Bootstrap tests for variance components in generalized linear mixed models

In many applications of generalized linear mixed models to clustered correlated or longitudinal data, often we are interested in testing whether a random effects variance component is zero. The usual asymptotic mixture of chi‐square distributions of the score statistic for testing constrained variance components does not necessarily hold. In this article, the author proposes and explores a parametric bootstrap test that appears to be valid based on its estimated level of significance under the null hypothesis. Results from a simulation study indicate that the bootstrap test has a level much closer to the nominal one while the asymptotic test is conservative, and is more powerful than the usual asymptotic score test based on a mixture of chi‐squares. The proposed bootstrap test is illustrated using two sets of real‐life data obtained from clinical trials. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Prediction for generalized linear models

Regression models are often used to make predictions. All the information needed is contained in the predictive distribution. However, this cannot be evaluated explicitly for most generalized linear models. We construct two approximations to this distribution and demonstrate their use on two sets of survival data, corresponding to the outcome of patients admitted to intensive care units and the survival times of leukaemia patients.Regression models are often used to make predictions. All the information needed is contained in the predictive distribution. However, this cannot be evaluated explicitly for most generalized linear models. We construct two approximations to this distribution and demonstrate their use on two sets of survival data, corresponding to the outcome of patients admitted to intensive care units and the survival times of leukaemia patients.Regression models are often used to make predictions. All the information needed is contained in the predictive distribution. However, this cannot be evaluated explicitly for most generalized linear models. We construct two approximations to this distribution and demonstrate their use on two sets of survival data, corresponding to the outcome of patients admitted to intensive care units and the survival times of leukaemia patients.Regression models are often used to make predictions. All the information needed is contained in the predictive distribution. However, this cannot be evaluated explicitly for most generalized linear models. We construct two approximations to this distribution and demonstrate their use on two sets of survival data, corresponding to the outcome of patients admitted to intensive care units and the survival times of leukaemia patients.     

Optimal designs for generalized linear models

Summary This paper solves some D-optimal design problems for certain Generalized Linear Models where the mean depends on two parameters and two explanatory variables. In all of the cases considered the support point of the optimal designs are found to be independent of the unknown parameters. While in some cases the optimal design measures are given by two points with equal weights, in others the support is given by three point with weights depending on the unknown parameters, hence the designs are locally optimal in general. Empirical results on the efficiency of the locally optimal designs are also given. Some of the designs found can also be used for planning D-optimal experiments for the normal linear model, where the mean must be positive. This research was carried out in part at University College, London as an M.Sc. project. Thanks are due to Prof. I. Ford (University of Glasgow) and Prof. A. Giovagnoli (University of Perugia) for their valuable suggestions and critical observations.     

Splice plots for generalized linear models

ABSTRACT

We develop splice plots as a diagnostic tool for parametric generalized linear models. Splice plots use the independence of the outcome and explanatory measures given the regression function. Plotting differences between the estimated parametric regression function and non-parametric estimates of the regression function computed in small neighborhoods of the fitted values from the parametric model can be used to assess model fit.     

Transformed generalized linear models

The estimation of data transformation is very useful to yield response variables satisfying closely a normal linear model. Generalized linear models enable the fitting of models to a wide range of data types. These models are based on exponential dispersion models. We propose a new class of transformed generalized linear models to extend the Box and Cox models and the generalized linear models. We use the generalized linear model framework to fit these models and discuss maximum likelihood estimation and inference. We give a simple formula to estimate the parameter that index the transformation of the response variable for a subclass of models. We also give a simple formula to estimate the r$r$th moment of the original dependent variable. We explore the possibility of using these models to time series data to extend the generalized autoregressive moving average models discussed by Benjamin et al. [Generalized autoregressive moving average models. J. Amer. Statist. Assoc. 98, 214–223]. The usefulness of these models is illustrated in a simulation study and in applications to three real data sets.     

Bootstrap variance and bias estimation in linear models

Let θ be a nonlinear function of the regression parameters and θ be its estimator based on the least-squares method. This paper studies the bootstrap estimators of the variance and bias of θ. The bootstrap estimators are shown to be consistent and asymptotically unbiased under some conditions. Asymptotic orders of the mean squared errors of the bootstrap estimators are also obtained. The bootstrap and the classical linearization method are compared in a simulation study. Discussions about when to use the bootstrap are given.     

Modified Wald statistics for generalized linear models

Summary: Wald statistics in generalized linear models are asymptotically ² distributed. The asymptotic chi–squared law of the corresponding quadratic form shows disadvantages with respect to the approximation of the finite–sample distribution. It is shown by means of a comprehensive simulation study that improvements can be achieved by applying simple finite–sample size approximations to the distribution of the quadratic form in generalized linear models. These approximations are based on a ² distribution with an estimated degree of freedom that generalizes an approach by Patnaik and Pearson. Simulation studies confirm that nominal level is maintained with higher accuracy compared to the Wald statistics.     

Variable selection for multivariate generalized linear models

Generalized linear models (GLMs) are widely studied to deal with complex response variables. For the analysis of categorical dependent variables with more than two response categories, multivariate GLMs are presented to build the relationship between this polytomous response and a set of regressors. Traditional variable selection approaches have been proposed for the multivariate GLM with a canonical link function when the number of parameters is fixed in the literature. However, in many model selection problems, the number of parameters may be large and grow with the sample size. In this paper, we present a new selection criterion to the model with a diverging number of parameters. Under suitable conditions, the criterion is shown to be model selection consistent. A simulation study and a real data analysis are conducted to support theoretical findings.     

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.     

Risk-reducing shrinkage estimation for generalized linear models

Summary.  Empirical Bayes techniques for normal theory shrinkage estimation are extended to generalized linear models in a manner retaining the original spirit of shrinkage estimation, which is to reduce risk. The investigation identifies two classes of simple, all-purpose prior distributions, which supplement such non-informative priors as Jeffreys's prior with mechanisms for risk reduction. One new class of priors is motivated as optimizers of a core component of asymptotic risk. The methodology is evaluated in a numerical exploration and application to an existing data set.     

Local influence for generalized linear mixed models

The authors describe a method for assessing model inadequacy in maximum likelihood estimation of a generalized linear mixed model. They treat the latent random effects in the model as missing data and develop the influence analysis on the basis of a Q‐function which is associated with the conditional expectation of the complete‐data log‐likelihood function in the EM algorithm. They propose a procedure to detect influential observations in six model perturbation schemes. They also illustrate their methodology in a hypothetical situation and in two real cases.     

Collinearity in generalized linear models

This paper defines collinearity for generalized linear models (GLMs), investigates its consequences and proposes diagnostic criteria. The relationship between collinearity in GLMs and standard linear models (SLMs) is explored and bounds which relate the degree of collinearity in these two models are given. Estimation based on ridge methods is discussed.     

Optimal allocation for linear combinations of linear predictors in generalized linear models

The problem of the allocation of experimental units to experimental groups is studied within the context of generalized linear models. Optimal designs for the estimation of linear combinations of linear predictors are characterized, using concepts from the theory of optimal design. If there is only one linear combination of interest, then the D-optimal allocation is equivalent to the well-known Neyman allocation of subsamples in stratified sampling. However, if the number of linear combinations equals the number of design points, or experimental groups, then the equal replication of all design points is D-optimal. For cases in between, there are no easily accessible general solutions to the problem, although some particular cases are solved, including: i estimation of the n- 1 possible comparisons with a control group in an n-point, one-factor design; and ii estimation of 2 one or two of the four natural parameters of a 2 factorial design. The A-optimal allocations are determined in general.     

Sequential D-optimal designs for generalized linear mixed models

We discuss the construction of D-optimal sequential designs for the analysis of longitudinal data or repeated measurements using generalized linear mixed models (GLMMs). We investigate the performance of the design through a simulation study, which indicates that the proposed design can be very successful in improving the efficiency of the ML estimators in GLMMs relative to some common competitors. Our simulations also suggest that the usual normal-theory inference procedures remain valid under the sequential sampling schemes. We also present an example using real data obtained from a clinical study.     

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.     

Comparison of designs for multivariate generalized linear models

The purpose of this paper is to discuss response surface designs for multivariate generalized linear models (GLMs). Such models are considered whenever several response variables can be measured for each setting of a group of control variables, and the response variables are adequately represented by GLMs. The mean-squared error of prediction (MSEP) matrix is used to assess the quality of prediction associated with a given design. The MSEP incorporates both the prediction variance and the prediction bias, which results from using maximum likelihood estimates of the parameters of the fitted linear predictor. For a given design, quantiles of a scalar-valued function of the MSEP are obtained within a certain region of interest. The quantiles depend on the unknown parameters of the linear predictor. The dispersion of these quantiles over the space of the unknown parameters is determined and then depicted by the so-called quantile dispersion graphs. An application of the proposed methodology is presented using the special case of the bivariate binary distribution.     

Overlapping group lasso for high-dimensional generalized linear models

Abstract

Structured sparsity has recently been a very popular technique to deal with the high-dimensional data. In this paper, we mainly focus on the theoretical problems for the overlapping group structure of generalized linear models (GLMs). Although the overlapping group lasso method for GLMs has been widely applied in some applications, the theoretical properties about it are still unknown. Under some general conditions, we presents the oracle inequalities for the estimation and prediction error of overlapping group Lasso method in the generalized linear model setting. Then, we apply these results to the so-called Logistic and Poisson regression models. It is shown that the results of the Lasso and group Lasso procedures for GLMs can be recovered by specifying the group structures in our proposed method. The effect of overlap and the performance of variable selection of our proposed method are both studied by numerical simulations. Finally, we apply our proposed method to two gene expression data sets: the p53 data and the lung cancer data.