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 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.     

An additive marginal regression model for clustered recurrent event in the presence of a terminal event

In this article, an additive rate model is proposed for clustered recurrent event with a terminal event. The subjects are clustered by some property. For the clustered subjects, the recurrent event is precluded by the death. An estimating equation is developed for the model parameter and the baseline rate function. The asymptotic properties of the resulting estimators are established. In addition, a goodness-of-fit test is presented to assess the adequacy of the model. The finite-sample behavior of the proposed estimators is evaluated through simulation studies, and an application to a bladder cancer data is illustrated.     

Non-parametric Tests for Recurrent Events under Competing Risks

Abstract.  We consider a data set on nosocomial infections of patients hospitalized in a French intensive care facility. Patients may suffer from recurrent infections of different types and they also have a high risk of death. To deal with such situations, a model of recurrent events with competing risks and a terminal event is introduced. Our aim was to compare the occurrence rates of two types of events. For this purpose, we propose two tests: one to detect if the occurrence rate of a given type of event increases with time; a second to detect if the instantaneous probability of experiencing an event of a given type is always greater than the one of another type. The asymptotic properties of the test statistics are derived and Monte Carlo methods are used to study the power of the tests. Finally, the procedures developed are applied to the French nosocomial infections data set.     

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.     

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.     

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.     

Semiparametric model for recurrent event data with excess zeros and informative censoring

Recurrent event data are often encountered in longitudinal follow-up studies in many important areas such as biomedical science, econometrics, reliability, criminology and demography. Multiplicative marginal rates models have been used extensively to analyze recurrent event data, but often fail to fit the data adequately. In addition, the analysis is complicated by excess zeros in the data as well as the presence of a terminal event that precludes further recurrence. To address these problems, we propose a semiparametric model with an additive rate function and an unspecified baseline to analyze recurrent event data, which includes a parameter to accommodate excess zeros and a frailty term to account for a terminal event. Local likelihood procedure is applied to estimate the parameters, and the asymptotic properties of the estimators are established. A simulation study is conducted to evaluate the performance of the proposed methods, and an example of their application is presented on a set of tumor recurrent data for bladder cancer.     

Semiparametric Methods for Clustered Recurrent Event Data

In biomedical studies, the event of interest is often recurrent and within-subject events cannot usually be assumed independent. In addition, individuals within a cluster might not be independent; for example, in multi-center or familial studies, subjects from the same center or family might be correlated. We propose methods of estimating parameters in two semi-parametric proportional rates/means models for clustered recurrent event data. The first model contains a baseline rate function which is common across clusters, while the second model features cluster-specific baseline rates. Dependence structures for patients-within-cluster and events-within-patient are both unspecified. Estimating equations are derived for the regression parameters. For the common baseline model, an estimator of the baseline mean function is proposed. The asymptotic distributions of the model parameters are derived, while finite-sample properties are assessed through a simulation study. Using data from a national organ failure registry, the proposed methods are applied to the analysis of technique failures among Canadian dialysis patients.     

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.     

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.     

Analyzing panel count data with a dependent observation process and a terminal event

Panel count data occur in many fields and a number of approaches have been developed. However, most of these approaches are for situations where there is no terminal event and the observation process is independent of the underlying recurrent event process unconditionally or conditional on the covariates. In this paper, we discuss a more general situation where the observation process is informative and there exists a terminal event which precludes further occurrence of the recurrent events of interest. For the analysis, a semiparametric transformation model is presented for the mean function of the underlying recurrent event process among survivors. To estimate the regression parameters, an estimating equation approach is proposed in which an inverse survival probability weighting technique is used. The asymptotic distribution of the proposed estimates is provided. Simulation studies are conducted and suggest that the proposed approach works well for practical situations. An illustrative example is provided. The Canadian Journal of Statistics 41: 174–191; 2013 © 2012 Statistical Society of Canada     

Modeling marginal features in studies of recurrent events in the presence of a terminal event

We study models for recurrent events with special emphasis on the situation where a terminal event acts as a competing risk for the recurrent events process and where there may be gaps between periods during which subjects are at risk for the recurrent event. We focus on marginal analysis of the expected number of events and show that an Aalen–Johansen type estimator proposed by Cook and Lawless is applicable in this situation. A motivating example deals with psychiatric hospital admissions where we supplement with analyses of the marginal distribution of time to the competing event and the marginal distribution of the time spent in hospital. Pseudo-observations are used for the latter purpose.

     

Mixture modelling of recurrent event times with long-term survivors: Analysis of Hutterite birth intervals

We propose a mixture model that combines a discrete-time survival model for analyzing the correlated times between recurrent events, e.g. births, with a logistic regression model for the probability of never experiencing the event of interest, i.e., being a long-term survivor. The proposed survival model incorporates both observed and unobserved heterogeneity in the probability of experiencing the event of interest. We use Gibbs sampling for the fitting of such mixture models, which leads to a computationally intensive solution to the problem of fitting survival models for multiple event time data with long-term survivors. We illustrate our Bayesian approach through an analysis of Hutterite birth histories.     

Partial sufficient dimension reduction on joint model of recurrent and terminal events

Joint modeling of recurrent and terminal events has attracted considerable interest and extensive investigations by many authors. The assumption of low-dimensional covariates has been usually applied in the existing studies, which is however inapplicable in many practical situations. In this paper, we consider a partial sufficient dimension reduction approach for a joint model with high-dimensional covariates. Some simulations as well as three real data applications are presented to confirm and assess the performance of the proposed model and approach.     

Estimating Marginal Effects in Accelerated Failure Time Models for Serial Sojourn Times Among Repeated Events

Recurrent event data are commonly encountered in longitudinal studies when events occur repeatedly over time for each study subject. An accelerated failure time (AFT) model on the sojourn time between recurrent events is considered in this article. This model assumes that the covariate effect and the subject-specific frailty are additive on the logarithm of sojourn time, and the covariate effect maintains the same over distinct episodes, while the distributions of the frailty and the random error in the model are unspecified. With the ordinal nature of recurrent events, two scale transformations of the sojourn times are derived to construct semiparametric methods of log-rank type for estimating the marginal covariate effects in the model. The proposed estimation approaches/inference procedures also can be extended to the bivariate events, which alternate themselves over time. Examples and comparisons are presented to illustrate the performance of the proposed methods.     

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.     

Bayesian regression model for recurrent event data with event-varying covariate effects and event effect

In the course of hypertension, cardiovascular disease events (e.g. stroke, heart failure) occur frequently and recurrently. The scientific interest in such study may lie in the estimation of treatment effect while accounting for the correlation among event times. The correlation among recurrent event times comes from two sources: subject-specific heterogeneity (e.g. varied lifestyles, genetic variations, and other unmeasurable effects) and event dependence (i.e. event incidences may change the risk of future recurrent events). Moreover, event incidences may change the disease progression so that there may exist event-varying covariate effects (the covariate effects may change after each event) and event effect (the effect of prior events on the future events). In this article, we propose a Bayesian regression model that not only accommodates correlation among recurrent events from both sources, but also explicitly characterizes the event-varying covariate effects and event effect. This model is especially useful in quantifying how the incidences of events change the effects of covariates and risk of future events. We compare the proposed model with several commonly used recurrent event models and apply our model to the motivating lipid-lowering trial (LLT) component of the Antihypertensive and Lipid-Lowering Treatment to Prevent Heart Attack Trial (ALLHAT) (ALLHAT-LLT).     

Estimating treatment effects on the marginal recurrent event mean in the presence of a terminating event

In biomedical studies where the event of interest is recurrent (e.g., hospitalization), it is often the case that the recurrent event sequence is subject to being stopped by a terminating event (e.g., death). In comparing treatment options, the marginal recurrent event mean is frequently of interest. One major complication in the recurrent/terminal event setting is that censoring times are not known for subjects observed to die, which renders standard risk set based methods of estimation inapplicable. We propose two semiparametric methods for estimating the difference or ratio of treatment-specific marginal mean numbers of events. The first method involves imputing unobserved censoring times, while the second methods uses inverse probability of censoring weighting. In each case, imbalances in the treatment-specific covariate distributions are adjusted out through inverse probability of treatment weighting. After the imputation and/or weighting, the treatment-specific means (then their difference or ratio) are estimated nonparametrically. Large-sample properties are derived for each of the proposed estimators, with finite sample properties assessed through simulation. The proposed methods are applied to kidney transplant data.