Incorporation of nested frailties into semiparametric multi‐state models

Proportional hazards models are among the most popular regression models in survival analysis. Multi‐state models generalize them by jointly considering different types of events and their interrelations, whereas frailty models incorporate random effects to account for unobserved risk factors, possibly shared by clusters of subjects. The integration of multi‐state and frailty methodology is an interesting way to control for unobserved heterogeneity in the presence of complex event history structures and is particularly appealing for multicenter clinical trials. We propose the incorporation of correlated frailties in the transition‐specific hazard function, thanks to a nested hierarchy. We studied a semiparametric estimation approach based on maximum integrated partial likelihood. We show in a simulation study that the nested frailty multi‐state model improves the estimation of the effect of covariates, as well as the coverage probability of their confidence intervals. We present a case study concerning a prostate cancer multicenter clinical trial. The multi‐state nature of the model allows us to evidence the effect of treatment on death taking into account intermediate events. Copyright © 2015 JohnWiley & Sons, Ltd.     

Investigating hospital heterogeneity with a competing risks frailty model

Survival analysis is used in the medical field to identify the effect of predictive variables on time to a specific event. Generally, not all variation of survival time can be explained by observed covariates. The effect of unobserved variables on the risk of a patient is called frailty. In multicenter studies, the unobserved center effect can induce frailty on its patients, which can lead to selection bias over time when ignored. For this reason, it is common practice in multicenter studies to include a random frailty term modeling center effect. In a more complex event structure, more than one type of event is possible. Independent frailty variables representing center effect can be incorporated in the model for each competing event. However, in the medical context, events representing disease progression are likely related and correlation is missed when assuming frailties to be independent. In this work, an additive gamma frailty model to account for correlation between frailties in a competing risks model is proposed, to model frailties at center level. Correlation indicates a common center effect on both events and measures how closely the risks are related. Estimation of the model using the expectation-maximization algorithm is illustrated. The model is applied to a data set from a multicenter clinical trial on breast cancer from the European Organisation for Research and Treatment of Cancer (EORTC trial 10854). Hospitals are compared by employing empirical Bayes estimates methodology together with corresponding confidence intervals.     

Joint modeling of recurrent events and a terminal event adjusted for zero inflation and a matched design

In longitudinal studies, matched designs are often employed to control the potential confounding effects in the field of biomedical research and public health. Because of clinical interest, recurrent time‐to‐event data are captured during the follow‐up. Meanwhile, the terminal event of death is always encountered, which should be taken into account for valid inference because of informative censoring. In some scenarios, a certain large portion of subjects may not have any recurrent events during the study period due to nonsusceptibility to events or censoring; thus, the zero‐inflated nature of data should be considered in analysis. In this paper, a joint frailty model with recurrent events and death is proposed to adjust for zero inflation and matched designs. We incorporate 2 frailties to measure the dependency between subjects within a matched pair and that among recurrent events within each individual. By sharing the random effects, 2 event processes of recurrent events and death are dependent with each other. The maximum likelihood based approach is applied for parameter estimation, where the Monte Carlo expectation‐maximization algorithm is adopted, and the corresponding R program is developed and available for public usage. In addition, alternative estimation methods such as Gaussian quadrature (PROC NLMIXED) and a Bayesian approach (PROC MCMC) are also considered for comparison to show our method's superiority. Extensive simulations are conducted, and a real data application on acute ischemic studies is provided in the end.     

General joint frailty model for recurrent event data with a dependent terminal event: Application to follicular lymphoma data

Many biomedical studies focus on delaying disease relapses and on prolonging survival. Usual methods only consider one event, often the first recurrence or death. However, ignoring the other recurrences may lead to biased results. The whole history of the disease should be considered for each patient. In addition, some diseases involve recurrences that can increase the risk of death. In this case, the death time may be dependent on the recurrent event history. We propose a joint frailty model to analyze recurrences and death simultaneously. Two gamma-distributed frailties take into account both the inter-recurrences dependence and the dependence between the recurrences and the survival times. We estimate separate parameters for disease recurrent event times and survival times in the joint frailty model to distinguish treatment effects and prognostic factors on these two types of events. We show how maximum penalized likelihood estimation can be applied to semiparametric estimation of the continuous hazard functions in the proposed joint frailty model with right censoring. We also propose parametrical approach. We evaluate the model by simulation studies and illustrate through a study of patients with follicular lymphoma.     

A relaxation of the gamma frailty (Burr) model

Frailty models are used in univariate data to account for individual heterogeneity. In the popular gamma frailty model the marginal hazard has the form of a Burr model. Although the Burr model is very useful and can offer insight on the data, it is far from perfect. The estimation of the covariate effects is linked to the baseline hazard and this makes the model coefficients hard to interpret. At the same time, the frailties are assumed constant over time, while biological reasoning in some cases may indicate that frailties may be time dependent. In this paper we present a relaxation of the Burr model which is based on loosening the link between the estimation of the covariate effects and the baseline hazard. This can be achieved by replacing the cumulative baseline hazard in the Burr model by a set of time functions, and the frailty variance by a vector of coefficients directly estimated from the data using a partial likelihood. We illustrate the similarities of the model with the Burr model and a further extension of the latter, a model with an autoregressive stochastic process for the frailty. We compare the models on simulated data sets with constant and time-dependent frailties and show how the relaxed Burr models performs on two different real data sets. We show that the relaxed Burr model serves as a good approximation to the Burr model when the frailty is constant, and furthermore it gives better results when the frailty is time dependent.     

Estimation of intervention effects using recurrent event time data in the presence of event dependence and a cured fraction

Recurrent event data with a fraction of subjects having zero event are often seen in randomized clinical trials. Those with zero event may belong to a cured (or non‐susceptible) fraction. Event dependence refers to the situation that a person's past event history affects his future event occurrences. In the presence of event dependence, an intervention may have an impact on the event rate in the non‐cured through two pathways—a primary effect directly on the outcome event and a secondary effect mediated through event dependence. The primary effect combined with the secondary effect is the total effect. We propose a frailty mixture model and a two‐step estimation procedure for the estimation of the effect of an intervention on the probability of cure and the total effect on event rate in the non‐cured. A summary measure of intervention effects is derived. The performance of the proposed model is evaluated by simulation. Data on respiratory exacerbations from a randomized, placebo‐controlled trial are re‐analyzed for illustration. Copyright © 2014 John Wiley & Sons, Ltd.     

Dispersion frailty models and HGLMs

In medical research recurrent event times can be analysed using a frailty model in which the frailties for different individuals are independent and identically distributed. However, such a homogeneous assumption about frailties could sometimes be suspect. For modelling heterogeneity in frailties we describe dispersion frailty models arising from a new class of models, namely hierarchical generalized linear models. Using the kidney infection data we illustrate how to detect and model heterogeneity among frailties. Stratification of frailty models is also investigated.     

Semiparametric regression analysis for alternating recurrent event data

Alternating recurrent event data arise frequently in clinical and epidemiologic studies, where 2 types of events such as hospital admission and discharge occur alternately over time. The 2 alternating states defined by these recurrent events could each carry important and distinct information about a patient's underlying health condition and/or the quality of care. In this paper, we propose a semiparametric method for evaluating covariate effects on the 2 alternating states jointly. The proposed methodology accounts for the dependence among the alternating states as well as the heterogeneity across patients via a frailty with unspecified distribution. Moreover, the estimation procedure, which is based on smooth estimating equations, not only properly addresses challenges such as induced dependent censoring and intercept sampling bias commonly confronted in serial event gap time data but also is more computationally tractable than the existing rank‐based methods. The proposed methods are evaluated by simulation studies and illustrated by analyzing psychiatric contacts from the South Verona Psychiatric Case Register.     

A joint frailty model to estimate the recurrence process and the disease‐specific mortality process without needing the cause of death

In chronic diseases, such as cancer, recurrent events (such as relapses) are commonly observed; these could be interrupted by death. With such data, a joint analysis of recurrence and mortality processes is usually conducted with a frailty parameter shared by both processes. We examined a joint modeling of these processes considering death under two aspects: ‘death due to the disease under study' and ‘death due to other causes', which enables estimating the disease‐specific mortality hazard. The excess hazard model was used to overcome the difficulties in determining the causes of deaths (unavailability or unreliability); this model allows estimating the disease‐specific mortality hazard without needing the cause of death but using the mortality hazards observed in the general population. We propose an approach to model jointly recurrence and disease‐specific mortality processes within a parametric framework. A correlation between the two processes is taken into account through a shared frailty parameter. This approach allows estimating unbiased covariate effects on the hazards of recurrence and disease‐specific mortality. The performance of the approach was evaluated by simulations with different scenarios. The method is illustrated by an analysis of a population‐based dataset on colon cancer with observations of colon cancer recurrences and deaths. The benefits of the new approach are highlighted by comparison with the ‘classical' joint model of recurrence and overall mortality. Moreover, we assessed the goodness of fit of the proposed model. Comparisons between the conditional hazard and the marginal hazard of the disease‐specific mortality are shown, and differences in interpretation are discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

Repeated events survival models: the conditional frailty model

Repeated events processes are ubiquitous across a great range of important health, medical, and public policy applications, but models for these processes have serious limitations. Alternative estimators often produce different inferences concerning treatment effects due to bias and inefficiency. We recommend a robust strategy for the estimation of effects in medical treatments, social conditions, individual behaviours, and public policy programs in repeated events survival models under three common conditions: heterogeneity across individuals, dependence across the number of events, and both heterogeneity and event dependence. We compare several models for analysing recurrent event data that exhibit both heterogeneity and event dependence. The conditional frailty model best accounts for the various conditions of heterogeneity and event dependence by using a frailty term, stratification, and gap time formulation of the risk set. We examine the performance of recurrent event models that are commonly used in applied work using Monte Carlo simulations, and apply the findings to data on chronic granulomatous disease and cystic fibrosis.     

Flexible modeling of competing risks in survival analysis

Prognostic studies often involve modeling competing risks, where an individual can experience only one of alternative events, and the goal is to estimate hazard functions and covariate effects associated with each event type. Lunn and McNeil proposed data manipulation that permits extending the Cox's proportional hazards model to estimate covariate effects on the hazard of each competing events. However, the hazard functions for competing events are assumed to remain proportional over the entire follow‐up period, implying the same shape of all event‐specific hazards, and covariate effects are restricted to also remain constant over time, even if such assumptions are often questionable. To avoid such limitations, we propose a flexible model to (i) obtain distinct estimates of the baseline hazard functions for each event type, and (ii) allow estimating time‐dependent covariate effects in a parsimonious model. Our flexible competing risks regression model uses smooth cubic regression splines to model the time‐dependent changes in (i) the ratio of event‐specific baseline hazards, and (ii) the covariate effects. In simulations, we evaluate the performance of the proposed estimators and likelihood ratio tests, under different assumptions. We apply the proposed flexible model in a prognostic study of colorectal cancer mortality, with two competing events: ‘death from colorectal cancer’ and ‘death from other causes’. Copyright © 2010 John Wiley & Sons, Ltd.     

Estimating and testing for center effects in competing risks

The problems of fitting Gaussian frailties proportional hazards models for the subdistribution of a competing risk and of testing for center effects are considered. In the analysis of competing risks data, Fine and Gray proposed a proportional hazards model for the subdistribution to directly assess the effects of covariates on the marginal failure probabilities of a given failure cause. Katsahianbiet al. extended their model to clustered time to event data, by including random center effects or frailties in the subdistribution hazard. We first introduce an alternate estimation procedure to the one proposed by Katsahian et al. This alternate estimation method is based on the penalized partial likelihood approach often used in fitting Gaussian frailty proportional hazards models in the standard survival analysis context, and has the advantage of using standard survival analysis software. Second, four hypothesis tests for the presence of center effects are given and compared via Monte-Carlo simulations. Statistical and numerical considerations lead us to formulate pragmatic guidelines as to which of the four tests is preferable. We also illustrate the proposed methodology with registry data from bone marrow transplantation for acute myeloid leukemia (AML).     

Practical implementation of frailty models in Mendelian risk prediction

There are numerous statistical models used to identify individuals at high risk of cancer due to inherited mutations. Mendelian models predict future risk of cancer by using family history with estimated cancer penetrances (age- and sex-specific risk of cancer given the genotype of the mutations) and mutation prevalences. However, there is often residual risk heterogeneity across families even after accounting for the mutations in the model, due to environmental or unobserved genetic risk factors. We aim to improve Mendelian risk prediction by incorporating a frailty model that contains a family-specific frailty vector, impacting the cancer hazard function, to account for this heterogeneity. We use a discrete uniform population frailty distribution and implement a marginalized approach that averages each family's risk predictions over the family's frailty distribution. We apply the proposed approach to improve breast cancer prediction in BRCAPRO, a Mendelian model that accounts for inherited mutations in the BRCA1 and BRCA2 genes to predict breast and ovarian cancer. We evaluate the proposed model's performance in simulations and real data from the Cancer Genetics Network and show improvements in model calibration and discrimination. We also discuss alternative approaches for incorporating frailties and their strengths and limitations.     

Power and sample size calculation for paired recurrent events data based on robust nonparametric tests

The purpose of this paper is to develop a formula for calculating the required sample size for paired recurrent events data. The developed formula is based on robust non‐parametric tests for comparing the marginal mean function of events between paired samples. This calculation can accommodate the associations among a sequence of paired recurrent event times with a specification of correlated gamma frailty variables for a proportional intensity model. We evaluate the performance of the proposed method with comprehensive simulations including the impacts of paired correlations, homogeneous or nonhomogeneous processes, marginal hazard rates, censoring rate, accrual and follow‐up times, as well as the sensitivity analysis for the assumption of the frailty distribution. The use of the formula is also demonstrated using a premature infant study from the neonatal intensive care unit of a tertiary center in southern Taiwan. Copyright © 2017 John Wiley & Sons, Ltd.     

Concordance measures in shared frailty models: application to clustered data in cancer prognosis

Frailty models are gaining interest in prognostic studies, especially because of the spread of multicenter studies. However, little research has been performed to extend prognostic tools to frailty models, including discrimination measures. As previously performed for the Harrell's c‐index, we extended two different discrimination measures (the model‐based concordance probability estimation of Gönen and Heller and the nonparametric Uno's c‐index) to take into account cluster membership. We calculate measures at three levels: between‐group, where only patients with different frailties are compared, within‐group, where only patients sharing the same frailty are compared, and overall. We performed simulations to study the impact of group size and the number of groups on these measures. Results showed that the two measures can be extended to frailty models while remaining independent from censoring distribution, provided that the group size is sufficient. We apply the extended measures to two real datasets, a meta‐analysis and a large multicenter trial. Copyright © 2013 John Wiley & Sons, Ltd.     

The correlated and shared gamma frailty model for bivariate current status data: An illustration for cross‐sectional serological data

Frailty models are often used to study the individual heterogeneity in multivariate survival analysis. Whereas the shared frailty model is widely applied, the correlated frailty model has gained attention because it elevates the restriction of unobserved factors to act similar within clusters. Estimating frailty models is not straightforward due to various types of censoring. In this paper, we study the behavior of the bivariate‐correlated gamma frailty model for type I interval‐censored data, better known as current status data. We show that applying a shared rather than a correlated frailty model to cross‐sectionally collected serological data on hepatitis A and B leads to biased estimates for the baseline hazard and variance parameters. Copyright © 2009 John Wiley & Sons, Ltd.     

The partly Aalen's model for recurrent event data with a dependent terminal event

Recurrent event data are commonly observed in biomedical longitudinal studies. In many instances, there exists a terminal event, which precludes the occurrence of additional repeated events, and usually there is also a nonignorable correlation between the terminal event and recurrent events. In this article, we propose a partly Aalen's additive model with a multiplicative frailty for the rate function of recurrent event process and assume a Cox frailty model for terminal event time. A shared gamma frailty is used to describe the correlation between the two types of events. Consequently, this joint model can provide the information of temporal influence of absolute covariate effects on the rate of recurrent event process, which is usually helpful in the decision‐making process for physicians. An estimating equation approach is developed to estimate marginal and association parameters in the joint model. The consistency of the proposed estimator is established. Simulation studies demonstrate that the proposed approach is appropriate for practical use. We apply the proposed method to a peritonitis cohort data set for illustration. Copyright © 2015 John Wiley & Sons, Ltd.     

Bayesian approach for flexible modeling of semicompeting risks data

Semicompeting risks data arise when two types of events, non‐terminal and terminal, are observed. When the terminal event occurs first, it censors the non‐terminal event, but not vice versa. To account for possible dependent censoring of the non‐terminal event by the terminal event and to improve prediction of the terminal event using the non‐terminal event information, it is crucial to model their association properly. Motivated by a breast cancer clinical trial data analysis, we extend the well‐known illness–death models to allow flexible random effects to capture heterogeneous association structures in the data. Our extension also represents a generalization of the popular shared frailty models that usually assume that the non‐terminal event does not affect the hazards of the terminal event beyond a frailty term. We propose a unified Bayesian modeling approach that can utilize existing software packages for both model fitting and individual‐specific event prediction. The approach is demonstrated via both simulation studies and a breast cancer data set analysis. Copyright © 2014 John Wiley & Sons, Ltd.     

Modelling survival data with a cured fraction using frailty models

Cure models have historically been utilized to analyse time-to-event data with a cured fraction. We consider the use of frailty models as an alternative approach to modelling such data. An attractive feature of the models is the allowance for heterogeneity in risk among those individuals experiencing the event of interest in addition to the incorporation of a cured component. Utilizing maximum likelihood techniques, we fit models to data concerning the recurrence of leukaemia among patients receiving autologous transplantation treatment. The analysis suggests that the gamma frailty mixture model and the compound Poisson improve on the fit of the leukaemia data as compared to the standard cure model.     

The trend‐renewal process: a useful model for medical recurrence data

Time‐to‐event data analysis has a long tradition in applied statistics. Many models have been developed for data where each subject or observation unit experiences at most one event during its life. In contrast, in some applications, the subjects may experience more than one event. Recurrent events appear in science, medicine, economy, and technology. Often the events are followed by a repair action in reliability or a treatment in life science. A model to deal with recurrent event times for incomplete repair of technical systems is the trend‐renewal process. It is composed of a trend and a renewal component. In the present paper, we use a Weibull process for both of these components. The model is extended to include a Cox type covariate term to account for observed heterogeneity. A further extension includes random effects to account for unobserved heterogeneity. We fit the suggested version of the trend‐renewal process to a data set of hospital readmission times of colon cancer patients to illustrate the method for application to clinical data. Copyright © 2012 John Wiley & Sons, Ltd.