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.     

Multiple Imputation Approaches for the Analysis of Dichotomized Responses in Longitudinal Studies with Missing Data

Summary Often a binary variable is generated by dichotomizing an underlying continuous variable measured at a specific time point according to a prespecified threshold value. In the event that the underlying continuous measurements are from a longitudinal study, one can use the repeated‐measures model to impute missing data on responder status as a result of subject dropout and apply the logistic regression model on the observed or otherwise imputed responder status. Standard Bayesian multiple imputation techniques ( Rubin, 1987 , in Multiple Imputation for Nonresponse in Surveys) that draw the parameters for the imputation model from the posterior distribution and construct the variance of parameter estimates for the analysis model as a combination of within‐ and between‐imputation variances are found to be conservative. The frequentist multiple imputation approach that fixes the parameters for the imputation model at the maximum likelihood estimates and construct the variance of parameter estimates for the analysis model using the results of Robins and Wang (2000, Biometrika 87, 113–124) is shown to be more efficient. We propose to apply ( Kenward and Roger, 1997 , Biometrics 53, 983–997) degrees of freedom to account for the uncertainty associated with variance–covariance parameter estimates for the repeated measures model.     

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.     

Multiple imputation and posterior simulation for multivariate missing data in longitudinal studies

This paper outlines a multiple imputation method for handling missing data in designed longitudinal studies. A random coefficients model is developed to accommodate incomplete multivariate continuous longitudinal data. Multivariate repeated measures are jointly modeled; specifically, an i.i.d. normal model is assumed for time-independent variables and a hierarchical random coefficients model is assumed for time-dependent variables in a regression model conditional on the time-independent variables and time, with heterogeneous error variances across variables and time points. Gibbs sampling is used to draw model parameters and for imputations of missing observations. An application to data from a study of startle reactions illustrates the model. A simulation study compares the multiple imputation procedure to the weighting approach of Robins, Rotnitzky, and Zhao (1995, Journal of the American Statistical Association 90, 106-121) that can be used to address similar data structures.     

Genome-wide linkage analysis of systolic blood pressure slope using the Genetic Analysis Workshop 13 data sets

Systolic blood pressure (SBP) is an age-dependent complex trait for which both environmental and genetic factors may play a role in explaining variability among individuals. We performed a genome-wide scan of the rate of change in SBP over time on the Framingham Heart Study data and one randomly selected replicate of the simulated data from the Genetic Analysis Workshop 13. We used a variance-component model to carry out linkage analysis and a Markov chain Monte Carlo-based multiple imputation approach to recover missing information. Furthermore, we adopted two selection strategies along with the multiple imputation to deal with subjects taking antihypertensive treatment. The simulated data were used to compare these two strategies, to explore the effectiveness of the multiple imputation in recovering varying degrees of missing information, and its impact on linkage analysis results. For the Framingham data, the marker with the highest LOD score for SBP slope was found on chromosome 7. Interestingly, we found that SBP slopes were not heritable in males but were for females; the marker with the highest LOD score was found on chromosome 18. Using the simulated data, we found that handling treated subjects using the multiple imputation improved the linkage results. We conclude that multiple imputation is a promising approach in recovering missing information in longitudinal genetic studies and hence in improving subsequent linkage analyses.     

Semiparametric regression analysis on longitudinal pattern of recurrent gap times

In longitudinal studies, individual subject may experience recurrent events of the same type over a relatively long period of time. The longitudinal pattern of gaps between successive recurrent events is often of great research interest. In this article, the probability structure of the recurrent gap times is first explored in the presence of censoring. According to the discovered structure, we introduce the stratified proportional reverse-time hazards models with unspecified baseline functions to accommodate individual heterogeneity, when the longitudinal pattern parameter is of main interest. Inference procedures are proposed and studied by way of proper riskset construction. The proposed methodology is demonstrated by the Monte Carlo simulations and an application to a well-known Denmark schizophrenia cohort study data set.     

Using a surrogate marker for early testing of a treatment effect

The development of methods to identify, validate, and use surrogate markers to test for a treatment effect has been an area of intense research interest given the potential for valid surrogate markers to reduce the required costs and follow‐up times of future studies. Several quantities and procedures have been proposed to assess the utility of a surrogate marker. However, few methods have been proposed to address how one might use the surrogate marker information to test for a treatment effect at an earlier time point, especially in settings where the primary outcome and the surrogate marker are subject to censoring. In this paper, we propose a novel test statistic to test for a treatment effect using surrogate marker information measured prior to the end of the study in a time‐to‐event outcome setting. We propose a robust nonparametric estimation procedure and propose inference procedures. In addition, we evaluate the power for the design of a future study based on surrogate marker information. We illustrate the proposed procedure and relative power of the proposed test compared to a test performed at the end of the study using simulation studies and an application to data from the Diabetes Prevention Program.     

Nonparametric estimation of asymptomatic duration from a randomized prospective cancer screening trial

We propose a nonparametric estimation of preclinical duration distribution in cancer based on data from a randomized early detection trial. In cancer screening studies, the preclinical duration of a disease is of great interest for better understanding the natural history of the disease, and for developing optimal screening strategies. To estimate the sojourn time distribution nonparametrically, we first estimate the distribution of the age at onset of preclinical disease nonparametrically using data from the screening arm in a randomized screening trial, and the distribution for the age at onset of clinical disease from the control arm of the randomized screening trial. Finally, by using deconvolution the two estimated distributions lead to a nonparametric estimate of the distribution for the gap time between the onset of preclinical disease and the onset of clinical disease. We illustrate the methodology using data from a randomized breast cancer screening trial.     

The effect of intruded events on peak time: the role of reinforcement history during the intruded event

Pigeons were studied in an extension of a study by Aum et al. [Aum, S., Brown, B.L., Hemmes, N.S. 2004. The effects of concurrent task and gap events on peak time in the peak procedure. Behav. Process. 65, 43-56] on timing behavior under a discrete-trial fixed-interval (FI) procedure during which 6-s intruded events were superimposed on peak-interval (PI) test trials. In Aum et al., one event consisted in termination of the timing cue (gap trial); the other was a stimulus in the presence of which subjects had been trained to respond under an independent random-interval (RI) schedule of reinforcement (concurrent task trial). Aum et al. found a disruption of timing on concurrent task trials that was greater than that on gap trials. The present study investigated history of reinforcement associated with intruded events as a possible explanation of this earlier finding. After training to peck a side key on a 30-s PI procedure, discrimination training was conducted on the center key in separate sessions; red or green 6-s stimuli were associated with RI 24s or EXT (extinction) schedules. During testing under the PI procedure, three types of intruded events were presented during probe trials--the stimulus associated with the RI (S+) or EXT (S-) schedule during discrimination training, or a gap (termination of the side-keylight). Intruded events occurred 3, 9, or 15s after PI trial onset. Effects of reinforcement history were revealed as substantial disruption of timing during the S+ event and relatively little disruption during the S- event. Intermediate effects were found for the gap event. Results indicate that postcue effects are at least partially responsible for the disruptive effects of the S+ event.     

Gap Dynamics in a Seagrass Landscape

We investigated gap dynamics within a shallow subtidal landscape characterized by seagrass vegetation and examined the relationship between gap formation and selected physical factors. The study was conducted over 2 y by using a biannual mapping of seagrass and water depth across an 48,800-m² area in Tampa Bay, Florida. In addition, monthly sediment deposition or erosion was recorded at 96 locations within the landscape. Gaps represented from 2.4% to 5.7% of the seagrass landscape, and all were within monospecific stands of Halodule wrightii. Gaps ranged in size from 10 to 305 m² and most frequently decreased in size over time. Most gaps were small and short lived (less than 6-mo duration), but the second age group most frequently recorded was at least 1.5 y old. No new species of seagrass invaded the gaps with Halodule replacing itself 100% of the time. Gaps were recorded over the entire range of water depths within the landscape. Neither gap area nor persistence of gaps was related to water depth. However gap area was associated positively with the number of extreme sedimentation events. Gaps originated not only from removal of interior vegetation (similar to classic gaps) but also from differential growth of the seagrass margin (similar to edaphic gaps). Distinct seasonal components to the mode of formation were detected with interior-produced gaps originating primarily in the winter and margin gaps most commonly during summer. These results combine to illustrate the importance of large-scale studies with fine-scale resolution for deciphering unique features of seagrass landscape dynamics. Our historical information suggests that a static enumeration of gaps may not provide an accurate assessment of disturbance intensity in this system, and the seagrass mosaic probably is explained best by a combination of disturbance regimes and edaphic factors, such as sediment stability. Moreover, we suggest that even in areas characterized by monospecific stands of vegetation and over short or moderate time periods, gaps indirectly may influence community structure and ecosystem function via modification of habitat arrangement. Received 17 September 1998; accepted 26 April 1999.     

Penalized survival models for the analysis of alternating recurrent event data

Recurrent event data are widely encountered in clinical and observational studies. Most methods for recurrent events treat the outcome as a point process and, as such, neglect any associated event duration. This generally leads to a less informative and potentially biased analysis. We propose a joint model for the recurrent event rate (of incidence) and duration. The two processes are linked through a bivariate normal frailty. For example, when the event is hospitalization, we can treat the time to admission and length-of-stay as two alternating recurrent events. In our method, the regression parameters are estimated through a penalized partial likelihood, and the variance-covariance matrix of the frailty is estimated through a recursive estimating formula. Moreover, we develop a likelihood ratio test to assess the dependence between the incidence and duration processes. Simulation results demonstrate that our method provides accurate parameter estimation, with a relatively fast computation time. We illustrate the methods through an analysis of hospitalizations among end-stage renal disease patients.     

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.     

Analysis of longitudinal data in the presence of informative observational times and a dependent terminal event, with application to medical cost data

Summary .   In longitudinal observational studies, repeated measures are often taken at informative observation times. Also, there may exist a dependent terminal event such as death that stops the follow-up. For example, patients in poorer health are more likely to seek medical treatment and their medical cost for each visit tends to be higher. They are also subject to a higher mortality rate. In this article, we propose a random effects model of repeated measures in the presence of both informative observation times and a dependent terminal event. Three submodels are used, respectively, for (1) the intensity of recurrent observation times, (2) the amount of repeated measure at each observation time, and (3) the hazard of death. Correlated random effects are incorporated to join the three submodels. The estimation can be conveniently accomplished by Gaussian quadrature techniques, e.g., SAS Proc NLMIXED . An analysis of the cost-accrual process of chronic heart failure patients from the clinical data repository at the University of Virginia Health System is presented to illustrate the proposed method.     

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.     

Models of sequence evolution for DNA sequences containing gaps

Most evolutionary tree estimation methods for DNA sequences ignore or inefficiently use the phylogenetic information contained within shared patterns of gaps. This is largely due to the computational difficulties in implementing models for insertions and deletions. A simple way to incorporate this information is to treat a gap as a fifth character (with the four nucleotides being the other four) and to incorporate it within a Markov model of nucleotide substitution. This idea has been dismissed in the past, since it treats a multiple-site insertion or deletion as a sequence of independent events rather than a single event. While this is true, we have found that under many circumstances it is better to incorporate gap information inadequately than to ignore it, at least for topology estimation. We propose an extension to a class of nucleotide substitution models to incorporate the gap character and show that, for data sets (both real and simulated) with short and medium gaps, these models do lead to effective use of the information contained within insertions and deletions. We also implement an ad hoc method in which the likelihood at columns containing multiple-site gaps is downweighted in order to avoid giving them undue influence. The precision of the estimated tree, assessed using Markov chain Monte Carlo techniques to find the posterior distribution over tree space, improves under these five-state models compared with standard methods which effectively ignore gaps.     

Joint Modelling of Repeated Measures and Survival Time Data

In many clinical trials both repeated measures data and event history data are simultaneously observed from the same subject. These two types of responses are usually correlated, because they are from the same subject. In this article, we propose a joint model for the combined analysis of repeated measures data and event history data in the framework of hierarchical generalized linear models. The correlation between repeated measures and event time is modelled by introducing a shared random effect. The model parameters are estimated using the hierarchical‐likelihood approach. The proposed model is illustrated using a real data set for the renal transplant patients.     

Semiparametric Analysis for Recurrent Event Data with Time-Dependent Covariates and Informative Censoring

Summary .  Recurrent event data analyses are usually conducted under the assumption that the censoring time is independent of the recurrent event process. In many applications the censoring time can be informative about the underlying recurrent event process, especially in situations where a correlated failure event could potentially terminate the observation of recurrent events. In this article, we consider a semiparametric model of recurrent event data that allows correlations between censoring times and recurrent event process via frailty. This flexible framework incorporates both time-dependent and time-independent covariates in the formulation, while leaving the distributions of frailty and censoring times unspecified. We propose a novel semiparametric inference procedure that depends on neither the frailty nor the censoring time distribution. Large sample properties of the regression parameter estimates and the estimated baseline cumulative intensity functions are studied. Numerical studies demonstrate that the proposed methodology performs well for realistic sample sizes. An analysis of hospitalization data for patients in an AIDS cohort study is presented to illustrate the proposed method.     

Time-sharing in pigeons: Independent effects of gap duration, position and discriminability from the timed signal

Previous data suggest that in a peak-interval procedure with gaps, memory for the pre-gap interval varies with the discriminability of the gap from the to-be-timed signal. Here we extend this finding by manipulating the pre-gap and gap intervals as well as the visual contrast between the gap and the to-be-timed signal. The delay in response function after the gap was found to vary with the duration and position of the gap. However, for each gap duration and position, the delay in response increased with the gap-signal contrast: at 60% gap-signal contrast pigeons continued to accumulate time during the gap, at 80% gap-signal contrast pigeons stopped timing during the gap, and at 100% gap-signal contrast pigeons reset their timing after the gap. Data are accounted for by a time-sharing model assuming two concurrent processes during the gap--time accumulation and memory decay controlled by the salience of the gap--whose interplay results in a continuum of responses in the gap procedure.     

Monitoring the rates of composite events with censored data in phase II clinical trials

In many phase II clinical trials, interim monitoring is based on the probability of a binary event, response, defined in terms of one or more time-to-event variables within a time period of fixed length. Such outcome-adaptive methods may require repeated interim suspension of accrual in order to follow each patient for the time period required to evaluate response. This may increase trial duration, and eligible patients arriving during such delays either must wait for accrual to reopen or be treated outside the trial. Alternatively, monitoring may be done continuously by ignoring censored data each time the stopping rule is applied, which wastes information. We propose an adaptive Bayesian method that eliminates these problems. At each patient's accrual time, an approximate posterior for the response probability based on all of the event-time data is used to compute an early stopping criterion. Application to a leukemia trial with a composite event shows that the method can reduce trial duration substantially while maintaining the reliability of interim decisions.