A joint modeling and estimation method for multivariate longitudinal data with mixed types of responses to analyze physical activity data generated by accelerometers

A mixed effect model is proposed to jointly analyze multivariate longitudinal data with continuous, proportion, count, and binary responses. The association of the variables is modeled through the correlation of random effects. We use a quasi‐likelihood type approximation for nonlinear variables and transform the proposed model into a multivariate linear mixed model framework for estimation and inference. Via an extension to the EM approach, an efficient algorithm is developed to fit the model. The method is applied to physical activity data, which uses a wearable accelerometer device to measure daily movement and energy expenditure information. Our approach is also evaluated by a simulation study.     

Marginalized random effects models for multivariate longitudinal binary data

Generalized linear models with random effects are often used to explain the serial dependence of longitudinal categorical data. Marginalized random effects models (MREMs) permit likelihood‐based estimations of marginal mean parameters and also explain the serial dependence of longitudinal data. In this paper, we extend the MREM to accommodate multivariate longitudinal binary data using a new covariance matrix with a Kronecker decomposition, which easily explains both the serial dependence and time‐specific response correlation. A maximum marginal likelihood estimation is proposed utilizing a quasi‐Newton algorithm with quasi‐Monte Carlo integration of the random effects. Our approach is applied to analyze metabolic syndrome data from the Korean Genomic Epidemiology Study for Korean adults. Copyright © 2009 John Wiley & Sons, Ltd.     

Real longitudinal data analysis for real people: Building a good enough mixed model

Mixed effects models have become very popular, especially for the analysis of longitudinal data. One challenge is how to build a good enough mixed effects model. In this paper, we suggest a systematic strategy for addressing this challenge and introduce easily implemented practical advice to build mixed effects models. A general discussion of the scientific strategies motivates the recommended five‐step procedure for model fitting. The need to model both the mean structure (the fixed effects) and the covariance structure (the random effects and residual error) creates the fundamental flexibility and complexity. Some very practical recommendations help to conquer the complexity. Centering, scaling, and full‐rank coding of all the predictor variables radically improve the chances of convergence, computing speed, and numerical accuracy. Applying computational and assumption diagnostics from univariate linear models to mixed model data greatly helps to detect and solve the related computational problems. Applying computational and assumption diagnostics from the univariate linear models to the mixed model data can radically improve the chances of convergence, computing speed, and numerical accuracy. The approach helps to fit more general covariance models, a crucial step in selecting a credible covariance model needed for defensible inference. A detailed demonstration of the recommended strategy is based on data from a published study of a randomized trial of a multicomponent intervention to prevent young adolescents' alcohol use. The discussion highlights a need for additional covariance and inference tools for mixed models. The discussion also highlights the need for improving how scientists and statisticians teach and review the process of finding a good enough mixed model. Copyright © 2009 John Wiley & Sons, Ltd.     

Bayesian consensus clustering for multivariate longitudinal data

In clinical and epidemiological studies, there is a growing interest in studying the heterogeneity among patients based on longitudinal characteristics to identify subtypes of the study population. Compared to clustering a single longitudinal marker, simultaneously clustering multiple longitudinal markers allow additional information to be incorporated into the clustering process, which reveals co-existing longitudinal patterns and generates deeper biological insight. In the current study, we propose a Bayesian consensus clustering (BCC) model for multivariate longitudinal data. Instead of arriving at a single overall clustering, the proposed model allows each marker to follow marker-specific local clustering and these local clusterings are aggregated to find a global (consensus) clustering. To estimate the posterior distribution of model parameters, a Gibbs sampling algorithm is proposed. We apply our proposed model to the primary biliary cirrhosis study to identify patient subtypes that may be associated with their prognosis. We also perform simulation studies to compare the clustering performance between the proposed model and existing models under several scenarios. The results demonstrate that the proposed BCC model serves as a useful tool for clustering multivariate longitudinal data.     

A latent factor linear mixed model for high‐dimensional longitudinal data analysis

High‐dimensional longitudinal data involving latent variables such as depression and anxiety that cannot be quantified directly are often encountered in biomedical and social sciences. Multiple responses are used to characterize these latent quantities, and repeated measures are collected to capture their trends over time. Furthermore, substantive research questions may concern issues such as interrelated trends among latent variables that can only be addressed by modeling them jointly. Although statistical analysis of univariate longitudinal data has been well developed, methods for modeling multivariate high‐dimensional longitudinal data are still under development. In this paper, we propose a latent factor linear mixed model (LFLMM) for analyzing this type of data. This model is a combination of the factor analysis and multivariate linear mixed models. Under this modeling framework, we reduced the high‐dimensional responses to low‐dimensional latent factors by the factor analysis model, and then we used the multivariate linear mixed model to study the longitudinal trends of these latent factors. We developed an expectation–maximization algorithm to estimate the model. We used simulation studies to investigate the computational properties of the expectation–maximization algorithm and compare the LFLMM model with other approaches for high‐dimensional longitudinal data analysis. We used a real data example to illustrate the practical usefulness of the model. Copyright © 2013 John Wiley & Sons, Ltd.     

A new classification rule for incomplete doubly multivariate data using mixed effects model with performance comparisons on the imputed data

A mixed effects model, enhanced by a Kronecker product structure for the residual variance-covariance matrix, is used in conjunction with a discriminant analysis technique, to devise a new statistical classification method on incomplete doubly multivariate data. The proposed method is efficient in small scale clinical trials that use relatively few patients. The new classification method is also applied to multiply imputed data sets. The misclassification error rates (MERs) are compared in order to investigate the effectiveness of the new classification rule on an incomplete data set. The classification method is applied to a real data set. The error rates on the incomplete data set are found to be much less than the median error rate on the multiply imputed data sets. Non-parametric methods, such as kernel method and k-nearest neighbourhood method, are also applied to multiply imputed data sets. Results illustrating the advantages of the new classification method over classic non-parametric classification methods are presented.     

Joint modelling of multivariate longitudinal profiles: pitfalls of the random-effects approach

Due to its flexibility, the random-effects approach for the joint modelling of multivariate longitudinal profiles received a lot of attention in recent publications. In this approach different mixed models are joined by specifying a common distribution for their random-effects. Parameter estimates of this common distribution can then be used to evaluate the relation between the different responses. Using bivariate longitudinal measurements on pure-tone hearing thresholds, it will be shown that such a random-effects approach can yield misleading results for evaluating this relationship.     

The mixed model for the analysis of a repeated-measurement multivariate count data

Clustered overdispersed multivariate count data are challenging to model due to the presence of correlation within and between samples. Typically, the first source of correlation needs to be addressed but its quantification is of less interest. Here, we focus on the correlation between time points. In addition, the effects of covariates on the multivariate counts distribution need to be assessed. To fulfill these requirements, a regression model based on the Dirichlet-multinomial distribution for association between covariates and the categorical counts is extended by using random effects to deal with the additional clustering. This model is the Dirichlet-multinomial mixed regression model. Alternatively, a negative binomial regression mixed model can be deployed where the corresponding likelihood is conditioned on the total count. It appears that these two approaches are equivalent when the total count is fixed and independent of the random effects. We consider both subject-specific and categorical-specific random effects. However, the latter has a larger computational burden when the number of categories increases. Our work is motivated by microbiome data sets obtained by sequencing of the amplicon of the bacterial 16S rRNA gene. These data have a compositional structure and are typically overdispersed. The microbiome data set is from an epidemiological study carried out in a helminth-endemic area in Indonesia. The conclusions are as follows: time has no statistically significant effect on microbiome composition, the correlation between subjects is statistically significant, and treatment has a significant effect on the microbiome composition only in infected subjects who remained infected.     

A Bayesian zero-inflated beta-binomial model for longitudinal data with group-specific changepoints

Timeline followback (TLFB) is often used in addiction research to monitor recent substance use, such as the number of abstinent days in the past week. TLFB data usually take the form of binomial counts that exhibit overdispersion and zero inflation. Motivated by a 12-week randomized trial evaluating the efficacy of varenicline tartrate for smoking cessation among adolescents, we propose a Bayesian zero-inflated beta-binomial model for the analysis of longitudinal, bounded TLFB data. The model comprises a mixture of a point mass that accounts for zero inflation and a beta-binomial distribution for the number of days abstinent in the past week. Because treatment effects appear to level off during the study, we introduce random changepoints for each study group to reflect group-specific changes in treatment efficacy over time. The model also includes fixed and random effects that capture group- and subject-level slopes before and after the changepoints. Using the model, we can accurately estimate the mean trend for each study group, test whether the groups experience changepoints simultaneously, and identify critical windows of treatment efficacy. For posterior computation, we propose an efficient Markov chain Monte Carlo algorithm that relies on easily sampled Gibbs and Metropolis–Hastings steps. Our application shows that the varenicline group has a short-term positive effect on abstinence that tapers off after week 9.     

A frailty model approach for regression analysis of multivariate current status data

This paper discusses regression analysis of multivariate current status failure time data (The Statistical Analysis of Interval‐censoring Failure Time Data. Springer: New York, 2006), which occur quite often in, for example, tumorigenicity experiments and epidemiologic investigations of the natural history of a disease. For the problem, several marginal approaches have been proposed that model each failure time of interest individually (Biometrics 2000; 56 :940–943; Statist. Med. 2002; 21 :3715–3726). In this paper, we present a full likelihood approach based on the proportional hazards frailty model. For estimation, an Expectation Maximization (EM) algorithm is developed and simulation studies suggest that the presented approach performs well for practical situations. The approach is applied to a set of bivariate current status data arising from a tumorigenicity experiment. Copyright © 2009 John Wiley & Sons, Ltd.     

An autoregressive linear mixed effects model for the analysis of unequally spaced longitudinal data with dose-modification

The assessment of the dose-response relationship is important but not straightforward when the therapeutic agent is administered repeatedly with dose-modification in each patient and a continuous response is measured repeatedly. We recently proposed an autoregressive linear mixed effects model for such data in which the current response is regressed on the previous response, fixed effects, and random effects. The model represents profiles approaching each patient's asymptote, takes into account the past dose history, and provides a dose-response relationship of the asymptote as a summary measure. In an autoregressive model, intermittent missing data mean the missing values in previous responses as covariates. We previously provided the marginal (unconditional on the previous response) form of the proposed model to deal with intermittent missing data. Irregular timings of dose-modification or measurement can also be treated as equally spaced data with intermittent missing values by selecting an adequately small unit of time. The likelihood is, however, expressed by matrices whose sizes depend on the number of observations for a patient, and the computational burden is large. In this study, we propose a state space form of the autoregressive linear mixed effects model to calculate the marginal likelihood without using large matrices. The regression coefficients of the fixed effects can be concentrated out of the likelihood in this model by the same way of a linear mixed effects model. As an illustration of the approach, we analyzed immunologic data from a clinical trial for multiple sclerosis patients and estimated the dose-response curves for each patient and the population mean.     

An autoregressive linear mixed effects model for the analysis of longitudinal data which show profiles approaching asymptotes

In longitudinal data, a continuous response sometimes shows a profile approaching an asymptote. For such data, we propose a new class of models, autoregressive linear mixed effects models in which the current response is regressed on the previous response, fixed effects, and random effects. Asymptotes can shift depending on treatment groups, individuals, and so on, and can be modelled by fixed and random effects. We also propose error structures that are useful in practice. The estimation methods of linear mixed effects models can be used as long as there is no intermittent missing.     

Variable selection for joint models of multivariate longitudinal measurements and event time data

Joint modeling of longitudinal and survival data has attracted a great deal of attention. Some research has been undertaken to extend the joint model to incorporate multivariate longitudinal measurements recently. However, there is a lack of variable selection methods in the joint modeling of multivariate longitudinal measurements and survival time. In this article, we develop penalized likelihood methods for the selection of longitudinal features in the survival submodel. A multivariate linear mixed effect model is used to model multiple longitudinal processes where random intercepts and slopes serve as essential features of the trajectories. We introduce L1 penalty functions to select both random effects in the survival submodel and off‐diagonal elements in the covariance matrix of random effects. An estimation procedure is developed based on Laplace approximation. Our simulations demonstrate excellent selection properties of the proposed procedure. We apply our methods to explore the relationship between mortality and multiple longitudinal processes for end stage renal disease patients on hemodialysis. We find that lower levels of albumin, higher levels of neutrophil‐to‐lymphocyte ratio, and higher levels of interdialytic weight gain at the beginning of the follow‐up time, as well as decrease in predialysis systolic blood pressure and increase of neutrophil‐to‐lymphocyte ratio over time are associated with higher mortality hazard rates.     

A mixed-effects regression model for three-level ordinal response data

Three-level data occur frequently in behaviour and medical sciences. For example, in a multi-centre trial, subjects within a given site are randomly assigned to treatments and then studied over time. In this example, the repeated observations (level-1) are nested within subjects (level-2) who are nested within sites (level-3). Similarly, in twin studies, repeated measurements (level-1) are taken on each twin (level-2) within each twin pair (level-3). A three-level mixed-effects regression model is described here. Random effects at the second and third level are included in the model. Additionally, both proportional odds and non-proportional odds models are developed. The latter allows the effects of explanatory variables to vary across the cumulative logits of the model. A maximum marginal likelihood (MML) solution is described and Gauss-Hermite numerical quadrature is used to integrate over the distribution of random effects. The random effects are normally distributed in this instance. Features of this model are illustrated using data from a school-based smoking prevention trial and an Alzheimer's disease clinical trial.     

A longitudinal model for magnetic resonance imaging lesion count data in multiple sclerosis patients

Magnetic resonance imaging (MRI) data are routinely collected at multiple time points during phase 2 clinical trials in multiple sclerosis. However, these data are typically summarized into a single response for each patient before analysis. Models based on these summary statistics do not allow the exploration of the trade-off between numbers of patients and numbers of scans per patient or the development of optimal schedules for MRI scanning. To address these limitations, in this paper, we develop a longitudinal model to describe one MRI outcome: the number of lesions observed on an individual MRI scan. We motivate our choice of a mixed hidden Markov model based both on novel graphical diagnostic methods applied to five real data sets and on conceptual considerations. Using this model, we compare the performance of a number of different tests of treatment effect. These include standard parametric and nonparametric tests, as well as tests based on the new model. We conduct an extensive simulation study using data generated from the longitudinal model to investigate the parameters that affect test performance and to assess size and power. We determine that the parameters of the hidden Markov chain do not substantially affect the performance of the tests. Furthermore, we describe conditions under which likelihood ratio tests based on the longitudinal model appreciably outperform the standard tests based on summary statistics. These results establish that the new model is a valuable practical tool for designing and analyzing multiple sclerosis clinical trials.     

A new estimation with minimum trace of asymptotic covariance matrix for incomplete longitudinal data with a surrogate process

Missing data is a very common problem in medical and social studies, especially when data are collected longitudinally. It is a challenging problem to utilize observed data effectively. Many papers on missing data problems can be found in statistical literature. It is well known that the inverse weighted estimation is neither efficient nor robust. On the other hand, the doubly robust (DR) method can improve the efficiency and robustness. As is known, the DR estimation requires a missing data model (i.e., a model for the probability that data are observed) and a working regression model (i.e., a model for the outcome variable given covariates and surrogate variables). Because the DR estimating function has mean zero for any parameters in the working regression model when the missing data model is correctly specified, in this paper, we derive a formula for the estimator of the parameters of the working regression model that yields the optimally efficient estimator of the marginal mean model (the parameters of interest) when the missing data model is correctly specified. Furthermore, the proposed method also inherits the DR property. Simulation studies demonstrate the greater efficiency of the proposed method compared with the standard DR method. A longitudinal dementia data set is used for illustration. Copyright © 2013 John Wiley & Sons, Ltd.     

A linear mixed model for predicting a binary event from longitudinal data under random effects misspecification

The use of longitudinal data for predicting a subsequent binary event is often the focus of diagnostic studies. This is particularly important in obstetrics, where ultrasound measurements taken during fetal development may be useful for predicting various poor pregnancy outcomes. We propose a modeling framework for predicting a binary event from longitudinal measurements where a shared random effect links the two processes together. Under a Gaussian random effects assumption, the approach is simple to implement with standard statistical software. Using asymptotic and simulation results, we show that estimates of predictive accuracy under a Gaussian random effects distribution are robust to severe misspecification of this distribution. However, under some circumstances, estimates of individual risk may be sensitive to severe random effects misspecification. We illustrate the methodology with data from a longitudinal fetal growth study.     

A high-dimensional joint model for longitudinal outcomes of different nature

In repeated dose-toxicity studies, many outcomes are repeatedly measured on the same animal to study the toxicity of a compound of interest. This is only one example in which one is confronted with the analysis of many outcomes, possibly of a different type. Probably the most common situation is that of an amalgamation of continuous and categorical outcomes. A possible approach towards the joint analysis of two longitudinal outcomes of a different nature is the use of random-effects models (Models for Discrete Longitudinal Data. Springer Series in Statistics. Springer: New York, 2005). Although a random-effects model can easily be extended to jointly model many outcomes of a different nature, computational problems arise as the number of outcomes increases. To avoid maximization of the full likelihood expression, Fieuws and Verbeke (Biometrics 2006; 62:424-431) proposed a pairwise modeling strategy in which all possible pairs are modeled separately, using a mixed model, yielding several different estimates for the same parameters. These latter estimates are then combined into a single set of estimates. Also inference, based on pseudo-likelihood principles, is indirectly derived from the separate analyses. In this paper, we extend the approach of Fieuws and Verbeke (Biometrics 2006; 62:424-431) in two ways: the method is applied to different types of outcomes and the full pseudo-likelihood expression is maximized at once, leading directly to unique estimates as well as direct application of pseudo-likelihood inference. This is very appealing when interested in hypothesis testing. The method is applied to data from a repeated dose-toxicity study designed for the evaluation of the neurofunctional effects of a psychotrophic drug. The relative merits of both methods are discussed. Copyright (c) 2008 John Wiley & Sons, Ltd.     

A residuals-based transition model for longitudinal analysis with estimation in the presence of missing data

We propose a transition model for analysing data from complex longitudinal studies. Because missing values are practically unavoidable in large longitudinal studies, we also present a two-stage imputation method for handling general patterns of missing values on both the outcome and the covariates by combining multiple imputation with stochastic regression imputation. Our model is a time-varying auto-regression on the past innovations (residuals), and it can be used in cases where general dynamics must be taken into account, and where the model selection is important. The entire estimation process was carried out using available procedures in statistical packages such as SAS and S-PLUS. To illustrate the viability of the proposed model and the two-stage imputation method, we analyse data collected in an epidemiological study that focused on various factors relating to childhood growth. Finally, we present a simulation study to investigate the behaviour of our two-stage imputation procedure.     

Latent time‐varying factors in longitudinal analysis: a linear mixed hidden Markov model for heart rates

Longitudinal data are often segmented by unobserved time‐varying factors, which introduce latent heterogeneity at the observation level, in addition to heterogeneity across subjects. We account for this latent structure by a linear mixed hidden Markov model. It integrates subject‐specific random effects and Markovian sequences of time‐varying effects in the linear predictor. We propose an expectation?‐maximization algorithm for maximum likelihood estimation, based on data augmentation. It reduces to the iterative maximization of the expected value of a complete likelihood function, derived from an augmented dataset with case weights, alternated with weights updating. In a case study of the Survey on Stress Aging and Health in Russia, the model is exploited to estimate the influence of the observed covariates under unobserved time‐varying factors, which affect the cardiovascular activity of each subject during the observation period. Copyright © 2014 John Wiley & Sons, Ltd.