Functional approach of flexibly modelling generalized longitudinal data and survival time

We propose a flexible functional approach for modelling generalized longitudinal data and survival time using principal components. In the proposed model the longitudinal observations can be continuous or categorical data, such as Gaussian, binomial or Poisson outcomes. We generalize the traditional joint models that treat categorical data as continuous data by using some transformations, such as CD4 counts. The proposed model is data-adaptive, which does not require pre-specified functional forms for longitudinal trajectories and automatically detects characteristic patterns. The longitudinal trajectories observed with measurement error or random error are represented by flexible basis functions through a possibly nonlinear link function, combining dimension reduction techniques resulting from functional principal component (FPC) analysis. The relationship between the longitudinal process and event history is assessed using a Cox regression model. Although the proposed model inherits the flexibility of non-parametric methods, the estimation procedure based on the EM algorithm is still parametric in computation, and thus simple and easy to implement. The computation is simplified by dimension reduction for random coefficients or FPC scores. An iterative selection procedure based on Akaike information criterion (AIC) is proposed to choose the tuning parameters, such as the knots of spline basis and the number of FPCs, so that appropriate degree of smoothness and fluctuation can be addressed. The effectiveness of the proposed approach is illustrated through a simulation study, followed by an application to longitudinal CD4 counts and survival data which were collected in a recent clinical trial to compare the efficiency and safety of two antiretroviral drugs.     

A modified two-stage approach for joint modelling of longitudinal and time-to-event data

Joint models for longitudinal and time-to-event data have been applied in many different fields of statistics and clinical studies. However, the main difficulty these models have to face with is the computational problem. The requirement for numerical integration becomes severe when the dimension of random effects increases. In this paper, a modified two-stage approach has been proposed to estimate the parameters in joint models. In particular, in the first stage, the linear mixed-effects models and best linear unbiased predictorsare applied to estimate parameters in the longitudinal submodel. In the second stage, an approximation of the fully joint log-likelihood is proposed using the estimated the values of these parameters from the longitudinal submodel. Survival parameters are estimated bymaximizing the approximation of the fully joint log-likelihood. Simulation studies show that the approach performs well, especially when the dimension of random effects increases. Finally, we implement this approach on AIDS data.     

Joint modelling of longitudinal binary data and survival data

The medical costs in an ageing society substantially increase when the incidences of chronic diseases, disabilities and inability to live independently are high. Healthy lifestyles not only affect elderly individuals but also influence the entire community. When assessing treatment efficacy, survival and quality of life should be considered simultaneously. This paper proposes the joint likelihood approach for modelling survival and longitudinal binary covariates simultaneously. Because some unobservable information is present in the model, the Monte Carlo EM algorithm and Metropolis-Hastings algorithm are used to find the estimators. Monte Carlo simulations are performed to evaluate the performance of the proposed model based on the accuracy and precision of the estimates. Real data are used to demonstrate the feasibility of the proposed model.     

A flexible link for joint modelling longitudinal and survival data accounting for individual longitudinal heterogeneity

Statistical Methods & Applications - This work aims at jointly modelling longitudinal and survival HIV data by considering the sharing of a set of parameters of interest. For the CD4...     

Comparing crossing hazard rate functions by joint modelling of survival and longitudinal data

Abstract

It is one of the important issues in survival analysis to compare two hazard rate functions to evaluate treatment effect. It is quite common that the two hazard rate functions cross each other at one or more unknown time points, representing temporal changes of the treatment effect. In certain applications, besides survival data, we also have related longitudinal data available regarding some time-dependent covariates. In such cases, a joint model that accommodates both types of data can allow us to infer the association between the survival and longitudinal data and to assess the treatment effect better. In this paper, we propose a modelling approach for comparing two crossing hazard rate functions by joint modelling survival and longitudinal data. Maximum likelihood estimation is used in estimating the parameters of the proposed joint model using the EM algorithm. Asymptotic properties of the maximum likelihood estimators are studied. To illustrate the virtues of the proposed method, we compare the performance of the proposed method with several existing methods in a simulation study. Our proposed method is also demonstrated using a real dataset obtained from an HIV clinical trial.     

Two-step and likelihood methods for joint models of longitudinal and survival data

We compare the commonly used two-step methods and joint likelihood method for joint models of longitudinal and survival data via extensive simulations. The longitudinal models include LME, GLMM, and NLME models, and the survival models include Cox models and AFT models. We find that the full likelihood method outperforms the two-step methods for various joint models, but it can be computationally challenging when the dimension of the random effects in the longitudinal model is not small. We thus propose an approximate joint likelihood method which is computationally efficient. We find that the proposed approximation method performs well in the joint model context, and it performs better for more “continuous” longitudinal data. Finally, a real AIDS data example shows that patients with higher initial viral load or lower initial CD4 are more likely to drop out earlier during an anti-HIV treatment.     

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.

     

Prediction of transplant-free survival in idiopathic pulmonary fibrosis patients using joint models for event times and mixed multivariate longitudinal data

We implement a joint model for mixed multivariate longitudinal measurements, applied to the prediction of time until lung transplant or death in idiopathic pulmonary fibrosis. Specifically, we formulate a unified Bayesian joint model for the mixed longitudinal responses and time-to-event outcomes. For the longitudinal model of continuous and binary responses, we investigate multivariate generalized linear mixed models using shared random effects. Longitudinal and time-to-event data are assumed to be independent conditional on available covariates and shared parameters. A Markov chain Monte Carlo algorithm, implemented in OpenBUGS, is used for parameter estimation. To illustrate practical considerations in choosing a final model, we fit 37 different candidate models using all possible combinations of random effects and employ a deviance information criterion to select a best-fitting model. We demonstrate the prediction of future event probabilities within a fixed time interval for patients utilizing baseline data, post-baseline longitudinal responses, and the time-to-event outcome. The performance of our joint model is also evaluated in simulation studies.     

A joint model for longitudinal data profiles and associated event risks with application to a depression study

Summary.  In many longitudinal studies, a subject's response profile is closely associated with his or her risk of experiencing a related event. Examples of such event risks include recurrence of disease, relapse, drop-out and non-compliance. When evaluating the effect of a treatment, it is sometimes of interest to consider the joint process consisting of both the response and the risk of an associated event. Motivated by a prevention of depression study among patients with malignant melanoma, we examine a joint model that incorporates the risk of discontinuation into the analysis of serial depression measures. We present a maximum likelihood estimator for the mean response and event risk vectors. We test hypotheses about functions of mean depression and withdrawal risk profiles from our joint model, predict depression from updated patient histories, characterize associations between components of the joint process and estimate the probability that a patient's depression and risk of withdrawal exceed specified levels. We illustrate the application of our joint model by using the depression data.     

Joint modelling of event counts and survival times

Summary.  In studies of recurrent events, such as epileptic seizures, there can be a large amount of information about a cohort over a period of time, but current methods for these data are often unable to utilize all of the available information. The paper considers data which include post-treatment survival times for individuals experiencing recurring events, as well as a measure of the base-line event rate, in the form of a pre-randomization event count. Standard survival analysis may treat this pre-randomization count as a covariate, but the paper proposes a parametric joint model based on an underlying Poisson process, which will give a more precise estimate of the treatment effect.     

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     

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.     

Frequentist and Bayesian approaches for a joint model for prostate cancer risk and longitudinal prostate-specific antigen data

The paper describes the use of frequentist and Bayesian shared-parameter joint models of longitudinal measurements of prostate-specific antigen (PSA) and the risk of prostate cancer (PCa). The motivating dataset corresponds to the screening arm of the Spanish branch of the European Randomized Screening for Prostate Cancer study. The results show that PSA is highly associated with the risk of being diagnosed with PCa and that there is an age-varying effect of PSA on PCa risk. Both the frequentist and Bayesian paradigms produced very close parameter estimates and subsequent 95% confidence and credibility intervals. Dynamic estimations of disease-free probabilities obtained using Bayesian inference highlight the potential of joint models to guide personalized risk-based screening strategies.     

A practical extension of the generalized estimating equation approach for longitudinal data

Liang and Zeger (1986) proposed an extension of generalized linear models to the analysis of longitudinal data. In their formulation, a common dispersion parameter assumption across observation times is required. However, this assumption is not expected to hold in most situations. Park (1993) proposed a simple extension of Liang and Zeger's formulation to allow for different dispersion parameters for each time point. The proposed model is easy to apply without heavy computations and useful to handle the cases when variations in over-dispersion over time exist. In this paper, we focus on evaluating the effect of additional dispersion parameters on the estimators of model parameters. Through a Monte Carlo simulation study, efficiency of Park's method is compared with the Liang and Zeger's method.     

A comparison of procedures to correct for base-line differences in the analysis of continuous longitudinal data: a case-study

Summary.  The main advantage of longitudinal studies is that they can distinguish changes over time within individuals (longitudinal effects) from differences between subjects at the start of the study (base-line characteristics; cross-sectional effects). Often, especially in observational studies, subjects are very heterogeneous at base-line, and one may want to correct for this, when doing inferences for the longitudinal trends. Three procedures for base-line correction are compared in the context of linear mixed models for continuous longitudinal data. All procedures are illustrated extensively by using data from an experiment which aimed at studying the relationship between the post-operative evolution of the functional status of elderly hip fracture patients and their preoperative neurocognitive status.     

Simultaneous Bayesian modeling of longitudinal and survival data in breast cancer patients

Abstract

Using simultaneous Bayesian modeling, an attempt is made to analyze data on the size of lymphedema occurring in the arms of breast cancer patients after breast cancer surgery (as the longitudinal data) and the time interval for disease progression (as the time-to-event occurrence). A model based on a multivariate skew t distribution is shown to provide the best fit. This outcome was confirmed by simulation studies too.     

A Bayesian hierarchical model for categorical longitudinal data from a social survey of immigrants

Summary.  The paper investigates a Bayesian hierarchical model for the analysis of categorical longitudinal data from a large social survey of immigrants to Australia. Data for each subject are observed on three separate occasions, or waves, of the survey. One of the features of the data set is that observations for some variables are missing for at least one wave. A model for the employment status of immigrants is developed by introducing, at the first stage of a hierarchical model, a multinomial model for the response and then subsequent terms are introduced to explain wave and subject effects. To estimate the model, we use the Gibbs sampler, which allows missing data for both the response and the explanatory variables to be imputed at each iteration of the algorithm, given some appropriate prior distributions. After accounting for significant covariate effects in the model, results show that the relative probability of remaining unemployed diminished with time following arrival in Australia.     

Bayesian analysis of joint mean and covariance models for longitudinal data

Efficient estimation of the regression coefficients in longitudinal data analysis requires a correct specification of the covariance structure. If misspecification occurs, it may lead to inefficient or biased estimators of parameters in the mean. One of the most commonly used methods for handling the covariance matrix is based on simultaneous modeling of the Cholesky decomposition. Therefore, in this paper, we reparameterize covariance structures in longitudinal data analysis through the modified Cholesky decomposition of itself. Based on this modified Cholesky decomposition, the within-subject covariance matrix is decomposed into a unit lower triangular matrix involving moving average coefficients and a diagonal matrix involving innovation variances, which are modeled as linear functions of covariates. Then, we propose a fully Bayesian inference for joint mean and covariance models based on this decomposition. A computational efficient Markov chain Monte Carlo method which combines the Gibbs sampler and Metropolis–Hastings algorithm is implemented to simultaneously obtain the Bayesian estimates of unknown parameters, as well as their standard deviation estimates. Finally, several simulation studies and a real example are presented to illustrate the proposed methodology.     

Joint modeling and estimation for longitudinal data with informative observation and terminal event times

ABSTRACT

In many longitudinal studies, there may exist informative observation times and a dependent terminal event that stops the follow-up. In this paper, we propose a joint model for analysis of longitudinal data with informative observation times and a dependent terminal event via two latent variables. Estimation procedures are developed for parameter estimation, and asymptotic properties of the proposed estimators are derived. Simulation studies demonstrate that the proposed method performs well for practical settings. An application to a bladder cancer study is illustrated.