Search

We compare the performance of nonparametric estimators for the mean number of recurrent events and provide a systematic overview for different recurrent event settings. The mean number of recurrent events is an easily interpreted marginal feature often used for treatment comparisons in clinical trials. Incomplete observations, dependencies between successive events, terminating events acting as competing risk, or gaps between at risk periods complicate the estimation. We use survival multistate models to represent different complex recurrent event situations, profiting from recent advances in nonparametric estimation for non-Markov multistate models, and explain several estimators by using multistate intensity processes, including the common Nelson-Aalen-type estimators with and without competing mortality. In addition to building on estimation of state occupation probabilities in non-Markov models, we consider a simple extension of the Nelson-Aalen estimator by allowing for dependence on the number of prior recurrent events. We pay particular attention to the assumptions required for the censoring mechanism, one issue being that some settings require the censoring process to be entirely unrelated while others allow for state-dependent or event-driven censoring. We conducted extensive simulation studies to compare the estimators in various complex situations with recurrent events. Our practical example deals with recurrent chronic obstructive pulmonary disease exacerbations in a clinical study, which will also be used to illustrate two-sample-inference using resampling., (© 2023 The Authors. Pharmaceutical Statistics published by John Wiley & Sons Ltd.)

A popular modeling approach for competing risks analysis in longitudinal studies is the proportional subdistribution hazards model by Fine and Gray (1999. A proportional hazards model for the subdistribution of a competing risk. Journal of the American Statistical Association94, 496-509). This model is widely used for the analysis of continuous event times in clinical and epidemiological studies. However, it does not apply when event times are measured on a discrete time scale, which is a likely scenario when events occur between pairs of consecutive points in time (e.g., between two follow-up visits of an epidemiological study) and when the exact lengths of the continuous time spans are not known. To adapt the Fine and Gray approach to this situation, we propose a technique for modeling subdistribution hazards in discrete time. Our method, which results in consistent and asymptotically normal estimators of the model parameters, is based on a weighted ML estimation scheme for binary regression. We illustrate the modeling approach by an analysis of nosocomial pneumonia in patients treated in hospitals., (© The Author 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com.)

A clinical hold order by the Food and Drug Administration (FDA) to the sponsor of a clinical trial is a measure to delay a proposed or to suspend an ongoing clinical investigation. The phase III clinical trial START serves as motivating data example to explore implications and potential statistical approaches for a trial continuing after a clinical hold is lifted. In spite of a modified intention-to-treat (ITT) analysis introduced to account for the clinical hold by excluding patients potentially affected most by the clinical hold, results of the trial did not show a significant improvement of overall survival duration, and the question remains whether the negative result was an effect of the clinical hold. In this paper, we propose a multistate model incorporating the clinical hold as well as disease progression as intermediate events to investigate the impact of the clinical hold on the treatment effect. Moreover, we consider a simple counterfactual censoring approach as alternative strategy to the modified ITT analysis to deal with a clinical hold. Using a realistic simulation study informed by the START data and with a design based on our multistate model, we show that the modified ITT analysis used in the START trial was reasonable. However, the censoring approach will be shown to have some benefits in terms of power and flexibility., (© 2019 The Authors. Pharmaceutical Statistics published by John Wiley & Sons Ltd.)

In this paper, we derive the joint distribution of progression-free and overall survival as a function of transition probabilities in a multistate model. No assumptions on copulae or latent event times are needed and the model is allowed to be non-Markov. From the joint distribution, statistics of interest can then be computed. As an example, we provide closed formulas and statistical inference for Pearson's correlation coefficient between progression-free and overall survival in a parametric framework. The example is inspired by recent approaches to quantify the dependence between progression-free survival, a common primary outcome in Phase 3 trials in oncology and overall survival. We complement these approaches by providing methods of statistical inference while at the same time working within a much more parsimonious modeling framework. Our approach is completely general and can be applied to other measures of dependence. We also discuss extensions to nonparametric inference. Our analytical results are illustrated using a large randomized clinical trial in breast cancer., (© 2019 John Wiley & Sons, Ltd.)

We suggest a wild bootstrap resampling technique for nonparametric inference on transition probabilities in a general time-inhomogeneous Markov multistate model. We first approximate the limiting distribution of the Nelson-Aalen estimator by repeatedly generating standard normal wild bootstrap variates, while the data is kept fixed. Next, a transformation using a functional delta method argument is applied. The approach is conceptually easier than direct resampling for the transition probabilities. It is used to investigate a non-standard time-to-event outcome, currently being alive without immunosuppressive treatment, with data from a recent study of prophylactic treatment in allogeneic transplanted leukemia patients. Due to non-monotonic outcome probabilities in time, neither standard survival nor competing risks techniques apply, which highlights the need for the present methodology. Finite sample performance of time-simultaneous confidence bands for the outcome probabilities is assessed in an extensive simulation study motivated by the clinical trial data. Example code is provided in the web-based Supplementary Materials., (© 2018, The International Biometric Society.)

Background: We evaluate three methods for competing risks analysis with time-dependent covariates in comparison with the corresponding methods with time-independent covariates., Methods: We used cause-specific hazard analysis and two summary approaches for in-hospital death: logistic regression and regression of the subdistribution hazard. We analysed real hospital data (n=1864) and considered pneumonia on admission / hospital-acquired pneumonia as time-independent / time-dependent covariates for the competing events 'discharge alive' and 'in-hospital death'. Several simulation studies with time-constant hazards were conducted., Results: All approaches capture the effect of time-independent covariates, whereas the approaches perform differently with time-dependent covariates. The subdistribution approach for time-dependent covariates detected effects in a simulated no-effects setting and provided counter-intuitive effects in other settings., Conclusions: The extension of the Fine and Gray model to time-dependent covariates is in general not a helpful synthesis of the cause-specific hazards. Cause-specific hazard analysis and, for uncensored data, the odds ratio are capable of handling competing risks data with time-dependent covariates but the use of the subdistribution approach should be neglected until the problems can be resolved. For general right-censored data, cause-specific hazard analysis is the method of choice.

Realistic power calculations for large cohort studies and nested case control studies are essential for successfully answering important and complex research questions in epidemiology and clinical medicine. For this, we provide a methodical framework for general realistic power calculations via simulations that we put into practice by means of an R-based template. We consider staggered recruitment and individual hazard rates, competing risks, interaction effects, and the misclassification of covariates. The study cohort is assembled with respect to given age-, gender-, and community distributions. Nested case-control analyses with a varying number of controls enable comparisons of power with a full cohort analysis. Time-to-event generation under competing risks, including delayed study-entry times, is realized on the basis of a six-state Markov model. Incidence rates, prevalence of risk factors and prefixed hazard ratios allow for the assignment of age-dependent transition rates given in the form of Cox models. These provide the basis for a central simulation-algorithm, which is used for the generation of sample paths of the underlying time-inhomogeneous Markov processes. With the inclusion of frailty terms into the Cox models the Markov property is specifically biased. An "individual Markov process given frailty" creates some unobserved heterogeneity between individuals. Different left-truncation- and right-censoring patterns call for the use of Cox models for data analysis. p-values are recorded over repeated simulation runs to allow for the desired power calculations. For illustration, we consider scenarios with a "testing" character as well as realistic scenarios. This enables the validation of a correct implementation of theoretical concepts and concrete sample size recommendations against an actual epidemiological background, here given with possible substudy designs within the German National Cohort., (© 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.)

Competing risks extend standard survival analysis to considering time-to-first-event and type-of-first-event, where the event types are called competing risks. The competing risks process is completely described by all cause-specific hazards, ie, the hazard marked by the event type. Separate Cox models for each cause-specific hazard are the standard approach to regression modelling, but they come with the interpretational challenge that there are as many regression coefficients as there are competing risks. An alternative approach is to directly model the cumulative event probabilities, but again, there will be as many models as there are competing risks. The aim of this paper is to investigate the usefulness of a third alternative. Proportional odds modelling of all cause-specific hazards summarizes the effect of one covariate on "opposing" competing outcomes in one regression coefficient. For instance, if the competing outcomes are hospital death and alive discharge from hospital, the modelling assumption is that a covariate affects both outcomes in opposing directions, but the effect size is of the same absolute magnitude. We will investigate the interpretational aspects of the approach analysing a data set on intensive care unit patients using parametric methods., (Copyright © 2017 John Wiley & Sons, Ltd.)

Competing risks endpoints are frequently encountered in hematopoietic stem cell transplantation where patients are exposed to relapse and treatment-related mortality. Both cause-specific hazards and direct models for the cumulative incidence functions have been used for analyzing such competing risks endpoints. For both approaches, the popular models are of a proportional hazards type. Such models have been used for studying prognostic factors in acute and chronic leukemias. We argue that a complete understanding of the event dynamics requires that both hazards and cumulative incidence be analyzed side by side, and that this is generally the most rigorous scientific approach to analyzing competing risks data. That is, understanding the effects of covariates on cause-specific hazards and cumulative incidence functions go hand in hand. A case study illustrates our proposal., (Copyright © 2013 Elsevier Inc. All rights reserved.)

Complex clinical endpoints are present in studies in cancer. Especially in studies on hematopoietic stem-cell transplantation (HSCT), various risks exist after HSCT. Patients can experience acute and chronic graft versus host disease (GVHD) or need to undergo immunosuppressive therapy (IST), a relapse can occur, or patients can die after relapse or without former relapse (nonrelapse mortality, NRM). Sometimes, endpoints can be reasonably combined in a composite endpoint, as, for example, relapse and NRM are combined into disease-free survival (DFS). In this case, standard survival techniques, as Kaplan-Meier estimation of the DFS probability, can be applied. Often, interest focuses on endpoints for which competing risks are present, as, for example, GVHD, with death without prior GVHD as competing risk. This results in a competing risks model, a special case of a multistate model. A more complex multistate model is required when the effects of events occurring in the course of the study on further disease process shall be investigated, as, for example, the effect of GVHD on relapse and NRM. Another endpoint of interest is time under IST. As patients usually experience multiple episodes of IST, thus switching back and forth between "IST" and "no IST" during follow-up, the multistate model used for analysis must be adapted for this event structure. The aim of this nontechnical report is to explain use and interpretation of Cox-type regression models suitable for the different situations in a randomized trial on the effects of anti-T-cell globulin as GVHD prophylaxis.

Prognostic models for time-to-event data play a prominent role in therapy assignment, risk stratification and inter-hospital quality assurance. The assessment of their prognostic value is vital not only for responsible resource allocation, but also for their widespread acceptance. The additional presence of competing risks to the event of interest requires proper handling not only on the model building side, but also during assessment. Research into methods for the evaluation of the prognostic potential of models accounting for competing risks is still needed, as most proposed methods measure either their discrimination or calibration, but do not examine both simultaneously. We adapt the prediction error proposal of Graf et al. (Statistics in Medicine 1999, 18, 2529–2545) and Gerds and Schumacher (Biometrical Journal 2006, 48, 1029–1040) to handle models with competing risks, i.e. more than one possible event type, and introduce a consistent estimator. A simulation study investigating the behaviour of the estimator in small sample size situations and for different levels of censoring together with a real data application follows.

The Aalen-Johansen estimator is the standard nonparametric estimator of the cumulative incidence function in competing risks. Estimating its variance in small samples has attracted some interest recently, together with a critique of the usual martingale-based estimators. We show that the preferred estimator equals a Greenwood-type estimator that has been derived as a recursion formula using counting processes and martingales in a more general multistate framework. We also extend previous simulation studies on estimating the variance of the Aalen-Johansen estimator in small samples to left-truncated observation schemes, which may conveniently be handled within the counting processes framework. This investigation is motivated by a real data example on spontaneous abortion in pregnancies exposed to coumarin derivatives, where both competing risks and left-truncation have recently been shown to be crucial methodological issues (Meister and Schaefer (2008), Reproductive Toxicology 26, 31-35). Multistate-type software and data are available online to perform the analyses. The Greenwood-type estimator is recommended for use in practice.

Purpose: To illustrate modern survival models with focus on the temporal dynamics of intensive care data. A typical situation is given in which time-dependent exposures and competing events are present., Methods: We briefly review the following established statistical methods: logistic regression, regression models for event-specific hazards and the subdistribution hazard. These approaches are compared by showing advantages as well as disadvantages. All methods are applied to real data from a study of day-by-day ICU surveillance., Results: Standard logistic regression ignores the time-dependent nature of the data and is only a crude approach. Cumulative hazards and probability plots add important information and provide a deep insight into the temporal dynamics., Conclusion: This paper might help to encourage researchers working in hospital epidemiology to apply adequate statistical models to complex medical questions.

Competing risks analysis considers time-to-first-event ('survival time') and the event type ('cause'), possibly subject to right-censoring. The cause-, i.e. event-specific hazards, completely determine the competing risk process, but simulation studies often fall back on the much criticized latent failure time model. Cause-specific hazard-driven simulation appears to be the exception; if done, usually only constant hazards are considered, which will be unrealistic in many medical situations. We explain simulating competing risks data based on possibly time-dependent cause-specific hazards. The simulation design is as easy as any other, relies on identifiable quantities only and adds to our understanding of the competing risks process. In addition, it immediately generalizes to more complex multistate models. We apply the proposed simulation design to computing the least false parameter of a misspecified proportional subdistribution hazard model, which is a research question of independent interest in competing risks. The simulation specifications have been motivated by data on infectious complications in stem-cell transplanted patients, where results from cause-specific hazards analyses were difficult to interpret in terms of cumulative event probabilities. The simulation illustrates that results from a misspecified proportional subdistribution hazard analysis can be interpreted as a time-averaged effect on the cumulative event probability scale.

Separate Cox analyses of all cause-specific hazards are the standard technique of choice to study the effect of a covariate in competing risks, but a synopsis of these results in terms of cumulative event probabilities is challenging. This difficulty has led to the development of the proportional subdistribution hazards model. If the covariate is known at baseline, the model allows for a summarizing assessment in terms of the cumulative incidence function. black Mathematically, the model also allows for including random time-dependent covariates, but practical implementation has remained unclear due to a certain risk set peculiarity. We use the intimate relationship of discrete covariates and multistate models to naturally treat time-dependent covariates within the subdistribution hazards framework. The methodology then straightforwardly translates to real-valued time-dependent covariates. As with classical survival analysis, including time-dependent covariates does not result in a model for probability functions anymore. Nevertheless, the proposed methodology provides a useful synthesis of separate cause-specific hazards analyses. We illustrate this with hospital infection data, where time-dependent covariates and competing risks are essential to the subject research question.