Semiparametric analysis for recurrent event data with time-dependent covariates and informative censoring

Tackling dynamic prediction of death in patients with recurrent cardiovascular events

In the field of cardiovascular disease, recurrent events such as stroke or myocardial infarction (MI) are often encountered, leading to an increase in the risk of death. Accurately evaluating the prognosis of patients and dynamically predicting the risk of death by considering the historical recurrent events can improve medical decisions and lead to better health care outcomes. Recently proposed joint modeling approaches within the Bayesian framework have inspired the development of a dynamic prediction tool, which can be applied for subject-level prediction of death with implementation in software packages. The prediction model incorporates subject heterogeneity with subject-level random effects that account for unobserved time-invariant factors and an extra copula function capturing the part caused by unmeasured time-dependent factors. Thereafter, given the prespecified landmark timet ' $$ {t}^{\prime } $$ , the survival probability for a prediction horizon time of interestt $$ t $$ can be estimated for each individual. The prediction accuracy is assessed by time-dependent receiving operating characteristic curve and the area under the curve and the Brier score with calibration plots is compared to traditional joint frailty models. Finally, the tool is applied to patients with multiple attacks of stroke or MI in the Cardiovascular Health study and the Atherosclerosis Risk in Communities study for illustration.

Estimation of accelerated hazards models based on case K informatively interval-censored failure time data

The accelerated hazards model is one of the most commonly used models for regression analysis of failure time data and this is especially the case when, for example, the hazard functions may have monotonicity property. Correspondingly a large literature has been established for its estimation or inference when right-censored data are observed. Although several methods have also been developed for its inference based on interval-censored data, they apply only to limited situations or rely on some assumptions such as independent censoring. In this paper, we consider the situation where one observes case K interval-censored data, the type of failure time data that occur most in, for example, medical research such as clinical trials or periodical follow-up studies. For inference, we propose a sieve borrow-strength method and in particular, it allows for informative censoring. The asymptotic properties of the proposed estimators are established. Simulation studies demonstrate that the proposed inference procedure performs well. The method is applied to a set of real data set arising from an AIDS clinical trial.

Regression Modeling for Recurrent Events Possibly with an Informative Terminal Event Using R Package reReg

Recurrent event analyses have found a wide range of applications in biomedicine, public health, and engineering, among others, where study subjects may experience a sequence of event of interest during follow-up. The R package reReg offers a comprehensive collection of practical and easy-to-use tools for regression analysis of recurrent events, possibly with the presence of an informative terminal event. The regression framework is a general scale-change model which encompasses the popular Cox-type model, the accelerated rate model, and the accelerated mean model as special cases. Informative censoring is accommodated through a subject-specific frailty without any need for parametric specification. Different regression models are allowed for the recurrent event process and the terminal event. Also included are visualization and simulation tools.

Joint modeling of generalized scale-change models for recurrent event and failure time data

Recurrent event and failure time data arise frequently in many clinical and observational studies. In this article, we propose a joint modeling of generalized scale-change models for the recurrent event process and the failure time, and allow the two processes to be correlated through a shared frailty. The proposed joint model is flexible in that it requires neither the Poisson assumption for the recurrent event process nor a parametric assumption on the frailty distribution. Estimating equation approaches are developed for parameter estimation, and the asymptotic properties of the resulting estimators are established. Simulation studies are conducted to evaluate the finite sample performances of the proposed method. An application to a medical cost study of chronic heart failure patients is provided.

Joint analysis of informatively interval-censored failure time and panel count data

Interval-censored failure time and panel count data, which frequently arise in medical studies and social sciences, are two types of important incomplete data. Although methods for their joint analysis have been available in the literature, they did not consider the observation process, which may depend on the failure time and/or panel count of interest. This study considers a three-component joint model to analyze interval-censored failure time, panel counts, and the observation process within a unique framework. Gamma and distribution-free frailties are introduced to jointly model the interdependency among the interval-censored data, panel count data, and the observation process. We propose a sieve maximum likelihood approach coupled with Bernstein polynomial approximation to estimate the unknown parameters and baseline hazard function. The asymptotic properties of the resulting estimators are established. An extensive simulation study suggests that the proposed procedure works well for practical situations. An application of the method to a real-life dataset collected from a cardiac allograft vasculopathy study is presented.

Statistical inference on shape and size indexes for counting processes

Single-index models have gained increased popularity in time-to-event analysis owing to their model flexibility and advantage in dimension reduction. We propose a semiparametric framework for the rate function of a recurrent event counting process by modelling its size and shape components with single-index models. With additional monotone constraints on the two link functions for the size and shape components, the proposed model possesses the desired directional interpretability of covariate effects and encompasses many commonly used models as special cases. To tackle the analytical challenges arising from leaving the two link functions unspecified, we develop a two-step rank-based estimation procedure to estimate the regression parameters with or without informative censoring. The proposed estimators are asymptotically normal, with a root-n convergence rate. To guide model selection, we develop hypothesis testing procedures for checking shape and size independence. Simulation studies and a data example on a hematopoietic stem cell transplantation study are presented to illustrate the proposed methodology.

Bayesian inference on the number of recurrent events: A joint model of recurrence and survival

The number of recurrent events before a terminating event is often of interest. For instance, death terminates an individual's process of rehospitalizations and the number of rehospitalizations is an important indicator of economic cost. We propose a model in which the number of recurrences before termination is a random variable of interest, enabling inference and prediction on it. Then, conditionally on this number, we specify a joint distribution for recurrence and survival. This novel conditional approach induces dependence between recurrence and survival, which is often present, for instance, due to frailty that affects both. Additional dependence between recurrence and survival is introduced by the specification of a joint distribution on their respective frailty terms. Moreover, through the introduction of an autoregressive model, our approach is able to capture the temporal dependence in the recurrent events trajectory. A non-parametric random effects distribution for the frailty terms accommodates population heterogeneity and allows for data-driven clustering of the subjects. A tailored Gibbs sampler involving reversible jump and slice sampling steps implements posterior inference. We illustrate our model on colorectal cancer data, compare its performance with existing approaches and provide appropriate inference on the number of recurrent events.

An additive hazards cure model with informative interval censoring

The existence of a cured subgroup happens quite often in survival studies and many authors considered this under various situations (Farewell in Biometrics 38:1041-1046, 1982; Kuk and Chen in Biometrika 79:531-541, 1992; Lam and Xue in Biometrika 92:573-586, 2005; Zhou et al. in J Comput Graph Stat 27:48-58, 2018). In this paper, we discuss the situation where only interval-censored data are available and furthermore, the censoring may be informative, for which there does not seem to exist an established estimation procedure. For the analysis, we present a three component model consisting of a logistic model for describing the cure rate, an additive hazards model for the failure time of interest and a nonhomogeneous Poisson model for the observation process. For estimation, we propose a sieve maximum likelihood estimation procedure and the asymptotic properties of the resulting estimators are established. Furthermore, an EM algorithm is developed for the implementation of the proposed estimation approach, and extensive simulation studies are conducted and suggest that the proposed method works well for practical situations. Also the approach is applied to a cardiac allograft vasculopathy study that motivated this investigation.

Joint modeling of recurrent events and survival: a Bayesian non-parametric approach

Heart failure (HF) is one of the main causes of morbidity, hospitalization, and death in the western world, and the economic burden associated with HF management is relevant and expected to increase in the future. We consider hospitalization data for HF in the most populated Italian Region, Lombardia. Data were extracted from the administrative data warehouse of the regional healthcare system. The main clinical outcome of interest is time to death and research focus is on investigating how recurrent hospitalizations affect the time to event. The main contribution of the article is to develop a joint model for gap times between consecutive rehospitalizations and survival time. The probability models for the gap times and for the survival outcome share a common patient specific frailty term. Using a flexible Dirichlet process model for %Bayesian nonparametric prior as the random-effects distribution accounts for patient heterogeneity in recurrent event trajectories. Moreover, the joint model allows for dependent censoring of gap times by death or administrative reasons and for the correlations between different gap times for the same individual. It is straightforward to include covariates in the survival and/or recurrence process through the specification of appropriate regression terms. The main advantages of the proposed methodology are wide applicability, ease of interpretation, and efficient computations. Posterior inference is implemented through Markov chain Monte Carlo methods.

Joint analysis of recurrence and termination: A Bayesian latent class approach

Like many other clinical and economic studies, each subject of our motivating transplant study is at risk of recurrent events of non-fatal tissue rejections as well as the terminating event of death due to total graft rejection. For such studies, our model and associated Bayesian analysis aim for some practical advantages over competing methods. Our semiparametric latent-class-based joint model has coherent interpretation of the covariate (including race and gender) effects on all functions and model quantities that are relevant for understanding the effects of covariates on future event trajectories. Our fully Bayesian method for estimation and prediction uses a complete specification of the prior process of the baseline functions. We also derive a practical and theoretically justifiable partial likelihood-based semiparametric Bayesian approach to deal with the analysis when there is a lack of prior information about baseline functions. Our model and method can accommodate fixed as well as time-varying covariates. Our Markov Chain Monte Carlo tools for both Bayesian methods are implementable via publicly available software. Our Bayesian analysis of transplant study and simulation study demonstrate practical advantages and improved performance of our approach.

Multiple event times in the presence of informative censoring: modeling and analysis by copulas

Motivated by a breast cancer research program, this paper is concerned with the joint survivor function of multiple event times when their observations are subject to informative censoring caused by a terminating event. We formulate the correlation of the multiple event times together with the time to the terminating event by an Archimedean copula to account for the informative censoring. Adapting the widely used two-stage procedure under a copula model, we propose an easy-to-implement pseudo-likelihood based procedure for estimating the model parameters. The approach yields a new estimator for the marginal distribution of a single event time with semicompeting-risks data. We conduct both asymptotics and simulation studies to examine the proposed approach in consistency, efficiency, and robustness. Data from the breast cancer program are employed to illustrate this research.

Semiparametric Regression Analysis of Panel Count Data: A Practical Review

Panel count data arise in many applications when the event history of a recurrent event process is only examined at a sequence of discrete time points. In spite of the recent methodological developments, the availability of their software implementations has been rather limited. Focusing on a practical setting where the effects of some time-independent covariates on the recurrent events are of primary interest, we review semiparametric regression modelling approaches for panel count data that have been implemented in R package spef. The methods are grouped into two categories depending on whether the examination times are associated with the recurrent event process after conditioning on covariates. The reviewed methods are illustrated with a subset of the data from a skin cancer clinical trial.

Semiparametric modeling and estimation of the terminal behavior of recurrent marker processes before failure events

Recurrent event processes with marker measurements are mostly and largely studied with forward time models starting from an initial event. Interestingly, the processes could exhibit important terminal behavior during a time period before occurrence of the failure event. A natural and direct way to study recurrent events prior to a failure event is to align the processes using the failure event as the time origin and to examine the terminal behavior by a backward time model. This paper studies regression models for backward recurrent marker processes by counting time backward from the failure event. A three-level semiparametric regression model is proposed for jointly modeling the time to a failure event, the backward recurrent event process, and the marker observed at the time of each backward recurrent event. The first level is a proportional hazards model for the failure time, the second level is a proportional rate model for the recurrent events occurring before the failure event, and the third level is a proportional mean model for the marker given the occurrence of a recurrent event backward in time. By jointly modeling the three components, estimating equations can be constructed for marked counting processes to estimate the target parameters in the three-level regression models. Large sample properties of the proposed estimators are studied and established. The proposed models and methods are illustrated by a community-based AIDS clinical trial to examine the terminal behavior of frequencies and severities of opportunistic infections among HIV infected individuals in the last six months of life.

Median Analysis of Repeated Measures Associated with Recurrent Events in Presence of Terminal Event

Recurrent events are often encountered in medical follow up studies. In addition, such recurrences have other quantities associated with them that are of considerable interest, for instance medical costs of the repeated hospitalizations and tumor size in cancer recurrences. These processes can be viewed as point processes, i.e. processes with arbitrary positive jump at each recurrence. An analysis of the mean function for such point processes have been proposed in the literature. However, such point processes are often skewed, leading to median as a more appropriate measure than the mean. Furthermore, the analysis of recurrent event data is often complicated by the presence of death. We propose a semiparametric model for assessing the effect of covariates on the quantiles of the point processes. We investigate both the finite sample as well as the large sample properties of the proposed estimators. We conclude with a real data analysis of the medical cost associated with the treatment of ovarian cancer.

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.

Joint Model for Mortality and Hospitalization

Modeling hospitalization is complicated because the follow-up time can be censored due to death. In this paper, we propose a shared frailty joint model for survival time and hospitalization. A random effect semi-parametric proportional hazard model is assumed for the survival time and conditional on the follow-up time, hospital admissions or total length of stay is modeled by a generalized linear model with a nonparametric offset function of the follow-up time. We assume that the hospitalization and the survival time are correlated through a latent subject-specific random frailty. The proposed model can be implemented using existing software such as SAS Proc NLMIXED. We demonstrate the feasibility through simulations. We apply our methods to study hospital admissions and total length of stay in a cohort of patients on hemodialysis. We identify age, albumin, neutrophil to lymphocyte ratio (NLR) and vintage as significant risk factors for mortality, and age, gender, race, albumin, NLR, pre-dialysis systolic blood pressure (preSBP), interdialytic weight gain (IDWG) and equilibrated Kt/V (eKt/V) as significant risk factors for both hospital admissions and total length of stay. In addition, hospitalization admissions is positively associated with vintage.

Who really gets strep sore throat? Confounding and effect modification of a time-varying exposure on recurrent events

Unmeasured confounding is the fundamental obstacle to drawing causal conclusions about the impact of an intervention from observational data. Typically, covariates are measured to eliminate or ameliorate confounding, but they may be insufficient or unavailable. In the special setting where a transient intervention or exposure varies over time within each individual and confounding is time constant, a different tack is possible. The key idea is to condition on either the overall outcome or the proportion of time in the intervention. These measures can eliminate the unmeasured confounding either by conditioning or by use of a proxy covariate. We evaluate existing methods and develop new models from which causal conclusions can be drawn from such observational data even if no baseline covariates are measured. Our motivation for this work was to determine the causal effect of Streptococcus bacteria in the throat on pharyngitis (sore throat) in Indian schoolchildren. Using our models, we show that existing methods can be badly biased and that sick children who are rarely colonized have a high probability that the Streptococcus bacteria are causing their disease. Published 2016. This article is a U.S. Government work and is in the public domain in the USA.

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.

Regression analysis of case K interval-censored failure time data in the presence of informative censoring

Interval-censored failure time data occur in many fields such as demography, economics, medical research, and reliability and many inference procedures on them have been developed (Sun, 2006; Chen, Sun, and Peace, 2012). However, most of the existing approaches assume that the mechanism that yields interval censoring is independent of the failure time of interest and it is clear that this may not be true in practice (Zhang et al., 2007; Ma, Hu, and Sun, 2015). In this article, we consider regression analysis of case K interval-censored failure time data when the censoring mechanism may be related to the failure time of interest. For the problem, an estimated sieve maximum-likelihood approach is proposed for the data arising from the proportional hazards frailty model and for estimation, a two-step procedure is presented. In the addition, the asymptotic properties of the proposed estimators of regression parameters are established and an extensive simulation study suggests that the method works well. Finally, we apply the method to a set of real interval-censored data that motivated this study.

Recurrent event data analysis with intermittently observed time-varying covariates

Although recurrent event data analysis is a rapidly evolving area of research, rigorous studies on estimation of the effects of intermittently observed time-varying covariates on the risk of recurrent events have been lacking. Existing methods for analyzing recurrent event data usually require that the covariate processes are observed throughout the entire follow-up period. However, covariates are often observed periodically rather than continuously. We propose a novel semiparametric estimator for the regression parameters in the popular proportional rate model. The proposed estimator is based on an estimated score function where we kernel smooth the mean covariate process. We show that the proposed semiparametric estimator is asymptotically unbiased, normally distributed, and derives the asymptotic variance. Simulation studies are conducted to compare the performance of the proposed estimator and the simple methods carrying forward the last covariates. The different methods are applied to an observational study designed to assess the effect of group A streptococcus on pharyngitis among school children in India. Copyright © 2016 John Wiley & Sons, Ltd.

Time-dependent prognostic score matching for recurrent event analysis to evaluate a treatment assigned during follow-up

Recurrent events often serve as the outcome in epidemiologic studies. In some observational studies, the goal is to estimate the effect of a new or "experimental" (i.e., less established) treatment of interest on the recurrent event rate. The incentive for accepting the new treatment may be that it is more available than the standard treatment. Given that the patient can choose between the experimental treatment and conventional therapy, it is of clinical importance to compare the treatment of interest versus the setting where the experimental treatment did not exist, in which case patients could only receive no treatment or the standard treatment. Many methods exist for the analysis of recurrent events and for the evaluation of treatment effects. However, methodology for the intersection of these two areas is sparse. Moreover, care must be taken in setting up the comparison groups in our setting; use of existing methods featuring time-dependent treatment indicators will generally lead to a biased treatment effect since the comparison group construction will not properly account for the timing of treatment initiation. We propose a sequential stratification method featuring time-dependent prognostic score matching to estimate the effect of a time-dependent treatment on the recurrent event rate. The performance of the method in moderate-sized samples is assessed through simulation. The proposed methods are applied to a prospective clinical study in order to evaluate the effect of living donor liver transplantation on hospitalization rates; in this setting, conventional therapy involves remaining on the wait list or receiving a deceased donor transplant.

Estimating the ratio of multivariate recurrent event rates with application to a blood transfusion study

In comparative effectiveness studies of multicomponent, sequential interventions like blood product transfusion (plasma, platelets, red blood cells) for trauma and critical care patients, the timing and dynamics of treatment relative to the fragility of a patient's condition is often overlooked and underappreciated. While many hospitals have established massive transfusion protocols to ensure that physiologically optimal combinations of blood products are rapidly available, the period of time required to achieve a specified massive transfusion standard (e.g. a 1:1 or 1:2 ratio of plasma or platelets:red blood cells) has been ignored. To account for the time-varying characteristics of transfusions, we use semiparametric rate models for multivariate recurrent events to estimate blood product ratios. We use latent variables to account for multiple sources of informative censoring (early surgical or endovascular hemorrhage control procedures or death). The major advantage is that the distributions of latent variables and the dependence structure between the multivariate recurrent events and informative censoring need not be specified. Thus, our approach is robust to complex model assumptions. We establish asymptotic properties and evaluate finite sample performance through simulations, and apply the method to data from the PRospective Observational Multicenter Major Trauma Transfusion study.

Regression modeling of longitudinal data with outcome-dependent observation times: extensions and comparative evaluation

Conventional longitudinal data analysis methods assume that outcomes are independent of the data-collection schedule. However, the independence assumption may be violated, for example, when a specific treatment necessitates a different follow-up schedule than the control arm or when adverse events trigger additional physician visits in between prescheduled follow-ups. Dependence between outcomes and observation times may introduce bias when estimating the marginal association of covariates on outcomes using a standard longitudinal regression model. We formulate a framework of outcome-observation dependence mechanisms to describe conditional independence given observed observation-time process covariates or shared latent variables. We compare four recently developed semi-parametric methods that accommodate one of these mechanisms. To allow greater flexibility, we extend these methods to accommodate a combination of mechanisms. In simulation studies, we show how incorrectly specifying the outcome-observation dependence may yield biased estimates of covariate-outcome associations and how our proposed extensions can accommodate a greater number of dependence mechanisms. We illustrate the implications of different modeling strategies in an application to bladder cancer data. In longitudinal studies with potentially outcome-dependent observation times, we recommend that analysts carefully explore the conditional independence mechanism between the outcome and observation-time processes to ensure valid inference regarding covariate-outcome associations.

Semiparametric estimation in generalized linear mixed models with auxiliary covariates: a pairwise likelihood approach

Auxiliary covariates are often encountered in biomedical research settings where the primary exposure variable is measured only for a subgroup of study subjects. This article is concerned with generalized linear mixed models in the presence of auxiliary covariate information for clustered data. We propose a novel semiparametric estimation method based on a pairwise likelihood function and develop an estimating equation-based inference procedure by treating both the error structure and random effects as nuisance parameters. This method is robust against misspecification of either error structure or random-effects distribution and allows for dependence between random effects and covariates. We show that the resulting estimators are consistent and asymptotically normal. Extensive simulation studies evaluate the finite sample performance of the proposed estimators and demonstrate their advantage over the validation set based method and the existing method. We illustrate the method with two real data examples.

A latent class model for defining severe hemorrhage: experience from the PROMMTT study

BACKGROUND

Several predictive models have been developed to identify trauma patients who have had severe hemorrhage (SH) and may need a massive transfusion (MT) protocol. However, almost all these models define SH as the transfusion of 10 or more units of red blood cells (RBCs) within 24 hours of emergency department admission (also known as MT). This definition excludes some patients with SH, especially those who die before a 10th unit of RBCs could be transfused, which calls the validity of these prediction models into question. We show how a latent class model could improve the accuracy of identifying the SH patients.

METHODS

Modeling SH classification as a latent variable, we estimate the posterior probability of a patient in SH based on emergency department admission variables (systolic blood pressure, heart rate, pH, hemoglobin), the 24-hour blood product use (plasma/RBC and platelet/RBC ratios), and 24-hour survival status. We define the SH subgroup as those having a posterior probability of 0.5 or greater. We compare our new classification of SH with that of the traditional MT using data from PROMMTT study.

RESULTS

Of the 1,245 patients, 913 had complete data, which were used in the latent class model. About 25.3% of patients were classified as SH. The overall agreement between the MT and SH classifications was 83.8%. However, among 49 patients who died before receiving the 10th unit of RBCs, 41 (84%) were classified as SH. Seven (87.5%) of the remaining eight patients who were not classified as SH had head injury.

CONCLUSION

Our definition of SH based on the aforementioned latent class model has an advantage of improving on the traditional MT definition by identifying SH patients who die before receiving the 10th unit of RBCs. We recommend further improvements to more accurately classify SH patients, which could replace the traditional definition of MT for use in developing prediction algorithms.