On familial Poisson mixed models with multi-dimensional random effects

When a generalized linear mixed model with multiple (two or more) sources of random effects is considered, the inferences may vary depending on the nature of the random effects. In this paper, we consider a familial Poisson mixed model where each of the count responses of a family are influenced by two independent unobservable familial random effects with two distinct components of dispersion. A generalized quasilikelihood (GQL) approach is discussed for the estimation of the dispersion components as well as the regression effects of the model. A simulation study is conducted to examine the relative performance of the GQL approach as opposed to a simpler method of moments. Furthermore, the GQL estimation methodology is illustrated by using health care utilization data that follow a Poisson mixed model with one component of dispersion and by using simulated asthma data that follow a Poisson mixed model with two sources of random effects with two distinct components of dispersion.     

Likelihood Inference for Unions of Interacting Discs

Abstract. This is probably the first paper which discusses likelihood inference for a random set using a germ‐grain model, where the individual grains are unobservable, edge effects occur and other complications appear. We consider the case where the grains form a disc process modelled by a marked point process, where the germs are the centres and the marks are the associated radii of the discs. We propose to use a recent parametric class of interacting disc process models, where the minimal sufficient statistic depends on various geometric properties of the random set, and the density is specified with respect to a given marked Poisson model (i.e. a Boolean model). We show how edge effects and other complications can be handled by considering a certain conditional likelihood. Our methodology is illustrated by analysing Peter Diggle's heather data set, where we discuss the results of simulation‐based maximum likelihood inference and the effect of specifying different reference Poisson models.     

Nonlinear mixed‐effects models with misspecified random‐effects distribution

Nonlinear mixed‐effects models are being widely used for the analysis of longitudinal data, especially from pharmaceutical research. They use random effects which are latent and unobservable variables so the random‐effects distribution is subject to misspecification in practice. In this paper, we first study the consequences of misspecifying the random‐effects distribution in nonlinear mixed‐effects models. Our study is focused on Gauss‐Hermite quadrature, which is now the routine method for calculation of the marginal likelihood in mixed models. We then present a formal diagnostic test to check the appropriateness of the assumed random‐effects distribution in nonlinear mixed‐effects models, which is very useful for real data analysis. Our findings show that the estimates of fixed‐effects parameters in nonlinear mixed‐effects models are generally robust to deviations from normality of the random‐effects distribution, but the estimates of variance components are very sensitive to the distributional assumption of random effects. Furthermore, a misspecified random‐effects distribution will either overestimate or underestimate the predictions of random effects. We illustrate the results using a real data application from an intensive pharmacokinetic study.     

Sampling from the posterior distribution in generalized linear mixed models

Generalized linear mixed models provide a unified framework for treatment of exponential family regression models, overdispersed data and longitudinal studies. These problems typically involve the presence of random effects and this paper presents a new methodology for making Bayesian inference about them. The approach is simulation-based and involves the use of Markov chain Monte Carlo techniques. The usual iterative weighted least squares algorithm is extended to include a sampling step based on the Metropolis–Hastings algorithm thus providing a unified iterative scheme. Non-normal prior distributions for the regression coefficients and for the random effects distribution are considered. Random effect structures with nesting required by longitudinal studies are also considered. Particular interests concern the significance of regression coefficients and assessment of the form of the random effects. Extensions to unknown scale parameters, unknown link functions, survival and frailty models are outlined.     

Robust linear mixed models for Small Area Estimation

Hierarchical models are popular in many applied statistics fields including Small Area Estimation. One well known model employed in this particular field is the Fay–Herriot model, in which unobservable parameters are assumed to be Gaussian. In Hierarchical models assumptions about unobservable quantities are difficult to check. For a special case of the Fay–Herriot model, Sinharay and Stern [2003. Posterior predictive model checking in Hierarchical models. J. Statist. Plann. Inference 111, 209–221] showed that violations of the assumptions about the random effects are difficult to detect using posterior predictive checks. In this present paper we consider two extensions of the Fay–Herriot model in which the random effects are assumed to be distributed according to either an exponential power (EP) distribution or a skewed EP distribution. We aim to explore the robustness of the Fay–Herriot model for the estimation of individual area means as well as the empirical distribution function of their ‘ensemble’. Our findings, which are based on a simulation experiment, are largely consistent with those of Sinharay and Stern as far as the efficient estimation of individual small area parameters is concerned. However, when estimating the empirical distribution function of the ‘ensemble’ of small area parameters, results are more sensitive to the failure of distributional assumptions.     

A Weibull Regression Model with Gamma Frailties for Multivariate Survival Data

Frequently in the analysis of survival data, survival times within the same group are correlated due to unobserved co-variates. One way these co-variates can be included in the model is as frailties. These frailty random block effects generate dependency between the survival times of the individuals which are conditionally independent given the frailty. Using a conditional proportional hazards model, in conjunction with the frailty, a whole new family of models is introduced. By considering a gamma frailty model, often the issue is to find an appropriate model for the baseline hazard function. In this paper a flexible baseline hazard model based on a correlated prior process is proposed and is compared with a standard Weibull model. Several model diagnostics methods are developed and model comparison is made using recently developed Bayesian model selection criteria. The above methodologies are applied to the McGilchrist and Aisbett (1991) kidney infection data and the analysis is performed using Markov Chain Monte Carlo methods. This revised version was published online in July 2006 with corrections to the Cover Date.     

Generalized Linear Factor Models: A New Local EM Estimation Algorithm

In this article, a general approach to latent variable models based on an underlying generalized linear model (GLM) with factor analysis observation process is introduced. We call these models Generalized Linear Factor Models (GLFM). The observations are produced from a general model framework that involves observed and latent variables that are assumed to be distributed in the exponential family. More specifically, we concentrate on situations where the observed variables are both discretely measured (e.g., binomial, Poisson) and continuously distributed (e.g., gamma). The common latent factors are assumed to be independent with a standard multivariate normal distribution. Practical details of training such models with a new local expectation-maximization (EM) algorithm, which can be considered as a generalized EM-type algorithm, are also discussed. In conjunction with an approximated version of the Fisher score algorithm (FSA), we show how to calculate maximum likelihood estimates of the model parameters, and to yield inferences about the unobservable path of the common factors. The methodology is illustrated by an extensive Monte Carlo simulation study and the results show promising performance.     

Generalized Score Test of Homogeneity Based on Correlated Random Effects Models

We present a family of tests based on correlated random effects models which provides a synthesis and a generalization of recent work on homogeneity testing. In these models each subject has a particular random effect, but the random effects between subjects are correlated. We derive the general form of the score statistic for testing that the random effects have a variance equal to 0. We apply this result to both parametric and semi-parametric models. In both cases we show that under certain conditions the score statistic has an asymptotic normal distribution. We consider several applications of this theory, including overdispersion, heterogeneity between groups, spatial correlations and genetic linkage.     

Joint modeling of hierarchically clustered and overdispersed non‐gaussian continuous outcomes for comet assay data

Multivariate longitudinal or clustered data are commonly encountered in clinical trials and toxicological studies. Typically, there is no single standard endpoint to assess the toxicity or efficacy of the compound of interest, but co‐primary endpoints are available to assess the toxic effects or the working of the compound. Modeling the responses jointly is thus appealing to draw overall inferences using all responses and to capture the association among the responses. Non‐Gaussian outcomes are often modeled univariately using exponential family models. To accommodate both the overdispersion and hierarchical structure in the data, Molenberghs et al. A family of generalized linear models for repeated measures with normal and conjugate random effects. Statistical Science 2010; 25:325–347 proposed using two separate sets of random effects. This papers considers a model for multivariate data with hierarchically clustered and overdispersed non‐Gaussian data. Gamma random effect for the over‐dispersion and normal random effects for the clustering in the data are being used. The two outcomes are jointly analyzed by assuming that the normal random effects for both endpoints are correlated. The association structure between the response is analytically derived. The fit of the joint model to data from a so‐called comet assay are compared with the univariate analysis of the two outcomes. Copyright © 2012 John Wiley & Sons, Ltd.     

Non-mixture cure correlated frailty models in Bayesian approach

In this article, we develop a Bayesian approach for the estimation of two cure correlated frailty models that have been extended to the cure frailty models introduced by Yin [34]. We used the two different type of frailty with bivariate log-normal distribution instead of gamma distribution. A likelihood function was constructed based on a piecewise exponential distribution function. The model parameters were estimated by the Markov chain Monte Carlo method. The comparison of models is based on the Cox correlated frailty model with log-normal distribution. A real data set of bilateral corneal graft rejection was used to compare these models. The results of this data, based on deviance information criteria, showed the advantage of the proposed models.     

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.     

Influence diagnostics in heteroscedastic and/or autoregressive nonlinear elliptical models for correlated data

In this paper, we propose nonlinear elliptical models for correlated data with heteroscedastic and/or autoregressive structures. Our aim is to extend the models proposed by Russo et al. 22 by considering a more sophisticated scale structure to deal with variations in data dispersion and/or a possible autocorrelation among measurements taken throughout the same experimental unit. Moreover, to avoid the possible influence of outlying observations or to take into account the non-normal symmetric tails of the data, we assume elliptical contours for the joint distribution of random effects and errors, which allows us to attribute different weights to the observations. We propose an iterative algorithm to obtain the maximum-likelihood estimates for the parameters and derive the local influence curvatures for some specific perturbation schemes. The motivation for this work comes from a pharmacokinetic indomethacin data set, which was analysed previously by Bocheng and Xuping 1 under normality.     

A frailty modeling approach for parental effects in animal breeding

Survival models involving frailties are commonly applied in studies where correlated event time data arise due to natural or artificial clustering. In this paper we present an application of such models in the animal breeding field. Specifically, a mixed survival model with a multivariate correlated frailty term is proposed for the analysis of data from over 3611 Brazilian Nellore cattle. The primary aim is to evaluate parental genetic effects on the trait length in days that their progeny need to gain a commercially specified standard weight gain. This trait is not measured directly but can be estimated from growth data. Results point to the importance of genetic effects and suggest that these models constitute a valuable data analysis tool for beef cattle breeding.     

Estimation of Dynamic Bivariate Mixture Models

This note compares a Bayesian Markov chain Monte Carlo approach implemented by Watanabe with a maximum likelihood ML approach based on an efficient importance sampling procedure to estimate dynamic bivariate mixture models. In these models, stock price volatility and trading volume are jointly directed by the unobservable number of price-relevant information arrivals, which is specified as a serially correlated random variable. It is shown that the efficient importance sampling technique is extremely accurate and that it produces results that differ significantly from those reported by Watanabe.     

Estimation procedures with delayed observations

The statistical model is considered in which the collection of data from several independent populations is available only at random times determined by order statistics of lifetimes of a given number of objects. Each of the populations is distributed according to a general multiparameter exponential family. The problem is to estimate the mean value vector parameter of the multiparameter exponential family of distributions of the forthcoming observations. Under the loss function involving a weighted squared error loss, the cost proportional to the events appeared and a cost of observing the process, a class of optimal sequential procedures is established. The procedures are derived in two situations: when the lifetime distribution is completely known and in the case when it is unknown but assumed to belong to an exponential subfamily with an unknown failure rate parameter.     

Testing for departures from nominal dispersion in generalized nonlinear models with varying dispersion and/or additive random effects

This paper discusses the tests for departures from nominal dispersion in the framework of generalized nonlinear models with varying dispersion and/or additive random effects. We consider two classes of exponential family distributions. The first is discrete exponential family distributions, such as Poisson, binomial, and negative binomial distributions. The second is continuous exponential family distributions, such as normal, gamma, and inverse Gaussian distributions. Correspondingly, we develop a unifying approach and propose several tests for testing for departures from nominal dispersion in two classes of generalized nonlinear models. The score test statistics are constructed and expressed in simple, easy to use, matrix formulas, so that the tests can easily be implemented using existing statistical software. The properties of test statistics are investigated through Monte Carlo simulations.     

A note on generating correlated matched-pair binary data through conditional linear family

Generating correlated binary data with specified marginal probabilities and correlation structure is often needed and useful in simulation studies to investigate the finite sample performance of statistical methods. Conditional linear family provides a powerful and flexible tool to generate correlated matched-pair binary data including the physician–patients and clustered match-pair data. To ensure the validity of the data generation process, constraints for parameters of the conditional linear family are needed. For the correlated matched-pair binary data with an exchangeable-type correlation structure, we derive the explicit expressions to check these constraints and it provides an efficient and convenient computational tool in validating the data generation process. The results are applied to check the constraints for two typical correlated matched-pair binary data.     

Time varying factor models with possibly strongly correlated noises

In factor models, noises are often assumed to be weakly correlated; otherwise, separation of factors from noises becomes difficult, if not impossible. This paper will address this problem. We utilize an econometric idea, the so called common correlated effects (CCE) to estimate time varying factor models. We first cross sectionally average the covariates and then project the responses to the space spanned by the averaged covariates. By doing so, noises are diminished while factors are distinguished. The advantages of our new estimators are two folds. First, the convergence rates of estimated factors and loadings are independent of cross sectional dimension. Second, our new estimators are robust to the correlation of noises. Hence our new estimators can, on one hand, separate market factors for the stock data set used in this paper even if noises exhibit strong correlations within industries due to industry-specific factors and on the other hand, avoid inappropriately absorbing industry-specific factors into market factors.     

Robustness of Parameter Estimation Procedures in Multilevel Models When Random Effects are MEP Distributed

In this paper we examine maximum likelihood estimation procedures in multilevel models for two level nesting structures. Usually, for fixed effects and variance components estimation, level-one error terms and random effects are assumed to be normally distributed. Nevertheless, in some circumstances this assumption might not be realistic, especially as concerns random effects. Thus we assume for random effects the family of multivariate exponential power distributions (MEP); subsequently, by means of Monte Carlo simulation procedures, we study robustness of maximum likelihood estimators under normal assumption when, actually, random effects are MEP distributed.     

The em algorithm for the quasi-likelihood regression model

The objective of this paper is to present a method which can accommodate certain types of missing data by using the quasi-likelihood function for the complete data. This method can be useful when we can make first and second moment assumptions only; in addition, it can be helpful when the EM algorithm applied to the actual likelihood becomes overly complicated. First we derive a loss function for the observed data using an exponential family density which has the same mean and variance structure of the complete data. This loss function is the counterpart of the quasi-deviance for the observed data. Then the loss function is minimized using the EM algorithm. The use of the EM algorithm guarantees a decrease in the loss function at every iteration. When the observed data can be expressed as a deterministic linear transformation of the complete data, or when data are missing completely at random, the proposed method yields consistent estimators. Examples are given for overdispersed polytomous data, linear random effects models, and linear regression with missing covariates. Simulation results for the linear regression model with missing covariates show that the proposed estimates are more efficient than estimates based on completely observed units, even when outcomes are bimodal or skewed.