Parametric bootstrap inferences for panel data models

This article presents parametric bootstrap (PB) approaches for hypothesis testing and interval estimation for the regression coefficients and the variance components of panel data regression models with complete panels. The PB pivot variables are proposed based on sufficient statistics of the parameters. On the other hand, we also derive generalized inferences and improved generalized inferences for variance components in this article. Some simulation results are presented to compare the performance of the PB approaches with the generalized inferences. Our studies show that the PB approaches perform satisfactorily for various sample sizes and parameter configurations, and the performance of PB approaches is mostly the same as that of generalized inferences with respect to the expected lengths and powers. The PB inferences have almost exact coverage probabilities and Type I error rates. Furthermore, the PB procedure can be simply carried out by a few simulation steps, and the derivation is easier to understand and to be extended to the incomplete panels. Finally, the proposed approaches are illustrated by using a real data example.     

Parametric bootstrap inferences for the growth curve models with intraclass correlation structure

This paper presents parametric bootstrap (PB) approaches for hypothesis testing and interval estimation of the fixed effects and the variance component in the growth curve models with intraclass correlation structure. The PB pivot variables are proposed based on the sufficient statistics of the parameters. Some simulation results are presented to compare the performance of the proposed approaches with the generalized inferences. Our studies show that the PB approaches perform satisfactorily for various cell sizes and parameter configurations, and tends to outperform the generalized inferences with respect to the coverage probabilities and powers. The PB approaches not only have almost exact coverage probabilities and Type I error rates, but also have the shorter expected lengths and the higher powers. Furthermore, the PB procedure can be simply carried out by a few simulation steps. Finally, the proposed approaches are illustrated by using a real data example.     

Parametric simultaneous robust inferences for regression coefficient under generalized linear models

In this article, the parametric robust regression approaches are proposed for making inferences about regression parameters in the setting of generalized linear models (GLMs). The proposed methods are able to test hypotheses on the regression coefficients in the misspecified GLMs. More specifically, it is demonstrated that with large samples, the normal and gamma regression models can be properly adjusted to become asymptotically valid for inferences about regression parameters under model misspecification. These adjusted regression models can provide the correct type I and II error probabilities and the correct coverage probability for continuous data, as long as the true underlying distributions have finite second moments.     

A Parametric Bootstrap Test for Comparing Heteroscedastic Regression Models

Testing equality of regression coefficients in several regression models is a common problem encountered in many applied fields. This article presents a parametric bootstrap (PB) approach and compares its performance to that of another simulation-based approach, namely, the generalized variable approach. Simulation studies indicate that the PB approach controls the Type I error rates satisfactorily regardless of the number of regression models and sample sizes whereas the generalized variable approach tends to be very liberal as the number of regression models goes up. The proposed PB approach is illustrated using a data set from stability study.     

A fiducial p-value approach for comparing heteroscedastic regression models

To study the equality of regression coefficients in several heteroscedastic regression models, we propose a fiducial-based test, and theoretically examine the frequency property of the proposed test. We numerically compare the performance of the proposed approach with the parametric bootstrap (PB) approach. Simulation results indicate that the fiducial approach controls the Type I error rates satisfactorily regardless of the number of regression models and sample sizes, whereas the PB approach tends to be a little of liberal in some scenarios. Finally, the proposed approach is applied to an analysis of a real dataset for illustration.     

Nonparametric Inference for Time-Varying Coefficient Quantile Regression

The article considers nonparametric inference for quantile regression models with time-varying coefficients. The errors and covariates of the regression are assumed to belong to a general class of locally stationary processes and are allowed to be cross-dependent. Simultaneous confidence tubes (SCTs) and integrated squared difference tests (ISDTs) are proposed for simultaneous nonparametric inference of the latter models with asymptotically correct coverage probabilities and Type I error rates. Our methodologies are shown to possess certain asymptotically optimal properties. Furthermore, we propose an information criterion that performs consistent model selection for nonparametric quantile regression models of nonstationary time series. For implementation, a wild bootstrap procedure is proposed, which is shown to be robust to the dependent and nonstationary data structure. Our method is applied to studying the asymmetric and time-varying dynamic structures of the U.S. unemployment rate since the 1940s. Supplementary materials for this article are available online.     

Estimation Bias in Complete-Case Analysis in Crossover Studies with Missing Data

Crossover designs are used often in clinical trials. It is not uncommon that subjects discontinue before completing all treatment periods in a crossover study. Despite availability of statistical methodologies utilizing all available data and software for obtaining valid inferences under the assumption of missing at random (MAR), naïve approaches, such as the complete case (CC) analysis, which is only valid with a strong assumption of missing completely at random are still widely used in practice. In this article, we obtain the analytical form of the estimation bias of treatment effects with CC for linear mixed models. We use simulation studies to examine the inflation of Type I error and efficiency loss in the inferences with CC under MAR. Invalidity and inefficiency of two other commonly used approaches for defining analyzed data in the presence of missing data, including data from at least two periods in three period crossover and available cases for a specific comparison of interest, are also demonstrated through simulation studies.     

A parametric bootstrap approach for two-way error component regression models

In this article, the two-way error component regression model is considered. For the nonhomogenous linear hypothesis testing of regression coefficients, a parametric bootstrap (PB) approach is proposed. Simulation results indicate that the PB test, regardless of the sample sizes, maintains the Type I error rates very well and outperforms the existing generalized variable test, which may far exceed the intended significance level when the sample sizes are small or moderate. Real data examples illustrate the proposed approach work quite satisfactorily.     

Parametric Bootstrap Tests for Unbalanced Three-factor Nested Designs under Heteroscedasticity

In this article, we consider the three-factor unbalanced nested design model without the assumption of equal error variance. For the problem of testing “main effects” of the three factors, we propose a parametric bootstrap (PB) approach and compare it with the existing generalized F (GF) test. The Type I error rates of the tests are evaluated using Monte Carlo simulation. Our studies show that the PB test performs better than the generalized F-test. The PB test performs very satisfactorily even for small samples while the GF test exhibits poor Type I error properties when the number of factorial combinations or treatments goes up. It is also noted that the same tests can be used to test the significance of the random effect variance component in a three-factor mixed effects nested model under unequal error variances.     

A parametric bootstrap solution to the MANOVA under heteroscedasticity

In this article, we consider the problem of comparing several multivariate normal mean vectors when the covariance matrices are unknown and arbitrary positive definite matrices. We propose a parametric bootstrap (PB) approach and develop an approximation to the distribution of the PB pivotal quantity for comparing two mean vectors. This approximate test is shown to be the same as the invariant test given in [Krishnamoorthy and Yu, Modified Nel and Van der Merwe test for the multivariate Behrens–Fisher problem, Stat. Probab. Lett. 66 (2004), pp. 161–169] for the multivariate Behrens–Fisher problem. Furthermore, we compare the PB test with two existing invariant tests via Monte Carlo simulation. Our simulation studies show that the PB test controls Type I error rates very satisfactorily, whereas other tests are liberal especially when the number of means to be compared is moderate and/or sample sizes are small. The tests are illustrated using an example.     

A Parametric Bootstrap Test for Two-Way ANOVA Model Without Interaction Under Heteroscedasticity

In this article we consider the two-way ANOVA model without interaction under heteroscedasticity. For the problem of testing equal effects of factors, we propose a parametric bootstrap (PB) approach and compare it with existing the generalized F (GF) test. The Type I error rates and powers of the tests are evaluated using Monte Carlo simulation. Our studies show that the PB test performs better than the GF test. The PB test performs very satisfactorily even for small samples while the GF test exhibits poor Type I error properties when the number of factorial combinations or treatments goes up. It is also noted that the same tests can be used to test the significance of random effect variance component in a two-way mixed-effects model under unequal error variances.     

Parametric bootstrap tests for unbalanced nested designs under heteroscedasticity

In this article, we consider the two-factor unbalanced nested design model without the assumption of equal error variance. For the problem of testing ‘main effects’ of both factors, we propose a parametric bootstrap (PB) approach and compare it with the existing generalized F (GF) test. The Type I error rates of the tests are evaluated using Monte Carlo simulation. Our studies show that the PB test performs better than the GF test. The PB test performs very satisfactorily even for small samples while the GF test exhibit poor Type I error properties when the number of factorial combinations or treatments goes up. It is also noted that the same tests can be used to test the significance of the random effect variance component in a two-factor mixed effects nested model under unequal error variances.     

Estimating individual-level interaction effects in multilevel models: a Monte Carlo simulation study with application*

Moderated multiple regression provides a useful framework for understanding moderator variables. These variables can also be examined within multilevel datasets, although the literature is not clear on the best way to assess data for significant moderating effects, particularly within a multilevel modeling framework. This study explores potential ways to test moderation at the individual level (level one) within a 2-level multilevel modeling framework, with varying effect sizes, cluster sizes, and numbers of clusters. The study examines five potential methods for testing interaction effects: the Wald test, F-test, likelihood ratio test, Bayesian information criterion (BIC), and Akaike information criterion (AIC). For each method, the simulation study examines Type I error rates and power. Following the simulation study, an applied study uses real data to assess interaction effects using the same five methods. Results indicate that the Wald test, F-test, and likelihood ratio test all perform similarly in terms of Type I error rates and power. Type I error rates for the AIC are more liberal, and for the BIC typically more conservative. A four-step procedure for applied researchers interested in examining interaction effects in multi-level models is provided.     

On the small sample properties of norm-restricted maximum likelihood estimators for logistic regression models

This paper develops alternatives to maximum likelihood estimators (MLE) for logistic regression models and compares the mean squared error (MSE) of the estimators. The MLE for the vector of underlying success probabilities has low MSE only when the true probabilities are extreme (i.e., near 0 or 1). Extreme probabilities correspond to logistic regression parameter vectors which are large in norm. A competing “restricted” MLE and an empirical version of it are suggested as estimators with better performance than the MLE for central probabilities. An approximate EM-algorithm for estimating the restriction is described. As in the case of normal theory ridge estimators, the proposed estimators are shown to be formally derivable by Bayes and empirical Bayes arguments. The small sample operating characteristics of the proposed estimators are compared to the MLE via a simulation study; both the estimation of individual probabilities and of logistic parameters are considered.     

Empirical study of some non-parametric tests for dispersion of correlated data

Results of a computer simulation study of power and robustness of three competitor tests for comparing scales, for use with correlated data: Rothstein, Richardson and Bell (RRB), Arvesen, and Pitman, are presented. It is found that unless one could ímprove the approximate null distributions for Arvesen's and Pitman's test, RRB's procedure is best, having simulated probabilities of Type I error closest to the test's nominal α and being reasonably robust and powerful, for all distributions considered.     

Analyzing Small Samples of Repeated Measures Data with the Mixed-Model Adjusted F Test

This research examines the Type I error rates obtained when using the mixed model with the Kenward-Roger correction and compares them with the Between–Within and Satterthwaite approaches in split-plot designs. A simulated study was conducted to generate repeated measures data with small samples under normal distribution conditions. The data were obtained via three covariance matrices (unstructured, heterogeneous first-order auto-regressive, and random coefficients), the one with the best fit being selected according to the Akaike criterion. The results of the simulation study showed the Kenward-Roger test to be more robust, particularly when the population covariance matrices were unstructured or heterogeneous first-order auto-regressive. The main contribution of this study lies in showing that the Kenward–Roger method corrects the liberal Type I error rates obtained with the Between–Within and Satterthwaite approaches, especially with positive pairings between group sizes and covariance matrices.     

A Bayesian Hierarchical Model for Multiple Comparisons in Mixed Models

We propose a Bayesian hierarchical model for multiple comparisons in mixed models where the repeated measures on subjects are described with the subject random effects. The model facilitates inferences in parameterizing the successive differences of the population means, and for them, we choose independent prior distributions that are mixtures of a normal distribution and a discrete distribution with its entire mass at zero. For the other parameters, we choose conjugate or vague priors. The performance of the proposed hierarchical model is investigated in the simulated and two real data sets, and the results illustrate that the proposed hierarchical model can effectively conduct a global test and pairwise comparisons using the posterior probability that any two means are equal. A simulation study is performed to analyze the type I error rate, the familywise error rate, and the test power. The Gibbs sampler procedure is used to estimate the parameters and to calculate the posterior probabilities.     

Properties of the rank transformation in factorial analysis of covariance

Real world data often fail to meet the underlying assumption of population normality. The Rank Transformation (RT) procedure has been recommended as an alternative to the parametric factorial analysis of covariance (ANCOVA). The purpose of this study was to compare the Type I error and power properties of the RT ANCOVA to the parametric procedure in the context of a completely randomized balanced 3 × 4 factorial layout with one covariate. This study was concerned with tests of homogeneity of regression coefficients and interaction under conditional (non)normality. Both procedures displayed erratic Type I error rates for the test of homogeneity of regression coefficients under conditional nonnormality. With all parametric assumptions valid, the simulation results demonstrated that the RT ANCOVA failed as a test for either homogeneity of regression coefficients or interaction due to severe Type I error inflation. The error inflation was most severe when departures from conditional normality were extreme. Also associated with the RT procedure was a loss of power. It is recommended that the RT procedure not be used as an alternative to factorial ANCOVA despite its encouragement from SAS, IMSL, and other respected sources.     

Comparison of Selected Methods for Modeling of Multi-State Disease Progression Processes: A Simulation Study

Prognostic studies are essential to understand the role of particular prognostic factors and, thus, improve prognosis. In most studies, disease progression trajectories of individual patients may end up with one of mutually exclusive endpoints or can involve a sequence of different events.

One challenge in such studies concerns separating the effects of putative prognostic factors on these different endpoints and testing the differences between these effects.

In this article, we systematically evaluate and compare, through simulations, the performance of three alternative multivariable regression approaches in analyzing competing risks and multiple-event longitudinal data. The three approaches are: (1) fitting separate event-specific Cox's proportional hazards models; (2) the extension of Cox's model to competing risks proposed by Lunn and McNeil; and (3) Markov multi-state model.

The simulation design is based on a prognostic study of cancer progression, and several simulated scenarios help investigate different methodological issues relevant to the modeling of multiple-event processes of disease progression. The results highlight some practically important issues. Specifically, the decreased precision of the observed timing of intermediary (non fatal) events has a strong negative impact on the accuracy of regression coefficients estimated with either the Cox's or Lunn-McNeil models, while the Markov model appears to be quite robust, under the same circumstances. Furthermore, the tests based on both Markov and Lunn-McNeil models had similar power for detecting a difference between the effects of the same covariate on the hazards of two mutually exclusive events. The Markov approach yields also accurate Type I error rate and good empirical power for testing the hypothesis that the effect of a prognostic factor on changes after an intermediary event, which cannot be directly tested with the Lunn-McNeil method. Bootstrap-based standard errors improve the coverage rates for Markov model estimates. Overall, the results of our simulations validate Markov multi-state model for a wide range of data structures encountered in prognostic studies of disease progression, and may guide end users regarding the choice of model(s) most appropriate for their specific application.     

A frequentist assessment of Bayesian inclusion probabilities for screening predictors

Bayesian inclusion probabilities have become a popular tool for variable assessment. From a frequentist perspective, it is often difficult to evaluate these probabilities as typically no Type I error rates are considered, neither are any explorations of power of the methods given. This paper considers how a frequentist may evaluate Bayesian inclusion probabilities for screening predictors. This evaluation looks at both unrestricted and restricted model spaces and develops a framework which a frequentist can utilize inclusion probabilities that preserve Type I error rates. Furthermore, this framework is applied to an analysis of the Arabidopsis thaliana with respect to determining quantitative trait loci associated with cotelydon opening angle.