Models for longitudinal data with censored changepoints

Summary.  In longitudinal studies of biological markers, different individuals may have different underlying patterns of response. In some applications, a subset of individuals experiences latent events, causing an instantaneous change in the level or slope of the marker trajectory. The paper presents a general mixture of hierarchical longitudinal models for serial biomarkers. Interest centres both on the time of the event and on levels of the biomarker before and after the event. In observational studies where marker series are incomplete, the latent event can be modelled by a survival distribution. Risk factors for the occurrence of the event can be investigated by including covariates in the survival distribution. A combination of Gibbs, Metropolis–Hastings and reversible jump Markov chain Monte Carlo sampling is used to fit the models to serial measurements of forced expiratory volume from lung transplant recipients.     

Identification of Longitudinal Biomarkers in Survival Analysis for Competing Risks Data

In recent years, joint analysis of longitudinal measurements and survival data has received much attention. However, previous work has primarily focused on a single failure type for the event time. In this article, we consider joint modeling of repeated measurements and competing risks failure time data to allow for more than one distinct failure type in the survival endpoint so we fit a cause-specific hazards sub-model to allow for competing risks, with a separate latent association between longitudinal measurements and each cause of failure. Besides, previous work does not focus on the hypothesis to test a separate latent association between longitudinal measurements and each cause of failure. In this article, we derive a score test to identify longitudinal biomarkers or surrogates for a time to event outcome in competing risks data. With a carefully chosen definition of complete data, the maximum likelihood estimation of the cause-specific hazard functions is performed via an EM algorithm. We extend this work and allow random effects to be present in both the longitudinal biomarker and underlying survival function. The random effects in the biomarker are introduced via an explicit term while the random effect in the underlying survival function is introduced by the inclusion of frailty into the model.

We use simulations to explore how the number of individuals, the number of time points per individual and the functional form of the random effects from the longitudinal biomarkers considering heterogeneous baseline hazards in individuals influence the power to detect the association of a longitudinal biomarker and the survival time.     

Starting Values for EM Estimation of Latent Class Joint Model

Latent class model is one of the important latent variable methods for joint modeling longitudinal and survival data. Latent class joint model can handle underlying heterogeneous population, discover subpopulation structure, and incorporate correlated non normally distributed outcomes. The maximum likelihood estimates of parameters in latent class joint model are generally obtained by the EM algorithm. Finding the starting values is one of the major issues to implement the EM algorithm successfully. In this article, initial value formulas are provided, a simulation study is conducted to show that the proposed starting values perform very well, and two illustrative examples are presented.     

Joint model-based clustering of nonlinear longitudinal trajectories and associated time-to-event data analysis,linked by latent class membership: with application to AIDS clinical studies

Longitudinal and time-to-event data are often observed together. Finite mixture models are currently used to analyze nonlinear heterogeneous longitudinal data, which, by releasing the homogeneity restriction of nonlinear mixed-effects (NLME) models, can cluster individuals into one of the pre-specified classes with class membership probabilities. This clustering may have clinical significance, and be associated with clinically important time-to-event data. This article develops a joint modeling approach to a finite mixture of NLME models for longitudinal data and proportional hazard Cox model for time-to-event data, linked by individual latent class indicators, under a Bayesian framework. The proposed joint models and method are applied to a real AIDS clinical trial data set, followed by simulation studies to assess the performance of the proposed joint model and a naive two-step model, in which finite mixture model and Cox model are fitted separately.     

Slope Estimation for Bivariate Longitudinal Outcomes Adjusting for Informative Right Censoring Using Discrete Survival Model: Application to the Renal Transplant Cohort

Patients undergoing renal transplantation are prone to graft failure which causes lost of follow-up measures on their blood urea nitrogen and serum creatinine levels. These two outcomes are measured repeatedly over time to assess renal function following transplantation. Loss of follow-up on these bivariate measures results in informative right censoring, a common problem in longitudinal data that should be adjusted for so that valid estimates are obtained. In this study, we propose a bivariate model that jointly models these two longitudinal correlated outcomes and generates population and individual slopes adjusting for informative right censoring using a discrete survival approach. The proposed approach is applied to the clinical dataset of patients who had undergone renal transplantation. A simulation study validates the effectiveness of the approach.     

Predictive comparison of joint longitudinal-survival modeling: a case study illustrating competing approaches

The joint modeling of longitudinal and survival data has received extraordinary attention in the statistics literature recently, with models and methods becoming increasingly more complex. Most of these approaches pair a proportional hazards survival with longitudinal trajectory modeling through parametric or nonparametric specifications. In this paper we closely examine one data set previously analyzed using a two parameter parametric model for Mediterranean fruit fly (medfly) egg-laying trajectories paired with accelerated failure time and proportional hazards survival models. We consider parametric and nonparametric versions of these two models, as well as a proportional odds rate model paired with a wide variety of longitudinal trajectory assumptions reflecting the types of analyses seen in the literature. In addition to developing novel nonparametric Bayesian methods for joint models, we emphasize the importance of model selection from among joint and non joint models. The default in the literature is to omit at the outset non joint models from consideration. For the medfly data, a predictive diagnostic criterion suggests that both the choice of survival model and longitudinal assumptions can grossly affect model adequacy and prediction. Specifically for these data, the simple joint model used in by Tseng et al. (Biometrika 92:587–603, 2005) and models with much more flexibility in their longitudinal components are predictively outperformed by simpler analyses. This case study underscores the need for data analysts to compare on the basis of predictive performance different joint models and to include non joint models in the pool of candidates under consideration.     

A model for markers and latent health status

We extend the bivariate Wiener process considered by Whitmore and co-workers and model the joint process of a marker and health status. The health status process is assumed to be latent or unobservable. The time to reach the primary end point or failure (death, onset of disease, etc.) is the time when the latent health status process first crosses a failure threshold level. Inferences for the model are based on two kinds of data: censored survival data and marker measurements. Covariates, such as treatment variables, risk factors and base-line conditions, are related to the model parameters through generalized linear regression functions. The model offers a much richer potential for the study of treatment efficacy than do conventional models. Treatment effects can be assessed in terms of their influence on both the failure threshold and the health status process parameters. We derive an explicit formula for the prediction of residual failure times given the current marker level. Also we discuss model validation. This model does not require the proportional hazards assumption and hence can be widely used. To demonstrate the usefulness of the model, we apply the methods in analysing data from the protocol 116a of the AIDS Clinical Trials Group.     

Joint analysis of nonlinear heterogeneous longitudinal data and binary outcome: an application to AIDS clinical studies

Finite mixture models are currently used to analyze heterogeneous longitudinal data. By releasing the homogeneity restriction of nonlinear mixed-effects (NLME) models, finite mixture models not only can estimate model parameters but also cluster individuals into one of the pre-specified classes with class membership probabilities. This clustering may have clinical significance, which might be associated with a clinically important binary outcome. This article develops a joint modeling of a finite mixture of NLME models for longitudinal data in the presence of covariate measurement errors and a logistic regression for a binary outcome, linked by individual latent class indicators, under a Bayesian framework. Simulation studies are conducted to assess the performance of the proposed joint model and a naive two-step model, in which finite mixture model and logistic regression are fitted separately, followed by an application to a real data set from an AIDS clinical trial, in which the viral dynamics and dichotomized time to the first decline of CD4/CD8 ratio are analyzed jointly.     

A Non-parametric Frailty Model for Temporally Clustered Multivariate Failure Times

Abstract A model is introduced here for multivariate failure time data arising from heterogenous populations. In particular, we consider a situation in which the failure times of individual subjects are often temporally clustered, so that many failures occur during a relatively short age interval. The clustering is modelled by assuming that the subjects can be divided into ‘internally homogenous’ latent classes, each such class being then described by a time‐dependent frailty profile function. As an example, we reanalysed the dental caries data presented earlier in Härkänen et al. [Scand. J. Statist. 27 (2000) 577], as it turned out that our earlier model could not adequately describe the observed clustering.     

New approaches for censored longitudinal data in joint modelling of longitudinal and survival data,with application to HIV vaccine studies

In HIV vaccine studies, longitudinal immune response biomarker data are often left-censored due to lower limits of quantification of the employed immunological assays. The censoring information is important for predicting HIV infection, the failure event of interest. We propose two approaches to addressing left censoring in longitudinal data: one that makes no distributional assumptions for the censored data—treating left censored values as a “point mass” subgroup—and the other makes a distributional assumption for a subset of the censored data but not for the remaining subset. We develop these two approaches to handling censoring for joint modelling of longitudinal and survival data via a Cox proportional hazards model fit by h-likelihood. We evaluate the new methods via simulation and analyze an HIV vaccine trial data set, finding that longitudinal characteristics of the immune response biomarkers are highly associated with the risk of HIV infection.

     

A multivariate finite mixture latent trajectory model with application to dementia studies

Dementia patients exhibit considerable heterogeneity in individual trajectories of cognitive decline, with some patients showing rapid decline following diagnoses while others exhibiting slower decline or remaining stable for several years. Dementia studies often collect longitudinal measures of multiple neuropsychological tests aimed to measure patients’ decline across a number of cognitive domains. We propose a multivariate finite mixture latent trajectory model to identify distinct longitudinal patterns of cognitive decline simultaneously in multiple cognitive domains, each of which is measured by multiple neuropsychological tests. EM algorithm is used for parameter estimation and posterior probabilities are used to predict latent class membership. We present results of a simulation study demonstrating adequate performance of our proposed approach and apply our model to the Uniform Data Set from the National Alzheimer's Coordinating Center to identify cognitive decline patterns among dementia patients.     

A new approach for joint modelling of longitudinal measurements and survival times with a cure fraction

The joint analysis of longitudinal measurements and survival data is useful in clinical trials and other medical studies. In this paper, we consider a joint model which assumes a linear mixed $tt$ model for longitudinal measurements and a promotion time cure model for survival data and links these two models through a latent variable. A semiparametric inference procedure with an EM algorithm implementation is developed for the parameters in the joint model. The proposed procedure is evaluated in a simulation study and applied to analyze the quality of life and time to recurrence data from a clinical trial on women with early breast cancer. The Canadian Journal of Statistics 40: 207–224; 2012 © 2012 Statistical Society of Canada     

A Joint Modeling Approach for Longitudinal Outcomes and Non-ignorable Dropout under Population Heterogeneity in Mental Health Studies

The paper proposes a joint mixture model to model non-ignorable drop-out in longitudinal cohort studies of mental health outcomes. The model combines a (non)-linear growth curve model for the time-dependent outcomes and a discrete-time survival model for the drop-out with random effects shared by the two sub-models. The mixture part of the model takes into account population heterogeneity by accounting for latent subgroups of the shared effects that may lead to different patterns for the growth and the drop-out tendency. A simulation study shows that the joint mixture model provides greater precision in estimating the average slope and covariance matrix of random effects. We illustrate its benefits with data from a longitudinal cohort study that characterizes depression symptoms over time yet is hindered by non-trivial participant drop-out.KEYWORDS: Latent growth curve, MNAR drop-out, survival analysis, finite mixture model, mental health     

Joint modeling of degradation and failure time data

This paper surveys some approaches to model the relationship between failure time data and covariate data like internal degradation and external environmental processes. These models which reflect the dependency between system state and system reliability include threshold models and hazard-based models. In particular, we consider the class of degradation–threshold–shock models (DTS models) in which failure is due to the competing causes of degradation and trauma. For this class of reliability models we express the failure time in terms of degradation and covariates. We compute the survival function of the resulting failure time and derive the likelihood function for the joint observation of failure times and degradation data at discrete times. We consider a special class of DTS models where degradation is modeled by a process with stationary independent increments and related to external covariates through a random time scale and extend this model class to repairable items by a marked point process approach. The proposed model class provides a rich conceptual framework for the study of degradation–failure issues.     

A general joint model for longitudinal measurements and competing risks survival data with heterogeneous random effects

This article studies a general joint model for longitudinal measurements and competing risks survival data. The model consists of a linear mixed effects sub-model for the longitudinal outcome, a proportional cause-specific hazards frailty sub-model for the competing risks survival data, and a regression sub-model for the variance–covariance matrix of the multivariate latent random effects based on a modified Cholesky decomposition. The model provides a useful approach to adjust for non-ignorable missing data due to dropout for the longitudinal outcome, enables analysis of the survival outcome with informative censoring and intermittently measured time-dependent covariates, as well as joint analysis of the longitudinal and survival outcomes. Unlike previously studied joint models, our model allows for heterogeneous random covariance matrices. It also offers a framework to assess the homogeneous covariance assumption of existing joint models. A Bayesian MCMC procedure is developed for parameter estimation and inference. Its performances and frequentist properties are investigated using simulations. A real data example is used to illustrate the usefulness of the approach.     

Nonparametric Bayesian Analysis of Competing Risks Problem with Masked Data

Bayesian analysis of system failure data under a competing-failure framework is considered when the failure causes have not been exactly identified but narrowed down to a subset of all potential failure causes. The usual assumption of independence of failure causes is relaxed. We obtain the posterior distribution of the joint survival function, assuming a Dirichlet process prior, and derive the limiting posterior distribution. We show that the posterior estimate of the reliability of the series system of interest in practice is consistent. A numerical example shows that our approach is feasible.     

Failure Inference From a Marker Process Based on a Bivariate Wiener Model

Many models have been proposed that relate failure times and stochastic time-varying covariates. In some of these models, failure occurs when a particular observable marker crosses a threshold level. We are interested in the more difficult, and often more realistic, situation where failure is not related deterministically to an observable marker. In this case, joint models for marker evolution and failure tend to lead to complicated calculations for characteristics such as the marginal distribution of failure time or the joint distribution of failure time and marker value at failure. This paper presents a model based on a bivariate Wiener process in which one component represents the marker and the second, which is latent (unobservable), determines the failure time. In particular, failure occurs when the latent component crosses a threshold level. The model yields reasonably simple expressions for the characteristics mentioned above and is easy to fit to commonly occurring data that involve the marker value at the censoring time for surviving cases and the marker value and failure time for failing cases. Parametric and predictive inference are discussed, as well as model checking. An extension of the model permits the construction of a composite marker from several candidate markers that may be available. The methodology is demonstrated by a simulated example and a case application.     

Failure time studies with intermittent observation and losses to follow-up

In health research interest often lies in modeling a failure time process but in many cohort studies failure status is only determined at scheduled assessment times. While the assessment times may be fixed upon study entry, individuals may become lost to follow-up and miss visits subsequent to the time of loss to follow-up. We consider a three-state model to characterize a joint failure and loss to follow-up process, and use it to investigate the impact of dependent loss to follow-up on standard parametric, nonparametric, and semiparametric analysis. The effect of dependent loss to follow-up is mitigated by fitting the joint model. The performance of standard methods is studied using the asymptotic theory of misspecified models, and the finite sample performance is examined for the standard and joint analyses through simulation studies. An application to data from a youth smoking prevention study is presented for illustration.     

Bayesian Analysis of Ordinal Survey Data Using the Dirichlet Process to Account for Respondent Personality Traits

This article presents a Bayesian latent variable model used to analyze ordinal response survey data by taking into account the characteristics of respondents. The ordinal response data are viewed as multivariate responses arising from continuous latent variables with known cut-points. Each respondent is characterized by two parameters that have a Dirichlet process as their joint prior distribution. The proposed mechanism adjusts for classes of personalities. The model is applied to student survey data in course evaluations. Goodness-of-fit (GoF) procedures are developed for assessing the validity of the model. The proposed GoF procedures are simple, intuitive, and do not seem to be a part of current Bayesian practice.     

The generalized log-gamma mixture model with covariates: local influence and residual analysis

In a sample of censored survival times, the presence of an immune proportion of individuals who are not subject to death, failure or relapse, may be indicated by a relatively high number of individuals with large censored survival times. In this paper the generalized log-gamma model is modified for the possibility that long-term survivors may be present in the data. The model attempts to separately estimate the effects of covariates on the surviving fraction, that is, the proportion of the population for which the event never occurs. The logistic function is used for the regression model of the surviving fraction. Inference for the model parameters is considered via maximum likelihood. Some influence methods, such as the local influence and total local influence of an individual are derived, analyzed and discussed. Finally, a data set from the medical area is analyzed under the log-gamma generalized mixture model. A residual analysis is performed in order to select an appropriate model. The authors would like to thank the editor and referees for their helpful comments. This work was supported by CNPq, Brazil.