Parametric Bootstrap Test Research Articles

Parametric test based on the bootstrapping approach for the MANOVA under a Behrens-Fisher problem

This article presents a comparison of multivariate normal mean vectors under covariance positive definite matrices. We introduce an improved parametric bootstrap (IPB) approach for addressing the multivariate Behrens-Fisher problem, specifically focusing on cases with unequal covariance matrices. Additionally, we evaluate the performance of the IPB test by comparing it with three existing tests: the parametric bootstrap (PB) test, the generalized variable (GV) test, and the Johansen test. Through Monte Carlo simulation, our results demonstrate that both the IPB test and the PB test exhibit superior control over Type I error rates compared to the GV and Johansen tests. Notably, the IPB test outperforms the PB test in terms of controlling Type I error rates. Consequently, our study concludes that the IPB test represents a preferred statistical method for testing the equality of mean vectors in the multivariate Behrens-Fisher problem.

Two approaches to extend classical MANOVA tests to the unequal covariances case

This article presents two approaches to extend classical MANOVA tests to the unequal covariances case. The first approach is illustrated by extending the classical Wilks test, which is valid only when covariances are equal. Such tests will be based on exact probabilities of certain extreme regions. We will also show how tests numerically equivalent to the parametric bootstrap tests could be easily obtained without using any bootstrap sampling arguments, so that resulting p-values are also based on exact probabilities of well defined extreme regions. Being systematic approaches, by taking similar approaches, researchers should be able to derive generalized tests in MANCOVA, higher-way MANOVA, and in RM MANOVA under heteroscedasticity, in which the parametric bootstrap type approaches run into difficulties.

Bootstrap Probability Errors of the Whittle MLE for Linear Regression Processes with Strongly Dependent Disturbances

This paper determines bounds on the asymptotic orders of the coverage probability errors of parametric bootstrap confidence intervals (CIs) and tests for the covariance parameters of a time series generated by a regression model with Gaussian, stationary, and strongly dependent errors. The CIs and tests are based on the plug-in Whittle maximum
 likelihood (PWML) estimators. It is shown that, under some sets of conditions on the regression coefficients, the spectral density function, and the parameter values, the bounds on the coverage probability errors of symmetric two-sided and one-sided parametric bootstrap confidence intervals on the plug-in Whittle log-likelihood function are shown to be O(n^{-3/2}\ln{n}) and O(n^{-1}\ln{n}), respectively. Apart from the \ln{n} term, the magnitudes of the coverage probability errors of the one-sided bootstrap confidence intervals for our model is shown to be essentially the same as that of the independent and identically distributed (iid) data. The error for the two-sided confidence intervals is not as small as the error O(n^{-2}) that has been established for many confidence intervals in the literature, see Hall (1992), pp 102-108.

Comparison of the several two-parameter exponential distributed group means in the presence of outliers

The two-parameter exponential distribution is often used to model the lifetime of a product. The comparison of the mean lifetimes of several products is a main concern in reliability applications. In this study, the performance of the methods to compare the mean lifetimes of several products based on generalized p-value, parametric bootstrap, and fiducial approach are compared in the presence of outliers. The results of Monte-Carlo simulations clearly indicate that there is no uniformly powerful test. The parametric bootstrap test is superior to the others except in the case of the lower number of groups and the presence of outliers. An illustrative example of testing the equality lifetimes of a component is given to perform the proposed tests. The considered tests are implemented in an R package doex.

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.

Heteroscedastic two-way ANOVA under constraints

In this article we derive two simple tests for two-way ANOVA under unequal variances requiring some constraints, which cannot be solved by classical regression formulas. We do so by taking the generalized p-value approach and provide explicit formulas to handle the constraints. The first test is an extension of fiducial one-way ANOVA test, which tends to assure the intended size of the test, but tends to be conservative, and the second test reduces that drawback, but may slightly exceed the intended size of the test. The second test is also a generalized test that is numerically equivalent to the parametric bootstrap (PB) test, which contains some unintended glitches in formulas, which we will rectify. Moreover, our approach does not require good estimators of parameters as PB approach does. Hence, one can take our approach dealing with such distributions as Weibull and Gamma that are used in analysis of lifetime data in engineering and analysis of Survival data in studies of public health. By taking similar approach, researchers are also encouraged to derive generalized tests in other ANOVA applications, such as higher-way ANOVA, ANCOVA and MANOVA under heteroscedasticity, especially in Mixed Effects Model applications such as repeated measures (RM) ANOVA, where the PB approach run into difficulties.

Estimating Age-specific Prevalence of Testosterone Deficiency in Men Using Normal Mixture Models

Testosterone levels decline as men age. There is little consensus on what testosterone levels are normal for aging men. In this paper, we estimate age-specific prevalence of testosterone deficiency in men using nor mal mixture models when no generally agreed on cut-off value for defining testosterone deficiency is available. The Box-Cox power transformation is used to determine which transformation is most appropriate for correcting skewness in data and best suits normal mixture distributions. Parametric bootstrap tests are used to determine the number of components in a normal mixture.

Testing for similarity of binary efficacy-toxicity responses.

Clinical trials often aim to compare two groups of patients for efficacy and/or toxicity depending on covariates such as dose. Examples include the comparison of populations from different geographic regions or age classes or, alternatively, of different treatment groups. Similarity of these groups can be claimed if the difference in average outcome is below a certain margin over the entire covariate range. In this article, we consider the problem of testing for similarity in the case that efficacy and toxicity are measured as binary outcome variables. We develop a new test for the assessment of similarity of two groups for a single binary endpoint. Our approach is based on estimating the maximal deviation between the curves describing the responses of the two groups, followed by a parametric bootstrap test. Further, using a two-dimensional Gumbel-type model we develop methodology to establish similarity for (correlated) binary efficacy-toxicity outcomes. We investigate the operating characteristics of the proposed methodology by means of a simulation study and present a case study as an illustration.

Statistical Significance Testing for Natural Language Processing. By Rotem Dror, Lotem Peled-Cohen, Segev Shlomov, and Roi Reichart (Technion Israel Institute of Technology)). Morgan & Claypool (Synthesis Lectures on Human Language Technologies, edited by Graeme Hirst, volume 45), 2020, xx+98 pp; paperback, ISBN 978-1-68173-795-9, $49.95; ebook, ISBN 978-1-68173-796-6, $39.96; hardcover, $69.95; doi:10.2200/S00994ED1V01Y202002HLT045

<i>Statistical Significance Testing for Natural Language Processing</i>. By Rotem Dror, Lotem Peled-Cohen, Segev Shlomov, and Roi Reichart (Technion Israel Institute of Technology)). Morgan &amp; Claypool (Synthesis Lectures on Human Language Technologies, edited by Graeme Hirst, volume 45), 2020, xx+98 pp; paperback, ISBN 978-1-68173-795-9, $49.95; ebook, ISBN 978-1-68173-796-6, $39.96; hardcover, $69.95; doi:10.2200/S00994ED1V01Y202002HLT045

HW_TEST, a program for comprehensive HARDY-WEINBERG equilibrium testing.

This article deals with a Windows (© Microsoft Inc.) executable, user-friendly program that tests the hypothesis of Hardy-Weinberg (HW) proportions from autosomal multiallelic data using different methods that include parametric, nonparametric and exact bootstrap tests, the latter obtained through computer simulations. The program can be obtained free of charge directly from the internet repository https://github.com/Lemes-RenanB/HardyWeinbergTesting.

Proposition of new alternative tests adapted to the traditional T 2 test

New alternative tests of the T 2 Hotelling’s test for testing hypotheses on the mean vector of a normal p-variate population were proposed. These tests were based on comedian robust estimator of the covariance matrix using an asymptotic T 2 distribution and a parametric bootstrap distribution to the null distribution of the statistical tests. The performance of these new tests was evaluated under normal and non-normal distributions through Monte Carlo simulations. The contaminated normal populations were also considered to evaluate the effects of outliers in performance of the tests. The type I error rats and power were computed in all Monte Carlo simulations by using the R software. The parametric bootstrap test based on the T 2 test statistic had equivalent performance of the T 2 original test. This test was recommended because it is easy to implement and computationally fast.

Equivalence of regression curves sharing common parameters.

In clinical trials, the comparison of two different populations is a common problem. Nonlinear (parametric) regression models are commonly used to describe the relationship between covariates, such as concentration or dose, and a response variable in the two groups. In some situations, it is reasonable to assume some model parameters to be the same, for instance, the placebo effect or the maximum treatment effect. In this paper, we develop a (parametric) bootstrap test to establish the similarity of two regression curves sharing some common parameters. We show by theoretical arguments and by means of a simulation study that the new test controls its significance level and achieves a reasonable power. Moreover, it is demonstrated that under the assumption of common parameters, a considerably more powerful test can be constructed compared with the test that does not use this assumption. Finally, we illustrate the potential applications of the new methodology by a clinical trial example.

Pernicious Polychorics: The Impact and Detection of Underlying Non-normality

Ordinal data in social science statistics are often modeled as discretizations of a multivariate normal vector. In contrast to the continuous case, where SEM estimation is also consistent under non-normality, violation of underlying normality in ordinal SEM may lead to inconsistent estimation. In this article, we illustrate how underlying non-normality induces bias in polychoric estimates and their standard errors. This bias is strongly affected by how we discretize. It is therefore important to consider tests of underlying multivariate normality. In this study we propose a parametric bootstrap test for this purpose. Its performance relative to the test of Maydeu-Olivares is evaluated in a Monte Carlo study. At realistic sample sizes, the bootstrap exhibited substantively better Type I error control and power than the Maydeu-Olivares test in ordinal data with ten dimensions or higher. R code for the bootstrap test is provided.

Testing the equality of several inverse Gaussian means under heterogeneity

We consider the problem of testing the equality of several inverse Gaussian means when the scale parameters and sample sizes are possibly unequal. We propose four parametric bootstrap (PB) tests based on the uniformly minimum variance unbiased estimators of parameters. We also compare our proposed tests with the existing ones via an extensive simulation study in terms of controlling the Type I error rate and power performance. Simulation results show the merits of the PB tests.

A note reconciling ANOVA tests under unequal error variances

ABSTRACTThe purpose of this note is to derive simple testing procedures for ANOVA under heteroscedasticity by a single approach that are equivalent to the prior art in the literature obtained by the Parametric Bootstrap and the Generalized Fiducial approach. By similar approach, researchers are encouraged to derive generalized tests in other applications, as alternative to parametric bootstrap tests and fiducial tests, including ANCOVA and MANOVA under heteroscedasticity, especially in Mixed Model applications, where the bootstrap approach fails.

Testing Process Factor Analysis Models Using the Parametric Bootstrap

ABSTRACTProcess factor analysis (PFA) is a latent variable model for intensive longitudinal data. It combines P-technique factor analysis and time series analysis. The goodness-of-fit test in PFA is currently unavailable. In the paper, we propose a parametric bootstrap method for assessing model fit in PFA. We illustrate the test with an empirical data set in which 22 participants rated their effects everyday over a period of 90 days. We also explore Type I error and power of the parametric bootstrap test with simulated data.

The accuracy of a non-invasive glucose monitoring device does not depend on clinical characteristics of people with type 2 diabetes mellitus

Objective: GlucoTrack is a non-invasive device that indirectly measures glucose fluctuation in the earlobe tissue. Thus, its accuracy may be subjected to a time lag between glucose concentration in blood and tissue. This time lag was shown to depend on individual characteristics related to microvascular complications, such as diabetes duration, HbA1c level, and smoking history. Therefore, the current study investigated the effects of these factors on GlucoTrack performance.Research design and methods: Clinical trials were conducted on 114 people with type 2 diabetes. Device performance was clinically evaluated using Clarke error grid (CEG) analysis and numerically evaluated using the distribution of absolute relative difference (ARD) values.Results: CEG analysis revealed that 98.0% of glucose readings were within the clinically acceptable CEG A + B zones. Total mean ARD was 22.7%. Clinical and numerical accuracies were comparable between never smokers and former/current smokers, but slightly reduced in the HbA1c ≥ 7.5% group and in the diabetes duration ≥15 years group. Yet, likelihood ratio and parametric bootstrap tests statistically demonstrated that ARD values did not depend on diabetes duration, HbA1c level, or smoking history.Conclusions: GlucoTrack performance does not depend on diabetes duration, HbA1c level, and smoking history, indicating the device is suitable for various people with type 2 diabetes.

Testing the equality of several multivariate normal mean vectors under heteroscedasticity: A fiducial approach and an approximate test

ABSTRACTWe consider the problem of testing the equality of several multivariate normal mean vectors under heteroscedasticity. We first construct a fiducial confidence region (FCR) for the differences between normal mean vectors and we then propose a fiducial test for comparing mean vectors by inverting the FCR. We also propose a simple approximate test that is based on a modification of the χ2 approximation. This simple test avoids the complications of simulation-based inference methods. We show that the proposed fiducial test has correct type one error rate asymptotically. We compare the proposed fiducial and approximate tests with the parametric bootstrap test in terms of controlling the type one error rate via an extensive simulation study. Our simulation results show that the proposed fiducial and approximate tests control the type one error rate, while there are cases that the parametric bootstrap test is out of control. We also discuss the power performance of the tests. Finally, we illustrate with a real example how our proposed methods are applicable in analyzing repeated measure designs including a single grouping variable.

A joint probability distribution model of directional extreme wind speeds based on the t-Copula function

The probabilistic information of directional extreme wind speeds is important for precisely estimating the design wind loads on structures. A new joint probability distribution model of directional extreme wind speeds is established based on observed wind-speed data using multivariate extreme value theory with the t-Copula function in the present study. At first, the theoretical deficiencies of the Gaussian-Copula and Gumbel-Copula models proposed by previous researchers for the joint probability distribution of directional extreme wind speeds are analysed. Then, the t-Copula model is adopted to solve this deficiency. Next, these three types of Copula models are discussed and evaluated with Spearman\'s rho, the parametric bootstrap test and the selection criteria based on the empirical Copula. Finally, the extreme wind speeds for a given return period are predicted by the t-Copula model with observed wind-speed records from several areas and the influence of dependence among directional extreme wind speeds on the predicted results is discussed.

Modified tests for comparison of group means under heteroskedasticity and non-normality caused by outlier(s)

There are several approximate tests proposed such as Welch's F-test (W), the Parametric Bootstrap Test (PB) and Generalized F-test (GF) for comparing several group means under heteroskedasticity. These tests are powerful and have nominal type 1 error rates but they are not performing satisfactorily under non-normality caused by outlier(s). To handle this problem, we investigate tests that are powerful and provide nominal type 1 error rates by using robust estimators both for location and scale parameters. The performance of the modi fied tests are examined with Monte-Carlo simulation studies. Results of simulations clearly indicate that Generalized F-test modi ed with Huber's M-estimators achieves the nominal type 1 error rate and provide higher power than alternative methods.