Robustness of the linear mixed model to misspecified error distribution

A simulation study is performed to investigate the robustness of the maximum likelihood estimator of fixed effects from a linear mixed model when the error distribution is misspecified. Inference for the fixed effects under the assumption of independent normally distributed errors with constant variance is shown to be robust when the errors are either non-gaussian or heteroscedastic, except when the error variance depends on a covariate included in the model with interaction with time. Inference is impaired when the errors are correlated. In the latter case, the model including a random slope in addition to the random intercept is more robust than the random intercept model. The use of Cholesky residuals and conditional residuals to evaluate the fit of a linear mixed model is also discussed.     

Influence diagnostics for random effect survival models: Application to a recurrent infection study for kidney patients on portable dialysis

In modeling multivariate failure time data, a class of survival model with random effects is applicable. It incorporates the random effect terms in the linear predictor and includes various random effect survival models as special cases, such as the random effect model assuming Cox's proportional hazards, with Weibull baseline hazards and with power family of transformation in the relative risk function. Residual maximum likelihood (REML) estimation of parameters is achieved by adopting the generalised linear mixed models (GLMM) approach. Accordingly, influence diagnostics are developed as sensitivity measures for the REML estimation of model parameters. A data set of recurrent infections of kidney patients on portable dialysis illustrates the usefulness of the influence diagnostics. A simulation study is carried out to examine the performance of the proposed influence diagnostics.     

A random effects four-part model, with application to correlated medical costs

In this paper we propose a four-part random effects model, with application to correlated medical cost data. Four joint equations are used to model respectively: (1) the probability of seeking medical treatment, (2) the probability of being hospitalized (conditional on seeking medical treatment), and the actual amount of (3) outpatient and (4) inpatient costs. Our model simultaneously takes account of the inter-temporal (or within-cluster) correlation of each patient and the cross-equation correlation of the four equations, by means of joint linear mixed models and generalized linear mixed models. The estimation is accomplished by the high-order Laplace approximation technique in Raudenbush et al. [Raudenbush, S.W., Yang, M., Yosef, M., 2000. Maximum likelihood for generalized linear models with nested random effects via high-order, multivariate Laplace approximation. Journal of Computational and Graphical Statistics 9, 141-157] and Olsen and Schafer [Olsen, M.K., Schafer, J.L., 2001. A two-part random effects model for semicontinuous longitudinal data. Journal of the American Statistical Association 96, 730-745]. Our model is used to analyze monthly medical costs of 1397 chronic heart failure patients from the clinical data repository (CDR) at the University of Virginia.     

Influence diagnostics for random effect survival models: Application to a recurrent infection study for kidney patients on portable dialysis

In modeling multivariate failure time data, a class of survival model with random effects is applicable. It incorporates the random effect terms in the linear predictor and includes various random effect survival models as special cases, such as the random effect model assuming Cox's proportional hazards, with Weibull baseline hazards and with power family of transformation in the relative risk function. Residual maximum likelihood (REML) estimation of parameters is achieved by adopting the generalised linear mixed models (GLMM) approach. Accordingly, influence diagnostics are developed as sensitivity measures for the REML estimation of model parameters. A data set of recurrent infections of kidney patients on portable dialysis illustrates the usefulness of the influence diagnostics. A simulation study is carried out to examine the performance of the proposed influence diagnostics.     

Estimation in the probit normal model for binary outcomes using the SAEM algorithm

Generalized linear mixed models (GLMM) form a very general class of random effects models for discrete and continuous responses in the exponential family. They are useful in a variety of applications. The traditional likelihood approach for GLMM usually involves high dimensional integrations which are computationally intensive. In this work, we investigate the case of binary outcomes analyzed under a two stage probit normal model with random effects. First, it is shown how ML estimates of the fixed effects and variance components can be computed using a stochastic approximation of the EM algorithm (SAEM). The SAEM algorithm can be applied directly, or in conjunction with a parameter expansion version of EM to speed up the convergence. A procedure is also proposed to obtain REML estimates of variance components and REML-based estimates of fixed effects. Finally an application to a real data set involving a clinical trial is presented, in which these techniques are compared to other procedures (penalized quasi-likelihood, maximum likelihood, Bayesian inference) already available in classical softwares (SAS Glimmix, SAS Nlmixed, WinBUGS), as well as to a Monte Carlo EM (MCEM) algorithm.     

Pairwise likelihood for the longitudinal mixed Rasch model

Inference in Generalized linear mixed models with multivariate random effects is often made cumbersome by the high-dimensional intractable integrals involved in the marginal likelihood. An inferential methodology based on the marginal pairwise likelihood approach is proposed. This method belonging to the broad class of composite likelihood involves marginal pairs probabilities of the responses which has analytical expression for the probit version of the model, from where we derived those of the logit version. The different results are illustrated with a simulation study and with an analysis of a real data from health-related quality of life.     

Semiparametric stochastic frontier models for clustered data

The mixed model approach to semiparametric regression is considered for stochastic frontier models, with focus on clustered data. Standard assumptions about the model component representing the inefficiency effect lead to a closed skew normal distribution for the response. Model parameters are estimated by a generalization of restricted maximum likelihood, and random effects are estimated by an orthodox best linear unbiased prediction procedure. The method is assessed by means of Monte Carlo studies, and illustrated by an empirical application on hospital productivity.     

A random effects epidemic-type aftershock sequence model

We consider an extension of the temporal epidemic-type aftershock sequence (ETAS) model with random effects as a special case of a well-known doubly stochastic self-exciting point process. The new model arises from a deterministic function that is randomly scaled by a nonnegative random variable, which is unobservable but assumed to follow either positive stable or one-parameter gamma distribution with unit mean. Both random effects models are of interest although the one-parameter gamma random effects model is more popular when modeling associated survival times. Our estimation is based on the maximum likelihood approach with marginalized intensity. The methods are shown to perform well in simulation experiments. When applied to an earthquake sequence on the east coast of Taiwan, the extended model with positive stable random effects provides a better model fit, compared to the original ETAS model and the extended model with one-parameter gamma random effects.     

Alternating imputation posterior estimation of models with crossed random effects

Generalized linear mixed models or latent variable models for categorical data are difficult to estimate if the random effects or latent variables vary at non-nested levels, such as persons and test items. Clayton and Rasbash (1999) suggested an Alternating Imputation Posterior (AIP) algorithm for approximate maximum likelihood estimation. For item response models with random item effects, the algorithm iterates between an item wing in which the item mean and variance are estimated for given person effects and a person wing in which the person mean and variance are estimated for given item effects. The person effects used for the item wing are sampled from the conditional posterior distribution estimated in the person wing and vice versa. Clayton and Rasbash (1999) used marginal quasi-likelihood (MQL) and penalized quasi-likelihood (PQL) estimation within the AIP algorithm, but this method has been shown to produce biased estimates in many situations, so we use maximum likelihood estimation with adaptive quadrature. We apply the proposed algorithm to the famous salamander mating data, comparing the estimates with many other methods, and to an educational testing dataset. We also present a simulation study to assess performance of the AIP algorithm and the Laplace approximation with different numbers of items and persons and a range of item and person variances.     

含时延和丢失数据不确定系统的FIR滤波器

针对状态空间模型中存在服从伯努利分布的时延和随机观测丢失的情况,基于极大似然法则,分别设计有限脉冲响应(finite impulse response, FIR)滤波器的慢速率批处理形式和快速率迭代形式.首先,将时延和数据丢失情况下的模型表述为服从伯努利分布的概率线性函数;然后,通过极大似然处理从而得到所提出极大似然FIR算法;最后,将在相同条件下的极大似然FIR估计、改进型卡尔曼滤波以及无偏FIR估计3种滤波方法进行对比,从估计误差、均方根误差和不确定性影响等角度进行比较分析.实验部分通过3-DOF直升机模型仿真,可发现所提出极大似然FIR估计方法在处理时延和数据丢失问题时更加有效,鲁棒性更高.     

Pairwise likelihood inference in spatial generalized linear mixed models

Spatial generalized linear mixed models are flexible models for a variety of applications, where spatially dependent and non-Gaussian random variables are observed. The focus is inference in spatial generalized linear mixed models for large data sets. Maximum likelihood or Bayesian Markov chain Monte Carlo approaches may in such cases be computationally very slow or even prohibitive. Alternatively, one may consider a composite likelihood, which is the product of likelihoods of subsets of data. In particular, a composite likelihood based on pairs of observations is adopted. In order to maximize the pairwise likelihood, a new expectation–maximization-type algorithm which uses numerical quadrature is introduced. The method is illustrated on simulated data and on data from air pollution effects for fish populations in Norwegian lakes. A comparison with alternative methods is given. The proposed algorithm is found to give reasonable parameter estimates and to be computationally efficient.     

A mixed effects least squares support vector machine model for classification of longitudinal data

A mixed effects least squares support vector machine (LS-SVM) classifier is introduced to extend the standard LS-SVM classifier for handling longitudinal data. The mixed effects LS-SVM model contains a random intercept and allows to classify highly unbalanced data, in the sense that there is an unequal number of observations for each case at non-fixed time points. The methodology consists of a regression modeling and a classification step based on the obtained regression estimates. Regression and classification of new cases are performed in a straightforward manner by solving a linear system. It is demonstrated that the methodology can be generalized to deal with multi-class problems and can be extended to incorporate multiple random effects. The technique is illustrated on simulated data sets and real-life problems concerning human growth.     

Stacked Laplace-EM algorithm for duration models with time-varying and random effects

An extension of the Cox proportional hazards model for clustered survival data is proposed. This allows both general random effects (frailties) and time-varying regression coefficients, the latter being smooth functions of time. The model is fitted using a mixed-model representation of penalized spline smoothing which offers a unified framework for estimation of the baseline hazard, the smooth effects and the random effects. The estimator is computed using a stacked laplace-EM (SLaEM) algorithm. More specifically, the smoothing parameters are integrated out in the log likelihood via a Laplace approximation. The approximation itself involves an integrated log-likelihood over the random cluster effects, for which the EM algorithm is used. A marginal Akaike information criterion is developed for selection among possible candidate models. The time-varying and mixed effects model is applied to unemployment data taken from the German Socio-Economic Panel. The duration of unemployment is modeled in a flexible way including smooth covariate effects and individual random effects.     

Inference in HIV dynamics models via hierarchical likelihood

HIV dynamical models are often based on non-linear systems of ordinary differential equations (ODE), which do not have an analytical solution. Introducing random effects in such models leads to very challenging non-linear mixed-effects models. To avoid the numerical computation of multiple integrals involved in the likelihood, a hierarchical likelihood (h-likelihood) approach, treated in the spirit of a penalized likelihood is proposed. The asymptotic distribution of the maximum h-likelihood estimators (MHLE) for fixed effects is given. The MHLE are slightly biased but the bias can be made negligible by using a parametric bootstrap procedure. An efficient algorithm for maximizing the h-likelihood is proposed. A simulation study, based on a classical HIV dynamical model, confirms the good properties of the MHLE. The method is applied to the analysis of a clinical trial.     

RE-EM trees: a data mining approach for longitudinal and clustered data

Longitudinal data refer to the situation where repeated observations are available for each sampled object. Clustered data, where observations are nested in a hierarchical structure within objects (without time necessarily being involved) represent a similar type of situation. Methodologies that take this structure into account allow for the possibilities of systematic differences between objects that are not related to attributes and autocorrelation within objects across time periods. A standard methodology in the statistics literature for this type of data is the mixed effects model, where these differences between objects are represented by so-called “random effects” that are estimated from the data (population-level relationships are termed “fixed effects,” together resulting in a mixed effects model). This paper presents a methodology that combines the structure of mixed effects models for longitudinal and clustered data with the flexibility of tree-based estimation methods. We apply the resulting estimation method, called the RE-EM tree, to pricing in online transactions, showing that the RE-EM tree is less sensitive to parametric assumptions and provides improved predictive power compared to linear models with random effects and regression trees without random effects. We also apply it to a smaller data set examining accident fatalities, and show that the RE-EM tree strongly outperforms a tree without random effects while performing comparably to a linear model with random effects. We also perform extensive simulation experiments to show that the estimator improves predictive performance relative to regression trees without random effects and is comparable or superior to using linear models with random effects in more general situations.     

A bivariate frailty model for events with a permanent survivor fraction and non-monotonic hazards; with an application to age at first maternity

For certain life cycle events a non-susceptible fraction of subjects will never undergo the event. In demographic applications, examples are provided by marriage and age at first maternity. A model for survival data allowing a permanent survival fraction, non-monotonic failure rates and unobserved frailty is considered here. Regressions are used to explain both the failure time and permanent survival mechanisms and additive correlated errors are included in the general linear models defining these regressions. A hierarchical Bayesian approach is adopted with likelihood conditional on the random frailty effects and a second stage prior defining the bivariate density of those effects. The gain in model fit, and potential effects on inference, from adding frailty is demonstrated in a case study application to age at first maternity in Germany.     

ECM-based maximum likelihood inference for multivariate linear mixed models with autoregressive errors

For the analysis of longitudinal data with multiple characteristics, we are devoted to providing additional tools for multivariate linear mixed models in which the errors are assumed to be serially correlated according to an autoregressive process. We present a computationally flexible ECM procedure for obtaining the maximum likelihood estimates of model parameters. A score test statistic for testing the existence of autocorrelation among within-subject errors of each characteristic is derived. The techniques for the estimation of random effects and the prediction of further responses given past repeated measures are also investigated. The methodology is illustrated through an application to a set of AIDS data and two small simulation studies.     

Nonignorable dropout models for longitudinal binary data with random effects: An application of Monte Carlo approximation through the Gibbs output

The analysis of longitudinal data with nonignorable dropout remains an active area in biostatistics research. Nonignorable dropout (ND) refers to the type of dropout when the probability of dropout depends on the missing observations at or after the time of dropout. Failure to account for such dependence may result in biased inference. Motivated by a methadone clinic data of longitudinal binary observations with dropouts, we propose a conditional first order autoregressive (AR1) logit model for the outcome measurements. The model is further extended to incorporate random effects in order to account for the population heterogeneity and intra-cluster correlation. The purposed models account for the dropout mechanism by a separate logit model in some covariates and missing outcomes for the binary dropout indicators. For model implementation, we proposed a likelihood approach through Monte Carlo approximation to the Gibbs output that evaluates the complicated likelihood function for the random effect ND model without tear. Finally simulation studies are performed to evaluate the biases on the parameter estimates of the outcome model for different dropout mechanisms.     

Quasi-Monte Carlo estimation in generalized linear mixed models

Generalized linear mixed models (GLMMs) are useful for modelling longitudinal and clustered data, but parameter estimation is very challenging because the likelihood may involve high-dimensional integrals that are analytically intractable. Gauss-Hermite quadrature (GHQ) approximation can be applied but is only suitable for low-dimensional random effects. Based on the Quasi-Monte Carlo (QMC) approximation, a heuristic approach is proposed to calculate the maximum likelihood estimates of parameters in the GLMM. The QMC points scattered uniformly on the high-dimensional integration domain are generated to replace the GHQ nodes. Compared to the GHQ approximation, the proposed method has many advantages such as its affordable computation, good approximation and fast convergence. Comparisons to the penalized quasi-likelihood estimation and Gibbs sampling are made using a real dataset and a simulation study. The real dataset is the salamander mating dataset whose modelling involves six 20-dimensional intractable integrals in the likelihood.     

Quasi-Monte Carlo estimation in generalized linear mixed models

Generalized linear mixed models (GLMMs) are useful for modelling longitudinal and clustered data, but parameter estimation is very challenging because the likelihood may involve high-dimensional integrals that are analytically intractable. Gauss–Hermite quadrature (GHQ) approximation can be applied but is only suitable for low-dimensional random effects. Based on the Quasi-Monte Carlo (QMC) approximation, a heuristic approach is proposed to calculate the maximum likelihood estimates of parameters in the GLMM. The QMC points scattered uniformly on the high-dimensional integration domain are generated to replace the GHQ nodes. Compared to the GHQ approximation, the proposed method has many advantages such as its affordable computation, good approximation and fast convergence. Comparisons to the penalized quasi-likelihood estimation and Gibbs sampling are made using a real dataset and a simulation study. The real dataset is the salamander mating dataset whose modelling involves six 20-dimensional intractable integrals in the likelihood.