screening problems to identify important factors (i.e., factors that can significantly affect system response) with high efficiency and effectiveness under the ‘sparse effects’ assumption. However, when the response is asymmetric, previous SB studies based on the assumption of normally distributed response have shown limitations in practice. In this paper, we develop a skew-normal distributed response model and propose a parametric bootstrap testing procedure that is incorporated into every ‘bifurcation’ stage of SB for testing the significance of factors. Numerical experiments are provided to demonstrate the effectiveness and efficiency of the screening methods.

Markov-switching state-space models with applications to neuroimaging

State-space models (SSM) with Markov switching offer a powerful framework for detecting multiple regimes in time series, analyzing mutual dependence and dynamics within regimes, and assessing transitions between regimes. These models however present considerable computational challenges due to the exponential number of possible regime sequences to account for. In addition, high dimensionality of time series can hinder likelihood-based inference. To address these challenges, novel statistical methods for Markov-switching SSMs are proposed using maximum likelihood estimation, Expectation-Maximization (EM), and parametric bootstrap. Solutions are developed for initializing the EM algorithm, accelerating convergence, and conducting inference. These methods, which are ideally suited to massive spatio-temporal data such as brain signals, are evaluated in simulations and applications to EEG studies of epilepsy and of motor imagery are presented.

Statistical Inference and Optimal Design of Accelerated Life Testing for the Chen Distribution under Progressive Type-II Censoring

This paper discusses statistical inference and optimal design of constant-stress accelerated life testing for the Chen distribution under progressive Type-II censoring. The scale parameter of the life distribution is assumed to be a logarithmic linear function of the stress level. The maximum likelihood estimates of the parameters are obtained. Then, the observed Fisher information matrix is derived and utilized to construct asymptotic confidence intervals. Meanwhile, the parametric bootstrap methods are provided for the interval estimation. In addition, the Bayes estimates under the squared error loss function are obtained by applying the Tierney and Kadane technique and Lindley’s approximation. As for the optimal design, D- and A-optimality criteria are considered to determine the optimal transformed stress level. Finally, the simulation is carried out to demonstrate the proposed estimation techniques and the optimal criteria, and a real data set is discussed.

Bootstrap Confidence Intervals for Common Signal-to-noise Ratio of Two-parameter Exponential Distributions

Signal-to-noise ratio (SNR) is a reciprocal of coefffficient of variation. The SNR is a measure of mean relative to the variability. Confidence procedures for common SNR of two-parameter exponential distributions were developed using generalized confidence interval (GCI) approach, large sample (LS) approach, adjusted method of variance estimates recovery (Adjusted MOVER) approach, and bootstrap approaches based on standard bootstrap (SB) and parametric bootstrap (PB). The performances of all approaches are measured by coverage probability and average length. Simulation studies show that all approaches have the coverage probabilities below the nominal confidence level of 0.95 when the common SNR is negative value. However, the coverage probabilities of all approaches are greater than the nominal confidence level of 0.95 when the common SNR is positive value. Moreover, the LS and AM approaches are the conservative confidence intervals. In addition, the GCI and PB approaches provide the confidence intervals with coverage probabilities close to the nominal confidence level of 0.95 when the sample sizes are large and the common SNR is positive value. The GCI and PB approaches are recommended to estimate the confidence intervals for the common SNR of two-parameter exponential distributions. Finally, all proposed approaches are employed in the data of the survival days of lung cancer patients for a demonstration.

AliSim: A Fast and Versatile Phylogenetic Sequence Simulator for the Genomic Era.

Sequence simulators play an important role in phylogenetics. Simulated data has many applications, such as evaluating the performance of different methods, hypothesis testing with parametric bootstraps, and, more recently, generating data for training machine-learning applications. Many sequence simulation programmes exist, but the most feature-rich programmes tend to be rather slow, and the fastest programmes tend to be feature-poor. Here, we introduce AliSim, a new tool that can efficiently simulate biologically realistic alignments under a large range of complex evolutionary models. To achieve high performance across a wide range of simulation conditions, AliSim implements an adaptive approach that combines the commonly used rate matrix and probability matrix approaches. AliSim takes 1.4 h and 1.3 GB RAM to simulate alignments with one million sequences or sites, whereas popular software Seq-Gen, Dawg, and INDELible require 2–5 h and 50–500 GB of RAM. We provide AliSim as an extension of the IQ-TREE software version 2.2, freely available at www.iqtree.org, and a comprehensive user tutorial at http://www.iqtree.org/doc/AliSim.