Describing complex disease progression using joint latent class models for multivariate longitudinal markers and clinical endpoints.

Neurodegenerative diseases are characterized by numerous markers of progression and clinical endpoints. For instance, multiple system atrophy (MSA), a rare neurodegenerative synucleinopathy, is characterized by various combinations of progressive autonomic failure and motor dysfunction, and a very poor prognosis. Describing the progression of such complex and multi-dimensional diseases is particularly difficult. One has to simultaneously account for the assessment of multivariate markers over time, the occurrence of clinical endpoints, and a highly suspected heterogeneity between patients. Yet, such description is crucial for understanding the natural history of the disease, staging patients diagnosed with the disease, unravelling subphenotypes, and predicting the prognosis. Through the example of MSA progression, we show how a latent class approach modeling multiple repeated markers and clinical endpoints can help describe complex disease progression and identify subphenotypes for exploring new pathological hypotheses. The proposed joint latent class model includes class-specific multivariate mixed models to handle multivariate repeated biomarkers possibly summarized into latent dimensions and class-and-cause-specific proportional hazard models to handle time-to-event data. Maximum likelihood estimation procedure, validated through simulations is available in the lcmm R package. In the French MSA cohort comprising data of 598 patients during up to 13 years, five subphenotypes of MSA were identified that differ by the sequence and shape of biomarkers degradation, and the associated risk of death. In posterior analyses, the five subphenotypes were used to explore the association between clinical progression and external imaging and fluid biomarkers, while properly accounting for the uncertainty in the subphenotypes membership.

Joint model for survival and multivariate sparse functional data with application to a study of Alzheimer's Disease.

Studies of Alzheimer's disease (AD) often collect multiple longitudinal clinical outcomes, which are correlated and predictive of AD progression. It is of great scientific interest to investigate the association between the outcomes and time to AD onset. We model the multiple longitudinal outcomes as multivariate sparse functional data and propose a functional joint model linking multivariate functional data to event time data. In particular, we propose a multivariate functional mixed model to identify the shared progression pattern and outcome-specific progression patterns of the outcomes, which enables more interpretable modeling of associations between outcomes and AD onset. The proposed method is applied to the Alzheimer's Disease Neuroimaging Initiative study (ADNI) and the functional joint model sheds new light on inference of five longitudinal outcomes and their associations with AD onset. Simulation studies also confirm the validity of the proposed model. Data used in preparation of this article were obtained from the ADNI database.

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.

Metrological properties of neuropsychological tests for measuring cognitive change in individuals with prodromal Alzheimer's disease.

OBJECTIVES: In Alzheimer's Disease (AD) research, choosing appropriate method for measuring change in cognitive function over time can be challenging. The aim for this study was to examine the sensitivity of four neuropsychological tests used to measure cognition during the transition from mild cognitive impairment (MCI) to AD, and the impacts of associated covariates. METHODS: We enrolled 223 patients with MCI who progressed to AD and had completed multiple follow-up assessments in the Alzheimer's Disease Neuroimaging Initiative (ADNI) database. We constructed nonlinear mixed model for multivariate longitudinal data assuming that multiple neuropsychological tests would exhibit nonlinear transformation of a common factor in the latent cognitive process underlying the progression from MCI to AD. RESULTS: The Clinical Dementia Rating-Sum of the Boxes (CDR-SB) and Alzheimer's Disease Assessment Scale (11 items; ADAS-11) were more sensitive to cognitive changes in individuals with higher cognitive function, the Functional Activities Questionnaire (FAQ) was more sensitive to cognitive changes in individuals with middle cognitive function, and the Mini-Mental State Examination (MMSE) was more sensitive to cognitive changes in individuals with lower cognitive function. Gender (p = 0.0139) and educational level (p = 0.0094) had varying effects on different tests, such that men performed better on the FAQ and CDR-SB, and individuals with higher educational level tended to perform better on the FAQ and MMSE. CONCLUSIONS: When choosing appropriate neuropsychological tests in cognitive measurements, the cognitive functional level of the patient as well as the impacts of covariates should be considered.

Latent Ornstein-Uhlenbeck models for Bayesian analysis of multivariate longitudinal categorical responses.

We propose a Bayesian latent Ornstein-Uhlenbeck (OU) model to analyze unbalanced longitudinal data of binary and ordinal variables, which are manifestations of fewer continuous latent variables. We focus on the evolution of such latent variables when they continuously change over time. Existing approaches are limited to data collected at regular time intervals. Our proposal makes use of an OU process for the latent variables to overcome this limitation. We show that assuming real eigenvalues for the drift matrix of the OU process, as is frequently done in practice, can lead to biased estimates and/or misleading inference when the true process is oscillating. In contrast, our proposal allows for both real and complex eigenvalues. We illustrate our proposed model with a motivating dataset, containing patients with amyotrophic lateral sclerosis disease. We were interested in how bulbar, cervical, and lumbar functions evolve over time.

Backward joint model and dynamic prediction of survival with multivariate longitudinal data.

An important approach to dynamic prediction of time-to-event outcomes using longitudinal data is based on modeling the joint distribution of longitudinal and time-to-event data. The widely used joint model for this purpose is the shared random effect model. Presumably, adding more longitudinal predictors improves the predictive accuracy. However, the shared random effect model can be computationally difficult or prohibitive when a large number of longitudinal variables are used. In this paper, we study an alternative way of modeling the joint distribution of longitudinal and time-to-event data. Under this formulation, the log-likelihood involves no more than one-dimensional integration, regardless of the number of longitudinal variables in the model. Therefore, this model is particularly suitable in dynamic prediction problems with large number of longitudinal predictors. The model fitting can be implemented with tractable and stable computation by using a combination of pseudo maximum likelihood estimation, Expectation-Maximization algorithm, and convex optimization. We evaluate the proposed methodology and its predictive accuracy with varying number of longitudinal variables using simulations and data from a primary biliary cirrhosis study.

Multivariate joint modeling to identify markers of growth and lung function decline that predict cystic fibrosis pulmonary exacerbation onset.

BACKGROUND: Attenuated decreases in lung function can signal the onset of acute respiratory events known as pulmonary exacerbations (PEs) in children and adolescents with cystic fibrosis (CF). Univariate joint modeling facilitates dynamic risk prediction of PE onset and accounts for measurement error of the lung function marker. However, CF is a multi-system disease and the extent to which simultaneously modeling growth and nutrition markers improves PE predictive accuracy is unknown. Furthermore, it is unclear which routinely collected clinical indicators of growth and nutrition in early life predict PE onset in CF. METHODS: Using a longitudinal cohort of 17,100 patients aged 6-20 years (US Cystic Fibrosis Foundation Patient Registry; 2003-2015), we fit a univariate joint model of lung-function decline and PE onset and contrasted its predictive performance with a class of multivariate joint models that included combinations of growth markers as additional submodels. Outcomes were longitudinal lung function (forced expiratory volume in 1 s of % predicted), percentiles of body mass index, weight-for-age and height-for-age and PE onset. Relevant demographic/clinical covariates were included in submodels. We implemented a univariate joint model of lung function and time-to-PE and four multivariate joint models including growth outcomes. RESULTS: All five joint models showed that declining lung function corresponded to slightly increased risk of PE onset (hazard ratio from univariate joint model: 0.97, P < 0.0001), and all had reasonable predictive accuracy (cross-validated area under the receiver-operator characteristic curve > 0.70). None of the growth markers alongside lung function as outcomes in multivariate joint modeling appeared to have an association with hazard of PE. Jointly modeling only lung function and PE onset yielded the most accurate (area under the receiver-operator characteristic curve = 0.75) and precise (narrowest interquartile range) predictions. Dynamic predictions were accurate across forecast horizons (0.5, 1 and 2 years) and precision improved with age. CONCLUSIONS: Including growth markers via multivariate joint models did not yield gains in prediction performance, compared to a univariate joint model with lung function. Individualized dynamic predictions from joint modeling could enhance physician monitoring of CF disease progression by providing PE risk assessment over a patient's clinical course.

Dynamic prediction of Alzheimer's disease progression using features of multiple longitudinal outcomes and time-to-event data.

This paper is motivated by combining serial neurocognitive assessments and other clinical variables for monitoring the progression of Alzheimer's disease (AD). We propose a novel framework for the use of multiple longitudinal neurocognitive markers to predict the progression of AD. The conventional joint modeling longitudinal and survival data approach is not applicable when there is a large number of longitudinal outcomes. We introduce various approaches based on functional principal component for dimension reduction and feature extraction from multiple longitudinal outcomes. We use these features to extrapolate the health outcome trajectories and use scores on these features as predictors in a Cox proportional hazards model to conduct predictions over time. We propose a personalized dynamic prediction framework that can be updated as new observations collected to reflect the patient's latest prognosis, and thus intervention could be initiated in a timely manner. Simulation studies and application to the Alzheimer's Disease Neuroimaging Initiative dataset demonstrate the robustness of the method for the prediction of future health outcomes and risks of target events under various scenarios.

A joint model for multiple dynamic processes and clinical endpoints: Application to Alzheimer's disease.

As other neurodegenerative diseases, Alzheimer's disease, the most frequent dementia in the elderly, is characterized by multiple progressive impairments in the brain structure and in clinical functions such as cognitive functioning and functional disability. Until recently, these components were mostly studied independently because no joint model for multivariate longitudinal data and time to event was available in the statistical community. Yet, these components are fundamentally interrelated in the degradation process toward dementia and should be analyzed together. We thus propose a joint model to simultaneously describe the dynamics of multiple correlated components. Each component, defined as a latent process, is measured by one or several continuous markers (not necessarily Gaussian). Rather than considering the associated time to diagnosis as in standard joint models, we assume diagnosis corresponds to the passing above a covariate-specific threshold (to be estimated) of a pathological process that is modeled as a combination of the component-specific latent processes. This definition captures the clinical complexity of diagnoses such as dementia diagnosis but also benefits from simplifications for the computation of maximum likelihood estimates. We show that the model and estimation procedure can also handle competing clinical endpoints. The estimation procedure, implemented in an R package, is validated by simulations and the method is illustrated on a large French population-based cohort of cerebral aging in which we focused on the dynamics of three clinical manifestations and the associated risk of dementia and death before dementia.

The Analysis of Multivariate Longitudinal Data: An Instructive Application of the Longitudinal Three-Mode Model.

Structural equation modeling is a common technique to assess change in longitudinal designs. However, these models can become of unmanageable size with many measurement occasions. One solution is the imposition of Kronecker product restrictions to model the multivariate longitudinal structure of the data. The resulting longitudinal three-mode models (L3MMs) are very parsimonious and have attractive interpretation. This paper provides an instructive description of L3MMs. The models are applied to health-related quality of life (HRQL) data obtained from 682 patients with painful bone metastasis, with eight measurements at 13 occasions; before and every week after treatment with radiotherapy. We explain (1) how the imposition of Kronecker product restrictions can be used to model the multivariate longitudinal structure of the data, (2) how to interpret the Kronecker product restrictions and the resulting L3MM parameters, and (3) how to test substantive hypotheses in L3MMs. In addition, we discuss the challenges for the evaluation of (differences in) fit of these complex and parsimonious models. The L3MM restrictions lead to parsimonious models and provide insight in the change patterns of relationships between variables in addition to the general patterns of change. The L3MM thus provides a convenient model for multivariate longitudinal data, as it not only facilitates the analysis of complex longitudinal data but also the substantive interpretation of the dynamics of change.

Hidden Markov latent variable models with multivariate longitudinal data.

Cocaine addiction is chronic and persistent, and has become a major social and health problem in many countries. Existing studies have shown that cocaine addicts often undergo episodic periods of addiction to, moderate dependence on, or swearing off cocaine. Given its reversible feature, cocaine use can be formulated as a stochastic process that transits from one state to another, while the impacts of various factors, such as treatment received and individuals' psychological problems on cocaine use, may vary across states. This article develops a hidden Markov latent variable model to study multivariate longitudinal data concerning cocaine use from a California Civil Addict Program. The proposed model generalizes conventional latent variable models to allow bidirectional transition between cocaine-addiction states and conventional hidden Markov models to allow latent variables and their dynamic interrelationship. We develop a maximum-likelihood approach, along with a Monte Carlo expectation conditional maximization (MCECM) algorithm, to conduct parameter estimation. The asymptotic properties of the parameter estimates and statistics for testing the heterogeneity of model parameters are investigated. The finite sample performance of the proposed methodology is demonstrated by simulation studies. The application to cocaine use study provides insights into the prevention of cocaine use.

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.

Estimating correlation between multivariate longitudinal data in the presence of heterogeneity.

BACKGROUND: Estimating correlation coefficients among outcomes is one of the most important analytical tasks in epidemiological and clinical research. Availability of multivariate longitudinal data presents a unique opportunity to assess joint evolution of outcomes over time. Bivariate linear mixed model (BLMM) provides a versatile tool with regard to assessing correlation. However, BLMMs often assume that all individuals are drawn from a single homogenous population where the individual trajectories are distributed smoothly around population average. METHODS: Using longitudinal mean deviation (MD) and visual acuity (VA) from the Ocular Hypertension Treatment Study (OHTS), we demonstrated strategies to better understand the correlation between multivariate longitudinal data in the presence of potential heterogeneity. Conditional correlation (i.e., marginal correlation given random effects) was calculated to describe how the association between longitudinal outcomes evolved over time within specific subpopulation. The impact of heterogeneity on correlation was also assessed by simulated data. RESULTS: There was a significant positive correlation in both random intercepts (ρ = 0.278, 95% CI: 0.121-0.420) and random slopes (ρ = 0.579, 95% CI: 0.349-0.810) between longitudinal MD and VA, and the strength of correlation constantly increased over time. However, conditional correlation and simulation studies revealed that the correlation was induced primarily by participants with rapid deteriorating MD who only accounted for a small fraction of total samples. CONCLUSION: Conditional correlation given random effects provides a robust estimate to describe the correlation between multivariate longitudinal data in the presence of unobserved heterogeneity (NCT00000125).

Predicting the multi-domain progression of Parkinson's disease: a Bayesian multivariate generalized linear mixed-effect model.

BACKGROUND: It is challenging for current statistical models to predict clinical progression of Parkinson's disease (PD) because of the involvement of multi-domains and longitudinal data. METHODS: Past univariate longitudinal or multivariate analyses from cross-sectional trials have limited power to predict individual outcomes or a single moment. The multivariate generalized linear mixed-effect model (GLMM) under the Bayesian framework was proposed to study multi-domain longitudinal outcomes obtained at baseline, 18-, and 36-month. The outcomes included motor, non-motor, and postural instability scores from the MDS-UPDRS, and demographic and standardized clinical data were utilized as covariates. The dynamic prediction was performed for both internal and external subjects using the samples from the posterior distributions of the parameter estimates and random effects, and also the predictive accuracy was evaluated based on the root of mean square error (RMSE), absolute bias (AB) and the area under the receiver operating characteristic (ROC) curve. RESULTS: First, our prediction model identified clinical data that were differentially associated with motor, non-motor, and postural stability scores. Second, the predictive accuracy of our model for the training data was assessed, and improved prediction was gained in particularly for non-motor (RMSE and AB: 2.89 and 2.20) compared to univariate analysis (RMSE and AB: 3.04 and 2.35). Third, the individual-level predictions of longitudinal trajectories for the testing data were performed, with ~80% observed values falling within the 95% credible intervals. CONCLUSIONS: Multivariate general mixed models hold promise to predict clinical progression of individual outcomes in PD. TRIAL REGISTRATION: The data was obtained from Dr. Xuemei Huang's NIH grant R01 NS060722 , part of NINDS PD Biomarker Program (PDBP). All data was entered within 24 h of collection to the Data Management Repository (DMR), which is publically available ( https://pdbp.ninds.nih.gov/data-management ).

Joint longitudinal data analysis in detecting determinants of CD4 cell count change and adherence to highly active antiretroviral therapy at Felege Hiwot Teaching and Specialized Hospital, North-west Ethiopia (Amhara Region).

BACKGROUND: Adherence and CD4 cell count change measure the progression of the disease in HIV patients after the commencement of HAART. Lack of information about associated factors on adherence to HAART and CD4 cell count reduction is a challenge for the improvement of cells in HIV positive adults. The main objective of adopting joint modeling was to compare separate and joint models of longitudinal repeated measures in identifying long-term predictors of the two longitudinal outcomes: CD4 cell count and adherence to HAART. METHODS: A longitudinal retrospective cohort study was conducted to examine the joint predictors of CD4 cell count change and adherence to HAART among HIV adult patients enrolled in the first 10 months of the year 2008 and followed-up to June 2012. Joint model was employed to determine joint predictors of two longitudinal response variables over time. Furthermore, the generalized linear mixed effect model had been used for specification of the marginal distribution, conditional to correlated random effect. RESULTS: A total of 792 adult HIV patients were studied to analyze the longitudinal joint model study. The result from this investigation revealed that age, weight, baseline CD4 cell count, ownership of cell phone, visiting times, marital status, residence area and level of disclosure of the disease to family members had significantly affected both outcomes. From the two-way interactions, time * owner of cell phone, time * sex, age * sex, age * level of education as well as time * level of education were significant for CD4 cell count change in the longitudinal data analysis. The multivariate joint model with linear predictor indicates that CD4 cell count change was positively correlated (p ≤ 0.0001) with adherence to HAART. Hence, as adherence to HAART increased, CD4 cell count also increased; and those patients who had significant CD4 cell count change at each visiting time had been encouraged to be good adherents. CONCLUSION: Joint model analysis was more parsimonious as compared to separate analysis, as it reduces type I error and subject-specific analysis improved its model fit. The joint model operates multivariate analysis simultaneously; and it has great power in parameter estimation. Developing joint model helps validate the observed correlation between the outcomes that have emerged from the association of intercepts. There should be a special attention and intervention for HIV positive adults, especially for those who had poor adherence and with low CD4 cell count change. The intervention may be important for pre-treatment counseling and awareness creation. The study also identified a group of patients who were with maximum risk of CD4 cell count change. It is suggested that this group of patients needs high intervention for counseling.

Two-stage model for multivariate longitudinal and survival data with application to nephrology research.

In many follow-up studies different types of outcomes are collected including longitudinal measurements and time-to-event outcomes. Commonly, it is of interest to study the association between them. Joint modeling approaches of a single longitudinal outcome and survival process have recently gained increasing attention from both frequentist and Bayesian perspective. However, in many studies several longitudinal biomarkers are of interest and instead of selecting one single biomarker, the relationships between all these outcomes and their association with survival needs to be investigated. Our motivating study comes from Peritoneal Dialysis Programme in Nephrology research from Nephrology Unit, CHP (Hospital de Santo António), Porto, Portugal in which the interest relies on the possible association between various biomarkers (calcium, phosphate, parathormone, and creatinine) and the patients' survival. To this aim, we propose a two-stage model-based approach for multivariate longitudinal and survival data that allowed us to study such complex association structure. The multivariate model suggested in this paper provided new insights in the area of nephrology research showing valid results in comparison with those models studying each longitudinal biomarker with survival separately.

Joint modeling of repeated multivariate cognitive measures and competing risks of dementia and death: a latent process and latent class approach.

Joint models initially dedicated to a single longitudinal marker and a single time-to-event need to be extended to account for the rich longitudinal data of cohort studies. Multiple causes of clinical progression are indeed usually observed, and multiple longitudinal markers are collected when the true latent trait of interest is hard to capture (e.g., quality of life, functional dependency, and cognitive level). These multivariate and longitudinal data also usually have nonstandard distributions (discrete, asymmetric, bounded, etc.). We propose a joint model based on a latent process and latent classes to analyze simultaneously such multiple longitudinal markers of different natures, and multiple causes of progression. A latent process model describes the latent trait of interest and links it to the observed longitudinal outcomes using flexible measurement models adapted to different types of data, and a latent class structure links the longitudinal and cause-specific survival models. The joint model is estimated in the maximum likelihood framework. A score test is developed to evaluate the assumption of conditional independence of the longitudinal markers and each cause of progression given the latent classes. In addition, individual dynamic cumulative incidences of each cause of progression based on the repeated marker data are derived. The methodology is validated in a simulation study and applied on real data about cognitive aging obtained from a large population-based study. The aim is to predict the risk of dementia by accounting for the competing death according to the profiles of semantic memory measured by two asymmetric psychometric tests.

A clustering algorithm for multivariate longitudinal data.

Latent growth modeling approaches, such as growth mixture models, are used to identify meaningful groups or classes of individuals in a larger heterogeneous population. But when applied to multivariate repeated measures computational problems are likely, due to the high dimension of the joint distribution of the random effects in these mixed-effects models. This article proposes a cluster algorithm for multivariate repeated data, using pseudo-likelihood and ideas based on k-means clustering, to reveal homogenous subgroups. The algorithm was demonstrated on an electro-encephalogram dataset set quantifying the effect of psychoactive compounds on the brain activity in rats.

Surveillance of cardiovascular diseases using a multivariate dynamic screening system.

In the SHARe Framingham Heart Study of the National Heart, Lung and Blood Institute, one major task is to monitor several health variables (e.g., blood pressure and cholesterol level) so that their irregular longitudinal pattern can be detected as soon as possible and some medical treatments applied in a timely manner to avoid some deadly cardiovascular diseases (e.g., stroke). To handle this kind of applications effectively, we propose a new statistical methodology called multivariate dynamic screening system (MDySS) in this paper. The MDySS method combines the major strengths of the multivariate longitudinal data analysis and the multivariate statistical process control, and it makes decisions about the longitudinal pattern of a subject by comparing it with other subjects cross sectionally and by sequentially monitoring it as well. Numerical studies show that MDySS works well in practice.

Multi-disease analysis of maternal antibody decay using non-linear mixed models accounting for censoring.

Biomedical studies often generate repeated measures of multiple outcomes on a set of subjects. It may be of interest to develop a biologically intuitive model for the joint evolution of these outcomes while assessing inter-subject heterogeneity. Even though it is common for biological processes to entail non-linear relationships, examples of multivariate non-linear mixed models (MNMMs) are still fairly rare. We contribute to this area by jointly analyzing the maternal antibody decay for measles, mumps, rubella, and varicella, allowing for a different non-linear decay model for each infectious disease. We present a general modeling framework to analyze multivariate non-linear longitudinal profiles subject to censoring, by combining multivariate random effects, non-linear growth and Tobit regression. We explore the hypothesis of a common infant-specific mechanism underlying maternal immunity using a pairwise correlated random-effects approach and evaluating different correlation matrix structures. The implied marginal correlation between maternal antibody levels is estimated using simulations. The mean duration of passive immunity was less than 4 months for all diseases with substantial heterogeneity between infants. The maternal antibody levels against rubella and varicella were found to be positively correlated, while little to no correlation could be inferred for the other disease pairs. For some pairs, computational issues occurred with increasing correlation matrix complexity, which underlines the importance of further developing estimation methods for MNMMs.