Analysis of Recurrent Events from Repairable Systems

Recurrent events arise in many longitudinal medical studies where time to a terminal event or failure is the primary endpoint. With incomplete follow-up data, the analysis of recurrent events is a challenge owing to their association with the failure. One specific quantity of interest rarely addressed in the statistical literature is the recurrence frequency at the failure time; an example is hospitalization frequency, which is often used as a rough measure of lifetime medical cost. In this article we show that a marginal model (e.g., the log-linear model) of the recurrence frequency, although desirable, is typically not identifiable. For this reason, we advocate modeling the recurrent events and the failure time jointly, and propose an approach to forming semiparametric joint models from prespecified marginal ones. We suggest two conceptually simple and nested regression models aiming at the recurrence frequency as a mark of the failure and at the process of recurrent events. We formulate monotone estimating functions and propose novel interval-estimation procedures to accommodate non-smooth estimating functions. The resulting estimators are consistent and asymptotically normal. Simulation studies and the application to an AIDS clinical trial exhibit that these proposals are easy to implement and reliable for practical use. Finally, we generalize our proposals to marked recurrent events, and also devise a global inference procedure for recurrent events of multiple types.

Many diseases recur after recovery, for example, recurrences in cancer and infections. However, research is often focused on analysing only time-to-first recurrence, thereby ignoring any subsequent recurrences that may occur after the first. Statistical models for the analysis of recurrent events are available, of which the extended Cox proportional hazards frailty model is the current state-of-the-art. However, this model is too statistically complex for computationally efficient application in high-dimensional data sets, including genome-wide association studies (GWAS). Here, we develop an application for fast and accurate recurrent event analysis in GWAS, called SPARE (SaddlePoint Approximation for Recurrent Event analysis). In SPARE, every DNA variant is tested for association with recurrence risk using a modified score statistic. A saddlepoint approximation is implemented to achieve statistical accuracy. SPARE controls the Type I error, and its statistical power is similar to existing recurrent event models, yet SPARE is significantly faster. An application of SPARE in a recurrent event GWAS on bladder cancer for 6.2 million DNA variants in 1,443 individuals required less than 15 min, whereas existing recurrent event methods would require several weeks.

CABG Improves Outcomes in Patients With Ischemic Cardiomyopathy: 10-Year Follow-Up of the STICH Trial

Economic Analysis and Prediction of Recurrent Events

This article is concerned with the analysis of recurrent events in the presence of a terminal event such as death. We consider the mean frequency function, defined as the marginal mean of the cumulative number of recurrent events over time. A simple nonparametric estimator for this quantity is presented. It is shown that the estimator, properly normalized, converges weakly to a zero-mean Gaussian process with an easily estimable covariance function. Nonparametric statistics for comparing two mean frequency functions and for combining data on recurrent events and death are also developed. The asymptotic null distributions of these statistics, together with consistent variance estimators, are derived. The small-sample properties of the proposed estimators and test statistics are examined through simulation studies. An application to a cancer clinical trial is provided.

Introduction: Recurrent event data are common in many longitudinal studies. Often, a terminating event such as death can be correlated with the recurrent event process. A shared frailty model applied to account for the association between recurrent and terminal events. In some situations, a fraction of subjects experience neither recurrent events nor death; these subjects are cured.
 Methods: In this paper, we discussed the Bayesian approach of a joint frailty model for recurrent and terminal events in the presence of cure fraction. We compared estimates of parameters in the Frequentist and Bayesian approaches via simulation studies in various sample sizes; we applied the joint frailty model in the presence of cure fraction with Frequentist and Bayesian approaches for breast cancer.
 Results: In small sample size Bayesian approach compared to Frequentist approach had a smaller standard error and mean square error, and the coverage probabilities close to nominal level of 95%. Also, in Bayesian approach, the sampling means of the estimated standard errors were close to the empirical standard error.
 Conclusion: The simulation results suggested that when sample size was small, the use of Bayesian joint frailty model in the presence of cure fraction led to more efficiency in parameter estimation and statistical inference.

111 Background: Local prostate radiotherapy (LPRT) is associated with improved overall survival in patients with low metastatic burden (MB) and is now standard of care. However, the role of LPRT in reducing symptomatic local events (SLE) in metastatic prostate cancer (MPC) remains unclear and requires long-term follow-up. The purpose of this study was two-fold: i) identify the risk factors associated with SLE, and ii) evaluate the association between LPRT and SLE in MPC. Methods: We conducted a retrospective, population-based cohort study of patients diagnosed with initial MPC between 2005 and 2016. Patients were identified through the Alberta Cancer Registry and patient, tumour, and treatment characteristics were collected by chart review. Data were linked to physician billing claims between 2004 and 2017 for diagnostic or therapeutic procedures potentially related to genitourinary (GU) and gastrointestinal (GI) SLE including percutaneous nephrostomy (PCN) and ureteric stent insertion (USI), cystoscopy, TURP, TURBT, colonoscopy and proctosigmoidoscopy. Both Andersen-Gill recurrent event and multivariable Cox regression time to first event analyses were conducted to evaluate the effect of LPRT on the occurrence of these procedures. Patients who underwent radical prostatectomy were excluded. LPRT was defined as 40 Gy or higher total dose to the prostate within one year of diagnosis. Patients with a SLE occurring after diagnosis but prior to LPRT were allocated to the control group. MB was defined as per STAMPEDE. Covariates for both models included MB, age at diagnosis, PSA at diagnosis, clinical T- and N-stage, and Gleason score (GS). Results: Of a total cohort of 1363 patients, 745 (54.7%) had high MB and 450 (33%) had low MB. Fifty-four (4%) received LPRT, of which 14.8% had high MB. Of those receiving LPRT, median PSA was 9.4, 79.6% had GS of 8-10, and 59.3% had T3-T4 disease. One or more SLE were observed in 43.5% and 37% of the control and LPRT groups, respectively. Among those with SLE, the median SLE frequency was 2 (interquartile range [IQR], 2-5) and 1 (IQR, 1-2.3) for the control and LPRT groups, respectively. On recurrent event analysis, LPRT was associated with lower risk of composite GU SLE (HR 0.34, 95% CI 0.17-0.67; p = 0.002), PCN and USI (HR 0.20, 95% CI 0.05-0.84; p = 0.027) and cystoscopy (HR 0.38, 95% CI 0.15-0.96; p = 0.041). Risk factors for GU SLE were T4 disease, GS of 8-10 and unknown GS. Risk factors for PCN and USI were T3, TX and N1 disease, GS of 8-10 and unknown GS. On time to first event analysis, there were no statistically significant differences for all outcomes between the control and LPRT groups. MB was not a risk factor for SLE in both analyses. Conclusions: LPRT was associated with lower risk for recurrent GU SLE, PCN and USI, and cystoscopy. The associated benefit in SLE reduction with LPRT warrants further study to determine if this effect is modified by MB and whether there may be a role for LPRT in patients with high MB.

This paper considers the analysis of a repeat event outcome in clinical trials of chronic diseases in the context of dependent censoring (e.g. mortality). It has particular application in the context of recurrent heart failure hospitalisations in trials of heart failure. Semi‐parametric joint frailty models (JFMs) simultaneously analyse recurrent heart failure hospitalisations and time to cardiovascular death, estimating distinct hazard ratios whilst individual‐specific latent variables induce associations between the two processes. A simulation study was carried out to assess the suitability of the JFM versus marginal analyses of recurrent events and cardiovascular death using standard methods. Hazard ratios were consistently overestimated when marginal models were used, whilst the JFM produced good, well‐estimated results. An application to the Candesartan in Heart failure: Assessment of Reduction in Mortality and morbidity programme was considered. The JFM gave unbiased estimates of treatment effects in the presence of dependent censoring. We advocate the use of the JFM for future trials that consider recurrent events as the primary outcome. © 2016 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.

Counting processes (point processes) constitute a useful tool for modelling random recurrent events. Numerous papers and books were published in recent decades on various aspects of modelling and analysis of recurrent events, describing also a wide range of applications. These applications include but not limited to: analysis of repairable items, queueing models, insurance risk analysis, biology, telecommunication, and so forth.

The Fundamentals of Survival and Event History Analysis Introduction: What Is Survival and Event History Analysis? Key Concepts and Terminology Censoring and Truncation Mathematical Expression and Relation of Basic Statistical Functions How Do the Survivor, Density and Hazard Function Relate? Why Use Survival and Event History Analysis? Overview of Survival and Event History Models Exercises Using R and Other Computer Programs for Survival and Event History Analysis Introduction: Computer Programs for Survival and Event History Analysis Conducting Serious Data Analysis: Life Lessons Why Use R? Downloading R on Your Personal Computer Add-On Packages Running R Determining and Setting your Working Directory Help and Documentation Importing Data Into R Working With Data: Opening and Accessing Variables from a Data Frame Saving Output as File, Workspace and History and Quitting R Exercises Your First Session: Using the Survival Package and Exploring Data Via Descriptive Statistics and Graphs Your First Session Using the 'Survival' Package In F Loading and Examining the Survival Package and Rcmdrplugin.Survival Plug-In Opening and Examining Data The Surv Object: Packaging the 'Survival Variable' Basic Descriptive Statistics Descriptive Data Exploration with Graphs Exercises Data and Data Reconstruction Introduction: Why Discuss Data and Data Preparation? Sources of Event History Data Single-Episode Data for Single Transition Analyses Multi-Episode Data for Recurrent Event and Frailty Analyses Subject-(Person)-Period Data for Discrete-Time Hazard Models The Counting Process and Episode Splitting A Note on Dates Exercises Non-Parametric Methods: Estimating and Comparing Survival Curves Using the Kaplan-Meier Estimator Introduction The Kaplan-Meier Estimator Producing Kaplan-Meier Estimates Plotting the Kaplan-Meier Survival Curve Testing Differences Between Two Groups Using Survdiff Stratifying the Analysis by a Covariate Exercises The Cox Proportional-Hazards Regression Introduction: Why is The Cox Model So Popular? The Cox Regression Model Estimating and Interpreting The Cox Model with Fixed Covariates The Cox Regression Model with Time-Varying Covariates Exercises Parametric Models Introduction: What are Parametric Models and Why Use Them? Proportional Hazards (Ph) Versus Accelerated Failure Time (Aft) Models The Path to Choosing a Model Estimating and Interpreting Parametric Survival Models Exponential and Piecewise Constant Exponential Model Weibull Model Log-Logistic and Log-Normal Models Additional Parametric Models Finding the Best Fitting Parametric Model Exercises Model Building and Diagnostics Introduction Model Building and Selection of Covariates Assessing the Overall Goodness of Fit of Your Model What is Residual Analysis? Testing Overall Model Adequacy: Cox-Snell Residuals Testing the Proportional Hazards Assumption: Schoenfeld Residuals Checking For Influential Observations: Score Residuals (Dfbeta Statistics) Assessing Nonlinearity: Martingale Residual and Component-Plus-Residual Plots Exercises Correlated and Discrete-Time Survival Data: Frailty, Recurrent Events and Discrete-Time Models Introduction Shared Frailty: Modeling Recurrent Events and Clustering In Groups Other Frailty Models: Unshared, Nested, Joint and Additive Models Estimating Frailty Models in R Example of Frailty Model Estimation and Interpretation Discrete-Time and Count Models Exercises Multiple Events and Entire Histories: Competing Risk, Multistate Models and Sequence Analysis Introduction Competing Risk Models Multistate Models Sequence Analysis: Modeling Entire Histories Exercises Appendix : Datasets Used in this Book

The incidence and determinants of severe morbidity recurrence in sub-Saharan African HIV-infected adults on antiretroviral therapy (ART) have never been reported. In a prospective cohort study of HIV-infected adults in Abidjan the association of severe morbidity occurrence and recurrence with follow-up CD4 counts and ART on/off status was analyzed by means of multivariate failure analysis for recurrent events (Prentice, Williams, and Peterson model). A total of 608 patients (median CD4 290/mm3 ) was followed off ART for 1824 person-years (PY). Of these 187 started HAART (median CD4 174/mm3 ) and were followed for 328 PY. The incidence of first, second, and third severe morbidity events was 40.6/100 PY, 68.4/100 PY, and 93.9/100 PY during the off-ART period, and 28.4/100 PY, 39.4/100 PY, and 37.6/100 PY during the on-ART period, respectively. The rates of recurrences were higher than the rates of first episodes for almost all diseases, even after stratifying by CD4 count and by ART on/off status. In multivariate analysis, the time-updated CD4 count was independently associated with increasing rates of morbidity first events and recurrences, after adjustment on other covariates (p > 10(4) ). By contrast, there was no association between the ART on/off status and the morbidity rates after adjustment for CD4 count (p = 0.37). Introducing ART led to a clear reduction in morbidity, mainly related to the ART-induced increase in CD4 count. In HIV-infected patients on ART, the incidence of severe morbidity varied with the past history of morbidity. The past history of morbidity should be taken into account when comparing HIV morbidity rates before and after ART initiation.

In modern biomedical datasets, it is common for recurrent outcomes data to be collected in an incomplete manner. More specifically, information on recurrent events is routinely recorded as a mixture of recurrent event data, panel count data, and panel binary data; we refer to this structure as general mixed recurrent event data. Although the aforementioned data types are individually well-studied, there does not appear to exist an established approach for regression analysis of the three component combination. Often, ad-hoc measures such as imputation or discarding of data are used to homogenize records prior to the analysis, but such measures lead to obvious concerns regarding robustness, loss of efficiency, and other issues. This work proposes a maximum likelihood regression estimation procedure for the combination of general mixed recurrent event data and establishes the asymptotic properties of the proposed estimators. In addition, we generalize the approach to allow for the existence of terminal events, a common complicating feature in recurrent event analysis. Numerical studies and application to the Childhood Cancer Survivor Study suggest that the proposed procedures work well in practical situations.

We propose a Bayesian semiparametric joint regression model for a recurrent event process and survival time. Assuming independent latent subject frailties, we define marginal models for the recurrent event process intensity and survival distribution as functions of the subject's frailty and baseline covariates. A robust Bayesian model, called Joint-DP, is obtained by assuming a Dirichlet process for the frailty distribution. We present a simulation study that compares posterior estimates under the Joint-DP model to a Bayesian joint model with lognormal frailties, a frequentist joint model, and marginal models for either the recurrent event process or survival time. The simulations show that the Joint-DP model does a good job of correcting for treatment assignment bias, and has favorable estimation reliability and accuracy compared with the alternative models. The Joint-DP model is applied to analyze an observational dataset from esophageal cancer patients treated with chemo-radiation, including the times of recurrent effusions of fluid to the heart or lungs, survival time, prognostic covariates, and radiation therapy modality.

The Cox proportional hazards model has been pervasively used in many social science areas to examine the effects of covariates on timing to an event. The standard Cox model is intended to study univariate survival data where there is a singular event of interest, which can only be experienced once. However, we may additionally wish to explore a number of other complexities that are prevalent in survival data. For example, an individual may experience events of the same type more than once or may experience multiple types of events. This study introduces innovations in recurrent (repeatable) event analysis, jointly modeling several endogenous survival processes. As an example and an application, we simultaneously model two types of recurrent events in the presence of a dependent terminal event. This model not only correctly handles different types of recurrent events but also explicitly estimates the direction and magnitude of relationships between recurrences and survival. This article concludes with an example of the model to examine how the timing of retirement is associated with the risks of multiple spells of employment and childbearing. The theoretical discussions and empirical analyses suggest that the multivariate joint models have much to offer to a wide variety of substantive research areas.

BackgroundIndividuals seeking alcohol and other drug (AOD) treatment consistently experience higher rates of suicidal behaviours and death by suicide when compared to the general population. By linking residential AOD treatment data to administrative healthcare and death datasets, we aimed to examine suicide-related behaviours and identify risk and protective factors for these events following discharge from residential treatment. MethodsParticipants included 1056 individuals aged 18–69 (M = 32.06, SD = 9.55, male = 696,65.9 %) admitted to three residential treatment facilities in Queensland, Australia from January 1, 2014 to December 31, 2016. Treatment data was linked to administrative hospital, emergency department (ED), mental health service, and Registry of Deaths data 2-years post-discharge. ICD-10 codes were used to identify and analyse suicide-related events. ResultsWithin 2-years post-discharge, 175 (16.6 %) individuals had a suicide-related event (n = 298 episodes). The highest proportion of episodes (11.1 %) occurred within 1-month of discharge. Higher risk of a recurrent suicide-related event was associated with receiving a Disability Support Pension (aHR = 1.69 (95%CI:1.10,2.59), two or more previous episodes of residential AOD treatment (aHR = 1.49 (95%CI:1.30,2.15). Completing residential treatment was associated with a lower risk of suicide-related events (aHR = 0.54 (95%CI:0.35,0.83). LimitationsThe amalgamation of suicidal ideation, attempts, and death into a single outcome oversimplifies their complex nature and interplay. The exclusive focus on one service provider limits generalisability, and data constraints and missingness preclude many analyses. ConclusionsUnderstanding suicidal behaviours and critical risk periods following discharge from residential treatment is crucial for improving continuing care, developing effective suicide prevention, and implementing targeted interventions among this high-risk population.