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 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 inferences for unbalanced panel data models

This article presents parametric bootstrap (PB) approaches for hypothesis testing and interval estimation for the regression coefficients of panel data regression models with incomplete panels. Some simulation results are presented to compare the performance of the PB approaches with the approximate 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 better than the approximate methods with respect to the coverage probabilities and the Type I error rates. 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 multi-way error component regression models with unbalanced panels. Finally, the proposed approaches are illustrated by using a real data example.     

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.     

Weighted quantile regression and testing for varying-coefficient models with randomly truncated data

This paper develops a varying-coefficient approach to the estimation and testing of regression quantiles under randomly truncated data. In order to handle the truncated data, the random weights are introduced and the weighted quantile regression (WQR) estimators for nonparametric functions are proposed. To achieve nice efficiency properties, we further develop a weighted composite quantile regression (WCQR) estimation method for nonparametric functions in varying-coefficient models. The asymptotic properties both for the proposed WQR and WCQR estimators are established. In addition, we propose a novel bootstrap-based test procedure to test whether the nonparametric functions in varying-coefficient quantile models can be specified by some function forms. The performance of the proposed estimators and test procedure are investigated through simulation studies and a real data example.     

A comparison of two approaches for power and sample size calculations in logistic regression models

Whittemore (1981) proposed an approach for calculating the sample size needed to test hypotheses with specified significance and power against a given alternative for logistic regression with small response probability. Based on the distribution of covariate, which could be either discrete or continuous, this approach first provides a simple closed-form approximation to the asymptotic covariance matrix of the maximum likelihood estimates, and then uses it to calculate the sample size needed to test a hypothesis about the parameter. Self et al. (1992) described a general approach for power and sample size calculations within the framework of generalized linear models, which include logistic regression as a special case. Their approach is based on an approximation to the distribution of the likelihood ratio statistic. Unlike the Whittemore approach, their approach is not limited to situations of small response probability. However, it is restricted to models with a finite number of covariate configurations. This study compares these two approaches to see how accurate they would be for the calculations of power and sample size in logistic regression models with various response probabilities and covariate distributions. The results indicate that the Whittemore approach has a slight advantage in achieving the nominal power only for one case with small response probability. It is outperformed for all other cases with larger response probabilities. In general, the approach proposed in Self et al. (1992) is recommended for all values of the response probability. However, its extension for logistic regression models with an infinite number of covariate configurations involves an arbitrary decision for categorization and leads to a discrete approximation. As shown in this paper, the examined discrete approximations appear to be sufficiently accurate for practical purpose.     

Chi-squared tests in kth-moment sufficient dimension reduction

We present a novel approach to sufficient dimension reduction for the conditional kth moments in regression. The approach provides a computationally feasible test for the dimension of the central kth-moment subspace. In addition, we can test predictor effects without assuming any models. All test statistics proposed in the novel approach have asymptotic chi-squared distributions.     

Empirical characteristic function tests for GARCH innovation distribution using multipliers

Goodness-of-fit tests for the innovation distribution in GARCH models based on measuring deviations between the empirical characteristic function of the residuals and the characteristic function under the null hypothesis have been proposed in the literature. The asymptotic distributions of these test statistics depend on unknown quantities, so their null distributions are usually estimated through parametric bootstrap (PB). Although easy to implement, the PB can become very computationally expensive for large sample sizes, which is typically the case in applications of these models. This work proposes to approximate the null distribution through a weighted bootstrap. The procedure is studied both theoretically and numerically. Its asymptotic properties are similar to those of the PB, but, from a computational point of view, it is more efficient.     

Extreme values identification in regression using a peaks-over-threshold approach

The problem of heavy tail in regression models is studied. It is proposed that regression models are estimated by a standard procedure and a statistical check for heavy tail using residuals is conducted as a tool for regression diagnostic. Using the peaks-over-threshold approach, the generalized Pareto distribution quantifies the degree of heavy tail by the extreme value index. The number of excesses is determined by means of an innovative threshold model which partitions the random sample into extreme values and ordinary values. The overall decision on a significant heavy tail is justified by both a statistical test and a quantile–quantile plot. The usefulness of the approach includes justification of goodness of fit of the estimated regression model and quantification of the occurrence of extremal events. The proposed methodology is supplemented by surface ozone level in the city center of Leeds.     

Goodness-of-link tests for multivariate regression models

ABSTRACT

This note presents an approximation to multivariate regression models which is obtained from a first-order series expansion of the multivariate link function. The proposed approach yields a variable-addition approximation of regression models that enables a multivariate generalization of the well-known goodness-of-link specification test, available for univariate generalized linear models. Application of this general methodology is illustrated with models of multinomial discrete choice and multivariate fractional data, in which context it is shown to lead to well-established approximation and testing procedures.     

Comparing Several Regression Models with Unequal Variances

We consider the problem of comparing k regression models, when the variances are not assumed to be equal. For this problem, the classical F test can lead to misleading results, and there is no simple test which adequately controls the size when the sample sizes are small. For k = 2, the most widely used test is the “weighted F test,” also known as the “asymptotic Chow test.” But this test does not work well for small samples, and various modifications have been proposed in the literature. For k > 2, few tests are available and only the parametric-bootstrap (PB) test of Tian et al. (2009) Tian, L., Ma, C., Vexler, A. (2009). A parametric bootstrap test for comparing heteroscedastic regression models. Communications in Statistics—Simulation and Computation, 38, 1026–1036.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar] controls the size fairly adequately. In this article, we propose three fairly simple F tests which can easily be applied in the general case, k ? 2, and avoid the complications of the PB test. Our simulations indicate that these tests have satisfactory performance. Also, our simulations confirm that the power properties of our proposed tests are similar to the PB test. Therefore, our proposed tests provide simple alternatives to the PB test, which can easily be used by practitioners who may not be familiar with the PB.     

A Parametric Bootstrap Solution to Two-way MANOVA without Interaction under Heteroscedasticity: Fixed and Mixed Models

In this article, we propose a parametric bootstrap (PB) test for heteroscedastic two-way multivariate analysis of variance without Interaction. For the problem of testing equal main effects of factors, we obtain a PB approach and compare it with existing modified Brown–Forsythe (MBF) test and approximate Hotelling T² (AHT) test by an extensive simulation study. The PB test is a symmetric function in samples, and does not depend on the chosen weights used to define the parameters uniquely. Simulation results indicate that the PB test performs satisfactorily for various cell sizes and parameter configurations when the homogeneity assumption is seriously violated, and tends to outperform the AHT test for moderate or larger samples in terms of power and controlling size. The MBF test, the AHT test, and the PB test have similar robustness to violations of underlying assumptions. It is also noted that the same PB test can be used to test the significance of random effect vector in a two-way multivariate mixed effects model with unequal cell covariance matrices.     

Errors-in-variables beta regression models

Beta regression models provide an adequate approach for modeling continuous outcomes limited to the interval (0, 1). This paper deals with an extension of beta regression models that allow for explanatory variables to be measured with error. The structural approach, in which the covariates measured with error are assumed to be random variables, is employed. Three estimation methods are presented, namely maximum likelihood, maximum pseudo-likelihood and regression calibration. Monte Carlo simulations are used to evaluate the performance of the proposed estimators and the naïve estimator. Also, a residual analysis for beta regression models with measurement errors is proposed. The results are illustrated in a real data set.     

Pseudo-empirical Bayes estimation of small area means under a nested error linear regression model with functional measurement errors

Small area estimation is studied under a nested error linear regression model with area level covariate subject to measurement error. Ghosh and Sinha (2007) obtained a pseudo-Bayes (PB) predictor of a small area mean and a corresponding pseudo-empirical Bayes (PEB) predictor, using the sample means of the observed covariate values to estimate the true covariate values. In this paper, we first derive an efficient PB predictor by using all the available data to estimate true covariate values. We then obtain a corresponding PEB predictor and show that it is asymptotically “optimal”. In addition, we employ a jackknife method to estimate the mean squared prediction error (MSPE) of the PEB predictor. Finally, we report the results of a simulation study on the performance of our PEB predictor and associated jackknife MSPE estimator. Our results show that the proposed PEB predictor can lead to significant gain in efficiency over the previously proposed PEB predictor. Area level models are also studied.     

Empirical smoothing lack-of-fit tests for variance function

This paper discusses a nonparametric empirical smoothing lack-of-fit test for the functional form of the variance in regression models. The proposed test can be treated as a nontrivial modification of Zheng's nonparametric smoothing test, Koul and Ni's minimum distance test for the mean function in the classic regression models. The paper establishes the asymptotic normality of the proposed test under the null hypothesis. Consistency at some fixed alternatives and asymptotic power under some local alternatives are also discussed. A simulation study is conducted to assess the finite sample performance of the proposed test. Simulation study also shows that the proposed test is more powerful and computationally more efficient than some existing tests.     

ESTIMATION AND TESTING FOR PARTIALLY LINEAR SINGLE-INDEX MODELS

In partially linear single-index models, we obtain the semiparametrically efficient profile least-squares estimators of regression coefficients. We also employ the smoothly clipped absolute deviation penalty (SCAD) approach to simultaneously select variables and estimate regression coefficients. We show that the resulting SCAD estimators are consistent and possess the oracle property. Subsequently, we demonstrate that a proposed tuning parameter selector, BIC, identifies the true model consistently. Finally, we develop a linear hypothesis test for the parametric coefficients and a goodness-of-fit test for the nonparametric component, respectively. Monte Carlo studies are also presented.     

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.     

Penalized likelihood ratio test for a biomarker threshold effect in clinical trials based on generalized linear models

In a clinical trial, the responses to the new treatment may vary among patient subsets with different characteristics in a biomarker. It is often necessary to examine whether there is a cutpoint for the biomarker that divides the patients into two subsets of those with more favourable and less favourable responses. More generally, we approach this problem as a test of homogeneity in the effects of a set of covariates in generalized linear regression models. The unknown cutpoint results in a model with nonidentifiability and a nonsmooth likelihood function to which the ordinary likelihood methods do not apply. We first use a smooth continuous function to approximate the indicator function defining the patient subsets. We then propose a penalized likelihood ratio test to overcome the model irregularities. Under the null hypothesis, we prove that the asymptotic distribution of the proposed test statistic is a mixture of chi-squared distributions. Our method is based on established asymptotic theory, is simple to use, and works in a general framework that includes logistic, Poisson, and linear regression models. In extensive simulation studies, we find that the proposed test works well in terms of size and power. We further demonstrate the use of the proposed method by applying it to clinical trial data from the Digitalis Investigation Group (DIG) on heart failure.     

Efficient Computation of Reduced Regression Models

We consider settings where it is of interest to fit and assess regression submodels that arise as various explanatory variables are excluded from a larger regression model. The larger model is referred to as the full model; the submodels are the reduced models. We show that a computationally efficient approximation to the regression estimates under any reduced model can be obtained from a simple weighted least squares (WLS) approach based on the estimated regression parameters and covariance matrix from the full model. This WLS approach can be considered an extension to unbiased estimating equations of a first-order Taylor series approach proposed by Lawless and Singhal. Using data from the 2010 Nationwide Inpatient Sample (NIS), a 20% weighted, stratified, cluster sample of approximately 8 million hospital stays from approximately 1000 hospitals, we illustrate the WLS approach when fitting interval censored regression models to estimate the effect of type of surgery (robotic versus nonrobotic surgery) on hospital length-of-stay while adjusting for three sets of covariates: patient-level characteristics, hospital characteristics, and zip-code level characteristics. Ordinarily, standard fitting of the reduced models to the NIS data takes approximately 10 hours; using the proposed WLS approach, the reduced models take seconds to fit.     

Testing the Significance of Categorical Predictor Variables in Nonparametric Regression Models

In this paper we propose a test for the significance of categorical predictors in nonparametric regression models. The test is fully data-driven and employs cross-validated smoothing parameter selection while the null distribution of the test is obtained via bootstrapping. The proposed approach allows applied researchers to test hypotheses concerning categorical variables in a fully nonparametric and robust framework, thereby deflecting potential criticism that a particular finding is driven by an arbitrary parametric specification. Simulations reveal that the test performs well, having significantly better power than a conventional frequency-based nonparametric test. The test is applied to determine whether OECD and non-OECD countries follow the same growth rate model or not. Our test suggests that OECD and non-OECD countries follow different growth rate models, while the tests based on a popular parametric specification and the conventional frequency-based nonparametric estimation method fail to detect any significant difference.