A semiparametric additive rate model for recurrent events with an informative terminal event

Dynamic semiparametric transformation models for recurrent event data with a terminal event

Recurrent event data with a terminal event commonly arise in many longitudinal follow-up studies. This article proposes a class of dynamic semiparametric transformation models for the marginal mean functions of the recurrent events with a terminal event, where some covariate effects may be time-varying. An estimation procedure is developed for the model parameters, and the asymptotic properties of the resulting estimators are established. In addition, relevant significance tests are suggested for examining whether or not covariate effects vary with time, and a model checking procedure is presented for assessing the adequacy of the proposed models. The finite sample performance of the proposed estimators is examined through simulation studies, and an application to a medical cost study of chronic heart failure patients is provided.

Sample size calculation for recurrent event data with additive rates models

This paper discusses the design of clinical trials where the primary endpoint is a recurrent event with the focus on the sample size calculation. For the problem, a few methods have been proposed but most of them assume a multiplicative treatment effect on the rate or mean number of recurrent events. In practice, sometimes the additive treatment effect may be preferred or more appealing because of its intuitive clinical meaning and straightforward interpretation compared to a multiplicative relationship. In this paper, new methods are presented and investigated for the sample size calculation based on the additive rates model for superiority, non-inferiority, and equivalence trials. They allow for flexible baseline rate function, staggered entry, random dropout, and overdispersion in event numbers, and simulation studies show that the proposed methods perform well in a variety of settings. We also illustrate how to use the proposed methods to design a clinical trial based on real data.

Doubly robust estimation and causal inference for recurrent event data

Many longitudinal databases record the occurrence of recurrent events over time. In this article, we propose a new method to estimate the average causal effect of a binary treatment for recurrent event data in the presence of confounders. We propose a doubly robust semiparametric estimator based on a weighted version of the Nelson-Aalen estimator and a conditional regression estimator under an assumed semiparametric multiplicative rate model for recurrent event data. We show that the proposed doubly robust estimator is consistent and asymptotically normal. In addition, a model diagnostic plot of residuals is presented to assess the adequacy of our proposed semiparametric model. We then evaluate the finite sample behavior of the proposed estimators under a number of simulation scenarios. Finally, we illustrate the proposed methodology via a database of circus artist injuries.

A semiparametric additive rates model for the weighted composite endpoint of recurrent and terminal events

Recurrent event data with a terminal event commonly arise in longitudinal follow-up studies. We use a weighted composite endpoint of all recurrent and terminal events to assess the overall effects of covariates on the two types of events. A semiparametric additive rates model is proposed to analyze the weighted composite event process and the dependence structure among recurrent and terminal events is left unspecified. An estimating equation approach is developed for inference, 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 illustrated.

GENERALIZED ACCELERATED RECURRENCE TIME MODEL IN THE PRESENCE OF A DEPENDENT TERMINAL EVENT

Recurrent events are commonly encountered in longitudinal studies. The observation of recurrent events is often stopped by a dependent terminal event in practice. For this data scenario, we propose two sensible adaptations of the generalized accelerated recurrence time (GART) model (Sun et al., 2016) to provide useful alternative analyses that can offer physical interpretations while rendering extra flexibility beyond the existing work based on the accelerated failure time model. Our modeling strategies align with the rationale underlying the use of the survivors' rate function or the adjusted rate function to account for the presence of the dependent terminal event. For the proposed models, we identify and develop estimation and inference procedures, which can be readily implemented based on existing software. We establish the asymptotic properties of the new estimator. Simulation studies demonstrate good finite-sample performance of the proposed methods. An application to a dataset from the Cystic Fibrosis Foundation Patient Registry (CFFPR) illustrates the practical utility of the new methods.

Causal inference for recurrent event data using pseudo-observations

Recurrent event data are commonly encountered in observational studies where each subject may experience a particular event repeatedly over time. In this article, we aim to compare cumulative rate functions (CRFs) of two groups when treatment assignment may depend on the unbalanced distribution of confounders. Several estimators based on pseudo-observations are proposed to adjust for the confounding effects, namely inverse probability of treatment weighting estimator, regression model-based estimators, and doubly robust estimators. The proposed marginal regression estimator and doubly robust estimators based on pseudo-observations are shown to be consistent and asymptotically normal. A bootstrap approach is proposed for the variance estimation of the proposed estimators. Model diagnostic plots of residuals are presented to assess the goodness-of-fit for the proposed regression models. A family of adjusted two-sample pseudo-score tests is proposed to compare two CRFs. Simulation studies are conducted to assess finite sample performance of the proposed method. The proposed technique is demonstrated through an application to a hospital readmission data set.

Semiparametric temporal process regression of survival-out-of-hospital

The recurrent/terminal event data structure has undergone considerable methodological development in the last 10-15 years. An example of the data structure that has arisen with increasing frequency involves the recurrent event being hospitalization and the terminal event being death. We consider the response Survival-Out-of-Hospital, defined as a temporal process (indicator function) taking the value 1 when the subject is currently alive and not hospitalized, and 0 otherwise. Survival-Out-of-Hospital is a useful alternative strategy for the analysis of hospitalization/survival in the chronic disease setting, with the response variate representing a refinement to survival time through the incorporation of an objective quality-of-life component. The semiparametric model we consider assumes multiplicative covariate effects and leaves unspecified the baseline probability of being alive-and-out-of-hospital. Using zero-mean estimating equations, the proposed regression parameter estimator can be computed without estimating the unspecified baseline probability process, although baseline probabilities can subsequently be estimated for any time point within the support of the censoring distribution. We demonstrate that the regression parameter estimator is asymptotically normal, and that the baseline probability function estimator converges to a Gaussian process. Simulation studies are performed to show that our estimating procedures have satisfactory finite sample performances. The proposed methods are applied to the Dialysis Outcomes and Practice Patterns Study (DOPPS), an international end-stage renal disease study.

A semiparametric additive rate model for a modulated renewal process

Recurrent event data from a long single realization are widely encountered in point process applications. Modeling and analyzing such data are different from those for independent and identical short sequences, and the development of statistical methods requires careful consideration of the underlying dependence structure of the long single sequence. In this paper, we propose a semiparametric additive rate model for a modulated renewal process, and develop an estimating equation approach for the model parameters. The asymptotic properties of the resulting estimators are established by applying the limit theory for stationary mixing sequences. A block-based bootstrap procedure is presented for the variance estimation. Simulation studies are conducted to assess the finite-sample performance of the proposed estimators. An application to a data set from a cardiovascular mortality study is provided.

Semiparametric estimation of the accelerated mean model with panel count data under informative examination times

Panel count data arise when the number of recurrent events experienced by each subject is observed intermittently at discrete examination times. The examination time process can be informative about the underlying recurrent event process even after conditioning on covariates. We consider a semiparametric accelerated mean model for the recurrent event process and allow the two processes to be correlated through a shared frailty. The regression parameters have a simple marginal interpretation of modifying the time scale of the cumulative mean function of the event process. A novel estimation procedure for the regression parameters and the baseline rate function is proposed based on a conditioning technique. In contrast to existing methods, the proposed method is robust in the sense that it requires neither the strong Poisson-type assumption for the underlying recurrent event process nor a parametric assumption on the distribution of the unobserved frailty. Moreover, the distribution of the examination time process is left unspecified, allowing for arbitrary dependence between the two processes. Asymptotic consistency of the estimator is established, and the variance of the estimator is estimated by a model-based smoothed bootstrap procedure. Numerical studies demonstrated that the proposed point estimator and variance estimator perform well with practical sample sizes. The methods are applied to data from a skin cancer chemoprevention trial.

Joint analysis of panel count data with an informative observation process and a dependent terminal event

Panel count data occur in many clinical and observational studies, and in many situations, the observation process may be informative and also there may exist a terminal event such as death which stops the follow-up. In this article, we propose a new joint model for the analysis of panel count data in the presence of both an informative observation process and a dependent terminal event via two latent variables. For the inference on the proposed models, a class of estimating equations is developed and the resulting estimators are shown to be consistent and asymptotically normal. In addition, a lack-of-fit test is provided for assessing the adequacy of the models. Simulation studies suggest that the proposed approach works well for practical situations. A real example from a bladder cancer clinical trial is used to illustrate the proposed methods.

Regression analysis of mixed panel count data with dependent terminal events

Event history studies are commonly conducted in many fields, and a great deal of literature has been established for the analysis of the two types of data commonly arising from these studies: recurrent event data and panel count data. The former arises if all study subjects are followed continuously, while the latter means that each study subject is observed only at discrete time points. In reality, a third type of data, a mixture of the two types of the data earlier, may occur and furthermore, as with the first two types of the data, there may exist a dependent terminal event, which may preclude the occurrences of recurrent events of interest. This paper discusses regression analysis of mixed recurrent event and panel count data in the presence of a terminal event and an estimating equation-based approach is proposed for estimation of regression parameters of interest. In addition, the asymptotic properties of the proposed estimator are established, and a simulation study conducted to assess the finite-sample performance of the proposed method suggests that it works well in practical situations. Finally, the methodology is applied to a childhood cancer study that motivated this study. Copyright © 2017 John Wiley & Sons, Ltd.

Joint scale-change models for recurrent events and failure time

Recurrent event data arise frequently in various fields such as biomedical sciences, public health, engineering, and social sciences. In many instances, the observation of the recurrent event process can be stopped by the occurrence of a correlated failure event, such as treatment failure and death. In this article, we propose a joint scale-change model for the recurrent event process and the failure time, where a shared frailty variable is used to model the association between the two types of outcomes. In contrast to the popular Cox-type joint modeling approaches, the regression parameters in the proposed joint scale-change model have marginal interpretations. The proposed approach is robust in the sense that no parametric assumption is imposed on the distribution of the unobserved frailty and that we do not need the strong Poisson-type assumption for the recurrent event process. We establish consistency and asymptotic normality of the proposed semiparametric estimators under suitable regularity conditions. To estimate the corresponding variances of the estimators, we develop a computationally efficient resampling-based procedure. Simulation studies and an analysis of hospitalization data from the Danish Psychiatric Central Register illustrate the performance of the proposed method.

Additive mixed effect model for recurrent gap time data

Gap times between recurrent events are often of primary interest in medical and observational studies. The additive hazards model, focusing on risk differences rather than risk ratios, has been widely used in practice. However, the marginal additive hazards model does not take the dependence among gap times into account. In this paper, we propose an additive mixed effect model to analyze gap time data, and the proposed model includes a subject-specific random effect to account for the dependence among the gap times. Estimating equation approaches are developed for parameter estimation, and the asymptotic properties of the resulting estimators are established. In addition, some graphical and numerical procedures are presented for model checking. The finite sample behavior of the proposed methods is evaluated through simulation studies, and an application to a data set from a clinic study on chronic granulomatous disease is provided.

Joint modeling of longitudinal, recurrent events and failure time data for survivor's population

Recurrent events together with longitudinal measurements are commonly observed in follow-up studies where the observation is terminated by censoring or a primary failure event. In this article, we developed a joint model where the dependence of longitudinal measurements, recurrent event process and time to failure event is modeled through rescaling the time index. The general idea is that the trajectories of all biology processes of subjects in the survivors' population are elongated or shortened by the rate identified from a model for the failure event. To avoid making disputing assumptions on recurrent events or biomarkers after the failure event (such as death), the model is constructed on the basis of survivors' population. The model also possesses a specific feature that, by aligning failure events as time origins, the backward-in-time model of recurrent events and longitudinal measurements shares the same parameter values with the forward time model. The statistical properties, simulation studies and real data examples are conducted. The proposed method can be generalized to analyze left-truncated data.

Longitudinal data subject to irregular observation: A review of methods with a focus on visit processes, assumptions, and study design

When data are collected longitudinally, measurement times often vary among patients. This is of particular concern in clinic-based studies, for example retrospective chart reviews. Here, typically no two patients will share the same set of measurement times and moreover, it is likely that the timing of the measurements is associated with disease course; for example, patients may visit more often when unwell. While there are statistical methods that can help overcome the resulting bias, these make assumptions about the nature of the dependence between visit times and outcome processes, and the assumptions differ across methods. The purpose of this paper is to review the methods available with a particular focus on how the assumptions made line up with visit processes encountered in practice. Through this we show that no one method can handle all plausible visit scenarios and suggest that careful analysis of the visit process should inform the choice of analytic method for the outcomes. Moreover, there are some commonly encountered visit scenarios that are not handled well by any method, and we make recommendations with regard to study design that would minimize the chances of these problematic visit scenarios arising.

Semiparametric regression for the weighted composite endpoint of recurrent and terminal events

Recurrent event data are commonly encountered in clinical and epidemiological studies. A major complication arises when recurrent events are terminated by death. To assess the overall effects of covariates on the two types of events, we define a weighted composite endpoint as the cumulative number of recurrent and terminal events properly weighted by the relative severity of each event. We propose a semiparametric proportional rates model which specifies that the (possibly time-varying) covariates have multiplicative effects on the rate function of the weighted composite endpoint while leaving the form of the rate function and the dependence among recurrent and terminal events completely unspecified. We construct appropriate estimators for the regression parameters and the cumulative frequency function. We show that the estimators are consistent and asymptotically normal with variances that can be consistently estimated. We also develop graphical and numerical procedures for checking the adequacy of the model. We then demonstrate the usefulness of the proposed methods in simulation studies. Finally, we provide an application to a major cardiovascular clinical trial.

Time-varying coefficients in a multivariate frailty model: Application to breast cancer recurrences of several types and death

During their follow-up, patients with cancer can experience several types of recurrent events and can also die. Over the last decades, several joint models have been proposed to deal with recurrent events with dependent terminal event. Most of them require the proportional hazard assumption. In the case of long follow-up, this assumption could be violated. We propose a joint frailty model for two types of recurrent events and a dependent terminal event to account for potential dependencies between events with potentially time-varying coefficients. For that, regression splines are used to model the time-varying coefficients. Baseline hazard functions (BHF) are estimated with piecewise constant functions or with cubic M-Splines functions. The maximum likelihood estimation method provides parameter estimates. Likelihood ratio tests are performed to test the time dependency and the statistical association of the covariates. This model was driven by breast cancer data where the maximum follow-up was close to 20 years.

A penalized algorithm for event-specific rate models for recurrent events

We introduce a covariate-specific total variation penalty in two semiparametric models for the rate function of recurrent event process. The two models are a stratified Cox model, introduced in Prentice and others (1981. On the regression analysis of multivariate failure time data. Biometrika 68: , 373-379.), and a stratified Aalen's additive model. We show the consistency and asymptotic normality of our penalized estimators. We demonstrate, through a simulation study, that our estimators outperform classical estimators for small-to-moderate sample sizes. Finally, an application to the bladder tumor data of Byar (1980. The veterans administration study of chemoprophylaxis for recurrent stage 1 bladder tumors: comparison of placebo, pyridoxine, and topical thiotepa. In Pavone-Macaluso, M. Smith, P. H. and F. Edsmyn (editors), Bladder Tumors and Others Topics in Urological Oncology, pp. 363-370.) is presented.

Multivariate frailty models for two types of recurrent events with a dependent terminal event: application to breast cancer data

Individuals may experience more than one type of recurrent event and a terminal event during the life course of a disease. Follow-up may be interrupted for several reasons, including the end of a study, or patients lost to follow-up, which are non informative censoring events. Death could also stop the follow-up, hence, it is considered as a dependent terminal event. We propose a multivariate frailty model that jointly analyzes two types of recurrent events with a dependent terminal event. Two estimation methods are proposed: a semiparametrical approach using penalized likelihood estimation where baseline hazard functions are approximated by M-s plines, and another one with piecewise constant baseline hazard functions. Finally, we derived martingale residuals to check the goodness-of-fit. We illustrate our proposals with a real dataset on breast cancer. The main objective was to model the dependency between the two types of recurrent events (locoregional and metastatic) and the terminal event (death) after a breast cancer.

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.

Semiparametric additive marginal regression models for multiple type recurrent events

Recurrent event data are often encountered in biomedical research, for example, recurrent infections or recurrent hospitalizations for patients after renal transplant. In many studies, there are more than one type of events of interest. Cai and Schaube (Lifetime Data Anal 10:121-138, 2004) advocated a proportional marginal rate model for multiple type recurrent event data. In this paper, we propose a general additive marginal rate regression model. Estimating equations approach is used to obtain the estimators of regression coefficients and baseline rate function. We prove the consistency and asymptotic normality of the proposed estimators. The finite sample properties of our estimators are demonstrated by simulations. The proposed methods are applied to the India renal transplant study to examine risk factors for bacterial, fungal and viral infections.

Reliability modeling and prediction using classical and Bayesian approach

Purpose

The purpose of this paper is to provide the maintenance personnel with a methodology for modeling and estimating the reliability of critical machine systems using the historical data of their inter‐failure times.

Design/methodology/approach

The failure patterns of five different machine systems were modeled with NHPP‐log linear process and HPP belonging to stochastic point process for predicting their reliability in future time frames. Besides the classical approach, Bayesian approach was also used involving Jeffreys's invariant non‐informative independent priors to derive the posterior densities of the model parameters of NHPP‐LLP and HPP with a view to estimating the reliability of the machine systems in future time intervals.

Findings

For at least three machine systems, Bayesian approach gave lower reliability estimates and a larger number of (expected) failures than those obtained by the classical approach. Again, Bayesian estimates of the probability that “ROCOF (rate of occurrence of failures) would exceed its upper threshold limit” in future time frames were uniformly higher for these machine systems than those obtained with the classical approach.

Practical implications

This study indicated that, the Bayesian approach would give more realistic estimates of reliability (in future time frames) of the machine systems, which had dependent inter‐failure times. Such information would be helpful to the maintenance team for deciding on appropriate maintenance strategy.

Originality/value

With the help of Bayesian approach, the posterior densities of the model parameters were found analytically by considering Jeffreys's invariant non‐informative independent prior. The case study would serve to motivate the maintenance teams to model the failure patterns of the repairable systems making use of the historical data on inter‐failure times and estimating their reliability in future time frames.