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.     

On variable selection in generalized linear and related regression models

This paper is concerned with selection of explanatory variables in generalized linear models (GLM). The class of GLM's is quite large and contains e.g. the ordinary linear regression, the binary logistic regression, the probit model and Poisson regression with linear or log-linear parameter structure. We show that, through an approximation of the log likelihood and a certain data transformation, the variable selection problem in a GLM can be converted into variable selection in an ordinary (unweighted) linear regression model. As a consequence no specific computer software for variable selection in GLM's is needed. Instead, some suitable variable selection program for linear regression can be used. We also present a simulation study which shows that the log likelihood approximation is very good in many practical situations. Finally, we mention briefly possible extensions to regression models outside the class of GLM's.     

Empirical likelihood for generalized partially linear varying-coefficient models

Generalized partially linear varying-coefficient models (GPLVCM) are frequently used in statistical modeling. However, the statistical inference of the GPLVCM, such as confidence region/interval construction, has not been very well developed. In this article, empirical likelihood-based inference for the parametric components in the GPLVCM is investigated. Based on the local linear estimators of the GPLVCM, an estimated empirical likelihood-based statistic is proposed. We show that the resulting statistic is asymptotically non-standard chi-squared. By the proposed empirical likelihood method, the confidence regions for the parametric components are constructed. In addition, when some components of the parameter are of particular interest, the construction of their confidence intervals is also considered. A simulation study is undertaken to compare the empirical likelihood and the other existing methods in terms of coverage accuracies and average lengths. The proposed method is applied to a real example.     

Nonnull asymptotic distributions of the LR,Wald, score and gradient statistics in generalized linear models with dispersion covariates

The class of generalized linear models with dispersion covariates, which allows us to jointly model the mean and dispersion parameters, is a natural extension to the classical generalized linear models. In this paper, we derive the asymptotic expansions under a sequence of Pitman alternatives (up to order n ^?1/2) for the nonnull distribution functions of the likelihood ratio, Wald, Rao score and gradient statistics in this class of models. The asymptotic distributions of these statistics are obtained for testing a subset of regression parameters and for testing a subset of dispersion parameters. Based on these nonnull asymptotic expansions, the power of all four tests, which are equivalent to first order, are compared. Furthermore, we consider Monte Carlo simulations in order to compare the finite-sample performance of these tests in this class of models. We present two empirical applications to two real data sets for illustrative purposes.     

Estimation and testing for semiparametric mixtures of partially linear models

In this paper, we study the estimation and inference for a class of semiparametric mixtures of partially linear models. We prove that the proposed models are identifiable under mild conditions, and then give a PL–EM algorithm estimation procedure based on profile likelihood. The asymptotic properties for the resulting estimators and the ascent property of the PL–EM algorithm are investigated. Furthermore, we develop a test statistic for testing whether the non parametric component has a linear structure. Monte Carlo simulations and a real data application highlight the interest of the proposed procedures.     

The effect of small sample on optimal designs for logistic regression models

Asymptotic methods are commonly used in statistical inference for unknown parameters in binary data models. These methods are based on large sample theory, a condition which may be in conflict with small sample size and hence leads to poor results in the optimal designs theory. In this paper, we apply the second order expansions of the maximum likelihood estimator and derive a matrix formula for the mean square error (MSE) to obtain more precise optimal designs based on the MSE. Numerical results indicate the new optimal designs are more efficient than the optimal designs based on the information matrix.     

Correcting data for measurement error in generalized linear models

This paper discusses a general strategy for reducing measurement-error-induced bias in statistical models. It is assumed that the measurement error is unbiased with a known variance although no other distributional assumptions on the measurement-error are employed,

Using a preliminary fit of the model to the observed data, a transformation of the variable measured with error is estimated. The transformation is constructed so that the estimates obtained by refitting the model to the ‘corrected’ data have smaller bias,

Whereas the general strategy can be applied in a number of settings, this paper focuses on the problem of covariate measurement error in generalized linear models, Two estimators are derived and their effectiveness at reducing bias is demonstrated in a Monte Carlo study.     

Parallel tempering for dynamic generalized linear models

ABSTRACT

Markov chain Monte Carlo (MCMC) methods can be used for statistical inference. The methods are time-consuming due to time-vary. To resolve these problems, parallel tempering (PT), as a parallel MCMC method, is tried, for dynamic generalized linear models (DGLMs), as well as the several optimal properties of our proposed method. In PT, two or more samples are drawn at the same time, and samples can exchange information with each other. We also present some simulations of the DGLMs in the case and provide two applications of Poisson-type DGLMs in financial research.     

Double hierarchical generalized linear models (with discussion)

Summary.  We propose a class of double hierarchical generalized linear models in which random effects can be specified for both the mean and dispersion. Heteroscedasticity between clusters can be modelled by introducing random effects in the dispersion model, as is heterogeneity between clusters in the mean model. This class will, among other things, enable models with heavy-tailed distributions to be explored, providing robust estimation against outliers. The h -likelihood provides a unified framework for this new class of models and gives a single algorithm for fitting all members of the class. This algorithm does not require quadrature or prior probabilities.     

A new multicollinearity diagnostic for generalized linear models

We propose a new collinearity diagnostic tool for generalized linear models. The new diagnostic tool is termed the weighted variance inflation factor (WVIF) behaving exactly the same as the traditional variance inflation factor in the context of regression diagnostic, given data matrix normalized. Compared to the use of condition number (CN), WVIF shows more reliable information on how severe the situation is, when data collinearity does exist. An alternative estimator, a by-product of the new diagnostic, outperforms the ridge estimator in the presence of data collinearity in both aspects of WVIF and CN. Evidences are given through analyzing various real-world numerical examples.     

Improved simultaneous intervals for linear combinations of parameters from generalized linear models

When employing generalized linear models, interest often focuses on estimation of odds ratios or relative risks. Additionally, researchers often make overall conclusions, requiring accurate estimation of a set of these quantities. Consequently, simultaneous estimation is warranted. Current simultaneous estimation methods only perform well in this setting when there are a very small number of comparisons and/or the sample size is relatively large. Additionally, the estimated quantities can have significant bias especially at small sample sizes. The proposed bounds: (1) perform well for a small or large number of comparisons, (2) exhibit improved performance over current methods for small to moderate sample sizes, (3) provide bias adjustment not reliant on asymptotics, and (4) avoid the infinite parameter estimates that can occur with maximum-likelihood estimators. Simulations demonstrate that the proposed bounds achieve the desired level of confidence at smaller sample sizes than previous methods.     

Robust quasi-likelihood inference in generalized linear mixed models with outliers

It is well known that in a traditional outlier-free situation, the generalized quasi-likelihood (GQL) approach [B.C. Sutradhar, On exact quasilikelihood inference in generalized linear mixed models, Sankhya: Indian J. Statist. 66 (2004), pp. 261–289] performs very well to obtain the consistent as well as the efficient estimates for the parameters involved in the generalized linear mixed models (GLMMs). In this paper, we first examine the effect of the presence of one or more outliers on the GQL estimation for the parameters in such GLMMs, especially in two important models such as count and binary mixed models. The outliers appear to cause serious biases and hence inconsistency in the estimation. As a remedy, we then propose a robust GQL (RGQL) approach in order to obtain the consistent estimates for the parameters in the GLMMs in the presence of one or more outliers. An extensive simulation study is conducted to examine the consistency performance of the proposed RGQL approach.     

Testing for constancy in varying coefficient models

We consider varying coefficient models, which are an extension of the classical linear regression models in the sense that the regression coefficients are replaced by functions in certain variables (for example, time), the covariates are also allowed to depend on other variables. Varying coefficient models are popular in longitudinal data and panel data studies, and have been applied in fields such as finance and health sciences. We consider longitudinal data and estimate the coefficient functions by the flexible B-spline technique. An important question in a varying coefficient model is whether an estimated coefficient function is statistically different from a constant (or zero). We develop testing procedures based on the estimated B-spline coefficients by making use of nice properties of a B-spline basis. Our method allows longitudinal data where repeated measurements for an individual can be correlated. We obtain the asymptotic null distribution of the test statistic. The power of the proposed testing procedures are illustrated on simulated data where we highlight the importance of including the correlation structure of the response variable and on real data.     

Estimation of group means when adjusting for covariates in generalized linear models

Generalized linear models are commonly used to analyze categorical data such as binary, count, and ordinal outcomes. Adjusting for important prognostic factors or baseline covariates in generalized linear models may improve the estimation efficiency. The model‐based mean for a treatment group produced by most software packages estimates the response at the mean covariate, not the mean response for this treatment group for the studied population. Although this is not an issue for linear models, the model‐based group mean estimates in generalized linear models could be seriously biased for the true group means. We propose a new method to estimate the group mean consistently with the corresponding variance estimation. Simulation showed the proposed method produces an unbiased estimator for the group means and provided the correct coverage probability. The proposed method was applied to analyze hypoglycemia data from clinical trials in diabetes. Copyright © 2014 John Wiley & Sons, Ltd.     

An asymptotic theory for semiparametric generalized least squares estimation in partially linear regression models

Consider a partially linear regression model with an unknown vector parameter β, an unknown functiong(·), and unknown heteroscedastic error variances. In this paper we develop an asymptotic semiparametric generalized least squares estimation theory under some weak moment conditions. These moment conditions are satisfied by many of the error distributions encountered in practice, and our theory does not require the number of replications to go to infinity.     

Robust testing with generalized partial linear models for longitudinal data

By approximating the nonparametric component using a regression spline in generalized partial linear models (GPLM), robust generalized estimating equations (GEE), involving bounded score function and leverage-based weighting function, can be used to estimate the regression parameters in GPLM robustly for longitudinal data or clustered data. In this paper, score test statistics are proposed for testing the regression parameters with robustness, and their asymptotic distributions under the null hypothesis and a class of local alternative hypotheses are studied. The proposed score tests reply on the estimation of a smaller model without the testing parameters involved, and perform well in the simulation studies and real data analysis conducted in this paper.     

Adaptive Lasso for generalized linear models with a diverging number of parameters

In this paper, we study the asymptotic properties of the adaptive Lasso estimators in high-dimensional generalized linear models. The consistency of the adaptive Lasso estimator is obtained. We show that, if a reasonable initial estimator is available, under appropriate conditions, the adaptive Lasso correctly selects covariates with non zero coefficients with probability converging to one, and that the estimators of non zero coefficients have the same asymptotic distribution they would have if the zero coefficients were known in advance. Thus, the adaptive Lasso has an Oracle property. The results are examined by some simulations and a real example.     

A comparison between bootstrap methods and generalized estimating equations for correlated outcomes in generalized linear models

We discuss and evaluate bootstrap algorithms for obtaining confidence intervals for parameters in Generalized Linear Models when the data are correlated. The methods are based on a stratified bootstrap and are suited to correlation occurring within “blocks” of data (e.g., individuals within a family, teeth within a mouth, etc.). Application of the intervals to data from a Dutch follow-up study on preterm infants shows the corroborative usefulness of the intervals, while the intervals are seen to be a powerful diagnostic in studying annual measles data. In a simulation study, we compare the coverage rates of the proposed intervals with existing methods (e.g., via Generalized Estimating Equations). In most cases, the bootstrap intervals are seen to perform better than current methods, and are produced in an automatic fashion, so that the user need not know (or have to guess) the dependence structure within a block.