Non-linear random effects model for multivariate responses with missing data

A sparse factor model for clustering high-dimensional longitudinal data

Recent advances in engineering technologies have enabled the collection of a large number of longitudinal features. This wealth of information presents unique opportunities for researchers to investigate the complex nature of diseases and uncover underlying disease mechanisms. However, analyzing such kind of data can be difficult due to its high dimensionality, heterogeneity and computational challenges. In this article, we propose a Bayesian nonparametric mixture model for clustering high-dimensional mixed-type (eg, continuous, discrete and categorical) longitudinal features. We employ a sparse factor model on the joint distribution of random effects and the key idea is to induce clustering at the latent factor level instead of the original data to escape the curse of dimensionality. The number of clusters is estimated through a Dirichlet process prior. An efficient Gibbs sampler is developed to estimate the posterior distribution of the model parameters. Analysis of real and simulated data is presented and discussed. Our study demonstrates that the proposed model serves as a useful analytical tool for clustering high-dimensional longitudinal data.

Joint clustering multiple longitudinal features: A comparison of methods and software packages with practical guidance

Clustering longitudinal features is a common goal in medical studies to identify distinct disease developmental trajectories. Compared to clustering a single longitudinal feature, integrating multiple longitudinal features allows additional information to be incorporated into the clustering process, which may reveal co-existing longitudinal patterns and generate deeper biological insight. Despite its increasing importance and popularity, there is limited practical guidance for implementing cluster analysis approaches for multiple longitudinal features and evaluating their comparative performance in medical datasets. In this paper, we provide an overview of several commonly used approaches to clustering multiple longitudinal features, with an emphasis on application and implementation through R software. These methods can be broadly categorized into two categories, namely model-based (including frequentist and Bayesian) approaches and algorithm-based approaches. To evaluate their performance, we compare these approaches using real-life and simulated datasets. These results provide practical guidance to applied researchers who are interested in applying these approaches for clustering multiple longitudinal features. Recommendations for applied researchers and suggestions for future research in this area are also discussed.

Flexible modeling of multiple nonlinear longitudinal trajectories with censored and non-ignorable missing outcomes

Multivariate nonlinear mixed-effects models (MNLMMs) have become a promising tool for analyzing multi-outcome longitudinal data following nonlinear trajectory patterns. However, such a classical analysis can be challenging due to censorship induced by detection limits of the quantification assay or non-response occurring when participants missed scheduled visits intermittently or discontinued participation. This article proposes an extension of the MNLMM approach, called the MNLMM-CM, by taking the censored and non-ignorable missing responses into account simultaneously. The non-ignorable missingness is described by the selection-modeling factorization to tackle the missing not at random mechanism. A Monte Carlo expectation conditional maximization algorithm coupled with the first-order Taylor approximation is developed for parameter estimation. The techniques for the calculation of standard errors of fixed effects, estimation of unobservable random effects, imputation of censored and missing responses and prediction of future values are also provided. The proposed methodology is motivated and illustrated by the analysis of a clinical HIV/AIDS dataset with censored RNA viral loads and the presence of missing CD4 and CD8 cell counts. The superiority of our method on the provision of more adequate estimation is validated by a simulation study.

Multivariate linear mixed models with censored and nonignorable missing outcomes, with application to AIDS studies

The analysis of multivariate longitudinal data could encounter some complications due to censorship induced by detection limits of the assay and nonresponse occurring when participants missed scheduled visits intermittently or discontinued participation. This paper establishes a generalization of the multivariate linear mixed model that can accommodate censored responses and nonignorable missing outcomes simultaneously. To account for the nonignorable missingness, the selection approach which decomposes the joint distribution as a marginal distribution for the primary outcome variables and a model describing the missing process conditional on the hypothetical complete data is used. A computationally feasible Monte Carlo expectation conditional maximization algorithm is developed for parameter estimation with the maximum likelihood (ML) method. Furthermore, a general information-based approach is presented to assess the variability of ML estimators. The techniques for the prediction of censored responses and imputation of missing outcomes are also discussed. The methodology is motivated and exemplified by a real dataset concerning HIV-AIDS clinical trials. A simulation study is conducted to examine the performance of the proposed method compared with other traditional approaches.

Repeated measures discriminant analysis using multivariate generalized estimation equations

Discriminant analysis procedures that assume parsimonious covariance and/or means structures have been proposed for distinguishing between two or more populations in multivariate repeated measures designs. However, these procedures rely on the assumptions of multivariate normality which is not tenable in multivariate repeated measures designs which are characterized by binary, ordinal, or mixed types of response distributions. This study investigates the accuracy of repeated measures discriminant analysis (RMDA) based on the multivariate generalized estimating equations (GEE) framework for classification in multivariate repeated measures designs with the same or different types of responses repeatedly measured over time. Monte Carlo methods were used to compare the accuracy of RMDA procedures based on GEE, and RMDA based on maximum likelihood estimators (MLE) under diverse simulation conditions, which included number of repeated measure occasions, number of responses, sample size, correlation structures, and type of response distribution. RMDA based on GEE exhibited higher average classification accuracy than RMDA based on MLE especially in multivariate non-normal distributions. Three repeatedly measured responses namely severity of epilepsy, current number of anti-epileptic drugs, and parent-reported quality of life in children with epilepsy were used to demonstrate the application of these procedures.

Multivariate-t linear mixed models with censored responses, intermittent missing values and heavy tails

Multivariate longitudinal data arisen in medical studies often exhibit complex features such as censored responses, intermittent missing values, and atypical or outlying observations. The multivariate-t linear mixed model (MtLMM) has been recognized as a powerful tool for robust modeling of multivariate longitudinal data in the presence of potential outliers or fat-tailed noises. This paper presents a generalization of MtLMM, called the MtLMM-CM, to properly adjust for censorship due to detection limits of the assay and missingness embodied within multiple outcome variables recorded at irregular occasions. An expectation conditional maximization either (ECME) algorithm is developed to compute parameter estimates using the maximum likelihood (ML) approach. The asymptotic standard errors of the ML estimators of fixed effects are obtained by inverting the empirical information matrix according to Louis' method. The techniques for the estimation of random effects and imputation of missing responses are also investigated. The proposed methodology is illustrated on two real-world examples from HIV-AIDS studies and a simulation study under a variety of scenarios.

A five-year longitudinal study of the relation between end-stage kidney disease as the outcomes

Background

Patients with end-stage kidney disease (ESKD) are required to undergo consecutive time-based blood and biochemical tests to determine the progression of the disease according to changes in their blood and biochemical data. This study employed a random intercept model to investigate whether time-based blood and biochemical data present any notable clinical meaning that can be used to track disease progression.

Methods

This study conducted a retrospective analysis on the dialytic data of 148 patients with ESKD, who received hemodialysis between January 2005 and December 2015. The patients were all at least 20 years old, and the data used included patient demographic information and results for at least 60 blood and biochemical tests. A random intercept model was used to analyze the relationships among blood and biochemical test results, explanatory variables of patient comorbidities, and time.

Results

The age range of patients was between 33 and 98 years, with an average of 66.1 years and those over 65 years old comprising 51.3% (n = 76) of the total. Furthermore, hypertension was found to be the most common comorbidity among patients (87.2%, n = 129), followed by anemia (48.6%, n = 72), diabetes (47.3%, n = 70), dyslipidemia (19.6%, n = 29), and peptic ulcer (19.6%, n = 29). Coronary atherosclerotic heart disease is a comorbidity that can serve as a strong and independent marker for prognosis in patients with ESKD. Serum creatinine level can serve as an alternative indicator because patients with ESKD and comorbid diabetes may exhibit increased creatinine levels.

Conclusions

The results of a parameter estimation for longitudinal data analysis suggested that comorbidity and time were critical variables influencing blood and biochemical test results. Furthermore, WBC and HBC, HCT, albumin, protein, and creatinine levels were recognized as variables of critical significance. The results obtained in this study indicate that multimorbidity increases the treatment burden on patients, leading to polypharmacy. For this reason, comprehensive care and treatment of ESKD cannot rely solely on data from one single time point; instead, longitudinal analysis and other data that can affect patient prognosis must also be considered.

Evolution of association between renal and liver functions while awaiting heart transplant: An application using a bivariate multiphase nonlinear mixed effects model

In many longitudinal follow-up studies, we observe more than one longitudinal outcome. Impaired renal and liver functions are indicators of poor clinical outcomes for patients who are on mechanical circulatory support and awaiting heart transplant. Hence, monitoring organ functions while waiting for heart transplant is an integral part of patient management. Longitudinal measurements of bilirubin can be used as a marker for liver function and glomerular filtration rate for renal function. We derive an approximation to evolution of association between these two organ functions using a bivariate nonlinear mixed effects model for continuous longitudinal measurements, where the two submodels are linked by a common distribution of time-dependent latent variables and a common distribution of measurement errors.

Multivariate longitudinal data analysis with censored and intermittent missing responses

The multivariate linear mixed model (MLMM) has emerged as an important analytical tool for longitudinal data with multiple outcomes. However, the analysis of multivariate longitudinal data could be complicated by the presence of censored measurements because of a detection limit of the assay in combination with unavoidable missing values arising when subjects miss some of their scheduled visits intermittently. This paper presents a generalization of the MLMM approach, called the MLMM-CM, for a joint analysis of the multivariate longitudinal data with censored and intermittent missing responses. A computationally feasible expectation maximization-based procedure is developed to carry out maximum likelihood estimation within the MLMM-CM framework. Moreover, the asymptotic standard errors of fixed effects are explicitly obtained via the information-based method. We illustrate our methodology by using simulated data and a case study from an AIDS clinical trial. Experimental results reveal that the proposed method is able to provide more satisfactory performance as compared with the traditional MLMM approach.

Error-rate estimation in discriminant analysis of non-linear longitudinal data: A comparison of resampling methods

Consider longitudinal observations across different subjects such that the underlying distribution is determined by a non-linear mixed-effects model. In this context, we look at the misclassification error rate for allocating future subjects using cross-validation, bootstrap algorithms (parametric bootstrap, leave-one-out, .632 and [Formula: see text]), and bootstrap cross-validation (which combines the first two approaches), and conduct a numerical study to compare the performance of the different methods. The simulation and comparisons in this study are motivated by real observations from a pregnancy study in which one of the main objectives is to predict normal versus abnormal pregnancy outcomes based on information gathered at early stages. Since in this type of studies it is not uncommon to have insufficient data to simultaneously solve the classification problem and estimate the misclassification error rate, we put special attention to situations when only a small sample size is available. We discuss how the misclassification error rate estimates may be affected by the sample size in terms of variability and bias, and examine conditions under which the misclassification error rate estimates perform reasonably well.

On estimating and testing associations between random coefficients from multivariate generalized linear mixed models of longitudinal outcomes

Different types of outcomes (e.g. binary, count, continuous) can be simultaneously modeled with multivariate generalized linear mixed models by assuming: (1) same or different link functions, (2) same or different conditional distributions, and (3) conditional independence given random subject effects. Others have used this approach for determining simple associations between subject-specific parameters (e.g. correlations between slopes). We demonstrate how more complex associations (e.g. partial regression coefficients between slopes adjusting for intercepts, time lags of maximum correlation) can be estimated. Reparameterizing the model to directly estimate coefficients allows us to compare standard errors based on the inverse of the Hessian matrix with more usual standard errors approximated by the delta method; a mathematical proof demonstrates their equivalence when the gradient vector approaches zero. Reparameterization also allows us to evaluate significance of coefficients with likelihood ratio tests and to compare this approach with more usual Wald-type t-tests and Fisher's z transformations. Simulations indicate that the delta method and inverse Hessian standard errors are nearly equivalent and consistently overestimate the true standard error. Only the likelihood ratio test based on the reparameterized model has an acceptable type I error rate and is therefore recommended for testing associations between stochastic parameters. Online supplementary materials include our medical data example, annotated code, and simulation details.

The nature of progression in Parkinson's disease: an application of non-linear, multivariate, longitudinal random effects modelling

Background

To date, statistical methods that take into account fully the non-linear, longitudinal and multivariate aspects of clinical data have not been applied to the study of progression in Parkinson’s disease (PD). In this paper, we demonstrate the usefulness of such methodology for studying the temporal and spatial aspects of the progression of PD. Extending this methodology further, we also explore the presymptomatic course of this disease.

Methods

Longitudinal Positron Emission Tomography (PET) measurements were collected on 78 PD patients, from 4 subregions on each side of the brain, using 3 different radiotracers. Non-linear, multivariate, longitudinal random effects modelling was applied to analyze and interpret these data.

Results

The data showed a non-linear decline in PET measurements, which we modelled successfully by an exponential function depending on two patient-related covariates duration since symptom onset and age at symptom onset. We found that the degree of damage was significantly greater in the posterior putamen than in the anterior putamen throughout the disease. We also found that over the course of the illness, the difference between the less affected and more affected sides of the brain decreased in the anterior putamen. Younger patients had significantly poorer measurements than older patients at the time of symptom onset suggesting more effective compensatory mechanisms delaying the onset of symptoms. Cautious extrapolation showed that disease onset had occurred some 8 to 17 years prior to symptom onset.

Conclusions

Our model provides important biological insights into the pathogenesis of PD, as well as its preclinical aspects. Our methodology can be applied widely to study many other chronic progressive diseases.

Multivariate t linear mixed models for irregularly observed multiple repeated measures with missing outcomes

Missing outcomes or irregularly timed multivariate longitudinal data frequently occur in clinical trials or biomedical studies. The multivariate t linear mixed model (MtLMM) has been shown to be a robust approach to modeling multioutcome continuous repeated measures in the presence of outliers or heavy-tailed noises. This paper presents a framework for fitting the MtLMM with an arbitrary missing data pattern embodied within multiple outcome variables recorded at irregular occasions. To address the serial correlation among the within-subject errors, a damped exponential correlation structure is considered in the model. Under the missing at random mechanism, an efficient alternating expectation-conditional maximization (AECM) algorithm is used to carry out estimation of parameters and imputation of missing values. The techniques for the estimation of random effects and the prediction of future responses are also investigated. Applications to an HIV-AIDS study and a pregnancy study involving analysis of multivariate longitudinal data with missing outcomes as well as a simulation study have highlighted the superiority of MtLMMs on the provision of more adequate estimation, imputation and prediction performances.

Longitudinal evolution of compensatory changes in striatal dopamine processing in Parkinson's disease

Parkinson's disease is a relentlessly progressive neurodegenerative disease. Breakdown of compensatory mechanisms influencing putaminal dopamine processing could contribute to the progressive motor symptoms. We studied a cohort of 78 subjects (at baseline) with sporadic Parkinson's disease and 35 healthy controls with multi-tracer positron emission tomography scans to investigate the evolution of adaptive mechanisms influencing striatal dopamine processing in Parkinson's disease progression. Presynaptic dopaminergic integrity was assessed with three radioligands: (i) [(11)C](±)dihydrotetrabenazine, to estimate the density of vesicular monoamine transporter type 2; (ii) [(11)C]d-threo-methylphenidate, to label the dopamine transporter; and (iii) 6-[(18)F]fluoro-L-DOPA, to assess the activity of aromatic amino acid decarboxylase and storage of 6-[(18)F]-fluorodopamine in synaptic vesicles. The subjects with Parkinson's disease and the healthy controls underwent positron emission tomography scans at the initial visit and after 4 and 8 years of follow-up. Non-linear multivariate regression analyses with random effects were utilized to model the longitudinal changes in tracer values in the putamen standardized relative to normal controls. We found evidence for possible upregulation of dopamine synthesis and downregulation of dopamine transporter in the more severely affected putamen in the early stage of Parkinson's disease. The standardized 6-[(18)F]fluoro-L-DOPA and [(11)C]d-threo-methylphenidate values tended to approach [(11)C](±)dihydrotetrabenazine values in the putamen in later stages of disease (i.e. for [(11)C](±)dihydrotetrabenazine values <25% of normal), when the rates of decline in the positron emission tomography measurements were similar for all the markers. Our data suggest that compensatory mechanisms decline as Parkinson's disease progresses. This breakdown of compensatory strategies in the putamen could contribute to the progression of motor symptoms in advanced disease.

Age-specific progression of nigrostriatal dysfunction in Parkinson's disease

OBJECTIVE

To investigate in vivo the impact of age on nigrostriatal dopamine dysfunction in Parkinson's disease (PD).

METHODS

PD patients (n = 78) and healthy control subjects (n = 35) underwent longitudinal positron emission tomography assessments using 3 presynaptic dopamine markers: (1) [¹¹C](±)dihydrotetrabenazine (DTBZ), to estimate the density of the vesicular monoamine transporter type 2; (2) [¹¹C]d-threo-methylphenidate, to estimate the density of the plasma membrane dopamine transporter; and (3) 6-[¹⁸F]-fluoro-L-dopa, to estimate the activity of the enzyme dopa-decarboxylase.

RESULTS

The study comprised 438 PD scans and 241 control scans (679 scans in total). At symptom onset, the loss of putamen DTBZ binding was substantially greater in younger compared to older PD patients (p = 0.015). Remarkably, however, the rate of progression of DTBZ binding loss was significantly slower in younger patients (p < 0.05). The estimated presymptomatic phase of the disease spanned more than 2 decades in younger patients, compared to 1 decade in older patients.

INTERPRETATION

Our results suggest that, compared to older patients, younger PD patients progress more slowly and are able to endure more damage to the dopaminergic system before the first motor symptoms appear. These observations suggest that younger PD patients have more efficient compensatory mechanisms.

Application of a joint multivariate longitudinal-survival analysis to examine the terminal decline hypothesis in the Swiss Interdisciplinary Longitudinal Study on the Oldest Old

In this work I aim at extending current knowledge on the terminal decline hypothesis by applying a joint multivariate longitudinal-survival analysis to the cognitive data of the Swiss Interdisciplinary Longitudinal Study on the Oldest Old. (In that study, 529 individuals between 79 and 85 years of age at study inception were assessed up to five times on a task of perceptual speed and one of verbal fluency.) I simultaneously estimated a multivariate, multilevel longitudinal model and a Weibull survival model to test whether individual performance and change in speed and fluency predict survival, controlling for retest effects, initial age, gender, overall health, socioeconomic status, and sensory functioning. Results revealed that age and performance level in fluency predicted survival, whereas level in speed and change in both cognitive variables did not. I discuss the relevance of fluency tasks in predicting mortality.

Discriminant analysis for longitudinal data with multiple continuous responses and possibly missing data

Multiple outcomes are often used to properly characterize an effect of interest. This article discusses model-based statistical methods for the classification of units into one of two or more groups where, for each unit, repeated measurements over time are obtained on each outcome. We relate the observed outcomes using multivariate nonlinear mixed-effects models to describe evolutions in different groups. Due to its flexibility, the random-effects approach for the joint modeling of multiple outcomes can be used to estimate population parameters for a discriminant model that classifies units into distinct predefined groups or populations. Parameter estimation is done via the expectation-maximization algorithm with a linear approximation step. We conduct a simulation study that sheds light on the effect that the linear approximation has on classification results. We present an example using data from a study in 161 pregnant women in Santiago, Chile, where the main interest is to predict normal versus abnormal pregnancy outcomes.