Robust Joint Modeling of Longitudinal Measurements and Competing Risks Failure Time Data

Existing methods for joint modeling of longitudinal measurements and survival data can be highly influenced by outliers in the longitudinal outcome. We propose a joint model for analysis of longitudinal measurements and competing risks failure time data which is robust in the presence of outlying longitudinal observations during follow‐up. Our model consists of a linear mixed effects sub‐model for the longitudinal outcome and a proportional cause‐specific hazards frailty sub‐model for the competing risks data, linked together by latent random effects. Instead of the usual normality assumption for measurement errors in the linear mixed effects sub‐model, we adopt a t ‐distribution which has a longer tail and thus is more robust to outliers. We derive an EM algorithm for the maximum likelihood estimates of the parameters and estimate their standard errors using a profile likelihood method. The proposed method is evaluated by simulation studies and is applied to a scleroderma lung study (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Joint Modeling of Multivariate Longitudinal Data and Competing Risks Using Multiphase Sub-models

In many clinical studies that involve follow-up, it is common to observe one or more sequences of longitudinal measurements, as well as one or more time to event outcomes. A competing risks situation arises when the probability of occurrence of one event is altered/hindered by another time to event. Recently, there has been much attention paid to the joint analysis of a single longitudinal response and a single time to event outcome, when the missing data mechanism in the longitudinal process is non-ignorable. We, in this paper, propose an extension where multiple longitudinal responses are jointly modeled with competing risks (multiple time to events). Our shared parameter joint model consists of a system of multiphase non-linear mixed effects sub-models for the multiple longitudinal responses, and a system of cause-specific non-proportional hazards frailty sub-models for competing risks, with associations among multiple longitudinal responses and competing risks modeled using latent parameters. The joint model is applied to a data set of patients who are on mechanical circulatory support and are awaiting heart transplant, using readily available software. While on the mechanical circulatory support, patient liver and renal functions may worsen and these in turn may influence one of the two possible competing outcomes: (i) death before transplant; (ii) transplant. In one application, we propose a system of multiphase cause-specific non-proportional hazard sub-model where frailty can be time varying. Performance under different scenarios was assessed using simulation studies. By using the proposed joint modeling of the multiphase sub-models, one can identify: (i) non-linear trends in multiple longitudinal outcomes; (ii) time-varying hazards and cumulative incidence functions of the competing risks; (iii) identify risk factors for the both types of outcomes, where the effect may or may not change with time; and (iv) assess the association between multiple longitudinal and competing risks outcomes, where the association may or may not change with time.     

Modeling longitudinal data with nonparametric multiplicative random effects jointly with survival data

Summary .   In clinical studies, longitudinal biomarkers are often used to monitor disease progression and failure time. Joint modeling of longitudinal and survival data has certain advantages and has emerged as an effective way to mutually enhance information. Typically, a parametric longitudinal model is assumed to facilitate the likelihood approach. However, the choice of a proper parametric model turns out to be more elusive than models for standard longitudinal studies in which no survival endpoint occurs. In this article, we propose a nonparametric multiplicative random effects model for the longitudinal process, which has many applications and leads to a flexible yet parsimonious nonparametric random effects model. A proportional hazards model is then used to link the biomarkers and event time. We use B-splines to represent the nonparametric longitudinal process, and select the number of knots and degrees based on a version of the Akaike information criterion (AIC). Unknown model parameters are estimated through maximizing the observed joint likelihood, which is iteratively maximized by the Monte Carlo Expectation Maximization (MCEM) algorithm. Due to the simplicity of the model structure, the proposed approach has good numerical stability and compares well with the competing parametric longitudinal approaches. The new approach is illustrated with primary biliary cirrhosis (PBC) data, aiming to capture nonlinear patterns of serum bilirubin time courses and their relationship with survival time of PBC patients.     

A Semiparametric Joint Model for Longitudinal and Survival Data with Application to Hemodialysis Study

Summary .  In many longitudinal clinical studies, the level and progression rate of repeatedly measured biomarkers on each subject quantify the severity of the disease and that subject's susceptibility to progression of the disease. It is of scientific and clinical interest to relate such quantities to a later time-to-event clinical endpoint such as patient survival. This is usually done with a shared parameter model. In such models, the longitudinal biomarker data and the survival outcome of each subject are assumed to be conditionally independent given subject-level severity or susceptibility (also called frailty in statistical terms). In this article, we study the case where the conditional distribution of longitudinal data is modeled by a linear mixed-effect model, and the conditional distribution of the survival data is given by a Cox proportional hazard model. We allow unknown regression coefficients and time-dependent covariates in both models. The proposed estimators are maximizers of an exact correction to the joint log likelihood with the frailties eliminated as nuisance parameters, an idea that originated from correction of covariate measurement error in measurement error models. The corrected joint log likelihood is shown to be asymptotically concave and leads to consistent and asymptotically normal estimators. Unlike most published methods for joint modeling, the proposed estimation procedure does not rely on distributional assumptions of the frailties. The proposed method was studied in simulations and applied to a data set from the Hemodialysis Study.     

Semiparametric approaches for joint modeling of longitudinal and survival data with time-varying coefficients

Summary .   We study joint modeling of survival and longitudinal data. There are two regression models of interest. The primary model is for survival outcomes, which are assumed to follow a time-varying coefficient proportional hazards model. The second model is for longitudinal data, which are assumed to follow a random effects model. Based on the trajectory of a subject's longitudinal data, some covariates in the survival model are functions of the unobserved random effects. Estimated random effects are generally different from the unobserved random effects and hence this leads to covariate measurement error. To deal with covariate measurement error, we propose a local corrected score estimator and a local conditional score estimator. Both approaches are semiparametric methods in the sense that there is no distributional assumption needed for the underlying true covariates. The estimators are shown to be consistent and asymptotically normal. However, simulation studies indicate that the conditional score estimator outperforms the corrected score estimator for finite samples, especially in the case of relatively large measurement error. The approaches are demonstrated by an application to data from an HIV clinical trial.     

A semi-parametric shared parameter model to handle nonmonotone nonignorable missingness

Summary .  Longitudinal studies often generate incomplete response patterns according to a missing not at random mechanism. Shared parameter models provide an appealing framework for the joint modelling of the measurement and missingness processes, especially in the nonmonotone missingness case, and assume a set of random effects to induce the interdependence. Parametric assumptions are typically made for the random effects distribution, violation of which leads to model misspecification with a potential effect on the parameter estimates and standard errors. In this article we avoid any parametric assumption for the random effects distribution and leave it completely unspecified. The estimation of the model is then made using a semi-parametric maximum likelihood method. Our proposal is illustrated on a randomized longitudinal study on patients with rheumatoid arthritis exhibiting nonmonotone missingness.     

Joint Analysis of Survival Time and Longitudinal Categorical Outcomes

In biomedical or public health research, it is common for both survival time and longitudinal categorical outcomes to be collected for a subject, along with the subject’s characteristics or risk factors. Investigators are often interested in finding important variables for predicting both survival time and longitudinal outcomes which could be correlated within the same subject. Existing approaches for such joint analyses deal with continuous longitudinal outcomes. New statistical methods need to be developed for categorical longitudinal outcomes. We propose to simultaneously model the survival time with a stratified Cox proportional hazards model and the longitudinal categorical outcomes with a generalized linear mixed model. Random effects are introduced to account for the dependence between survival time and longitudinal outcomes due to unobserved factors. The Expectation–Maximization (EM) algorithm is used to derive the point estimates for the model parameters, and the observed information matrix is adopted to estimate their asymptotic variances. Asymptotic properties for our proposed maximum likelihood estimators are established using the theory of empirical processes. The method is demonstrated to perform well in finite samples via simulation studies. We illustrate our approach with data from the Carolina Head and Neck Cancer Study (CHANCE) and compare the results based on our simultaneous analysis and the separately conducted analyses using the generalized linear mixed model and the Cox proportional hazards model. Our proposed method identifies more predictors than by separate analyses.     

On the Statistical Inference from Survival Experiments with Two Types of Failure

Formulated in terms of latent failure times a survival model with two dependent competing risks is considered. Using this model, it is possible to introduce the probabilistic characteristics of each type of failure contribution to the joint effect of two damaging agents. Confidence bounds for such characteristics are given.     

A Bayesian multi-risks survival (MRS) model in the presence of double censorings

Semi-competing risks data include the time to a nonterminating event and the time to a terminating event, while competing risks data include the time to more than one terminating event. Our work is motivated by a prostate cancer study, which has one nonterminating event and two terminating events with both semi-competing risks and competing risks present as well as two censoring times. In this paper, we propose a new multi-risks survival (MRS) model for this type of data. In addition, the proposed MRS model can accommodate noninformative right-censoring times for nonterminating and terminating events. Properties of the proposed MRS model are examined in detail. Theoretical and empirical results show that the estimates of the cumulative incidence function for a nonterminating event may be biased if the information on a terminating event is ignored. A Markov chain Monte Carlo sampling algorithm is also developed. Our methodology is further assessed using simulations and also an analysis of the real data from a prostate cancer study. As a result, a prostate-specific antigen velocity greater than 2.0 ng/mL per year and higher biopsy Gleason scores are positively associated with a shorter time to death due to prostate cancer.     

Competing Risks and Time‐Dependent Covariates

Time‐dependent covariates are frequently encountered in regression analysis for event history data and competing risks. They are often essential predictors, which cannot be substituted by time‐fixed covariates. This study briefly recalls the different types of time‐dependent covariates, as classified by Kalbfleisch and Prentice [The Statistical Analysis of Failure Time Data, Wiley, New York, 2002] with the intent of clarifying their role and emphasizing the limitations in standard survival models and in the competing risks setting. If random (internal) time‐dependent covariates are to be included in the modeling process, then it is still possible to estimate cause‐specific hazards but prediction of the cumulative incidences and survival probabilities based on these is no longer feasible. This article aims at providing some possible strategies for dealing with these prediction problems. In a multi‐state framework, a first approach uses internal covariates to define additional (intermediate) transient states in the competing risks model. Another approach is to apply the landmark analysis as described by van Houwelingen [Scandinavian Journal of Statistics 2007, 34 , 70–85] in order to study cumulative incidences at different subintervals of the entire study period. The final strategy is to extend the competing risks model by considering all the possible combinations between internal covariate levels and cause‐specific events as final states. In all of those proposals, it is possible to estimate the changes/differences of the cumulative risks associated with simple internal covariates. An illustrative example based on bone marrow transplant data is presented in order to compare the different methods.     

Joint modeling of survival and longitudinal data: likelihood approach revisited

The maximum likelihood approach to jointly model the survival time and its longitudinal covariates has been successful to model both processes in longitudinal studies. Random effects in the longitudinal process are often used to model the survival times through a proportional hazards model, and this invokes an EM algorithm to search for the maximum likelihood estimates (MLEs). Several intriguing issues are examined here, including the robustness of the MLEs against departure from the normal random effects assumption, and difficulties with the profile likelihood approach to provide reliable estimates for the standard error of the MLEs. We provide insights into the robustness property and suggest to overcome the difficulty of reliable estimates for the standard errors by using bootstrap procedures. Numerical studies and data analysis illustrate our points.     

An estimator for the proportional hazards model with multiple longitudinal covariates measured with error

In many longitudinal studies, it is of interest to characterize the relationship between a time-to-event (e.g. survival) and several time-dependent and time-independent covariates. Time-dependent covariates are generally observed intermittently and with error. For a single time-dependent covariate, a popular approach is to assume a joint longitudinal data-survival model, where the time-dependent covariate follows a linear mixed effects model and the hazard of failure depends on random effects and time-independent covariates via a proportional hazards relationship. Regression calibration and likelihood or Bayesian methods have been advocated for implementation; however, generalization to more than one time-dependent covariate may become prohibitive. For a single time-dependent covariate, Tsiatis and Davidian (2001) have proposed an approach that is easily implemented and does not require an assumption on the distribution of the random effects. This technique may be generalized to multiple, possibly correlated, time-dependent covariates, as we demonstrate. We illustrate the approach via simulation and by application to data from an HIV clinical trial.     

Robust joint modeling of longitudinal measurements and time to event data using normal/independent distributions: A Bayesian approach

Joint modeling of longitudinal data and survival data has been used widely for analyzing AIDS clinical trials, where a biological marker such as CD4 count measurement can be an important predictor of survival. In most of these studies, a normal distribution is used for modeling longitudinal responses, which leads to vulnerable inference in the presence of outliers in longitudinal measurements. Powerful distributions for robust analysis are normal/independent distributions, which include univariate and multivariate versions of the Student's t, the slash and the contaminated normal distributions in addition to the normal. In this paper, a linear‐mixed effects model with normal/independent distribution for both random effects and residuals and Cox's model for survival time are used. For estimation, a Bayesian approach using Markov Chain Monte Carlo is adopted. Some simulation studies are performed for illustration of the proposed method. Also, the method is illustrated on a real AIDS data set and the best model is selected using some criteria.     

A joint model for repeated events of different types and multiple longitudinal outcomes with application to a follow‐up study of patients after kidney transplant

This paper presents an extension of the joint modeling strategy for the case of multiple longitudinal outcomes and repeated infections of different types over time, motivated by postkidney transplantation data. Our model comprises two parts linked by shared latent terms. On the one hand is a multivariate mixed linear model with random effects, where a low‐rank thin‐plate spline function is incorporated to collect the nonlinear behavior of the different profiles over time. On the other hand is an infection‐specific Cox model, where the dependence between different types of infections and the related times of infection is through a random effect associated with each infection type to catch the within dependence and a shared frailty parameter to capture the dependence between infection types. We implemented the parameterization used in joint models which uses the fitted longitudinal measurements as time‐dependent covariates in a relative risk model. Our proposed model was implemented in OpenBUGS using the MCMC approach.     

Joint models for multivariate longitudinal and multivariate survival data

Joint modeling of longitudinal and survival data is becoming increasingly essential in most cancer and AIDS clinical trials. We propose a likelihood approach to extend both longitudinal and survival components to be multidimensional. A multivariate mixed effects model is presented to explicitly capture two different sources of dependence among longitudinal measures over time as well as dependence between different variables. For the survival component of the joint model, we introduce a shared frailty, which is assumed to have a positive stable distribution, to induce correlation between failure times. The proposed marginal univariate survival model, which accommodates both zero and nonzero cure fractions for the time to event, is then applied to each marginal survival function. The proposed multivariate survival model has a proportional hazards structure for the population hazard, conditionally as well as marginally, when the baseline covariates are specified through a specific mechanism. In addition, the model is capable of dealing with survival functions with different cure rate structures. The methodology is specifically applied to the International Breast Cancer Study Group (IBCSG) trial to investigate the relationship between quality of life, disease-free survival, and overall survival.     

Nonparametric maximum likelihood estimation for competing risks survival data subject to interval censoring and truncation

We derive the nonparametric maximum likelihood estimate (NPMLE) of the cumulative incidence functions for competing risks survival data subject to interval censoring and truncation. Since the cumulative incidence function NPMLEs give rise to an estimate of the survival distribution which can be undefined over a potentially larger set of regions than the NPMLE of the survival function obtained ignoring failure type, we consider an alternative pseudolikelihood estimator. The methods are then applied to data from a cohort of injecting drug users in Thailand susceptible to infection from HIV-1 subtypes B and E.     

Latent-Model Robustness in Joint Models for a Primary Endpoint and a Longitudinal Process

Summary .  Joint modeling of a primary response and a longitudinal process via shared random effects is widely used in many areas of application. Likelihood-based inference on joint models requires model specification of the random effects. Inappropriate model specification of random effects can compromise inference. We present methods to diagnose random effect model misspecification of the type that leads to biased inference on joint models. The methods are illustrated via application to simulated data, and by application to data from a study of bone mineral density in perimenopausal women and data from an HIV clinical trial.     

Sample size determination for hierarchical longitudinal designs with differential attrition rates

Summary .   We consider the problem of sample size determination for three-level mixed-effects linear regression models for the analysis of clustered longitudinal data. Three-level designs are used in many areas, but in particular, multicenter randomized longitudinal clinical trials in medical or health-related research. In this case, level 1 represents measurement occasion, level 2 represents subject, and level 3 represents center. The model we consider involves random effects of the time trends at both the subject level and the center level. In the most common case, we have two random effects (constant and a single trend), at both subject and center levels. The approach presented here is general with respect to sampling proportions, number of groups, and attrition rates over time. In addition, we also develop a cost model, as an aid in selecting the most parsimonious of several possible competing models (i.e., different combinations of centers, subjects within centers, and measurement occasions). We derive sample size requirements (i.e., power characteristics) for a test of treatment-by-time interaction(s) for designs based on either subject-level or cluster-level randomization. The general methodology is illustrated using two characteristic examples.     

Joint inference for competing risks data using multiple endpoints

Competing risks data are commonly encountered in randomized clinical trials and observational studies. This paper considers the situation where the ending statuses of competing events have different clinical interpretations and/or are of simultaneous interest. In clinical trials, often more than one competing event has meaningful clinical interpretations even though the trial effects of different events could be different or even opposite to each other. In this paper, we develop estimation procedures and inferential properties for the joint use of multiple cumulative incidence functions (CIFs). Additionally, by incorporating longitudinal marker information, we develop estimation and inference procedures for weighted CIFs and related metrics. The proposed methods are applied to a COVID-19 in-patient treatment clinical trial, where the outcomes of COVID-19 hospitalization are either death or discharge from the hospital, two competing events with completely different clinical implications.     

Joint Modeling for Cognitive Trajectory and Risk of Dementia in the Presence of Death

Summary .  Dementia is characterized by accelerated cognitive decline before and after diagnosis as compared to normal aging. It has been known that cognitive impairment occurs long before the diagnosis of dementia. For individuals who develop dementia, it is important to determine the time when the rate of cognitive decline begins to accelerate and the subsequent gap time to dementia diagnosis. For normal aging individuals, it is also useful to understand the trajectory of cognitive function until their death. A Bayesian change-point model is proposed to fit the trajectory of cognitive function for individuals who develop dementia. In real life, people in older ages are subject to two competing risks, e.g., dementia and dementia-free death. Because the majority of people do not develop dementia, a mixture model is used for survival data with competing risks, which consists of dementia onset time after the change point of cognitive function decline for demented individuals and death time for nondemented individuals. The cognitive trajectories and the survival process are modeled jointly and the parameters are estimated using the Markov chain Monte Carlo method. Using data from the Honolulu Asia Aging Study, we show the trajectories of cognitive function and the effect of education, apolipoprotein E 4 genotype, and hypertension on cognitive decline and the risk of dementia.