Joint modeling of longitudinal data with a dependent terminal event

ABSTRACT

Longitudinal data often arise in longitudinal follow-up studies, and there may exist a dependent terminal event such as death that stops the follow-up. In this article, we propose a new joint modeling for the analysis of longitudinal data with informative observation times via a dependent terminal event and two latent variables. Estimating equations are developed for parameter estimation, and asymptotic properties of the resulting estimators are established. In addition, a generalization of the joint model with time-varying coefficients for the longitudinal response variable is considered, and goodness-of-fit methods for assessing the adequacy of the model are also provided. The proposed method works well in our simulation studies, and is applied to a data set from a bladder cancer study.     

Simulated maximum likelihood estimation in joint models for multiple longitudinal markers and recurrent events of multiple types,in the presence of a terminal event

In medical studies we are often confronted with complex longitudinal data. During the follow-up period, which can be ended prematurely by a terminal event (e.g. death), a subject can experience recurrent events of multiple types. In addition, we collect repeated measurements from multiple markers. An adverse health status, represented by ‘bad’ marker values and an abnormal number of recurrent events, is often associated with the risk of experiencing the terminal event. In this situation, the missingness of the data is not at random and, to avoid bias, it is necessary to model all data simultaneously using a joint model. The correlations between the repeated observations of a marker or an event type within an individual are captured by normally distributed random effects. Because the joint likelihood contains an analytically intractable integral, Bayesian approaches or quadrature approximation techniques are necessary to evaluate the likelihood. However, when the number of recurrent event types and markers is large, the dimensionality of the integral is high and these methods are too computationally expensive. As an alternative, we propose a simulated maximum-likelihood approach based on quasi-Monte Carlo integration to evaluate the likelihood of joint models with multiple recurrent event types and markers.     

A new test for single against competing risks models in duration analysis

This paper shows that the single-risk duration model with two event types is a limiting case of bivariate dependent competing risks model, where the joint distribution of event times are degenerate. Then a new test is proposed for the null hypothesis of single risk against dependent competing risks model under the proportional hazard model assumption.     

Joint model for bivariate zero-inflated recurrent event data with terminal events

Bivariate recurrent event data are observed when subjects are at risk of experiencing two different type of recurrent events. In this paper, our interest is to suggest statistical model when there is a substantial portion of subjects not experiencing recurrent events but having a terminal event. In a context of recurrent event data, zero events can be related with either the risk free group or a terminal event. For simultaneously reflecting both a zero inflation and a terminal event in a context of bivariate recurrent event data, a joint model is implemented with bivariate frailty effects. Simulation studies are performed to evaluate the suggested models. Infection data from AML (acute myeloid leukemia) patients are analyzed as an application.     

Joint covariate-adjusted score test statistics for recurrent events and a terminal event

Recurrent events data are frequently encountered and could be stopped by a terminal event in clinical trials. It is of interest to assess the treatment efficacy simultaneously with respect to both the recurrent events and the terminal event in many applications. In this paper we propose joint covariate-adjusted score test statistics based on joint models of recurrent events and a terminal event. No assumptions on the functional form of the covariates are needed. Simulation results show that the proposed tests can improve the efficiency over tests based on covariate unadjusted model. The proposed tests are applied to the SOLVD data for illustration.     

Bayesian Modeling of Recurrent Event Data with Dependent Censoring

The recurrent-event setting, where the subjects experience multiple occurrences of the event of interest, are encountered in many biomedical applications. In analyzing recurrent event data, non informative censoring is often assumed for the implementation of statistical methods. However, when a terminating event such as death serves as part of the censoring mechanism, validity of the censoring assumption may be violated because recurrence can be a powerful risk factor for death. We consider joint modeling of recurrent event process and terminating event under a Bayesian framework in which a shared frailty is used to model the association between the intensity of the recurrent event process and the hazard of the terminating event. Our proposed model is implemented on data from a well-known cancer study.     

Regression analysis of multivariate recurrent event data with a dependent terminal event

Recurrent event data occur in many clinical and observational studies (Cook and Lawless, Analysis of recurrent event data, 2007) and in these situations, there may exist a terminal event such as death that is related to the recurrent event of interest (Ghosh and Lin, Biometrics 56:554–562, 2000; Wang et al., J Am Stat Assoc 96:1057–1065, 2001; Huang and Wang, J Am Stat Assoc 99:1153–1165, 2004; Ye et al., Biometrics 63:78–87, 2007). In addition, sometimes there may exist more than one type of recurrent events, that is, one faces multivariate recurrent event data with some dependent terminal event (Chen and Cook, Biostatistics 5:129–143, 2004). It is apparent that for the analysis of such data, one has to take into account the dependence both among different types of recurrent events and between the recurrent and terminal events. In this paper, we propose a joint modeling approach for regression analysis of the data and both finite and asymptotic properties of the resulting estimates of unknown parameters are established. The methodology is applied to a set of bivariate recurrent event data arising from a study of leukemia patients.     

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.     

Recurrent Events Analysis in the Presence of Terminal Event and Zero-recurrence Subjects

Recurrent event data are often encountered in longitudinal follow-up studies related to biomedical science, econometrics, reliability, and demography. In many situations, a terminal event such as death can happen during the follow-up period that precludes further recurrences. In this article, we will review some existing models for recurrent event with information censoring, and then extend them to allow zero-recurrence subjects as well as a terminal event. Estimating equations and partial likelihood are employed to estimate the coefficients of covariates, accumulative rate functions and the proportions of zero-recurrence subjects. The large-sample properties ofthe estimators are established as well. Simulations are performed to evaluate the estimationprocedure and an example of application on a set of migration data is provided to illustrateour proposed models and methods.     

Maximum likelihood analysis of semicompeting risks data with semiparametric regression models

The “semicompeting risks” include a terminal event and a non-terminal event. The terminal event may censor the non-terminal event but not vice versa. Because times to the two events are usually correlated, the non-terminal event is subject to dependent/informative censoring by the terminal event. We seek to conduct marginal regressions and joint association analyses for the two event times under semicompeting risks. The proposed method is based on the modeling setup where the semiparametric transformation models are assumed for marginal regressions, and a copula model is assumed for the joint distribution. We propose a nonparametric maximum likelihood approach for inferences, which provides a martingale representation for the score function and an analytical expression for the information matrix. Direct theoretical developments and computational implementation are allowed for the proposed approach. Simulations and a real data application demonstrate the utility of the proposed methodology.     

Marginal Regression Analysis for Semi-Competing Risks Data Under Dependent Censoring

Abstract.  Multiple events data are commonly seen in medical applications. There are two types of events, namely terminal and non-terminal. Statistical analysis for non-terminal events is complicated due to dependent censoring. Consequently, joint modelling and inference are often needed to avoid the problem of non-identifiability. This article considers regression analysis for multiple events data with major interest in a non-terminal event such as disease progression. We generalize the technique of artificial censoring, which is a popular way to handle dependent censoring, under flexible model assumptions on the two types of events. The proposed method is applied to analyse a data set of bone marrow transplantation.     

A Marginal Additive Rates Model for Recurrent Event Data with a Terminal Event

Recurrent events data with a terminal event often arise in many longitudinal studies. Most of existing models assume multiplicative covariate effects and model the conditional recurrent event rate given survival. In this article, we propose a marginal additive rates model for recurrent events with a terminal event, and develop two procedures for estimating the model parameters. The asymptotic properties of the resulting estimators are established. In addition, some numerical procedures are presented for model checking. The finite-sample behavior of the proposed methods is examined through simulation studies, and an application to a bladder cancer study is also illustrated.     

Bayesian hierarchical joint modeling using skew-normal/independent distributions

The multiple longitudinal outcomes collected in many clinical trials are often analyzed by multilevel item response theory (MLIRT) models. The normality assumption for the continuous outcomes in the MLIRT models can be violated due to skewness and/or outliers. Moreover, patients’ follow-up may be stopped by some terminal events (e.g., death or dropout), which are dependent on the multiple longitudinal outcomes. We proposed a joint modeling framework based on the MLIRT model to account for three data features: skewness, outliers, and dependent censoring. Our method development was motivated by a clinical study for Parkinson’s disease.     

The analysis of multivariate recurrent events with partially missing event types

In many clinical studies, subjects are at risk of experiencing more than one type of potentially recurrent event. In some situations, however, the occurrence of an event is observed, but the specific type is not determined. We consider the analysis of this type of incomplete data when the objectives are to summarize features of conditional intensity functions and associated treatment effects, and to study the association between different types of event. Here we describe a likelihood approach based on joint models for the multi-type recurrent events where parameter estimation is obtained from a Monte-Carlo EM algorithm. Simulation studies show that the proposed method gives unbiased estimators for regression coefficients and variance–covariance parameters, and the coverage probabilities of confidence intervals for regression coefficients are close to the nominal level. When the distribution of the frailty variable is misspecified, the method still provides estimators of the regression coefficients with good properties. The proposed method is applied to a motivating data set from an asthma study in which exacerbations were to be sub-typed by cellular analysis of sputum samples as eosinophilic or non-eosinophilic.     

Regression analysis based on conditional likelihood approach under semi-competing risks data

Medical studies often involve semi-competing risks data, which consist of two types of events, namely terminal event and non-terminal event. Because the non-terminal event may be dependently censored by the terminal event, it is not possible to make inference on the non-terminal event without extra assumptions. Therefore, this study assumes that the dependence structure on the non-terminal event and the terminal event follows a copula model, and lets the marginal regression models of the non-terminal event and the terminal event both follow time-varying effect models. This study uses a conditional likelihood approach to estimate the time-varying coefficient of the non-terminal event, and proves the large sample properties of the proposed estimator. Simulation studies show that the proposed estimator performs well. This study also uses the proposed method to analyze AIDS Clinical Trial Group (ACTG 320).     

An additive-multiplicative rates model for recurrent event data with informative terminal event

In this article, we propose an additive-multiplicative rates model for recurrent event data in the presence of a terminal event such as death. The association between recurrent and terminal events is nonparametric. For inference on the model parameters, estimating equation approaches are developed, and the asymptotic properties of the resulting estimators are established. The finite sample behavior of the proposed estimators is evaluated through simulation studies, and an application to a bladder cancer study is provided.     

A joint model of cancer incidence,metastasis, and mortality

Many diseases, especially cancer, are not static, but rather can be summarized by a series of events or stages (e.g. diagnosis, remission, recurrence, metastasis, death). Most available methods to analyze multi-stage data ignore intermediate events and focus on the terminal event or consider (time to) multiple events as independent. Competing-risk or semi-competing-risk models are often deficient in describing the complex relationship between disease progression events which are driven by a shared progression stochastic process. A multi-stage model can only examine two stages at a time and thus fails to capture the effect of one stage on the time spent between other stages. Moreover, most models do not account for latent stages. We propose a semi-parametric joint model of diagnosis, latent metastasis, and cancer death and use nonparametric maximum likelihood to estimate covariate effects on the risks of intermediate events and death and the dependence between them. We illustrate the model with Monte Carlo simulations and analysis of real data on prostate cancer from the SEER database.     

Using recurrent time-to-event models with multinomial outcomes to generate toxicity profiles

Most clinical studies, which investigate the impact of therapy simultaneously, record the frequency of adverse events in order to monitor safety of the intervention. Study reports typically summarise adverse event data by tabulating the frequencies of the worst grade experienced but provide no details of the temporal profiles of specific types of adverse events. Such 'toxicity profiles' are potentially important tools in disease management and in the assessment of newer therapies including targeted treatments and immunotherapy where different types of toxicity may be more common at various times during long-term drug exposure. Toxicity profiles of commonly experienced adverse events occurring due to exposure to long-term treatment could assist in evaluating the costs of the health care benefits of therapy. We show how to generate toxicity profiles using an adaptation of the ordinal time-to-event model comprising of a two-step process, involving estimation of the multinomial response probabilities using multinomial logistic regression and combining these with recurrent time to event hazard estimates to produce cumulative event probabilities for each of the multinomial adverse event response categories. Such a model permits the simultaneous assessment of the risk of events over time and provides cumulative risk probabilities for each type of adverse event response. The method can be applied more generally by using different models to estimate outcome/response probabilities. The method is illustrated by developing toxicity profiles for three distinct types of adverse events associated with two treatment regimens for patients with advanced breast cancer.     

A seasonal mixed-POT model to estimate high flood quantiles from different event types and seasons

Flood events can be caused by several different meteorological circumstances. For example, heavy rain events often lead to short flood events with high peaks, whereas snowmelt normally results in events of very long duration with a high volume. Both event types have to be considered in the design of flood protection systems. Unfortunately, all these different event types are often included in annual maximum series (AMS) leading to inhomogeneous samples. Moreover, certain event types are underrepresented in the AMS. This is especially unsatisfactory if the most extreme events result from such an event type. Therefore, monthly maximum data are used to enlarge the information spectrum on the different event types. Of course, not all events can be included in the flood statistics because not every monthly maximum can be declared as a flood. To take this into account, a mixture Peak-over-threshold model is applied, with thresholds specifying flood events of several types that occur in a season of the year. This model is then extended to cover the seasonal type of the data. The applicability is shown in a German case study, where the impact of the single event types in different parts of a year is evaluated.     

Delayed treatment effects,treatment switching and heterogeneous patient populations: How to design and analyze RCTs in oncology

In the analysis of survival times, the logrank test and the Cox model have been established as key tools, which do not require specific distributional assumptions. Under the assumption of proportional hazards, they are efficient and their results can be interpreted unambiguously. However, delayed treatment effects, disease progression, treatment switchers or the presence of subgroups with differential treatment effects may challenge the assumption of proportional hazards. In practice, weighted logrank tests emphasizing either early, intermediate or late event times via an appropriate weighting function may be used to accommodate for an expected pattern of non-proportionality. We model these sources of non-proportional hazards via a mixture of survival functions with piecewise constant hazard. The model is then applied to study the power of unweighted and weighted log-rank tests, as well as maximum tests allowing different time dependent weights. Simulation results suggest a robust performance of maximum tests across different scenarios, with little loss in power compared to the most powerful among the considered weighting schemes and huge power gain compared to unfavorable weights. The actual sources of non-proportional hazards are not obvious from resulting populationwise survival functions, highlighting the importance of detailed simulations in the planning phase of a trial when assuming non-proportional hazards.We provide the required tools in a software package, allowing to model data generating processes under complex non-proportional hazard scenarios, to simulate data from these models and to perform the weighted logrank tests.