Analysis of Panel Count Data with Dependent Observation Times

In this article, a semiparametric approach is proposed for the regression analysis of panel count data. Panel count data commonly arise in clinical trials and demographical studies where the response variable is the number of multiple recurrences of the event of interest and observation times are not fixed, varying from subject to subject. It is assumed that two processes exist in this data: the first is for a recurrent event and the second is for observation time. Many studies have been done to estimate mean function and regression parameters under the independency between recurrent event process and observation time process. In this article, the same statistical inference is studied, but the situation where these two processes may be related is also considered. The mixed Poisson process is applied for the recurrent event processes, and a frailty intensity function for the observation time is also used, respectively. Simulation studies are conducted to study the performance of the suggested methods. The bladder tumor data are applied to compare previous studie' results.     

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.     

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.     

Sample size calculation for recurrent events data in one‐arm studies

In some exceptional circumstances, as in very rare diseases, nonrandomized one‐arm trials are the sole source of evidence to demonstrate efficacy and safety of a new treatment. The design of such studies needs a sound methodological approach in order to provide reliable information, and the determination of the appropriate sample size still represents a critical step of this planning process. As, to our knowledge, no method exists for sample size calculation in one‐arm trials with a recurrent event endpoint, we propose here a closed sample size formula. It is derived assuming a mixed Poisson process, and it is based on the asymptotic distribution of the one‐sample robust nonparametric test recently developed for the analysis of recurrent events data. The validity of this formula in managing a situation with heterogeneity of event rates, both in time and between patients, and time‐varying treatment effect was demonstrated with exhaustive simulation studies. Moreover, although the method requires the specification of a process for events generation, it seems to be robust under erroneous definition of this process, provided that the number of events at the end of the study is similar to the one assumed in the planning phase. The motivating clinical context is represented by a nonrandomized one‐arm study on gene therapy in a very rare immunodeficiency in children (ADA‐SCID), where a major endpoint is the recurrence of severe infections. Copyright © 2012 John Wiley & Sons, Ltd.     

Estimation of the joint survival function for three successive duration times under double truncation and right censoring

In incident cohort studies, it is common to include subjects who have experienced a certain event within a calendar time window. For all the included individuals, the time of the previous events is retrospectively confirmed and the occurrence of subsequent events is observed during the follow-up periods. During the follow-up periods, subjects may undergo three successive events. Since the second/third duration process becomes observable only if the first/second event has occurred, the data is subject to double truncation and right censoring. We consider two cases: the case when the first event time is subject to double truncation and the case when the second event time is subject to double truncation. Using the inverse-probability-weighted approach, we propose nonparametric and semiparametric estimators for the estimation of the joint survival function of three successive duration times. We establish the asymptotic properties of the proposed estimators and conduct a simulation study to investigate the finite sample properties of the proposed estimators.     

Gaussian models for degradation processes-part I: Methods for the analysis of biomarker data

We present two stochastic models that describe the relationship between biomarker process values at random time points, event times, and a vector of covariates. In both models the biomarker processes are degradation processes that represent the decay of systems over time. In the first model the biomarker process is a Wiener process whose drift is a function of the covariate vector. In the second model the biomarker process is taken to be the difference between a stationary Gaussian process and a time drift whose drift parameter is a function of the covariates. For both models we present statistical methods for estimation of the regression coefficients. The first model is useful for predicting the residual time from study entry to the time a critical boundary is reached while the second model is useful for predicting the latency time from the infection until the time the presence of the infection is detected. We present our methods principally in the context of conducting inference in a population of HIV infected individuals.     

Semiparametric inference for the recurrent events process by means of a single-index model

In this paper, we introduce new parametric and semiparametric regression techniques for a recurrent event process subject to random right censoring. We develop models for the cumulative mean function and provide asymptotically normal estimators. Our semiparametric model which relies on a single-index assumption can be seen as a dimension reduction technique that, contrary to a fully nonparametric approach, is not stroke by the curse of dimensionality when the number of covariates is high. We discuss data-driven techniques to choose the parameters involved in the estimation procedures and provide a simulation study to support our theoretical results.     

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.

     

Event dependent sampling of recurrent events

The effect of event-dependent sampling of processes consisting of recurrent events is investigated when analyzing whether the risk of recurrence increases with event count. We study the situation where processes are selected for study if an event occurs in a certain selection interval. Motivation comes from psychiatric epidemiology where repeated hospital admissions are studied for patients with affective disease, as seen in Kessing et al. (Acta Psychiatr Scand 109:339–344, 2004b). For the selected processes, either only disease course from selection and onwards is used in the analysis, or, both retrospective and prospective disease course histories are used. We examine two methods to correct for the selection depending on which data are used in the analysis. In the first case, the conditional distribution of the process given the pre-selection history is determined. In the second case, an inverse-probability-of-selection weighting scheme is suggested. The ability of the methods to correct for the bias due to selection is investigated with simulations. Furthermore, the methods are applied to affective disease data from a register-based study (Kessing et al. Br J Psychiatry 185:372–377, 2004a) and from a long-term clinical study (Kessing et al. Acta Psychiatr Scand 109:339–344, 2004b).     

Estimation of the joint survival function for successive duration times under double-truncation and right-censoring

ABSTRACT

In incident cohort studies, survival data often include subjects who have had an initiate event at recruitment and may potentially experience two successive events (first and second) during the follow-up period. When disease registries or surveillance systems collect data based on incidence occurring within a specific calendar time interval, the initial event is usually subject to double truncation. Furthermore, since the second duration process is observable only if the first event has occurred, double truncation and dependent censoring arise. In this article, under the two sampling biases with an unspecified distribution of truncation variables, we propose a nonparametric estimator of the joint survival function of two successive duration times using the inverse-probability-weighted (IPW) approach. The consistency of the proposed estimator is established. Based on the estimated marginal survival functions, we also propose a two-stage estimation procedure for estimating the parameters of copula model. The bootstrap method is used to construct confidence interval. Numerical studies demonstrate that the proposed estimation approaches perform well with moderate sample sizes.     

Recurrent first hitting times in Wiener diffusion under several observation schemes

Recurrent events are commonly encountered in the natural sciences, engineering, and medicine. The theory of renewal and regenerative processes provides an elegant mathematical foundation for idealized recurrent event processes. In real-world applications, however, the contexts tend to be complicated by a variety of practical intricacies, including observation schemes with different phase and data structures. This paper formulates a recurrent event process as a succession of independent and identically distributed first hitting times for a Wiener sample path as it passes through successive equally-spaced levels. We develop exact mathematical results for statistical inferences based on several observation schemes that include observation initiated at a renewal point, observation of a stationary process over a finite window, and other variants. We also consider inferences drawn from different data structures, including gap times between renewal points (or fragments thereof) and counts of renewal events occurring within an observation window. We explore the precision of estimates using simulated scenarios and develop empirical regression functions for planning the sample size of a recurrent event study. We demonstrate our results using data from a clinical trial for chronic obstructive pulmonary disease in which the recurrent events are successive exacerbations of the condition. The case study demonstrates how covariates can be incorporated into the analysis using threshold regression.     

Sojourning With the Homogeneous Poisson Process

In this pedagogical article, distributional properties, some surprising, pertaining to the homogeneous Poisson process (HPP), when observed over a possibly random window, are presented. Properties of the gap-time that covered the termination time and the correlations among gap-times of the observed events are obtained. Inference procedures, such as estimation and model validation, based on event occurrence data over the observation window, are also presented. We envision that through the results in this article, a better appreciation of the subtleties involved in the modeling and analysis of recurrent events data will ensue, since the HPP is arguably one of the simplest among recurrent event models. In addition, the use of the theorem of total probability, Bayes’ theorem, the iterated rules of expectation, variance and covariance, and the renewal equation could be illustrative when teaching distribution theory, mathematical statistics, and stochastic processes at both the undergraduate and graduate levels. This article is targeted toward both instructors and students.     

Evolution of recurrent asthma event rate over time in frailty models

Summary. To model the time evolution of the event rate in recurrent event data a crucial role is played by the timescale that is used. Depending on the timescale selected the interpretation of the time evolution will be entirely different, both in parametric and semiparametric frailty models. The gap timescale is more appropriate when studying the recurrent event rate as a function of time since the last event, whereas the calendar timescale keeps track of actual time. We show both timescales in action on data from an asthma prevention trial in young children. The frailty model is further extended to include both timescales simultaneously as this might be most relevant in practice.     

Jointly modeling time-to-event and longitudinal data: a Bayesian approach

This article explores Bayesian joint models of event times and longitudinal measures with an attempt to overcome departures from normality of the longitudinal response, measurement errors, and shortages of confidence in specifying a parametric time-to-event model. We allow the longitudinal response to have a skew distribution in the presence of measurement errors, and assume the time-to-event variable to have a nonparametric prior distribution. Posterior distributions of the parameters are attained simultaneously for inference based on Bayesian approach. An example from a recent AIDS clinical trial illustrates the methodology by jointly modeling the viral dynamics and the time to decrease in CD4/CD8 ratio in the presence of CD4 counts with measurement errors to compare potential models with various scenarios and different distribution specifications. The analysis outcome indicates that the time-varying CD4 covariate is closely related to the first-phase viral decay rate, but the time to CD4/CD8 decrease is not highly associated with either the two viral decay rates or the CD4 changing rate over time. These findings may provide some quantitative guidance to better understand the relationship of the virological and immunological responses to antiretroviral treatments.     

Nonparametric Inference of Doubly Stochastic Poisson Process Data via the Kernel Method

Doubly stochastic Poisson processes, also known as the Cox processes, frequently occur in various scientific fields. In this article, motivated primarily by analyzing Cox process data in biophysics, we propose a nonparametric kernel-based inference method. We conduct a detailed study, including an asymptotic analysis, of the proposed method, and provide guidelines for its practical use, introducing a fast and stable regression method for bandwidth selection. We apply our method to real photon arrival data from recent single-molecule biophysical experiments, investigating proteins' conformational dynamics. Our result shows that conformational fluctuation is widely present in protein systems, and that the fluctuation covers a broad range of time scales, highlighting the dynamic and complex nature of proteins' structure.     

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.     

Robust inference for bivariate point processes

In the analysis of recurrent events where the primary interest lies in studying covariate effects on the expected number of events occurring over a period of time, it is appealing to base models on the cumulative mean function (CMF) of the processes (Lawless & Nadeau 1995). In many chronic diseases, however, more than one type of event is manifested. Here we develop a robust inference procedure for joint regression models for the CMFs arising from a bivariate point process. Consistent parameter estimates with robust variance estimates are obtained via unbiased estimating functions for the CMFs. In most situations, the covariance structure of the bivariate point processes is difficult to specify correctly, but when it is known, an optimal estimating function for the CMFs can be obtained. As a convenient model for more general settings, we suggest the use of the estimating functions arising from bivariate mixed Poisson processes. Simulation studies demonstrate that the estimators based on this working model are practically unbiased with robust variance estimates. Furthermore, hypothesis tests may be based on the generalized Wald or generalized score tests. Data from a trial of patients with bronchial asthma are analyzed to illustrate the estimation and inference procedures.     

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.     

Nonparametric estimation of the mean function of a stochastic process with missing observations

In an attempt to identify similarities between methods for estimating a mean function with different types of response or observation processes, we explore a general theoretical framework for nonparametric estimation of the mean function of a response process subject to incomplete observations. Special cases of the response process include quantitative responses and discrete state processes such as survival processes, counting processes and alternating binary processes. The incomplete data are assumed to arise from a general response-independent observation process, which includes right- censoring, interval censoring, periodic observation, and mixtures of these as special cases. We explore two criteria for defining nonparametric estimators, one based on the sample mean of available data and the other inspired by the construction of Kaplan-Meier (or product-limit) estimator [J. Am. Statist. Assoc. 53 (1958) 457] for right-censored survival data. We show that under regularity conditions the estimated mean functions resulting from both criteria are consistent and converge weakly to Gaussian processes, and provide consistent estimators of their covariance functions. We then evaluate these general criteria for specific responses and observation processes, and show how they lead to familiar estimators for some response and observation processes and new estimators for others. We illustrate the latter with data from an recently completed AIDS clinical trial.     

MULTILEVEL DISCRETE‐TIME EVENT HISTORY MODELS WITH APPLICATIONS TO THE ANALYSIS OF RECURRENT EMPLOYMENT TRANSITIONS

Data on the timing of events such as births, residential moves and changes in employment status are collected in many longitudinal surveys. These data often have a highly complex structure, with events of several types occurring repeatedly over time to an individual and interdependences between different event processes (e.g. births and employment transitions). The aim of this paper is to review a general class of multilevel discrete‐time event history models for handling recurrent events and transitions between multiple states. It is also shown how standard methods can be extended to allow for time‐varying covariates that are outcomes of an event process that is jointly determined with the process of interest. The considerable potential of these methods for studying transitions through the life course is illustrated in analyses of the effect of the presence and age of children on women's employment transitions, using data from the British Household Panel Survey.