A joint latent class changepoint model to improve the prediction of time to graft failure

Summary.  The reciprocal of serum creatinine concentration, RC, is often used as a biomarker to monitor renal function. It has been observed that RC trajectories remain relatively stable after transplantation until a certain moment, when an irreversible decrease in the RC levels occurs. This decreasing trend commonly precedes failure of a graft. Two subsets of individuals can be distinguished according to their RC trajectories: a subset of individuals having stable RC levels and a subset of individuals who present an irrevocable decrease in their RC levels. To describe such data, the paper proposes a joint latent class model for longitudinal and survival data with two latent classes. RC trajectories within latent class one are modelled by an intercept-only random-effects model and RC trajectories within latent class two are modelled by a segmented random changepoint model. A Bayesian approach is used to fit this joint model to data from patients who had their first kidney transplantation in the Leiden University Medical Center between 1983 and 2002. The resulting model describes the kidney transplantation data very well and provides better predictions of the time to failure than other joint and survival models.     

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.     

A Bayesian approach to mixture cure models with spatial frailties for population‐based cancer relative survival data

As the treatments of cancer progress, a certain number of cancers are curable if diagnosed early. In population‐based cancer survival studies, cure is said to occur when mortality rate of the cancer patients returns to the same level as that expected for the general cancer‐free population. The estimates of cure fraction are of interest to both cancer patients and health policy makers. Mixture cure models have been widely used because the model is easy to interpret by separating the patients into two distinct groups. Usually parametric models are assumed for the latent distribution for the uncured patients. The estimation of cure fraction from the mixture cure model may be sensitive to misspecification of latent distribution. We propose a Bayesian approach to mixture cure model for population‐based cancer survival data, which can be extended to county‐level cancer survival data. Instead of modeling the latent distribution by a fixed parametric distribution, we use a finite mixture of the union of the lognormal, loglogistic, and Weibull distributions. The parameters are estimated using the Markov chain Monte Carlo method. Simulation study shows that the Bayesian method using a finite mixture latent distribution provides robust inference of parameter estimates. The proposed Bayesian method is applied to relative survival data for colon cancer patients from the Surveillance, Epidemiology, and End Results (SEER) Program to estimate the cure fractions. The Canadian Journal of Statistics 40: 40–54; 2012 © 2012 Statistical Society of Canada     

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.     

Penalized spline joint models for longitudinal and time-to-event data

The joint models for longitudinal data and time-to-event data have recently received numerous attention in clinical and epidemiologic studies. Our interest is in modeling the relationship between event time outcomes and internal time-dependent covariates. In practice, the longitudinal responses often show non linear and fluctuated curves. Therefore, the main aim of this paper is to use penalized splines with a truncated polynomial basis to parameterize the non linear longitudinal process. Then, the linear mixed-effects model is applied to subject-specific curves and to control the smoothing. The association between the dropout process and longitudinal outcomes is modeled through a proportional hazard model. Two types of baseline risk functions are considered, namely a Gompertz distribution and a piecewise constant model. The resulting models are referred to as penalized spline joint models; an extension of the standard joint models. The expectation conditional maximization (ECM) algorithm is applied to estimate the parameters in the proposed models. To validate the proposed algorithm, extensive simulation studies were implemented followed by a case study. In summary, the penalized spline joint models provide a new approach for joint models that have improved the existing standard joint models.     

A family of models for uniform and serial dependence in repeated measurements studies

Data arising from a randomized double-masked clinical trial for multiple sclerosis have provided particularly variable longitudinal repeated measurements responses. Specific models for such data, other than those based on the multivariate normal distribution, would be a valuable addition to the applied statistician's toolbox. A useful family of multivariate distributions can be generated by substituting the integrated intensity of one distribution into a second (outer) distribution. The parameters in the second distribution are then used to create a dependence structure among observations on a unit. These may either be a form of serial dependence for longitudinal data or of uniform dependence within clusters. These are respectively analogous to the Kalman filter of state space models and to copulas, but they have the major advantage that they do not require any explicit integration. One useful outer distribution for constructing such multivariate distributions is the Pareto distribution. Certain special models based on it have previously been used in event history analysis, but those considered here have much wider application.     

Application of trajectories from growth curve in identification of longitudinal biomarker for the multivariate survival data

In clinical studies, the researchers measure the patients' response longitudinally. In recent studies, Mixed models are used to determine effects in the individual level. In the other hand, Henderson et al. [3,4] developed a joint likelihood function which combines likelihood functions of longitudinal biomarkers and survival times. They put random effects in the longitudinal component to determine if a longitudinal biomarker is associated with time to an event. In this paper, we deal with a longitudinal biomarker as a growth curve and extend Henderson's method to determine if a longitudinal biomarker is associated with time to an event for the multivariate survival data.     

Joint modeling of multivariate longitudinal mixed measurements and time to event data using a Bayesian approach

In many longitudinal studies multiple characteristics of each individual, along with time to occurrence of an event of interest, are often collected. In such data set, some of the correlated characteristics may be discrete and some of them may be continuous. In this paper, a joint model for analysing multivariate longitudinal data comprising mixed continuous and ordinal responses and a time to event variable is proposed. We model the association structure between longitudinal mixed data and time to event data using a multivariate zero-mean Gaussian process. For modeling discrete ordinal data we assume a continuous latent variable follows the logistic distribution and for continuous data a Gaussian mixed effects model is used. For the event time variable, an accelerated failure time model is considered under different distributional assumptions. For parameter estimation, a Bayesian approach using Markov Chain Monte Carlo is adopted. The performance of the proposed methods is illustrated using some simulation studies. A real data set is also analyzed, where different model structures are used. Model comparison is performed using a variety of statistical criteria.     

BAYESIAN HIDDEN MARKOV MODELS FOR LONGITUDINAL COUNTS

This paper describes a Bayesian approach to modelling carcinogenity in animal studies where the data consist of counts of the number of tumours present over time. It compares two autoregressive hidden Markov models. One of them models the transitions between three latent states: an inactive transient state, a multiplying state for increasing counts and a reducing state for decreasing counts. The second model introduces a fourth tied state to describe non‐zero observations that are neither increasing nor decreasing. Both these models can model the length of stay upon entry of a state. A discrete constant hazards waiting time distribution is used to model the time to onset of tumour growth. Our models describe between‐animal‐variability by a single hierarchy of random effects and the within‐animal variation by first‐order serial dependence. They can be extended to higher‐order serial dependence and multi‐level hierarchies. Analysis of data from animal experiments comparing the influence of two genes leads to conclusions that differ from those of Dunson (2000). The observed data likelihood defines an information criterion to assess the predictive properties of the three‐ and four‐state models. The deviance information criterion is appropriately defined for discrete parameters.     

A joint-modeling approach to assess the impact of biomarker variability on the risk of developing clinical outcome

In some clinical trials and epidemiologic studies, investigators are interested in knowing whether the variability of a biomarker is independently predictive of clinical outcomes. This question is often addressed via a naïve approach where a sample-based estimate (e.g., standard deviation) is calculated as a surrogate for the “true” variability and then used in regression models as a covariate assumed to be free of measurement error. However, it is well known that the measurement error in covariates causes underestimation of the true association. The issue of underestimation can be substantial when the precision is low because of limited number of measures per subject. The joint analysis of survival data and longitudinal data enables one to account for the measurement error in longitudinal data and has received substantial attention in recent years. In this paper we propose a joint model to assess the predictive effect of biomarker variability. The joint model consists of two linked sub-models, a linear mixed model with patient-specific variance for longitudinal data and a full parametric Weibull distribution for survival data, and the association between two models is induced by a latent Gaussian process. Parameters in the joint model are estimated under Bayesian framework and implemented using Markov chain Monte Carlo (MCMC) methods with WinBUGS software. The method is illustrated in the Ocular Hypertension Treatment Study to assess whether the variability of intraocular pressure is an independent risk of primary open-angle glaucoma. The performance of the method is also assessed by simulation studies.     

Multilevel models for longitudinal data

Summary.  Repeated measures and repeated events data have a hierarchical structure which can be analysed by using multilevel models. A growth curve model is an example of a multilevel random-coefficients model, whereas a discrete time event history model for recurrent events can be fitted as a multilevel logistic regression model. The paper describes extensions to the basic growth curve model to handle auto-correlated residuals, multiple-indicator latent variables and correlated growth processes, and event history models for correlated event processes. The multilevel approach to the analysis of repeated measures data is contrasted with structural equation modelling. The methods are illustrated in analyses of children's growth, changes in social and political attitudes, and the interrelationship between partnership transitions and childbearing.     

From distance sampling to spatial capture–recapture

Distance sampling and capture–recapture are the two most widely used wildlife abundance estimation methods. capture–recapture methods have only recently incorporated models for spatial distribution and there is an increasing tendency for distance sampling methods to incorporated spatial models rather than to rely on partly design-based spatial inference. In this overview we show how spatial models are central to modern distance sampling and that spatial capture–recapture models arise as an extension of distance sampling methods. Depending on the type of data recorded, they can be viewed as particular kinds of hierarchical binary regression, Poisson regression, survival or time-to-event models, with individuals’ locations as latent variables and a spatial model as the latent variable distribution. Incorporation of spatial models in these two methods provides new opportunities for drawing explicitly spatial inferences. Areas of likely future development include more sophisticated spatial and spatio-temporal modelling of individuals’ locations and movements, new methods for integrating spatial capture–recapture and other kinds of ecological survey data, and methods for dealing with the recapture uncertainty that often arise when “capture” consists of detection by a remote device like a camera trap or microphone.     

Mixture models in survival analysis: Are they worth the risk?

There has been a recurring interest in models for survival data which hypothesize subpopulations of individuals highly susceptible to some type of adverse event. Other individuals are assumed to be at much less risk. Most commonly, in clinical trials, these models attempt to estimate the fraction of patients cured of disease. The use of such models is examined, and the likelihood function is advocated as an informative inference tool.     

Bayesian variable selection in a finite mixture of linear mixed-effects models

Mixture of linear mixed-effects models has received considerable attention in longitudinal studies, including medical research, social science and economics. The inferential question of interest is often the identification of critical factors that affect the responses. We consider a Bayesian approach to select the important fixed and random effects in the finite mixture of linear mixed-effects models. To accomplish our goal, latent variables are introduced to facilitate the identification of influential fixed and random components and to classify the membership of observations in the longitudinal data. A spike-and-slab prior for the regression coefficients is adopted to sidestep the potential complications of highly collinear covariates and to handle large p and small n issues in the variable selection problems. Here we employ Markov chain Monte Carlo (MCMC) sampling techniques for posterior inferences and explore the performance of the proposed method in simulation studies, followed by an actual psychiatric data analysis concerning depressive disorder.     

A Comparative Study of Observation- and Parameter-driven Zero-inflated Poisson Models for Longitudinal Count Data

Longitudinal count data with excessive zeros frequently occur in social, biological, medical, and health research. To model such data, zero-inflated Poisson (ZIP) models are commonly used, after separating zero and positive responses. As longitudinal count responses are likely to be serially correlated, such separation may destroy the underlying serial correlation structure. To overcome this problem recently observation- and parameter-driven modelling approaches have been proposed. In the observation-driven model, the response at a specific time point is modelled through the responses at previous time points after incorporating serial correlation. One limitation of the observation-driven model is that it fails to accommodate the presence of any possible over-dispersion, which frequently occurs in the count responses. This limitation is overcome in a parameter-driven model, where the serial correlation is captured through the latent process using random effects. We compare the results obtained by the two models. A quasi-likelihood approach has been developed to estimate the model parameters. The methodology is illustrated with analysis of two real life datasets. To examine model performance the models are also compared through a simulation study.     

Nonparametric estimators for a survivor function of paired recurrent events

Recurrent event data arise in longitudinal studies where each study subject may experience multiple events during the follow-up. In many situations in survival studies, pairs of individuals can potentially experience recurrent events. The analysis of such data is not straightforward as it involves two kinds of dependences, namely, dependence between the individuals in the same pair and dependence among a sequence of pairs. In the present paper, we introduce a new stochastic model for the analysis of such recurrent event data. Nonparametric estimators for a bivariate survivor function are developed. Asymptotic properties of the estimators are discussed. Simulation studies are carried out to assess the finite sample properties of the estimator. We illustrate the procedure with real life data on eye disease.     

Jointly modeling skew longitudinal survival data with missingness and mismeasured covariates

Jointly modeling longitudinal and survival data has been an active research area. Most researches focus on improving the estimating efficiency but ignore many data features frequently encountered in practice. In the current study, we develop the joint models that concurrently accounting for longitudinal and survival data with multiple features. Specifically, the proposed model handles skewness, missingness and measurement errors in covariates which are typically observed in the collection of longitudinal survival data from many studies. We employ a Bayesian inferential method to make inference on the proposed model. We applied the proposed model to an real data study. A few alternative models under different conditions are compared. We conduct extensive simulations in order to evaluate how the method works.     

Testing for constancy in varying coefficient models

We consider varying coefficient models, which are an extension of the classical linear regression models in the sense that the regression coefficients are replaced by functions in certain variables (for example, time), the covariates are also allowed to depend on other variables. Varying coefficient models are popular in longitudinal data and panel data studies, and have been applied in fields such as finance and health sciences. We consider longitudinal data and estimate the coefficient functions by the flexible B-spline technique. An important question in a varying coefficient model is whether an estimated coefficient function is statistically different from a constant (or zero). We develop testing procedures based on the estimated B-spline coefficients by making use of nice properties of a B-spline basis. Our method allows longitudinal data where repeated measurements for an individual can be correlated. We obtain the asymptotic null distribution of the test statistic. The power of the proposed testing procedures are illustrated on simulated data where we highlight the importance of including the correlation structure of the response variable and on real data.     

Longitudinal data: different approaches in the context of item-response theory models

In this paper, some extended Rasch models are analyzed in the presence of longitudinal measurements of a latent variable. Two main approaches, multidimensional and multilevel, are compared: we investigate the different information that can be obtained from the latent variable, and we give advice on the use of the different kinds of models. The multidimensional and multilevel approaches are illustrated with a simulation study and with a longitudinal study on the health-related quality of life in terminal cancer patients.     

Joint Modelling of Survival and Longitudinal Data with Informative Observation Times

In this paper, we consider the joint modelling of survival and longitudinal data with informative observation time points. The survival model and the longitudinal model are linked via random effects, for which no distribution assumption is required under our estimation approach. The estimator is shown to be consistent and asymptotically normal. The proposed estimator and its estimated covariance matrix can be easily calculated. Simulation studies and an application to a primary biliary cirrhosis study are also provided.