Nonparametric and semiparametric trend analysis for stratified recurrence times

Recurrent event data are frequently encountered in longitudinal follow-up studies when the occurrences of multiple events are considered as the major outcomes. Suppose that the recurrent events are of the same type and the variable of interest is the recurrence time between successive events. In many applications, the distributional pattern of recurrence times can be used as an index for the progression of a disease. Such a distributional pattern is important for understanding the natural history of a disease or for confirming long-term treatment effect. In this article, we discuss and define the comparability of recurrence times. Nonparametric and semiparametric methods are developed for testing trend of recurrence time distributions and estimating trend parameters in regression models. The construction of the methods is based on comparable recurrence times from stratified data. A real data example is presented to illustrate the use of methodology.     

A joint modeling approach for analyzing marker data in the presence of a terminal event

In many medical studies, markers are contingent on recurrent events and the cumulative markers are usually of interest. However, the recurrent event process is often interrupted by a dependent terminal event, such as death. In this article, we propose a joint modeling approach for analyzing marker data with informative recurrent and terminal events. This approach introduces a shared frailty to specify the explicit dependence structure among the markers, the recurrent, and terminal events. Estimation procedures are developed for the model parameters and the degree of dependence, and a prediction of the covariate‐specific cumulative markers is provided. The finite sample performance of the proposed estimators is examined through simulation studies. An application to a medical cost study of chronic heart failure patients from the University of Virginia Health System is illustrated.     

Clustered mixed nonhomogeneous Poisson process spline models for the analysis of recurrent event panel data

Summary .   A flexible semiparametric model for analyzing longitudinal panel count data arising from mixtures is presented. Panel count data refers here to count data on recurrent events collected as the number of events that have occurred within specific follow-up periods. The model assumes that the counts for each subject are generated by mixtures of nonhomogeneous Poisson processes with smooth intensity functions modeled with penalized splines. Time-dependent covariate effects are also incorporated into the process intensity using splines. Discrete mixtures of these nonhomogeneous Poisson process spline models extract functional information from underlying clusters representing hidden subpopulations. The motivating application is an experiment to test the effectiveness of pheromones in disrupting the mating pattern of the cherry bark tortrix moth. Mature moths arise from hidden, but distinct, subpopulations and monitoring the subpopulation responses was of interest. Within-cluster random effects are used to account for correlation structures and heterogeneity common to this type of data. An estimating equation approach to inference requiring only low moment assumptions is developed and the finite sample properties of the proposed estimating functions are investigated empirically by simulation.     

Analysis of multivariate recurrent event data with time‐dependent covariates and informative censoring

Multivariate recurrent event data are usually encountered in many clinical and longitudinal studies in which each study subject may experience multiple recurrent events. For the analysis of such data, most existing approaches have been proposed under the assumption that the censoring times are noninformative, which may not be true especially when the observation of recurrent events is terminated by a failure event. In this article, we consider regression analysis of multivariate recurrent event data with both time‐dependent and time‐independent covariates where the censoring times and the recurrent event process are allowed to be correlated via a frailty. The proposed joint model is flexible where both the distributions of censoring and frailty variables are left unspecified. We propose a pairwise pseudolikelihood approach and an estimating equation‐based approach for estimating coefficients of time‐dependent and time‐independent covariates, respectively. The large sample properties of the proposed estimates are established, while the finite‐sample properties are demonstrated by simulation studies. The proposed methods are applied to the analysis of a set of bivariate recurrent event data from a study of platelet transfusion reactions.     

Semiparametric analysis of recurrent events data in the presence of dependent censoring

Dependent censoring occurs in longitudinal studies of recurrent events when the censoring time depends on the potentially unobserved recurrent event times. To perform regression analysis in this setting, we propose a semiparametric joint model that formulates the marginal distributions of the recurrent event process and dependent censoring time through scale-change models, while leaving the distributional form and dependence structure unspecified. We derive consistent and asymptotically normal estimators for the regression parameters. We also develop graphical and numerical methods for assessing the adequacy of the proposed model. The finite-sample behavior of the new inference procedures is evaluated through simulation studies. An application to recurrent hospitalization data taken from a study of intravenous drug users is provided.     

A hot-deck multiple imputation procedure for gaps in longitudinal recurrent event histories

We propose a regression-based hot-deck multiple imputation method for gaps of missing data in longitudinal studies, where subjects experience a recurrent event process and a terminal event. Examples are repeated asthma episodes and death, or menstrual periods and menopause, as in our motivating application. Research interest concerns the onset time of a marker event, defined by the recurrent event process, or the duration from this marker event to the final event. Gaps in the recorded event history make it difficult to determine the onset time of the marker event, and hence, the duration from onset to the final event. Simple approaches such as jumping gap times or dropping cases with gaps have obvious limitations. We propose a procedure for imputing information in the gaps by substituting information in the gap from a matched individual with a completely recorded history in the corresponding interval. Predictive mean matching is used to incorporate information on longitudinal characteristics of the repeated process and the final event time. Multiple imputation is used to propagate imputation uncertainty. The procedure is applied to an important data set for assessing the timing and duration of the menopausal transition. The performance of the proposed method is assessed by a simulation study.     

Methods for Estimating Center Effects on Recurrent Events

In this article, we develop methods for quantifying center effects with respect to recurrent event data. In the models of interest, center effects are assumed to act multiplicatively on the recurrent event rate function. When the number of centers is large, traditional estimation methods that treat centers as categorical variables have many parameters and are sometimes not feasible to implement, especially with large numbers of distinct recurrent event times. We propose a new estimation method for center effects which avoids including indicator variables for centers. We then show that center effects can be consistently estimated by the center-specific ratio of observed to expected cumulative numbers of events. We also consider the case where the recurrent event sequence can be stopped permanently by a terminating event. Large-sample results are developed for the proposed estimators. We assess the finite-sample properties of the proposed estimators through simulation studies. The methods are then applied to national hospital admissions data for end stage renal disease patients.     

Flexible Estimation of Differences in Treatment-Specific Recurrent Event Means in the Presence of a Terminating Event

Summary .  In this article, we consider the setting where the event of interest can occur repeatedly for the same subject (i.e., a recurrent event; e.g., hospitalization) and may be stopped permanently by a terminating event (e.g., death). Among the different ways to model recurrent/terminal event data, the marginal mean (i.e., averaging over the survival distribution) is of primary interest from a public health or health economics perspective. Often, the difference between treatment-specific recurrent event means will not be constant over time, particularly when treatment-specific differences in survival exist. In such cases, it makes more sense to quantify treatment effect based on the cumulative difference in the recurrent event means, as opposed to the instantaneous difference in the rates. We propose a method that compares treatments by separately estimating the survival probabilities and recurrent event rates given survival, then integrating to get the mean number of events. The proposed method combines an additive model for the conditional recurrent event rate and a proportional hazards model for the terminating event hazard. The treatment effects on survival and on recurrent event rate among survivors are estimated in constructing our measure and explain the mechanism generating the difference under study. The example that motivates this research is the repeated occurrence of hospitalization among kidney transplant recipients, where the effect of expanded criteria donor (ECD) compared to non-ECD kidney transplantation on the mean number of hospitalizations is of interest.     

Regression analysis of multivariate panel count data

We consider panel count data which are frequently obtained in prospective studies involving recurrent events that are only detected and recorded at periodic assessment times. The data take the form of counts of the cumulative number of events detected at each inspection time, along with explanatory covariates. Examples arise in diverse areas such as epidemiological studies, medical follow-up studies, reliability studies, and tumorigenicity experiments. This article is concerned with regression analysis of multivariate panel count data which arise if more than one type of recurrent event is of interest and individuals are only observed intermittently. We present a class of marginal mean models which leave the dependence structures for related types of recurrent events completely unspecified. Estimating equations are developed for regression parameters, and the resulting estimates are shown to be consistent and asymptotically normal. Simulation studies show that the proposed estimation procedures work well for practical situations. The methodology is applied to a motivating study of patients with psoriatic arthritis in which the events of interest are the onset of joint damage according to 2 different criteria.     

A two-sample comparison for multiple ordered event data

A longitudinal study is conducted to compare the process of particular disease between two groups. The process of the disease is monitored according to which of several ordered events occur. In the paper, the sojourn time between two successive events is considered as the outcome of interest. The group effects on the sojourn times of the multiple events are parameterized by scale changes in a semiparametric accelerated failure time model where the dependence structure among the multivariate sojourn times is unspecified. Suppose that the sojourn times are subject to dependent censoring and the censoring times are observed for all subjects. A log-rank-type estimating approach by rescaling the sojourn times and the dependent censoring times into the same distribution is constructed to estimate the group effects and the corresponding estimators are consistent and asymptotically normal. Without the dependent censoring, the independent censoring times in general are not available for the uncensored data. In order to complete the censoring information, pseudo-censoring times are generated from the corresponding nonparametrically estimated survival function in each group, and we can still obtained unbiased estimating functions for the group effects. A real application and a simulation study are conducted to illustrate the proposed methods.     

New Data on Axial Locomotion in Fishes: How Speed Affects Diversity of Kinematics and Motor Patterns

Despite considerable recent progress in understanding the functionof the axial muscles and skeleton in fishes, generalizing fromthese results has been hindered by the great phylogenetic diversityof taxa, the lack of quantitative morphometric data on axialmusculoskeletal structure, and limited analysis of the fullrange of locomotor behaviors exhibited within any one taxon.This paper reviews novel results from our studies of two taxawithin a single monophyletic clade, the sunfish family Centrarchidae.Integrated analyses of lateral displacement, lateral bending,and axial muscle activity reveal widespread effects of swimmingspeed both within a particular mode of swimming and among differentbehavioral modes. The longitudinal position along the body ofthe fish also commonly affects kinematics, muscle activity andthe timing of electromyograms (EMGs) relative to kinematics.EMGs and kinematic events propagate from head to tail for bothsteady and kick and glide swimming. In contrast, during theescape response, the onset of EMGs forms a standing wave pattern,whereas kinematic events are propagated. Several novel featuresof the axial motor pattern are summarized for the kick and glidemode of unsteady swimming. For example, the onset of white fiberEMGs lags significantly behind that of the red fibers at thesame longitudinal position, and red fibers are inactivated athigher unsteady swimming speeds. Muscle activity propagatesposteriorly via the sequential activation of myomeres, but thereare statistically significant differences in the timing of EMGsfrom the contractile tissue opposite a single vertebra. Duringrelatively slow kick and glide swimming, the extreme dorsaland ventral portions of myomeres are not active. Estimates ofthe longitudinal extent of the fish with simultaneous muscleactivity indicate that EMGs from an individual myomere usuallyhave temporal overlap with more than 20 neighboring myomereson the same side of the fish. Consequently, the functional unitsfor axial locomotion of fishes do not correspond simply to theanatomical units of individual myomeres. Rather the in vivomotor pattern is a primary determinant of the functional unitsinvolved in swimming.     

Joint analysis of longitudinal data and recurrent episodes data with application to medical cost analysis

This paper discusses regression analysis of longitudinal data in which the observation process may be related to the longitudinal process of interest. Such data have recently attracted a great deal of attention and some methods have been developed. However, most of those methods treat the observation process as a recurrent event process, which assumes that one observation can immediately follow another. Sometimes, this is not the case, as there may be some delay or observation duration. Such a process is often referred to as a recurrent episode process. One example is the medical cost related to hospitalization, where each hospitalization serves as a single observation. For the problem, we present a joint analysis approach for regression analysis of both longitudinal and observation processes and a simulation study is conducted that assesses the finite sample performance of the approach. The asymptotic properties of the proposed estimates are also given and the method is applied to the medical cost data that motivated this study.     

Shared frailty models for recurrent events and a terminal event

There has been an increasing interest in the analysis of recurrent event data (Cook and Lawless, 2002, Statistical Methods in Medical Research 11, 141-166). In many situations, a terminating event such as death can happen during the follow-up period to preclude further occurrence of the recurrent events. Furthermore, the death time may be dependent on the recurrent event history. In this article we consider frailty proportional hazards models for the recurrent and terminal event processes. The dependence is modeled by conditioning on a shared frailty that is included in both hazard functions. Covariate effects can be taken into account in the model as well. Maximum likelihood estimation and inference are carried out through a Monte Carlo EM algorithm with Metropolis-Hastings sampler in the E-step. An analysis of hospitalization and death data for waitlisted dialysis patients is presented to illustrate the proposed methods. Methods to check the validity of the proposed model are also demonstrated. This model avoids the difficulties encountered in alternative approaches which attempt to specify a dependent joint distribution with marginal proportional hazards and yields an estimate of the degree of dependence.     

Two‐stage estimation for multivariate recurrent event data with a dependent terminal event

Recurrent event data arise in longitudinal follow‐up studies, where each subject may experience the same type of events repeatedly. The work in this article is motivated by the data from a study of repeated peritonitis for patients on peritoneal dialysis. Due to the aspects of medicine and cost, the peritonitis cases were classified into two types: Gram‐positive and non‐Gram‐positive peritonitis. Further, since the death and hemodialysis therapy preclude the occurrence of recurrent events, we face multivariate recurrent event data with a dependent terminal event. We propose a flexible marginal model, which has three characteristics: first, we assume marginal proportional hazard and proportional rates models for terminal event time and recurrent event processes, respectively; second, the inter‐recurrences dependence and the correlation between the multivariate recurrent event processes and terminal event time are modeled through three multiplicative frailties corresponding to the specified marginal models; third, the rate model with frailties for recurrent events is specified only on the time before the terminal event. We propose a two‐stage estimation procedure for estimating unknown parameters. We also establish the consistency of the two‐stage estimator. Simulation studies show that the proposed approach is appropriate for practical use. The methodology is applied to the peritonitis cohort data that motivated this study.     

Hierarchical Bayesian methods for estimation of parameters in a longitudinal HIV dynamic system

HIV dynamics studies have significantly contributed to the understanding of HIV infection and antiviral treatment strategies. But most studies are limited to short-term viral dynamics due to the difficulty of establishing a relationship of antiviral response with multiple treatment factors such as drug exposure and drug susceptibility during long-term treatment. In this article, a mechanism-based dynamic model is proposed for characterizing long-term viral dynamics with antiretroviral therapy, described by a set of nonlinear differential equations without closed-form solutions. In this model we directly incorporate drug concentration, adherence, and drug susceptibility into a function of treatment efficacy, defined as an inhibition rate of virus replication. We investigate a Bayesian approach under the framework of hierarchical Bayesian (mixed-effects) models for estimating unknown dynamic parameters. In particular, interest focuses on estimating individual dynamic parameters. The proposed methods not only help to alleviate the difficulty in parameter identifiability, but also flexibly deal with sparse and unbalanced longitudinal data from individual subjects. For illustration purposes, we present one simulation example to implement the proposed approach and apply the methodology to a data set from an AIDS clinical trial. The basic concept of the longitudinal HIV dynamic systems and the proposed methodologies are generally applicable to any other biomedical dynamic systems.     

Logistic regression when covariates are random effects from a non‐linear mixed model

In many studies, the association of longitudinal measurements of a continuous response and a binary outcome are often of interest. A convenient framework for this type of problems is the joint model, which is formulated to investigate the association between a binary outcome and features of longitudinal measurements through a common set of latent random effects. The joint model, which is the focus of this article, is a logistic regression model with covariates defined as the individual‐specific random effects in a non‐linear mixed‐effects model (NLMEM) for the longitudinal measurements. We discuss different estimation procedures, which include two‐stage, best linear unbiased predictors, and various numerical integration techniques. The proposed methods are illustrated using a real data set where the objective is to study the association between longitudinal hormone levels and the pregnancy outcome in a group of young women. The numerical performance of the estimating methods is also evaluated by means of simulation.     

A nonparametric approach to the analysis of longitudinal data via a set of level crossing problems with application to the analysis of microarray time course experiments

Here we develop a completely nonparametric method for comparing two groups on a set of longitudinal measurements. No assumptions are made about the form of the mean response function, the covariance structure or the distributional form of disturbances around the mean response function. The solution proposed here is based on the realization that every longitudinal data set can also be thought of as a collection of survival data sets where the events of interest are level crossings. The method for testing for differences in the longitudinal measurements then is as follows: for an arbitrarily large set of levels, for each subject determine the first time the subject has an upcrossing and a downcrossing for each level. For each level one then computes the log rank statistic and uses the maximum in absolute value of all these statistics as the test statistic. By permuting group labels we obtain a permutation test of the hypothesis that the joint distribution of the measurements over time does not depend on group membership. Simulations are performed to investigate the power and it is applied to the area that motivated the method-the analysis of microarrays. In this area small sample sizes, few time points and far too many genes to consider genuine gene level longitudinal modeling have created a need for a simple, model free test to screen for interesting features in the data.     

Nonparametric estimation of the survival function for ordered multivariate failure time data: A comparative study

In longitudinal studies of disease, patients may experience several events through a follow‐up period. In these studies, the sequentially ordered events are often of interest and lead to problems that have received much attention recently. Issues of interest include the estimation of bivariate survival, marginal distributions, and the conditional distribution of gap times. In this work, we consider the estimation of the survival function conditional to a previous event. Different nonparametric approaches will be considered for estimating these quantities, all based on the Kaplan–Meier estimator of the survival function. We explore the finite sample behavior of the estimators through simulations. The different methods proposed in this article are applied to a dataset from a German Breast Cancer Study. The methods are used to obtain predictors for the conditional survival probabilities as well as to study the influence of recurrence in overall survival.     

Joint Models of Longitudinal Data and Recurrent Events with Informative Terminal Event

This article presents semiparametric joint models to analyze longitudinal data with recurrent events (e.g. multiple tumors, repeated hospital admissions) and a terminal event such as death. A broad class of transformation models for the cumulative intensity of the recurrent events and the cumulative hazard of the terminal event is considered, which includes the proportional hazards model and the proportional odds model as special cases. We propose to estimate all the parameters using the nonparametric maximum likelihood estimators (NPMLE). We provide the simple and efficient EM algorithms to implement the proposed inference procedure. Asymptotic properties of the estimators are shown to be asymptotically normal and semiparametrically efficient. Finally, we evaluate the performance of the method through extensive simulation studies and a real-data application.     

Mixed Poisson likelihood regression models for longitudinal interval count data

In many longitudinal studies it is desired to estimate and test the rate over time of a particular recurrent event. Often only the event counts corresponding to the elapsed time intervals between each subject's successive observation times, and baseline covariate data, are available. The intervals may vary substantially in length and number between subjects, so that the corresponding vectors of counts are not directly comparable. A family of Poisson likelihood regression models incorporating a mixed random multiplicative component in the rate function of each subject is proposed for this longitudinal data structure. A related empirical Bayes estimate of random-effect parameters is also described. These methods are illustrated by an analysis of dyspepsia data from the National Cooperative Gallstone Study.