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.     

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.     

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-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.     

Proportional Rate Model with Incomplete Covariate and Complete Auxiliary Information

This paper deals with the analysis of proportional rate model for recurrent event data when covariates are subject to missing. The true covariate is measured only on a randomly chosen validation set, whereas auxiliary information is available for all cohort subjects. To further utilize the auxiliary information to improve study efficiency, we propose an estimated estimating equation for the regression parameters. The resulting estimators are shown to be consistent and asymptotically normal. Both graphical and numerical techniques for checking the adequacy of the model are presented. Simulations are conducted to evaluate the finite sample performance of the proposed estimators. Illustration with a real medical study is provided.     

Semiparametric proportional means model for marker data contingent on recurrent event

In many biomedical studies with recurrent events, some markers can only be measured when events happen. For example, medical cost attributed to hospitalization can only incur when patients are hospitalized. Such marker data are contingent on recurrent events. In this paper, we present a proportional means model for modelling the markers using the observed covariates contingent on the recurrent event. We also model the recurrent event via a marginal rate model. Estimating equations are constructed to derive the point estimators for the parameters in the proposed models. The estimators are shown to be asymptotically normal. Simulation studies are conducted to examine the finite-sample properties of the proposed estimators and the proposed method is applied to a data set from the Vitamin A Community Trial.     

A semiparametric additive rates model for recurrent event data

Recurrent event data often arise in biomedical studies, with examples including hospitalizations, infections, and treatment failures. In observational studies, it is often of interest to estimate the effects of covariates on the marginal recurrent event rate. The majority of existing rate regression methods assume multiplicative covariate effects. We propose a semiparametric model for the marginal recurrent event rate, wherein the covariates are assumed to add to the unspecified baseline rate. Covariate effects are summarized by rate differences, meaning that the absolute effect on the rate function can be determined from the regression coefficient alone. We describe modifications of the proposed method to accommodate a terminating event (e.g., death). Proposed estimators of the regression parameters and baseline rate are shown to be consistent and asymptotically Gaussian. Simulation studies demonstrate that the asymptotic approximations are accurate in finite samples. The proposed methods are applied to a state-wide kidney transplant 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.     

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.     

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.     

Cox regression of clustered event times with covariates missing not at random

Motivated by a recent tuberculosis (TB) study, this paper is concerned with covariates missing not at random (MNAR) and models the potential intracluster correlation by a frailty. We consider the regression analysis of right‐censored event times from clustered subjects under a Cox proportional hazards frailty model and present the semiparametric maximum likelihood estimator (SPMLE) of the model parameters. An easy‐to‐implement pseudo‐SPMLE is then proposed to accommodate more realistic situations using readily available supplementary information on the missing covariates. Algorithms are provided to compute the estimators and their consistent variance estimators. We demonstrate that both the SPMLE and the pseudo‐SPMLE are consistent and asymptotically normal by the arguments based on the theory of modern empirical processes. The proposed approach is examined numerically via simulation and illustrated with an analysis of the motivating TB study data.     

Multiple imputation methods for recurrent event data with missing event category

Frequently in clinical and epidemiologic studies, the event of interest is recurrent (i.e., can occur more than once per subject). When the events are not of the same type, an analysis which accounts for the fact that events fall into different categories will often be more informative. Often, however, although event times may always be known, information through which events are categorized may potentially be missing. Complete‐case methods (whose application may require, for example, that events be censored when their category cannot be determined) are valid only when event categories are missing completely at random. This assumption is rather restrictive. The authors propose two multiple imputation methods for analyzing multiple‐category recurrent event data under the proportional means/rates model. The use of a proper or improper imputation technique distinguishes the two approaches. Both methods lead to consistent estimation of regression parameters even when the missingness of event categories depends on covariates. The authors derive the asymptotic properties of the estimators and examine their behaviour in finite samples through simulation. They illustrate their approach using data from an international study on dialysis.     

Treatment policy estimands for recurrent event data using data collected after cessation of randomised treatment

In the past, many clinical trials have withdrawn subjects from the study when they prematurely stopped their randomised treatment and have therefore only collected ‘on‐treatment’ data. Thus, analyses addressing a treatment policy estimand have been restricted to imputing missing data under assumptions drawn from these data only. Many confirmatory trials are now continuing to collect data from subjects in a study even after they have prematurely discontinued study treatment as this event is irrelevant for the purposes of a treatment policy estimand. However, despite efforts to keep subjects in a trial, some will still choose to withdraw. Recent publications for sensitivity analyses of recurrent event data have focused on the reference‐based imputation methods commonly applied to continuous outcomes, where imputation for the missing data for one treatment arm is based on the observed outcomes in another arm. However, the existence of data from subjects who have prematurely discontinued treatment but remained in the study has now raised the opportunity to use this ‘off‐treatment’ data to impute the missing data for subjects who withdraw, potentially allowing more plausible assumptions for the missing post‐study‐withdrawal data than reference‐based approaches. In this paper, we introduce a new imputation method for recurrent event data in which the missing post‐study‐withdrawal event rate for a particular subject is assumed to reflect that observed from subjects during the off‐treatment period. The method is illustrated in a trial in chronic obstructive pulmonary disease (COPD) where the primary endpoint was the rate of exacerbations, analysed using a negative binomial model.     

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.     

Non parametric analysis of gap times for bivariate recurrent event data with cure fraction

Recurrent event data often arise in longitudinal studies. In many applications, subjects may experience two different types of events alternatively over time or a pair of subjects may experience recurrent events of the same type. Medical advances have made it possible for some patients to be cured such that the disease of interest does not recur. In this article, we consider non parametric analysis of bivariate recurrent event data with cure fraction. Using the inverse-probability weighted (IPW) approach, we propose non parametric estimators for the proportion of cured patients and for the joint distribution functions of bivariate recurrence times of the uncured ones. The asymptotic properties of the proposed estimators are established. Simulation study indicates that the proposed estimators perform well in finite samples.     

A class of mixed models for recurrent event data

In this article, we propose a class of mixed models for recurrent event data. The new models include the proportional rates model and Box–Cox transformation rates models as special cases, and allow the effects of covariates on the rate functions of counting processes to be proportional or convergent. For inference on the model parameters, estimating equation approaches are developed. The asymptotic properties of the resulting estimators are established and the finite sample performance of the proposed procedure is evaluated through simulation studies. A real example with data taken from a clinic study on chronic granulomatous disease (CGD) is also illustrated for the use of the proposed methodology. The Canadian Journal of Statistics 39: 578–590; 2011. © 2011 Statistical Society of Canada     

A class of accelerated means regression models for recurrent event data

In this article, we propose a general class of accelerated means regression models for recurrent event data. The class includes the proportional means model, the accelerated failure time model and the accelerated rates model as special cases. The new model offers great flexibility in formulating the effects of covariates on the mean functions of counting processes while leaving the stochastic structure completely unspecified. For the inference on the model parameters, estimating equation approaches are developed and both large and final sample properties of the proposed estimators are established. In addition, some graphical and numerical procedures are presented for model checking. An illustration with multiple-infection data from a clinic study on chronic granulomatous disease is also provided.     

Marginal Regression of Multivariate Event Times Based on Linear Transformation Models

Multivariate event time data are common in medical studies and have received much attention recently. In such data, each study subject may potentially experience several types of events or recurrences of the same type of event, or event times may be clustered. Marginal distributions are specified for the multivariate event times in multiple events and clustered events data, and for the gap times in recurrent events data, using the semiparametric linear transformation models while leaving the dependence structures for related events unspecified. We propose several estimating equations for simultaneous estimation of the regression parameters and the transformation function. It is shown that the resulting regression estimators are asymptotically normal, with variance–covariance matrix that has a closed form and can be consistently estimated by the usual plug-in method. Simulation studies show that the proposed approach is appropriate for practical use. An application to the well-known bladder cancer tumor recurrences data is also given to illustrate the methodology.     

Additive Transformation Models for Recurrent Events

In this article, we propose a class of additive transformation models for recurrent event data, which includes the additive rates model as a special case. The new models offer great flexibility in formulating the effects of covariates on the mean function of recurrent events. Estimating equation approaches are developed for the model parameters, and asymptotic properties of the resulting estimators are established. In addition, a model checking procedure is presented to assess the adequacy of the model. The finite sample performance of the proposed estimators is examined through simulation studies, and an application to a bladder cancer study is presented.     

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.