Multivariate models for correlated count data

In this study, we deal with the problem of overdispersion beyond extra zeros for a collection of counts that can be correlated. Poisson, negative binomial, zero-inflated Poisson and zero-inflated negative binomial distributions have been considered. First, we propose a multivariate count model in which all counts follow the same distribution and are correlated. Then we extend this model in a sense that correlated counts may follow different distributions. To accommodate correlation among counts, we have considered correlated random effects for each individual in the mean structure, thus inducing dependency among common observations to an individual. The method is applied to real data to investigate variation in food resources use in a species of marsupial in a locality of the Brazilian Cerrado biome.     

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.     

A review of the CTP distribution: a comparison with other over- and underdispersed count data models

The complex triparametric Pearson (CTP) distribution is a flexible model belonging to the Gaussian hypergeometric family that can account for over- and underdispersion. However, despite its good properties, not much attention has been paid to it. So, we revive the CTP comparing it with some well-known distributions that cope with overdispersion (negative binomial, generalized Poisson and univariate generalized Waring) as well as underdispersion (Conway–Maxwell–Poisson (CMP) and hyper-Poisson (HP)). We make a simulation study that reveals the performance of the CTP and shows that it has its own space among count data models. In this sense, we also explore some overdispersed datasets which seem to be more appropriately modelled by the CTP than by other usual models. Moreover, we include two underdispersed examples to illustrate that the CTP can provide similar fits to the CMP or HP (sometimes even more accurate) without the computational problems of these models.     

Comparison of methods for analyzing binary repeated measures data: A simulation-based study (comparison of methods for binary repeated measures)

In this study, some methods suggested for binary repeated measures, namely, Weighted Least Squares (WLS), Generalized Estimating Equations (GEE), and Generalized Linear Mixed Models (GLMM) are compared with respect to power, type 1 error, and properties of estimates. The results indicate that with adequate sample size, no missing data, the only covariate being time effect, and a relatively limited number of time points, the WLS method performs well. The GEE approach performs well only for large sample sizes. The GLMM method is satisfactory with respect to type I error, but its estimates have poorer properties than the other methods.     

A comparison of univariate and multivariate multilevel models for repeated measures of use of antenatal care in Uttar Pradesh

Summary.  We compare two different multilevel modelling approaches to the analysis of repeated measures data to assess the effect of mother level characteristics on women's use of prenatal care services in Uttar Pradesh, India. We apply univariate multilevel models to our data and find that the model assumptions are severely violated and the parameter estimates are not stable, particularly for the mother level random effect. To overcome this we apply a multivariate multilevel model. The correlation structure shows that, once the decision has been made regarding use of antenatal care by the mother for her first observed birth in the data, she does not tend to change this decision for higher order births.     

Selection of linear mixed-effects models for clustered data

We consider model selection for linear mixed-effects models with clustered structure, where conditional Kullback–Leibler (CKL) loss is applied to measure the efficiency of the selection. We estimate the CKL loss by substituting the empirical best linear unbiased predictors (EBLUPs) into random effects with model parameters estimated by maximum likelihood. Although the BLUP approach is commonly used in predicting random effects and future observations, selecting random effects to achieve asymptotic loss efficiency concerning CKL loss is challenging and has not been well studied. In this paper, we propose addressing this difficulty using a conditional generalized information criterion (CGIC) with two tuning parameters. We further consider a challenging but practically relevant situation where the number, $m$, of clusters does not go to infinity with the sample size. Hence the random-effects variances are not consistently estimable. We show that via a novel decomposition of the CKL risk, the CGIC achieves consistency and asymptotic loss efficiency, whether $m$ is fixed or increases to infinity with the sample size. We also conduct numerical experiments to illustrate the theoretical findings.     

Testing for heteroscedasticity of exponential correlation mixed-effects linear models based on M-estimation

Homogeneity of variance is a basic assumption in longitudinal data analysis. However, the assumption is not necessarily appropriate. In this paper, Fisher scoring method is applied to get M-estimator in the exponential correlation mixed-effects linear model. The score tests for heteroscedasticity and correlation coefficient based on M-estimator are then studied. Monte Carlo method is applied to investigate the properties of test statistics. At last, the methods and properties are illustrated by an actual data example.     

Mean-Squared errors of small area estimators under a multivariate linear model for repeated measures data

In this paper, we discuss the derivation of the first and second moments for the proposed small area estimators under a multivariate linear model for repeated measures data. The aim is to use these moments to estimate the mean-squared errors (MSE) for the predicted small area means as a measure of precision. At the first stage, we derive the MSE when the covariance matrices are known. At the second stage, a method based on parametric bootstrap is proposed for bias correction and for prediction error that reflects the uncertainty when the unknown covariance is replaced by its suitable estimator.     

Bayesian estimation and influence diagnostics of generalized partially linear mixed-effects models for longitudinal data

This paper develops a Bayesian approach to obtain the joint estimates of unknown parameters, nonparametric functions and random effects in generalized partially linear mixed models (GPLMMs), and presents three case deletion influence measures to identify influential observations based on the φ-divergence, Cook's posterior mean distance and Cook's posterior mode distance of parameters. Fisher's iterative scoring algorithm is developed to evaluate the posterior modes of parameters in GPLMMs. The first-order approximation to Cook's posterior mode distance is presented. The computationally feasible formulae for the φ-divergence diagnostic and Cook's posterior mean distance are given. Several simulation studies and an example are presented to illustrate our proposed methodologies.     

Variance component models for longitudinal count data with baseline information: epilepsy data revisited

Random effect models have often been used in longitudinal data analysis since they allow for association among repeated measurements due to unobserved heterogeneity. Various approaches have been proposed to extend mixed models for repeated count data to include dependence on baseline counts. Dependence between baseline counts and individual-specific random effects result in a complex form of the (conditional) likelihood. An approximate solution can be achieved ignoring this dependence, but this approach could result in biased parameter estimates and in wrong inferences. We propose a computationally feasible approach to overcome this problem, leaving the random effect distribution unspecified. In this context, we show how the EM algorithm for nonparametric maximum likelihood (NPML) can be extended to deal with dependence of repeated measures on baseline counts.     

A test of the missing data mechanism for repeated measures data

The occurrence of missing data is an often unavoidable consequence of repeated measures studies. Fortunately, multivariate general linear models such as growth curve models and linear mixed models with random effects have been well developed to analyze incomplete normally-distributed repeated measures data. Most statistical methods have assumed that the missing data occur at random. This assumption may include two types of missing data mechanism: missing completely at random (MCAR) and missing at random (MAR) in the sense of Rubin (1976). In this paper, we develop a test procedure for distinguishing these two types of missing data mechanism for incomplete normally-distributed repeated measures data. The proposed test is similar in spiril to the test of Park and Davis (1992). We derive the test for incomplete normally-distribrlted repeated measures data using linear mixed models. while Park and Davis (1992) cleirved thr test for incomplete repeatctl categorical data in the framework of Grizzle Starmer. and Koch (1969). Thr proposed procedure can be applied easily to any other multivariate general linear model which allow for missing data. The test is illustrated using the hip-replacernent patient.data from Crowder and Hand (1990).     

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.     

Goodness-of-fit tests of generalized linear mixed models for repeated ordinal responses

Categorical longitudinal data are frequently applied in a variety of fields, and are commonly fitted by generalized linear mixed models (GLMMs) and generalized estimating equations models. The cumulative logit is one of the useful link functions to deal with the problem involving repeated ordinal responses. To check the adequacy of the GLMMs with cumulative logit link function, two goodness-of-fit tests constructed by the unweighted sum of squared model residuals using numerical integration and bootstrap resampling technique are proposed. The empirical type I error rates and powers of the proposed tests are examined by simulation studies. The ordinal longitudinal studies are utilized to illustrate the application of the two proposed tests.     

A test for lack-of-fit of zero-inflated negative binomial models

When a count data set has excessive zero counts, nonzero counts are overdispersed, and the effect of a continuous covariate might be nonlinear, for analysis a semiparametric zero-inflated negative binomial (ZINB) regression model is proposed. The unspecified smooth functional form for the continuous covariate effect is approximated by a cubic spline. The semiparametric ZINB regression model is fitted by maximizing the likelihood function. The likelihood ratio procedure is used to evaluate the adequacy of a postulated parametric functional form for the continuous covariate effect. An extensive simulation study is conducted to assess the finite-sample performance of the proposed test. The practicality of the proposed methodology is demonstrated with data of a motorcycle survey of traffic regulations conducted in 2007 in Taiwan by the Ministry of Transportation and Communication.     

Modification of GEE1 and linear mixed-effects models for heteroscedastic longitudinal Gaussian data

A characterization of GLMs is given. Modification of the Gaussian GEE1, modified GEE1, was applied to heteroscedastic longitudinal data, to which linear mixed-effects models are usually applied. The modified GEE1 models scale multivariate data to homoscedastic data maintaining the correlation structure and apply usual GEE1 to homoscedastic data, which needs no-diagnostics for diagonal variances. Relationships among multivariate linear regression methods, ordinary/generalized LS, naïve/modified GEE1, and linear mixed-effects models were discussed. An application showed modified GEE1 gave most efficient parameter estimation. Correct specification of the main diagonals of heteroscedastic data variance appears to be more important for efficient mean parameter estimation.     

On implementation of a test for Kronecker product covariance structure for multivariate repeated measures data

Under the assumption of multivariate normality the likelihood ratio test is derived to test a hypothesis for Kronecker product structure on a covariance matrix in the context of multivariate repeated measures data. Although the proposed hypothesis testing can be computationally performed by indirect use of Proc Mixed of SAS, the Proc Mixed algorithm often fails to converge. We provide an alternative algorithm. The algorithm is illustrated with two real data sets. A simulation study is also conducted for the purpose of sample size consideration.     

A cautionary note on generalized linear models for covariance of unbalanced longitudinal data

Missing data in longitudinal studies can create enormous challenges in data analysis when coupled with the positive-definiteness constraint on a covariance matrix. For complete balanced data, the Cholesky decomposition of a covariance matrix makes it possible to remove the positive-definiteness constraint and use a generalized linear model setup to jointly model the mean and covariance using covariates (Pourahmadi, 2000). However, this approach may not be directly applicable when the longitudinal data are unbalanced, as coherent regression models for the dependence across all times and subjects may not exist. Within the existing generalized linear model framework, we show how to overcome this and other challenges by embedding the covariance matrix of the observed data for each subject in a larger covariance matrix and employing the familiar EM algorithm to compute the maximum likelihood estimates of the parameters and their standard errors. We illustrate and assess the methodology using real data sets and simulations.     

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.     

On the practical application of mixed effects models for repeated measures to clinical trial data

The use of mixed effects models for repeated measures (MMRM) for clinical trial analyses has recently gained broad support as a primary analysis methodology. Some questions of practical implementation detail remain, however. For example, whether and how to incorporate clinical trial data that is collected at nonprotocol‐specified timepoints or clinic visits has not been systematically studied. In this paper, we compare different methods for applying MMRM to trials wherein data is available at protocol‐specified timepoints, as well as nonprotocol‐specified timepoints due to patient early discontinuation. The methods under consideration included observed case MMRM, per protocol visits MMRM, interval last observation carried forward (LOCF) MMRM, and a hybrid of the per protocol visits and interval LOCF MMRM approaches. Simulation results reveal that the method that best controls the type I error rate is the per protocol visits method. This method is also associated with the least precision among the competing methods. Thus, in confirmatory clinical trials wherein control of type I error rates is critical, per protocol visits MMRM is recommended. However, in exploratory trials where strict type I error control is not as critical, one may prefer interval LOCF MMRM due to its increased precision. Points to consider with respect to both study design (e.g., assigning schedule of events) and subsequent analysis are offered. Copyright © 2012 John Wiley & Sons, Ltd.