Nonparametric inference for panel count data with competing risks

In survival and reliability studies, panel count data arise when we investigate a recurrent event process and each study subject is observed only at discrete time points. If recurrent events of several types are possible, we obtain panel count data with competing risks. Such data arise frequently from transversal studies on recurrent events in demography, epidemiology and reliability experiments where the individuals cannot be observed continuously. In the present paper, we propose an isotonic regression estimator for the cause specific mean function of the underlying recurrent event process of a competing risks panel count data. Further, a nonparametric test is proposed to compare the cause specific mean functions of the panel count competing risks data. Asymptotic properties of the proposed estimator and test statistic are studied. A simulation study is conducted to assess the finite sample behaviour of the proposed estimator and test statistic. Finally, the procedures developed are applied to a real data arising from skin cancer chemo prevention trial.     

A flexible parametric survival model for fitting time to event data in clinical trials

Time‐to‐event data are common in clinical trials to evaluate survival benefit of a new drug, biological product, or device. The commonly used parametric models including exponential, Weibull, Gompertz, log‐logistic, log‐normal, are simply not flexible enough to capture complex survival curves observed in clinical and medical research studies. On the other hand, the nonparametric Kaplan Meier (KM) method is very flexible and successful on catching the various shapes in the survival curves but lacks ability in predicting the future events such as the time for certain number of events and the number of events at certain time and predicting the risk of events (eg, death) over time beyond the span of the available data from clinical trials. It is obvious that neither the nonparametric KM method nor the current parametric distributions can fulfill the needs in fitting survival curves with the useful characteristics for predicting. In this paper, a full parametric distribution constructed as a mixture of three components of Weibull distribution is explored and recommended to fit the survival data, which is as flexible as KM for the observed data but have the nice features beyond the trial time, such as predicting future events, survival probability, and hazard function.     

Semiparametric transformation models for multivariate panel count data with dependent observation process

This article discusses regression analysis of multivariate panel count data in which the observation process may contain relevant information about or be related to the underlying recurrent event processes of interest. Such data occur if a recurrent event study involves several related types of recurrent events and the observation scheme or process may be subject-specific. For the problem, a class of semiparametric transformation models is presented, which provides a great flexibility for modelling the effects of covariates on the recurrent event processes. For estimation of regression parameters, an estimating equation-based inference procedure is developed and the asymptotic properties of the resulting estimates are established. Also the proposed approach is evaluated by simulation studies and applied to the data arising from a skin cancer chemoprevention trial.     

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     

Joint modelling of pre-randomisation event counts and multiple post-randomisation survival times with cure rates: application to data for early epilepsy and single seizures

In this paper, we consider the analysis of recurrent event data that examines the differences between two treatments. The outcomes that are considered in the analysis are the pre-randomisation event count and post-randomisation times to first and second events with associated cure fractions. We develop methods that allow pre-randomisation counts and two post-randomisation survival times to be jointly modelled under a Poisson process framework, assuming that outcomes are predicted by (unobserved) event rates. We apply these methods to data that examine the difference between immediate and deferred treatment policies in patients presenting with single seizures or early epilepsy. We find evidence to suggest that post-randomisation seizure rates change at randomisation and following a first seizure after randomisation. We also find that there are cure rates associated with the post-randomisation times to first and second seizures. The increase in power over standard survival techniques, offered by the joint models that we propose, resulted in more precise estimates of the treatment effect and the ability to detect interactions with covariate effects.     

A parametric model fitting time to first event for overdispersed data: application to time to relapse in multiple sclerosis

In this article, we propose a parametric model for the distribution of time to first event when events are overdispersed and can be properly fitted by a Negative Binomial distribution. This is a very common situation in medical statistics, when the occurrence of events is summarized as a count for each patient and the simple Poisson model is not adequate to account for overdispersion of data. In this situation, studying the time of occurrence of the first event can be of interest. From the Negative Binomial distribution of counts, we derive a new parametric model for time to first event and apply it to fit the distribution of time to first relapse in multiple sclerosis (MS). We develop the regression model with methods for covariate estimation. We show that, as the Negative Binomial model properly fits relapse counts data, this new model matches quite perfectly the distribution of time to first relapse, as tested in two large datasets of MS patients. Finally we compare its performance, when fitting time to first relapse in MS, with other models widely used in survival analysis (the semiparametric Cox model and the parametric exponential, Weibull, log-logistic and log-normal models).     

Nonparametric Estimation of Sojourn Time Distributions for Truncated Serial Event Data—a Weight-adjusted Approach

In follow-up studies, survival data often include subjects who have had a certain event at recruitment and may potentially experience a series of subsequent events during the follow-up period. This kind of survival data collected under a cross-sectional sampling criterion is called truncated serial event data. The outcome variables of interest in this paper are serial sojourn times between successive events. To analyze the sojourn times in truncated serial event data, we need to confront two potential sampling biases arising simultaneously from a sampling criterion and induced informative censoring. In this study, nonparametric estimation of the joint probability function of serial sojourn times is developed by using inverse probabilities of the truncation and censoring times as weight functions to accommodate these two sampling biases under various situations of truncation and censoring. Relevant statistical properties of the proposed estimators are also discussed. Simulation studies and two real data are presented to illustrate the proposed methods.     

Rate/Mean Regression for Multiple-Sequence Recurrent Event Data with Missing Event Category

Abstract.  Censored recurrent event data frequently arise in biomedical studies. Often, the events are not homogenous, and may be categorized. We propose semiparametric regression methods for analysing multiple-category recurrent event data and consider the setting where event times are always known, but the information used to categorize events may be missing. Application of existing methods after censoring events of unknown category (i.e. 'complete-case' methods) produces consistent estimators only when event types are missing completely at random, an assumption which will frequently fail in practice. We propose methods, based on weighted estimating equations, which are applicable when event category missingness is missing at random. Parameter estimators are shown to be consistent and asymptotically normal. Finite sample properties are examined through simulations and the proposed methods are applied to an end-stage renal disease data set obtained from a national organ failure registry.     

Control‐based imputation for sensitivity analyses in informative censoring for recurrent event data

In clinical trials, missing data commonly arise through nonadherence to the randomized treatment or to study procedure. For trials in which recurrent event endpoints are of interests, conventional analyses using the proportional intensity model or the count model assume that the data are missing at random, which cannot be tested using the observed data alone. Thus, sensitivity analyses are recommended. We implement the control‐based multiple imputation as sensitivity analyses for the recurrent event data. We model the recurrent event using a piecewise exponential proportional intensity model with frailty and sample the parameters from the posterior distribution. We impute the number of events after dropped out and correct the variance estimation using a bootstrap procedure. We apply the method to an application of sitagliptin study.     

Nonparametric Test for Doubly Interval-Censored Failure Time Data

This paper considers comparison of discrete failure time distributions when the survival time of interest measures elapsed time between two related events and observations on the occurrences of both events could be interval-censored. This kind of data is often referred to as doubly interval-censored failure time data. If the occurrence of the first event defining the survival time can be exactly observed, the data are usually referred to as interval-censored data. For the comparison problem based on interval-censored failure time data, Sun (1996) proposed a nonparametric test procedure. In this paper we generalize the procedure given in Sun (1996) to doubly interval-censored data case and the generalized test is evaluated by simulations.     

Nonparametric estimators for a survivor function of paired recurrent events

Recurrent event data arise in longitudinal studies where each study subject may experience multiple events during the follow-up. In many situations in survival studies, pairs of individuals can potentially experience recurrent events. The analysis of such data is not straightforward as it involves two kinds of dependences, namely, dependence between the individuals in the same pair and dependence among a sequence of pairs. In the present paper, we introduce a new stochastic model for the analysis of such recurrent event data. Nonparametric estimators for a bivariate survivor function are developed. Asymptotic properties of the estimators are discussed. Simulation studies are carried out to assess the finite sample properties of the estimator. We illustrate the procedure with real life data on eye disease.     

Sensitivity analyses for partially observed recurrent event data

Recurrent events involve the occurrences of the same type of event repeatedly over time and are commonly encountered in longitudinal studies. Examples include seizures in epileptic studies or occurrence of cancer tumors. In such studies, interest lies in the number of events that occur over a fixed period of time. One considerable challenge in analyzing such data arises when a large proportion of patients discontinues before the end of the study, for example, because of adverse events, leading to partially observed data. In this situation, data are often modeled using a negative binomial distribution with time‐in‐study as offset. Such an analysis assumes that data are missing at random (MAR). As we cannot test the adequacy of MAR, sensitivity analyses that assess the robustness of conclusions across a range of different assumptions need to be performed. Sophisticated sensitivity analyses for continuous data are being frequently performed. However, this is less the case for recurrent event or count data. We will present a flexible approach to perform clinically interpretable sensitivity analyses for recurrent event data. Our approach fits into the framework of reference‐based imputations, where information from reference arms can be borrowed to impute post‐discontinuation data. Different assumptions about the future behavior of dropouts dependent on reasons for dropout and received treatment can be made. The imputation model is based on a flexible model that allows for time‐varying baseline intensities. We assess the performance in a simulation study and provide an illustration with a clinical trial in patients who suffer from bladder cancer. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

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.     

Landmark estimation of survival and treatment effects in observational studies

Clinical studies aimed at identifying effective treatments to reduce the risk of disease or death often require long term follow-up of participants in order to observe a sufficient number of events to precisely estimate the treatment effect. In such studies, observing the outcome of interest during follow-up may be difficult and high rates of censoring may be observed which often leads to reduced power when applying straightforward statistical methods developed for time-to-event data. Alternative methods have been proposed to take advantage of auxiliary information that may potentially improve efficiency when estimating marginal survival and improve power when testing for a treatment effect. Recently, Parast et al. (J Am Stat Assoc 109(505):384–394, 2014) proposed a landmark estimation procedure for the estimation of survival and treatment effects in a randomized clinical trial setting and demonstrated that significant gains in efficiency and power could be obtained by incorporating intermediate event information as well as baseline covariates. However, the procedure requires the assumption that the potential outcomes for each individual under treatment and control are independent of treatment group assignment which is unlikely to hold in an observational study setting. In this paper we develop the landmark estimation procedure for use in an observational setting. In particular, we incorporate inverse probability of treatment weights (IPTW) in the landmark estimation procedure to account for selection bias on observed baseline (pretreatment) covariates. We demonstrate that consistent estimates of survival and treatment effects can be obtained by using IPTW and that there is improved efficiency by using auxiliary intermediate event and baseline information. We compare our proposed estimates to those obtained using the Kaplan–Meier estimator, the original landmark estimation procedure, and the IPTW Kaplan–Meier estimator. We illustrate our resulting reduction in bias and gains in efficiency through a simulation study and apply our procedure to an AIDS dataset to examine the effect of previous antiretroviral therapy on survival.     

Alternating event processes during lifetimes: population dynamics and statistical inference

In the literature studying recurrent event data, a large amount of work has been focused on univariate recurrent event processes where the occurrence of each event is treated as a single point in time. There are many applications, however, in which univariate recurrent events are insufficient to characterize the feature of the process because patients experience nontrivial durations associated with each event. This results in an alternating event process where the disease status of a patient alternates between exacerbations and remissions. In this paper, we consider the dynamics of a chronic disease and its associated exacerbation-remission process over two time scales: calendar time and time-since-onset. In particular, over calendar time, we explore population dynamics and the relationship between incidence, prevalence and duration for such alternating event processes. We provide nonparametric estimation techniques for characteristic quantities of the process. In some settings, exacerbation processes are observed from an onset time until death; to account for the relationship between the survival and alternating event processes, nonparametric approaches are developed for estimating exacerbation process over lifetime. By understanding the population dynamics and within-process structure, the paper provide a new and general way to study alternating event processes.     

Rejoinder

Kaplan–Meier graphs of survival and other timed events are an extremely effective way to summarize outcome data from some types of clinical studies, but information on dates and the interaction among events is lost. A new mode for presenting such data is proposed, an Eventchart, that aids in looking at the study as a whole by simultaneously displaying the timing of 2–3 different types of events, including censoring and time-dependent covariate events. The chart helps to monitor accrual and can be used to “guesstimate” the amount of information resulting from additional enrollment and/or follow-up. Eventcharts are designed to supplement, not replace, quantitative methods of analysis or conventional survival graphs. Data from bone marrow transplantation and AIDS studies are used to illustrate the method. Terminology is suggested for describing the concepts and distinguishing among the complex of events studied in modern clinical trials, including timed events, loss and date censoring, and time-braided events.     

Estimation of the joint survival function for successive duration times

In incident cohort studies, survival data often include subjects who have experienced an initiate event but have not experienced a subsequent event at the calendar time of recruitment. During the follow-up periods, subjects may undergo a series of successive events. Since the second/third duration process becomes observable only if the first/second event has occurred, the data are subject to left-truncation and dependent censoring. In this article, using the inverse-probability-weighted (IPW) approach, we propose nonparametric estimators for the estimation of the joint survival function of three successive duration times. The asymptotic properties of the proposed estimators are established. The simple bootstrap methods are used to estimate standard deviations and construct interval estimators. A simulation study is conducted to investigate the finite sample properties of the proposed estimators.     

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.     

An investigation into the two‐stage meta‐analytic copula modelling approach for evaluating time‐to‐event surrogate endpoints which comprise of one or more events of interest

Clinical trials of experimental treatments must be designed with primary endpoints that directly measure clinical benefit for patients. In many disease areas, the recognised gold standard primary endpoint can take many years to mature, leading to challenges in the conduct and quality of clinical studies. There is increasing interest in using shorter‐term surrogate endpoints as substitutes for costly long‐term clinical trial endpoints; such surrogates need to be selected according to biological plausibility, as well as the ability to reliably predict the unobserved treatment effect on the long‐term endpoint. A number of statistical methods to evaluate this prediction have been proposed; this paper uses a simulation study to explore one such method in the context of time‐to‐event surrogates for a time‐to‐event true endpoint. This two‐stage meta‐analytic copula method has been extensively studied for time‐to‐event surrogate endpoints with one event of interest, but thus far has not been explored for the assessment of surrogates which have multiple events of interest, such as those incorporating information directly from the true clinical endpoint. We assess the sensitivity of the method to various factors including strength of association between endpoints, the quantity of data available, and the effect of censoring. In particular, we consider scenarios where there exist very little data on which to assess surrogacy. Results show that the two‐stage meta‐analytic copula method performs well under certain circumstances and could be considered useful in practice, but demonstrates limitations that may prevent universal use.