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     

Goodness-of-fit tests for parametric models in censored regression

The authors propose a goodness-of-fit test for parametric regression models when the response variable is right-censored. Their test compares an estimation of the error distribution based on parametric residuals to another estimation relying on nonparametric residuals. They call on a bootstrap mechanism in order to approximate the critical values of tests based on Kolmogorov-Smirnov and Cramér-von Mises type statistics. They also present the results of Monte Carlo simulations and use data from a study about quasars to illustrate their work.     

A test of separate hypotheses for comparing linear mixed models with non nested fixed effects

As researchers increasingly rely on linear mixed models to characterize longitudinal data, there is a need for improved techniques for selecting among this class of models which requires specification of both fixed and random effects via a mean model and variance-covariance structure. The process is further complicated when fixed and/or random effects are non nested between models. This paper explores the development of a hypothesis test to compare non nested linear mixed models based on extensions of the work begun by Sir David Cox. We assess the robustness of this approach for comparing models containing correlated measures of body fat for predicting longitudinal cardiometabolic risk.     

Leverage analysis for linear mixed models

We consider a generalized leverage matrix useful for the identification of influential units and observations in linear mixed models and show how a decomposition of this matrix may be employed to identify high leverage points for both the marginal fitted values and the random effect component of the conditional fitted values. We illustrate the different uses of the two components of the decomposition with a simulated example as well as with a real data set.     

Goodness-of-fit measures of R 2 for repeated measures mixed effect models

Linear mixed effects model (LMEM) is efficient in modeling repeated measures longitudinal data. However, little research has been done in developing goodness-of-fit measures that can evaluate the models, particularly those that can be interpreted in an absolute sense without referencing a null model. This paper proposes three coefficient of determination (R ²) as goodness-of-fit measures for LMEM with repeated measures longitudinal data. Theorems are presented describing the properties of R ² and relationships between the R ² statistics. A simulation study was conducted to evaluate and compare the R ² along with other criteria from literature. Finally, we applied the proposed R ² to a real virologic response data of an HIV-patient cohort. We conclude that our proposed R ² statistics have more advantages than other goodness-of-fit measures in the literature, in terms of robustness to sample size, intuitive interpretation, well-defined range, and unnecessary to determine a null model.     

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.     

Score tests for zero inflation in generalized linear models

The authors develop score tests of goodness of fit for discrete generalized linear models against zero inflation. The binomial and Poisson models are treated as examples, and in the latter case the proposed test reduces to that of Broek (1995). Some simulation results and an illustrative example are presented.     

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.     

Robust inference in generalized linear models for longitudinal data

The author develops a robust quasi‐likelihood method, which appears to be useful for down‐weighting any influential data points when estimating the model parameters. He illustrates the computational issues of the method in an example. He uses simulations to study the behaviour of the robust estimates when data are contaminated with outliers, and he compares these estimates to those obtained by the ordinary quasi‐likelihood method.     

Approximate composite marginal likelihood inference in spatial generalized linear mixed models

Non-Gaussian spatial responses are usually modeled using spatial generalized linear mixed model with spatial random effects. The likelihood function of this model cannot usually be given in a closed form, thus the maximum likelihood approach is very challenging. There are numerical ways to maximize the likelihood function, such as Monte Carlo Expectation Maximization and Quadrature Pairwise Expectation Maximization algorithms. They can be applied but may in such cases be computationally very slow or even prohibitive. Gauss–Hermite quadrature approximation only suitable for low-dimensional latent variables and its accuracy depends on the number of quadrature points. Here, we propose a new approximate pairwise maximum likelihood method to the inference of the spatial generalized linear mixed model. This approximate method is fast and deterministic, using no sampling-based strategies. The performance of the proposed method is illustrated through two simulation examples and practical aspects are investigated through a case study on a rainfall data set.     

Diagnostics for repeated measurements in generalized linear mixed effects models

ABSTRACT

As there is an extensive body of research on diagnostics in regression models, various outlier detection methods have been developed. These methods have been extended to mixed effects models and generalized linear models, but there exist intrinsic drawbacks and limitations. This paper presents two-dimensional plots to identify discordant subjects and observations in generalized linear mixed effects models, displaying discordance in two directions. The sTudentized Residual Sum of Squares is not an extension of any regression tools but a new approach designed to efficiently reflect the characteristics of repeated measures. And this noteworthy clustering of outliers is identified in the plot. Applications to real-life examples are presented to illustrate the favorable/beneficial performance of the new tool.     

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.     

A Simulation-based Goodness-of-fit Test for Random Effects in Generalized Linear Mixed Models

Abstract.  The goodness-of-fit of the distribution of random effects in a generalized linear mixed model is assessed using a conditional simulation of the random effects conditional on the observations. Provided that the specified joint model for random effects and observations is correct, the marginal distribution of the simulated random effects coincides with the assumed random effects distribution. In practice, the specified model depends on some unknown parameter which is replaced by an estimate. We obtain a correction for this by deriving the asymptotic distribution of the empirical distribution function obtained from the conditional sample of the random effects. The approach is illustrated by simulation studies and data examples.     

Empirical likelihood for generalized linear models with missing responses

The paper uses the empirical likelihood method to study the construction of confidence intervals and regions for regression coefficients and response mean in generalized linear models with missing response. By using the inverse selection probability weighted imputation technique, the proposed empirical likelihood ratios are asymptotically chi-squared. Our approach is to directly calibrate the empirical likelihood ratio, which is called as a bias-correction method. Also, a class of estimators for the parameters of interest is constructed, and the asymptotic distributions of the proposed estimators are obtained. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths/areas of confidence intervals/regions. An example of a real data set is used for illustrating our methods.     

Goodness-of-fit tests for general linear models with covariates missed at random

In this paper, we consider a model checking problem for general linear models with randomly missing covariates. Two types of score type tests with inverse probability weight, which is estimated by parameter and nonparameter methods respectively, are proposed to this goodness of fit problem. The asymptotic properties of the test statistics are developed under the null and local alternative hypothesis. Simulation study is carried out to present the performance of the sizes and powers of the tests. We illustrate the proposed method with a data set on monozygotic twins.     

GEEs for repeated categorical responses based on generalized residuals

Clustered or correlated samples of categorical response data arise frequently in many fields of application. The method of generalized estimating equations (GEEs) introduced in Liang and Zeger [Longitudinal data analysis using generalized linear models, Biometrika 73 (1986), pp. 13–22] is often used to analyse this type of data. GEEs give consistent estimates of the regression parameters and their variance based upon the Pearson residuals. Park et al. [Alternative GEE estimation procedures for discrete longitudinal data, Comput. Stat. Data Anal. 28 (1998), pp. 243–256] considered a modification of the GEE approach using the Anscombe residual and the deviance residual. In this work, we propose to extend this idea to a family of generalized residuals. A wide simulation study is conducted for binary and Poisson correlated outcomes and also two numerical illustrations are presented.     

Spatial generalized linear mixed models in small area estimation

In survey sampling, policy decisions regarding the allocation of resources to sub‐groups of a population depend on reliable predictors of their underlying parameters. However, in some sub‐groups, called small areas due to small sample sizes relative to the population, the information needed for reliable estimation is typically not available. Consequently, data on a coarser scale are used to predict the characteristics of small areas. Mixed models are the primary tools in small area estimation (SAE) and also borrow information from alternative sources (e.g., previous surveys and administrative and census data sets). In many circumstances, small area predictors are associated with location. For instance, in the case of chronic disease or cancer, it is important for policy makers to understand spatial patterns of disease in order to determine small areas with high risk of disease and establish prevention strategies. The literature considering SAE with spatial random effects is sparse and mostly in the context of spatial linear mixed models. In this article, small area models are proposed for the class of spatial generalized linear mixed models to obtain small area predictors and corresponding second‐order unbiased mean squared prediction errors via Taylor expansion and a parametric bootstrap approach. The performance of the proposed approach is evaluated through simulation studies and application of the models to a real esophageal cancer data set from Minnesota, U.S.A. The Canadian Journal of Statistics 47: 426–437; 2019 © 2019 Statistical Society of Canada     

Testing for generalized linear mixed models with cluster correlated data under linear inequality constraints

Generalized linear mixed models (GLMMs) are often used for analyzing cluster correlated data, including longitudinal data and repeated measurements. Full unrestricted maximum likelihood (ML) approaches for inference on both fixed‐and random‐effects parameters in GLMMs have been extensively studied in the literature. However, parameter orderings or constraints may occur naturally in practice, and in such cases, the efficiency of a statistical method is improved by incorporating the parameter constraints into the ML estimation and hypothesis testing. In this paper, inference for GLMMs under linear inequality constraints is considered. The asymptotic properties of the constrained ML estimators and constrained likelihood ratio tests for GLMMs have been studied. Simulations investigated the empirical properties of the constrained ML estimators, compared to their unrestricted counterparts. An application to a recent survey on Canadian youth smoking patterns is also presented. As these survey data exhibit natural parameter orderings, a constrained GLMM has been considered for data analysis. The Canadian Journal of Statistics 40: 243–258; 2012 © 2012 Crown in the right of Canada     

Analysis of generalized linear mixed models via a stochastic approximation algorithm with Markov chain Monte-Carlo method

In recent years much effort has been devoted to maximum likelihood estimation of generalized linear mixed models. Most of the existing methods use the EM algorithm, with various techniques in handling the intractable E-step. In this paper, a new implementation of a stochastic approximation algorithm with Markov chain Monte Carlo method is investigated. The proposed algorithm is computationally straightforward and its convergence is guaranteed. A simulation and three real data sets, including the challenging salamander data, are used to illustrate the procedure and to compare it with some existing methods. The results indicate that the proposed algorithm is an attractive alternative for problems with a large number of random effects or with high dimensional intractable integrals in the likelihood function.