Hierarchical likelihood inference on clustered competing risks data

The frailty model, an extension of the proportional hazards model, is often used to model clustered survival data. However, some extension of the ordinary frailty model is required when there exist competing risks within a cluster. Under competing risks, the underlying processes affecting the events of interest and competing events could be different but correlated. In this paper, the hierarchical likelihood method is proposed to infer the cause‐specific hazard frailty model for clustered competing risks data. The hierarchical likelihood incorporates fixed effects as well as random effects into an extended likelihood function, so that the method does not require intensive numerical methods to find the marginal distribution. Simulation studies are performed to assess the behavior of the estimators for the regression coefficients and the correlation structure among the bivariate frailty distribution for competing events. The proposed method is illustrated with a breast cancer dataset. 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.     

Meta‐analysis of pairs of survival curves under heterogeneity: A Poisson correlated gamma‐frailty approach

We address the problem of meta‐analysis of pairs of survival curves under heterogeneity. Starting point for the meta‐analysis is a set of studies, each comparing the same two treatments, containing information about multiple survival outcomes. Under heterogeneity, we model the number of events using an extension of the Poisson correlated gamma‐frailty model with serial within‐arm and positive between‐arm correlations. The parameters of the models are estimated following a two‐stage estimation procedure. In the first stage the underlying hazards and between‐study variance are estimated using the marginals, while a second stage is used to estimate both within‐arm and between‐arm correlations. The methodology is illustrated with an observational study on breast cancer. Copyright © 2009 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.     

Score test for age at onset genetic linkage analysis in selected sibling pairs

A new score statistic is derived, which uses information from registries (age‐specific incidences) and family studies (sib–sib marginal correlation) to weight affected sibling pairs according to their age at onset. Age at onset of sibling pairs is modelled by a gamma frailty model. From this model we derive a bivariate survival function, which depends on the marginal survival and on the marginal correlation. The score statistic for linkage is a classical nonparametric linkage (NPL) statistic where the identical by descent sharing is weighted by a particular function of the age at onset data. Since the statistic is based on survival models, it can also be applied to discordant and healthy sibling pairs. Simulation studies show that the proposed method is robust and more powerful than standard NPL methods. As illustration we apply the new score statistic to data from a breast cancer study. Copyright © 2009 John Wiley & Sons, Ltd.     

A general frailty model to accommodate individual heterogeneity in the acquisition of multiple infections: An application to bivariate current status data

The analysis of multivariate time-to-event (TTE) data can become complicated due to the presence of clustering, leading to dependence between multiple event times. For a long time, (conditional) frailty models and (marginal) copula models have been used to analyze clustered TTE data. In this article, we propose a general frailty model employing a copula function between the frailty terms to construct flexible (bivariate) frailty distributions with the application to current status data. The model has the advantage to impose a less restrictive correlation structure among latent frailty variables as compared to traditional frailty models. Specifically, our model uses a copula function to join the marginal distributions of the frailty vector. In this article, we considered different copula functions, and we relied on marginal gamma distributions due to their mathematical convenience. Based on a simulation study, our novel model outperformed the commonly used additive correlated gamma frailty model, especially in the case of a negative association between the frailties. At the end of the article, the new methodology is illustrated on real-life data applications entailing bivariate serological survey data.     

Dealing with competing risks in clinical trials: How to choose the primary efficacy analysis?

We investigate different primary efficacy analysis approaches for a 2‐armed randomized clinical trial when interest is focused on a time to event primary outcome that is subject to a competing risk. We extend the work of Friedlin and Korn (2005) by considering estimation as well as testing and by simulating the primary and competing events' times from both a cause‐specific hazards model as well as a joint subdistribution–cause‐specific hazards model. We show that the cumulative incidence function can provide useful prognostic information for a particular patient but is not advisable for the primary efficacy analysis. Instead, it is preferable to fit a Cox model for the primary event which treats the competing event as an independent censoring. This is reasonably robust for controlling type I error and treatment effect bias with respect to the true primary and competing events' cause‐specific hazards model, even when there is a shared, moderately prognostic, unobserved baseline frailty for the primary and competing events in that model. However, when it is plausible that a strongly prognostic frailty exists, combining the primary and competing events into a composite event should be considered. Finally, when there is an a priori interest in having both the primary and competing events in the primary analysis, we compare a bivariate approach for establishing overall treatment efficacy to the composite event approach. The ideas are illustrated by analyzing the Women's Health Initiative clinical trials sponsored by the National Heart, Lung, and Blood Institute.     

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.     

Bayesian joint modeling of longitudinal and spatial survival AIDS data

Joint analysis of longitudinal and survival data has received increasing attention in the recent years, especially for analyzing cancer and AIDS data. As both repeated measurements (longitudinal) and time‐to‐event (survival) outcomes are observed in an individual, a joint modeling is more appropriate because it takes into account the dependence between the two types of responses, which are often analyzed separately. We propose a Bayesian hierarchical model for jointly modeling longitudinal and survival data considering functional time and spatial frailty effects, respectively. That is, the proposed model deals with non‐linear longitudinal effects and spatial survival effects accounting for the unobserved heterogeneity among individuals living in the same region. This joint approach is applied to a cohort study of patients with HIV/AIDS in Brazil during the years 2002–2006. Our Bayesian joint model presents considerable improvements in the estimation of survival times of the Brazilian HIV/AIDS patients when compared with those obtained through a separate survival model and shows that the spatial risk of death is the same across the different Brazilian states. Copyright © 2016 John Wiley & Sons, Ltd.     

Generalized survival models for correlated time‐to‐event data

Our aim is to develop a rich and coherent framework for modeling correlated time‐to‐event data, including (1) survival regression models with different links and (2) flexible modeling for time‐dependent and nonlinear effects with rich postestimation. We extend the class of generalized survival models, which expresses a transformed survival in terms of a linear predictor, by incorporating a shared frailty or random effects for correlated survival data. The proposed approach can include parametric or penalized smooth functions for time, time‐dependent effects, nonlinear effects, and their interactions. The maximum (penalized) marginal likelihood method is used to estimate the regression coefficients and the variance for the frailty or random effects. The optimal smoothing parameters for the penalized marginal likelihood estimation can be automatically selected by a likelihood‐based cross‐validation criterion. For models with normal random effects, Gauss‐Hermite quadrature can be used to obtain the cluster‐level marginal likelihoods. The Akaike Information Criterion can be used to compare models and select the link function. We have implemented these methods in the R package rstpm2. Simulating for both small and larger clusters, we find that this approach performs well. Through 2 applications, we demonstrate (1) a comparison of proportional hazards and proportional odds models with random effects for clustered survival data and (2) the estimation of time‐varying effects on the log‐time scale, age‐varying effects for a specific treatment, and two‐dimensional splines for time and age.     

Assessing heterogeneity and correlation of paired failure times with the bivariate frailty model.

We consider bivariate survival times for heterogeneous populations, where heterogeneity induces deviations in an individual's risk of an event as well as associations between survival times. The heterogeneity is characterized by a bivariate frailty model. We measure the heterogeneity effects through deviations associated with hazard functions and an association function defined through the conditional hazard functions: the cross-ratio function proposed by Oakes. We show how the deviation and association measures are determined by the frailty distribution. A Gibbs sampling method is developed for Bayesian inferences on regression coefficients, frailty parameters and the heterogeneity measures. The method is applied to a mental health care data set.     

A frailty model approach for regression analysis of multivariate current status data

This paper discusses regression analysis of multivariate current status failure time data (The Statistical Analysis of Interval‐censoring Failure Time Data. Springer: New York, 2006), which occur quite often in, for example, tumorigenicity experiments and epidemiologic investigations of the natural history of a disease. For the problem, several marginal approaches have been proposed that model each failure time of interest individually (Biometrics 2000; 56 :940–943; Statist. Med. 2002; 21 :3715–3726). In this paper, we present a full likelihood approach based on the proportional hazards frailty model. For estimation, an Expectation Maximization (EM) algorithm is developed and simulation studies suggest that the presented approach performs well for practical situations. The approach is applied to a set of bivariate current status data arising from a tumorigenicity experiment. Copyright © 2009 John Wiley & Sons, Ltd.     

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.     

Heterogeneity in survival analysis

I discuss the impact of individual heterogeneity in survival analysis. It is well known that this phenomenon may distort what is observed. A general class of mixing (or frailty) distributions is applied, extending a model of Hougaard. The extension allows part of the population to be non-susceptible, and contains the traditional gamma distribution as a special case. I consider the mixing of both a constant and a Weibull individual rate, and also discuss the comparison of rates from two populations. A number of practical examples are mentioned. Finally, I analyse two data sets, the main one containing data from the Norwegian Cancer Registry on the survival of breast cancer patients. The statistical analysis is of necessity speculative, but may still provide some insight.     

Joint model for a diagnostic test without a gold standard in the presence of a dependent terminal event

Breast cancer patients after breast conservation therapy often develop ipsilateral breast tumor relapse (IBTR), whose classification (true local recurrence versus new ipsilateral primary tumor) is subject to error, and there is no available gold standard. Some patients may die because of breast cancer before IBTR develops. Because this terminal event may be related to the individual patient's unobserved disease status and time to IBTR, the terminal mechanism is non‐ignorable. This article presents a joint analysis framework to model the binomial regression with misclassified binary outcome and the correlated time to IBTR, subject to a dependent terminal event and in the absence of a gold standard. Shared random effects are used to link together two survival times. The proposed approach is evaluated by a simulation study and is applied to a breast cancer data set consisting of 4477 breast cancer patients. The proposed joint model can be conveniently fit using adaptive Gaussian quadrature tools implemented in SAS 9.3 (SAS Institute Inc., Cary, NC, USA) procedure NLMIXED . Copyright © 2014 John Wiley & Sons, Ltd.     

Multilevel mixed effects parametric survival models using adaptive Gauss–Hermite quadrature with application to recurrent events and individual participant data meta‐analysis

Multilevel mixed effects survival models are used in the analysis of clustered survival data, such as repeated events, multicenter clinical trials, and individual participant data (IPD) meta‐analyses, to investigate heterogeneity in baseline risk and covariate effects. In this paper, we extend parametric frailty models including the exponential, Weibull and Gompertz proportional hazards (PH) models and the log logistic, log normal, and generalized gamma accelerated failure time models to allow any number of normally distributed random effects. Furthermore, we extend the flexible parametric survival model of Royston and Parmar, modeled on the log‐cumulative hazard scale using restricted cubic splines, to include random effects while also allowing for non‐PH (time‐dependent effects). Maximum likelihood is used to estimate the models utilizing adaptive or nonadaptive Gauss–Hermite quadrature. The methods are evaluated through simulation studies representing clinically plausible scenarios of a multicenter trial and IPD meta‐analysis, showing good performance of the estimation method. The flexible parametric mixed effects model is illustrated using a dataset of patients with kidney disease and repeated times to infection and an IPD meta‐analysis of prognostic factor studies in patients with breast cancer. User‐friendly Stata software is provided to implement the methods. Copyright © 2014 John Wiley & Sons, Ltd.     

Accounting for competing risks in randomized controlled trials: a review and recommendations for improvement

In studies with survival or time‐to‐event outcomes, a competing risk is an event whose occurrence precludes the occurrence of the primary event of interest. Specialized statistical methods must be used to analyze survival data in the presence of competing risks. We conducted a review of randomized controlled trials with survival outcomes that were published in high‐impact general medical journals. Of 40 studies that we identified, 31 (77.5%) were potentially susceptible to competing risks. However, in the majority of these studies, the potential presence of competing risks was not accounted for in the statistical analyses that were described. Of the 31 studies potentially susceptible to competing risks, 24 (77.4%) reported the results of a Kaplan–Meier survival analysis, while only five (16.1%) reported using cumulative incidence functions to estimate the incidence of the outcome over time in the presence of competing risks. The former approach will tend to result in an overestimate of the incidence of the outcome over time, while the latter approach will result in unbiased estimation of the incidence of the primary outcome over time. We provide recommendations on the analysis and reporting of randomized controlled trials with survival outcomes in the presence of competing risks. © 2017 The Authors. Statistics in Medicine published by John Wiley & 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.     

Genetic analysis of durations: Correlated frailty model applied to survival of Danish twins

Population studies of changes in human morbidity and mortality require models which take into account the influence of genetic and environmental factors on life-related durations, such as age at onset of the disease or disability, age at death, etc. In this paper we show how a bivariate survival model based on the concept of correlated individual frailty can be used for the genetic analysis of durations. Six genetic models of frailty are considered and applied to Danish twin survival data. The results of statistical analysis allow us to conclude that at least 50% of variability in individual frailty is determined by environmental factors. The approach suggests a method of estimation of the lower bound for the biological limit of human longevity. Directions for further research are discussed. ©1995 Wiley-Liss, Inc.     

Cancer incidence for Swedish twins studied by means of bivariate frailty models

Cancer incidence rates for Swedish twins born between 1928 and 1965 and who both were alive at age 30 are studied by means of bivariate frailty models. Altogether, 7,280 fraternal (DZ) and 4,699 identical (MZ) twin pairs were followed up through December 31, 1995, for cancer status. The association between cancer incidence rates was statistically greater among the MZ than among the DZ pairs and stronger between women than between men; however, the magnitude of this association is relatively small and decreases over time. The relative decrease in dependency (association) is most easily detected using shared frailty models but may also be demonstrated, at least for women, using correlated frailty models. We also demonstrate that estimates of the correlation coefficient are similar when using any correlated frailty models derived from the power variance family but that these estimates disagree regarding the age at which the dependence is most important. The relative importance of dependence across age may sometimes be more interesting than the correlation coefficient itself. The latter may usually be estimated using alternative methods. Furthermore, when estimating correlation coefficients close to the boundary of the parameter space, simulation studies indicate that the correlated inverse Gaussian frailty model is more robust than the gamma frailty model.