A joint model of recurrent events and a terminal event with a nonparametric covariate function

We extend the shared frailty model of recurrent events and a dependent terminal event to allow for a nonparametric covariate function. We include a Gaussian random effect (frailty) in the intensity functions of both the recurrent and terminal events to capture correlation between the two processes. We employ the penalized cubic spline method to describe the nonparametric covariate function in the recurrent events model. We use Laplace approximation to evaluate the marginal penalized partial likelihood without a closed form. We also propose the variance estimates for regression coefficients. Numerical analysis results show that the proposed estimates perform well for both the nonparametric and parametric components. We apply this method to analyze the hospitalization rate of patients with heart failure in the presence of death.     

Joint analysis of multi-level repeated measures data and survival: an application to the end stage renal disease (ESRD) data

Shared random effects models have been increasingly common in the joint analyses of repeated measures (e.g. CD4 counts, hemoglobin levels) and a correlated failure time such as death. In this paper we study several shared random effects models in the multi-level repeated measures data setting with dependent failure times. Distinct random effects are used to characterize heterogeneity in repeated measures at different levels. The hazard of death may be dependent on random effects from various levels. To simplify the estimation procedure, we adopt the Gaussian quadrature technique with a piecewise log-linear baseline hazard for the death process, which can be conveniently implemented in the freely available software aML. As an example, we analyze repeated measures of hematocrit level and survival for end stage renal disease patients clustered within a randomly selected 126 dialysis centers in the U.S. renal data system data set. Our model is very comprehensive yet easy to implement, making it appealing to general statistical practitioners.     

General joint frailty model for recurrent event data with a dependent terminal event: Application to follicular lymphoma data

Many biomedical studies focus on delaying disease relapses and on prolonging survival. Usual methods only consider one event, often the first recurrence or death. However, ignoring the other recurrences may lead to biased results. The whole history of the disease should be considered for each patient. In addition, some diseases involve recurrences that can increase the risk of death. In this case, the death time may be dependent on the recurrent event history. We propose a joint frailty model to analyze recurrences and death simultaneously. Two gamma-distributed frailties take into account both the inter-recurrences dependence and the dependence between the recurrences and the survival times. We estimate separate parameters for disease recurrent event times and survival times in the joint frailty model to distinguish treatment effects and prognostic factors on these two types of events. We show how maximum penalized likelihood estimation can be applied to semiparametric estimation of the continuous hazard functions in the proposed joint frailty model with right censoring. We also propose parametrical approach. We evaluate the model by simulation studies and illustrate through a study of patients with follicular lymphoma.     

A comparison of frailty models for multivariate survival data

This paper reviews some of the main approaches to the analysis of multivariate censored survival data. Such data typically have correlated failure times. The correlation can be a consequence of the observational design, for example with clustered sampling and matching, or it can be a focus of interest as in genetic studies, longitudinal studies of recurrent events and other studies involving multiple measurements. We assume that the correlation between the failure or survival times can be accounted for by fixed or random frailty effects. We then compare the performance of conditional and mixture likelihood approaches to estimating models with these frailty effects in censored bivariate survival data. We find that the mixture methods are surprisingly robust to misspecification of the frailty distribution. The paper also contains an illustrative example on the times to onset of chest pain brought on by three endurance exercise tests during a drug treatment trial of heart patients.     

Nonproportional hazards and unobserved heterogeneity in clustered survival data: When can we tell the difference?

Multivariate survival data are frequently encountered in biomedical applications in the form of clustered failures (or recurrent events data). A popular way of analyzing such data is by using shared frailty models, which assume that the proportional hazards assumption holds conditional on an unobserved cluster-specific random effect. Such models are often incorporated in more complicated joint models in survival analysis. If the random effect distribution has finite expectation, then the conditional proportional hazards assumption does not carry over to the marginal models. It has been shown that, for univariate data, this makes it impossible to distinguish between the presence of unobserved heterogeneity (eg, due to missing covariates) and marginal nonproportional hazards. We show that time-dependent covariate effects may falsely appear as evidence in favor of a frailty model also in the case of clustered failures or recurrent events data, when the cluster size or number of recurrent events is small. When true unobserved heterogeneity is present, the presence of nonproportional hazards leads to overestimating the frailty effect. We show that this phenomenon is somewhat mitigated as the cluster size grows. We carry out a simulation study to assess the behavior of test statistics and estimators for frailty models in such contexts. The gamma, inverse Gaussian, and positive stable shared frailty models are contrasted using a novel software implementation for estimating semiparametric shared frailty models. Two main questions are addressed in the contexts of clustered failures and recurrent events: whether covariates with a time-dependent effect may appear as indication of unobserved heterogeneity and whether the additional presence of unobserved heterogeneity can be detected in this case. Finally, the practical implications are illustrated in a real-world data analysis example.     

Design and analysis of nested case–control studies for recurrent events subject to a terminal event

The process by which patients experience a series of recurrent events, such as hospitalizations, may be subject to death. In cohort studies, one strategy for analyzing such data is to fit a joint frailty model for the intensities of the recurrent event and death, which estimates covariate effects on the two event types while accounting for their dependence. When certain covariates are difficult to obtain, however, researchers may only have the resources to subsample patients on whom to collect complete data: one way is using the nested case–control (NCC) design, in which risk set sampling is performed based on a single outcome. We develop a general framework for the design of NCC studies in the presence of recurrent and terminal events and propose estimation and inference for a joint frailty model for recurrence and death using data arising from such studies. We propose a maximum weighted penalized likelihood approach using flexible spline models for the baseline intensity functions. Two standard error estimators are proposed: a sandwich estimator and a perturbation resampling procedure. We investigate operating characteristics of our estimators as well as design considerations via a simulation study and illustrate our methods using two studies: one on recurrent cardiac hospitalizations in patients with heart failure and the other on local recurrence and metastasis in patients with breast cancer.     

Regression analysis of multivariate current status data with dependent censoring: application to ankylosing spondylitis data

Multivariate current‐status failure time data consist of several possibly related event times of interest, in which the status of each event is determined at a single examination time. If the examination time is intrinsically related to the event times, the examination is referred to as dependent censoring and needs to be taken into account. Such data often occur in clinical studies and animal carcinogenicity experiments. To accommodate for possible dependent censoring, this paper proposes a joint frailty model for event times and dependent censoring time. We develop a likelihood approach using Gaussian quadrature techniques for obtaining maximum likelihood estimates. We conduct extensive simulation studies for investigating finite‐sample properties of the proposed method. We illustrate the proposed method with an analysis of patients with ankylosing spondylitis, where the examination time may be dependent on the event times of interest. Copyright © 2013 John Wiley & Sons, Ltd.     

Bayesian transformation cure frailty models with multivariate failure time data

We propose a class of transformation cure frailty models to accommodate a survival fraction in multivariate failure time data. Established through a general power transformation, this family of cure frailty models includes the proportional hazards and the proportional odds modeling structures as two special cases. Within the Bayesian paradigm, we obtain the joint posterior distribution and the corresponding full conditional distributions of the model parameters for the implementation of Gibbs sampling. Model selection is based on the conditional predictive ordinate statistic and deviance information criterion. As an illustration, we apply the proposed method to a real data set from dentistry.     

Joint modeling of recurrent events and a terminal event adjusted for zero inflation and a matched design

In longitudinal studies, matched designs are often employed to control the potential confounding effects in the field of biomedical research and public health. Because of clinical interest, recurrent time‐to‐event data are captured during the follow‐up. Meanwhile, the terminal event of death is always encountered, which should be taken into account for valid inference because of informative censoring. In some scenarios, a certain large portion of subjects may not have any recurrent events during the study period due to nonsusceptibility to events or censoring; thus, the zero‐inflated nature of data should be considered in analysis. In this paper, a joint frailty model with recurrent events and death is proposed to adjust for zero inflation and matched designs. We incorporate 2 frailties to measure the dependency between subjects within a matched pair and that among recurrent events within each individual. By sharing the random effects, 2 event processes of recurrent events and death are dependent with each other. The maximum likelihood based approach is applied for parameter estimation, where the Monte Carlo expectation‐maximization algorithm is adopted, and the corresponding R program is developed and available for public usage. In addition, alternative estimation methods such as Gaussian quadrature (PROC NLMIXED) and a Bayesian approach (PROC MCMC) are also considered for comparison to show our method's superiority. Extensive simulations are conducted, and a real data application on acute ischemic studies is provided in the end.     

Analysis of recurrent events with an associated informative dropout time: Application of the joint frailty model

This paper considers the analysis of a repeat event outcome in clinical trials of chronic diseases in the context of dependent censoring (e.g. mortality). It has particular application in the context of recurrent heart failure hospitalisations in trials of heart failure. Semi‐parametric joint frailty models (JFMs) simultaneously analyse recurrent heart failure hospitalisations and time to cardiovascular death, estimating distinct hazard ratios whilst individual‐specific latent variables induce associations between the two processes. A simulation study was carried out to assess the suitability of the JFM versus marginal analyses of recurrent events and cardiovascular death using standard methods. Hazard ratios were consistently overestimated when marginal models were used, whilst the JFM produced good, well‐estimated results. An application to the Candesartan in Heart failure: Assessment of Reduction in Mortality and morbidity programme was considered. The JFM gave unbiased estimates of treatment effects in the presence of dependent censoring. We advocate the use of the JFM for future trials that consider recurrent events as the primary outcome. © 2016 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

Mixture cure model with random effects for the analysis of a multi-center tonsil cancer study

Cure models for clustered survival data have the potential for broad applicability. In this paper, we consider the mixture cure model with random effects and propose several estimation methods based on Gaussian quadrature, rejection sampling, and importance sampling to obtain the maximum likelihood estimates of the model for clustered survival data with a cure fraction. The methods are flexible to accommodate various correlation structures. A simulation study demonstrates that the maximum likelihood estimates of parameters in the model tend to have smaller biases and variances than the estimates obtained from the existing methods. We apply the model to a study of tonsil cancer patients clustered by treatment centers to investigate the effect of covariates on the cure rate and on the failure time distribution of the uncured patients. The maximum likelihood estimates of the parameters demonstrate strong correlation among the failure times of the uncured patients and weak correlation among cure statuses in the same center.     

Non-parametric estimation and model checking procedures for marginal gap time distributions for recurrent events

For recurrent events there is evidence that misspecification of the frailty distribution can cause severe bias in estimated regression coefficients (Am. J. Epidemiol 1998; 149:404-411; Statist. Med. 2006; 25:1672-1684). In this paper we adapt a procedure originally suggested in (Biometrika 1999; 86:381-393) for parallel data for checking the gamma frailty to recurrent events. To apply the model checking procedure, a consistent non-parametric estimator for the marginal gap time distributions is needed. This is in general not possible due to induced dependent censoring in the recurrent events setting, however, in (Biometrika 1999; 86:59-70) a non-parametric estimator for the joint gap time distributions based on the principle of inverse probability of censoring weights is suggested. Here, we attempt to apply this estimator in the model checking procedure and the performance of the method is investigated with simulations and applied to Danish registry data. The method is further investigated using the usual Kaplan-Meier estimator and a marginalized estimator for the marginal gap time distributions. We conclude that the procedure only works when the recurrent event is common and when the intra-individual association between gap times is weak.     

Multivariate joint frailty model for the analysis of nonlinear tumor kinetics and dynamic predictions of death

The Response Evaluation Criteria in Solid Tumors are used as standard guidelines for the clinical evaluation of cancer treatments. The assessment is based on the anatomical tumor burden: change in size of target lesions and evolution of nontarget lesions (NTL). Despite unquestionable advantages of this standard tool, Response Evaluation Criteria in Solid Tumors are subject to some limitations such as categorization of continuous tumor size or negligence of its longitudinal trajectory. In particular, it is of interest to capture its nonlinear shape and model it simultaneously with recurrent progressions of NTL and overall survival. We propose a multivariate nonlinear mechanistic joint frailty model for longitudinal data, recurrent events, and a terminal event. In the model, the tumor size trajectory is described using an ordinary differential equation that accounts for the natural growth and treatment‐induced decline. We perform a simulation study to validate the method and apply the model to a phase III clinical trial in colorectal cancer. In the results of the analysis, we determine on which component, tumor size, NTL, or death the treatment acts mostly and perform dynamic predictions of death. We compare the model with other models that consider parametric functions or splines for the tumor size trajectory in terms of goodness of fit and predictive accuracy.     

Analysis of childhood brain tumour data in New York City using frailty models.

This paper uses frailty models to analyse overall survival and progression-free survival times for children with a brain tumour. We are interested in how surgery resection affects survival times. We are also interested in how strong a child's progression-free survival correlates with his/her overall survival and if the association differs with age. Traditionally the frailty is modelled parametrically and a maximum likelihood approach is used to estimate the parameters of interest. However, the result is sensitive to the misspecification of the frailty distribution and the currently developed algorithms for the maximum likelihood approach do not allow the association parameter to depend on covariates. Xue formulates a Poisson regression model and applies an estimating equation approach to obtain a consistent estimate of the covariate effect on survival. This paper extends that approach to obtain consistent and efficient estimates of the association parameter as well as the covariate effect and to allow the association parameter to depend on the covariates. The approach does not require the specification of the frailty distribution. The performance of the method is evaluated through simulation studies. We apply this method to a childhood brain tumour data set in New York City.     

Regression analysis of discrete time survival data under heterogeneity

This paper concerns the regression analysis of discrete time survival data for heterogeneous populations by means of frailty models. We express the survival time for each individual as a sequence of binary variables that indicate if the individual survived at each time point. The main result is that the likelihood for these indicators can be factored into contributions that involve the conditional survival probabilities integrated over the frailty distribution of the risk set (population-averaged). We then model these population-averaged conditional probabilities as a function of covariates. The result justifies the practice of treating the failure indicators as independent Bernoulli trials and fitting binary regression models for the conditional failure probabilities at each time point. However, we must interpret the regression coefficients as population-averaged rather than subject-specific parameters. We apply the method to the Framingham Heart Study on risk factors for cardiovascular disease. © 1997 by John Wiley & Sons, Ltd.     

How to analyze and interpret recurrent events data in the presence of a terminal event: An application on readmission after colorectal cancer surgery

Recurrent events arise when an event occurs many times for a subject. Many models have been developed to analyze these kind of data: the Andersen-Gill's model is one of them as well as the Prentice-William and the Peterson's model, the Wei Lee and Weissfeld's model, or even frailty models, all assuming an independent and noninformative censoring. However, in practice, these assumptions may be violated by the existence of a terminal event that permanently stops the recurrent process (eg, death). Indeed, a patient who experiences an early terminal event is more likely to have a lower number of recurrent events than a patient who experiences a terminal event later. Thus, ignoring terminal events in the analysis may lead to biased results. Many methods have been developed to handle terminal events. In this paper, we describe the existing methods classifying into conditional or marginal methods and compare them in a simulation study to highlight bias in results if an inappropriate method is used, when recurrent events and terminal event are correlated. In addition, we apply the different models on a real dataset to show how results should be interpreted. Finally, we provide recommendations for choosing the appropriate method for analyzing recurrent events in the presence of a terminal event.     

Methodology for analyzing censored correlated data: application of marginal and frailty approaches in human genetics. The European Community Alport Syndrome Concerted Action Group (ECASCA)

BACKGROUND: Statistical analysis for correlated censored data allows to study censored events in clustered structure designs. Considering a possible correlation among failure times of the same group, standard methodology is no longer applicable. We investigated proposed models in this context to study familial data about a genetic disease, Alport syndrome. Alport syndrome is a severe hereditary disease due to abnormal collagenous chains. Renal failure is the main symptom of the disease. It progresses toward end-stage renal failure (IRT) according to a high time variability. As shown by genetic studies, mutations of COL4A5 gene are involved in the X-linked Alport Syndrome. Due to the large range of the mutation types, the aim of this study was to search for a possible genetic origin of the heterogeneity of the disease severity. METHODS: Marginal survival models and mixed effects survival models (so-called frailty models) were proposed to take into account the possible non independence of the observations. In this study, time until end-stage renal failure is a rightly censored end point. Possible intra-familial correlations due to shared environmental and/or genetic factors could induce dependence among familial failure times. In this paper, we fit marginal and frailty proportional hazards models to evaluate the effect of mutation type on the risk of IRT and an interfamilial heterogeneity of failure times. RESULTS: In this study, the use of these models allows to show the presence of an interfamilial heterogeneity of the failure times to IRT. Moreover, the results suggest that some mutation types are linked to a higher risk of fast evolution to IRT, which explains partially the interfamilial heterogeneity of the failure times. CONCLUSIONS: This paper shows the interest of marginal and frailty models to evaluate the heterogeneity of censored responses and to study relationships between a censored criterion and covariables. This study puts forward the importance of characterizing the mutation at a molecular level to understand the relationship between genotype and phenotype.     

Generalized survival models for correlated time‐to‐event data

Our aim is to develop a rich and coherent framework for modeling correlated time‐to‐event data, including (1) survival regression models with different links and (2) flexible modeling for time‐dependent and nonlinear effects with rich postestimation. We extend the class of generalized survival models, which expresses a transformed survival in terms of a linear predictor, by incorporating a shared frailty or random effects for correlated survival data. The proposed approach can include parametric or penalized smooth functions for time, time‐dependent effects, nonlinear effects, and their interactions. The maximum (penalized) marginal likelihood method is used to estimate the regression coefficients and the variance for the frailty or random effects. The optimal smoothing parameters for the penalized marginal likelihood estimation can be automatically selected by a likelihood‐based cross‐validation criterion. For models with normal random effects, Gauss‐Hermite quadrature can be used to obtain the cluster‐level marginal likelihoods. The Akaike Information Criterion can be used to compare models and select the link function. We have implemented these methods in the R package rstpm2. Simulating for both small and larger clusters, we find that this approach performs well. Through 2 applications, we demonstrate (1) a comparison of proportional hazards and proportional odds models with random effects for clustered survival data and (2) the estimation of time‐varying effects on the log‐time scale, age‐varying effects for a specific treatment, and two‐dimensional splines for time and age.     

A Bayesian multivariate joint frailty model for disease recurrences and survival

Motivated by a study for soft tissue sarcoma, this article considers the analysis of diseases recurrence and survival. A multivariate frailty hazard model is established for joint modeling of three correlated time‐to‐event outcomes: local disease recurrence, distant disease recurrence (metastasis), and death. The goals are to find out (i) the effects of treatments on local and distant disease recurrences, and death, (ii) the effects of local and distant disease recurrences on death, and (iii) the correlation between local and distant recurrences. By our approach, all these three important questions, which are commonly asked in similar medical research studies, can be answered by a single model. We put the proposed joint frailty model in a Bayesian framework and use a hybrid Monte Carlo algorithm for the computation of posterior distributions. This hybrid algorithm relies on the evaluation of the gradient of target log density and a guided walk progress, and it combines these two strategies to suppress random walk behavior. A further distinction is that the hybrid algorithm can update all the components of a multivariate state vector simultaneously. Simulation studies are conducted to assess the proposed joint frailty model and the computation algorithm. The motivating soft tissue sarcoma data set is analyzed for illustration purpose. Copyright © 2016 John Wiley & Sons, Ltd.     

Joint modeling of longitudinal ordinal data and competing risks survival times and analysis of the NINDS rt‐PA stroke trial

Existing joint models for longitudinal and survival data are not applicable for longitudinal ordinal outcomes with possible non‐ignorable missing values caused by multiple reasons. We propose a joint model for longitudinal ordinal measurements and competing risks failure time data, in which a partial proportional odds model for the longitudinal ordinal outcome is linked to the event times by latent random variables. At the survival endpoint, our model adopts the competing risks framework to model multiple failure types at the same time. The partial proportional odds model, as an extension of the popular proportional odds model for ordinal outcomes, is more flexible and at the same time provides a tool to test the proportional odds assumption. We use a likelihood approach and derive an EM algorithm to obtain the maximum likelihood estimates of the parameters. We further show that all the parameters at the survival endpoint are identifiable from the data. Our joint model enables one to make inference for both the longitudinal ordinal outcome and the failure times simultaneously. In addition, the inference at the longitudinal endpoint is adjusted for possible non‐ignorable missing data caused by the failure times. We apply the method to the NINDS rt‐PA stroke trial. Our study considers the modified Rankin Scale only. Other ordinal outcomes in the trial, such as the Barthel and Glasgow scales, can be treated in the same way. Copyright © 2009 John Wiley & Sons, Ltd.