Cause‐Specific Cumulative Incidence Estimation and the Fine and Gray Model Under Both Left Truncation and Right Censoring

Summary The standard estimator for the cause‐specific cumulative incidence function in a competing risks setting with left truncated and/or right censored data can be written in two alternative forms. One is a weighted empirical cumulative distribution function and the other a product‐limit estimator. This equivalence suggests an alternative view of the analysis of time‐to‐event data with left truncation and right censoring: individuals who are still at risk or experienced an earlier competing event receive weights from the censoring and truncation mechanisms. As a consequence, inference on the cumulative scale can be performed using weighted versions of standard procedures. This holds for estimation of the cause‐specific cumulative incidence function as well as for estimation of the regression parameters in the Fine and Gray proportional subdistribution hazards model. We show that, with the appropriate filtration, a martingale property holds that allows deriving asymptotic results for the proportional subdistribution hazards model in the same way as for the standard Cox proportional hazards model. Estimation of the cause‐specific cumulative incidence function and regression on the subdistribution hazard can be performed using standard software for survival analysis if the software allows for inclusion of time‐dependent weights. We show the implementation in the R statistical package. The proportional subdistribution hazards model is used to investigate the effect of calendar period as a deterministic external time varying covariate, which can be seen as a special case of left truncation, on AIDS related and non‐AIDS related cumulative mortality.     

Semiparametric analysis of correlated recurrent and terminal events

In clinical and observational studies, recurrent event data (e.g., hospitalization) with a terminal event (e.g., death) are often encountered. In many instances, the terminal event is strongly correlated with the recurrent event process. In this article, we propose a semiparametric method to jointly model the recurrent and terminal event processes. The dependence is modeled by a shared gamma frailty that is included in both the recurrent event rate and terminal event hazard function. Marginal models are used to estimate the regression effects on the terminal and recurrent event processes, and a Poisson model is used to estimate the dispersion of the frailty variable. A sandwich estimator is used to achieve additional robustness. An analysis of hospitalization data for patients in the peritoneal dialysis study is presented to illustrate the proposed method.     

Marginal hazard ratio estimates in joint frailty models for heart failure trials

This work is motivated by clinical trials in chronic heart failure disease, where treatment has effects both on morbidity (assessed as recurrent non‐fatal hospitalisations) and on mortality (assessed as cardiovascular death, CV death). Recently, a joint frailty proportional hazards model has been proposed for these kind of efficacy outcomes to account for a potential association between the risk rates for hospital admissions and CV death. However, more often clinical trial results are presented by treatment effect estimates that have been derived from marginal proportional hazards models, that is, a Cox model for mortality and an Andersen–Gill model for recurrent hospitalisations. We show how these marginal hazard ratios and their estimates depend on the association between the risk processes, when these are actually linked by shared or dependent frailty terms. First we derive the marginal hazard ratios as a function of time. Then, applying least false parameter theory, we show that the marginal hazard ratio estimate for the hospitalisation rate depends on study duration and on parameters of the underlying joint frailty model. In particular, we identify parameters, for example the treatment effect on mortality, that determine if the marginal hazard ratio estimate for hospitalisations is smaller, equal or larger than the conditional one. How this affects rejection probabilities is further investigated in simulation studies. Our findings can be used to interpret marginal hazard ratio estimates in heart failure trials and are illustrated by the results of the CHARM‐Preserved trial (where CHARM is the ‘Candesartan in Heart failure Assessment of Reduction in Mortality and morbidity’ programme).     

Nonparametric and semiparametric estimation with sequentially truncated survival data

In observational cohort studies with complex sampling schemes, truncation arises when the time to event of interest is observed only when it falls below or exceeds another random time, that is, the truncation time. In more complex settings, observation may require a particular ordering of event times; we refer to this as sequential truncation. Estimators of the event time distribution have been developed for simple left-truncated or right-truncated data. However, these estimators may be inconsistent under sequential truncation. We propose nonparametric and semiparametric maximum likelihood estimators for the distribution of the event time of interest in the presence of sequential truncation, under two truncation models. We show the equivalence of an inverse probability weighted estimator and a product limit estimator under one of these models. We study the large sample properties of the proposed estimators and derive their asymptotic variance estimators. We evaluate the proposed methods through simulation studies and apply the methods to an Alzheimer's disease study. We have developed an R package, seqTrun , for implementation of our method.     

Regression models for infant mortality data in Norwegian siblings, using a compound Poisson frailty distribution with random scale

The power variance function distributions, which include the gamma and compound Poisson (CP) distributions among others, are commonly used in frailty models for family data. In a previous paper, we presented a frailty model constructed by randomizing the scale parameter in a CP distribution. When combined with a parametric baseline hazard, this yields a model with heterogeneity on both the individual and the family level and a subgroup with zero frailty, corresponding to people not experiencing the event. In this paper, we discuss covariates in the model. Depending on where the covariates are inserted in the model, one may have proportional hazards at the individual level, the family level, and a larger group level (for covariates shared by many families, e.g. ethnic groups) or get accelerated failure times. Each of these alternatives gives a specific interpretation of the covariate effects. An application to data infant mortality in siblings from the Medical Birth Registry of Norway is included. We compare the results for some of the different covariate modeling options.     

Analysis of multivariate recurrent event data with time‐dependent covariates and informative censoring

Multivariate recurrent event data are usually encountered in many clinical and longitudinal studies in which each study subject may experience multiple recurrent events. For the analysis of such data, most existing approaches have been proposed under the assumption that the censoring times are noninformative, which may not be true especially when the observation of recurrent events is terminated by a failure event. In this article, we consider regression analysis of multivariate recurrent event data with both time‐dependent and time‐independent covariates where the censoring times and the recurrent event process are allowed to be correlated via a frailty. The proposed joint model is flexible where both the distributions of censoring and frailty variables are left unspecified. We propose a pairwise pseudolikelihood approach and an estimating equation‐based approach for estimating coefficients of time‐dependent and time‐independent covariates, respectively. The large sample properties of the proposed estimates are established, while the finite‐sample properties are demonstrated by simulation studies. The proposed methods are applied to the analysis of a set of bivariate recurrent event data from a study of platelet transfusion reactions.     

A Positive Stable Frailty Model for Clustered Failure Time Data with Covariate‐Dependent Frailty

Summary In this article, we propose a positive stable shared frailty Cox model for clustered failure time data where the frailty distribution varies with cluster‐level covariates. The proposed model accounts for covariate‐dependent intracluster correlation and permits both conditional and marginal inferences. We obtain marginal inference directly from a marginal model, then use a stratified Cox‐type pseudo‐partial likelihood approach to estimate the regression coefficient for the frailty parameter. The proposed estimators are consistent and asymptotically normal and a consistent estimator of the covariance matrix is provided. Simulation studies show that the proposed estimation procedure is appropriate for practical use with a realistic number of clusters. Finally, we present an application of the proposed method to kidney transplantation data from the Scientific Registry of Transplant Recipients.     

A semiparametric mixture cure survival model for left‐truncated and right‐censored data

In follow‐up studies, the disease event time can be subject to left truncation and right censoring. Furthermore, medical advancements have made it possible for patients to be cured of certain types of diseases. In this article, we consider a semiparametric mixture cure model for the regression analysis of left‐truncated and right‐censored data. The model combines a logistic regression for the probability of event occurrence with the class of transformation models for the time of occurrence. We investigate two techniques for estimating model parameters. The first approach is based on martingale estimating equations (EEs). The second approach is based on the conditional likelihood function given truncation variables. The asymptotic properties of both proposed estimators are established. Simulation studies indicate that the conditional maximum‐likelihood estimator (cMLE) performs well while the estimator based on EEs is very unstable even though it is shown to be consistent. This is a special and intriguing phenomenon for the EE approach under cure model. We provide insights into this issue and find that the EE approach can be improved significantly by assigning appropriate weights to the censored observations in the EEs. This finding is useful in overcoming the instability of the EE approach in some more complicated situations, where the likelihood approach is not feasible. We illustrate the proposed estimation procedures by analyzing the age at onset of the occiput‐wall distance event for patients with ankylosing spondylitis.     

Modeling clustered, discrete, or grouped time survival data with covariates

We have developed methods for modeling discrete or grouped time, right-censored survival data collected from correlated groups or clusters. We assume that the marginal hazard of failure for individual items within a cluster is specified by a linear log odds survival model and the dependence structure is based on a gamma frailty model. The dependence can be modeled as a function of cluster-level covariates. Likelihood equations for estimating the model parameters are provided. Generalized estimating equations for the marginal hazard regression parameters and pseudolikelihood methods for estimating the dependence parameters are also described. Data from two clinical trials are used for illustration purposes.     

Nonparametric estimation of the survival function for ordered multivariate failure time data: A comparative study

In longitudinal studies of disease, patients may experience several events through a follow‐up period. In these studies, the sequentially ordered events are often of interest and lead to problems that have received much attention recently. Issues of interest include the estimation of bivariate survival, marginal distributions, and the conditional distribution of gap times. In this work, we consider the estimation of the survival function conditional to a previous event. Different nonparametric approaches will be considered for estimating these quantities, all based on the Kaplan–Meier estimator of the survival function. We explore the finite sample behavior of the estimators through simulations. The different methods proposed in this article are applied to a dataset from a German Breast Cancer Study. The methods are used to obtain predictors for the conditional survival probabilities as well as to study the influence of recurrence in overall survival.     

Joint models for multivariate longitudinal and multivariate survival data

Joint modeling of longitudinal and survival data is becoming increasingly essential in most cancer and AIDS clinical trials. We propose a likelihood approach to extend both longitudinal and survival components to be multidimensional. A multivariate mixed effects model is presented to explicitly capture two different sources of dependence among longitudinal measures over time as well as dependence between different variables. For the survival component of the joint model, we introduce a shared frailty, which is assumed to have a positive stable distribution, to induce correlation between failure times. The proposed marginal univariate survival model, which accommodates both zero and nonzero cure fractions for the time to event, is then applied to each marginal survival function. The proposed multivariate survival model has a proportional hazards structure for the population hazard, conditionally as well as marginally, when the baseline covariates are specified through a specific mechanism. In addition, the model is capable of dealing with survival functions with different cure rate structures. The methodology is specifically applied to the International Breast Cancer Study Group (IBCSG) trial to investigate the relationship between quality of life, disease-free survival, and overall survival.     

A class of Bayesian shared gamma frailty models with multivariate failure time data

For multivariate failure time data, we propose a new class of shared gamma frailty models by imposing the Box-Cox transformation on the hazard function, and the product of the baseline hazard and the frailty. This novel class of models allows for a very broad range of shapes and relationships between the hazard and baseline hazard functions. It includes the well-known Cox gamma frailty model and a new additive gamma frailty model as two special cases. Due to the nonnegative hazard constraint, this shared gamma frailty model is computationally challenging in the Bayesian paradigm. The joint priors are constructed through a conditional-marginal specification, in which the conditional distribution is univariate, and it absorbs the nonlinear parameter constraints. The marginal part of the prior specification is free of constraints. The prior distributions allow us to easily compute the full conditionals needed for Gibbs sampling, while incorporating the constraints. This class of shared gamma frailty models is illustrated with a real dataset.     

A Bayesian joint model of recurrent events and a terminal event

Recurrent events could be stopped by a terminal event, which commonly occurs in biomedical and clinical studies. In this situation, dependent censoring is encountered because of potential dependence between these two event processes, leading to invalid inference if analyzing recurrent events alone. The joint frailty model is one of the widely used approaches to jointly model these two processes by sharing the same frailty term. One important assumption is that recurrent and terminal event processes are conditionally independent given the subject‐level frailty; however, this could be violated when the dependency may also depend on time‐varying covariates across recurrences. Furthermore, marginal correlation between two event processes based on traditional frailty modeling has no closed form solution for estimation with vague interpretation. In order to fill these gaps, we propose a novel joint frailty‐copula approach to model recurrent events and a terminal event with relaxed assumptions. Metropolis–Hastings within the Gibbs Sampler algorithm is used for parameter estimation. Extensive simulation studies are conducted to evaluate the efficiency, robustness, and predictive performance of our proposal. The simulation results show that compared with the joint frailty model, the bias and mean squared error of the proposal is smaller when the conditional independence assumption is violated. Finally, we apply our method into a real example extracted from the MarketScan database to study the association between recurrent strokes and mortality.     

Estimation of multivariate frailty models using penalized partial likelihood

There exists a growing literature on the estimation of gamma distributed multiplicative shared frailty models. There is, however, often a need to model more complicated frailty structures, but attempts to extend gamma frailties run into complications. Motivated by hip replacement data with a more complicated dependence structure, we propose a model based on multiplicative frailties with a multivariate log-normal joint distribution. We give a justification and an estimation procedure for this generally structured frailty model, which is a generalization of the one presented by McGilchrist (1993, Biometrics 49, 221-225). The estimation is based on Laplace approximation of the likelihood function. This leads to estimating equations based on a penalized fixed effects partial likelihood, where the marginal distribution of the frailty terms determines the penalty term. The tuning parameters of the penalty function, i.e., the frailty variances, are estimated by maximizing an approximate profile likelihood. The performance of the approximation is evaluated by simulation, and the frailty model is fitted to the hip replacement data.     

Bayesian Semiparametric Frailty Selection in Multivariate Event Time Data

Biomedical studies often collect multivariate event time data from multiple clusters (either subjects or groups) within each of which event times for individuals are correlated and the correlation may vary in different classes. In such survival analyses, heterogeneity among clusters for shared and specific classes can be accommodated by incorporating parametric frailty terms into the model. In this article, we propose a Bayesian approach to relax the parametric distribution assumption for shared and specific‐class frailties by using a Dirichlet process prior while also allowing for the uncertainty of heterogeneity for different classes. Multiple cluster‐specific frailty selections rely on variable selection‐type mixture priors by applying mixtures of point masses at zero and inverse gamma distributions to the variance of log frailties. This selection allows frailties with zero variance to effectively drop out of the model. A reparameterization of log‐frailty terms is performed to reduce the potential bias of fixed effects due to variation of the random distribution and dependence among the parameters resulting in easy interpretation and faster Markov chain Monte Carlo convergence. Simulated data examples and an application to a lung cancer clinical trial are used for illustration.     

The heritability of mortality due to heart diseases: a correlated frailty model applied to Danish twins.

Data of the Danish Twin Registry on monozygotic and dizygotic twins are used to analyse genetic and environmental influences on susceptibility to heart diseases for males and females, respectively. The sample includes 7955 like-sexed twin pairs born between 1870 and 1930. Follow-up was from 1 January 1943 to 31 December 1993 which results in truncation (twin pairs were included in the study if both individuals were still alive at the beginning of the follow-up) and censoring (nearly 40% of the study population was still alive at the end of the follow-up). We use the correlated gamma-frailty model for the genetic analysis of frailty to account for this censoring and truncation. During the follow-up 9370 deaths occurred, 3393 deaths were due to heart diseases in general, including 2476 deaths due to coronary heart disease (CHD). Proportions of variance of frailty attributable to genetic and environmental factors were analyzed using the structural equation model approach. Different standard biometric models are fitted to the data to evaluate the magnitude and nature of genetic and environmental factors on mortality. Using the best fitting model heritability of frailty (liability to death) was found to be 0.55 (0.07) and 0.53 (0.11) with respect to heart diseases and CHD, respectively, for males and 0.52 (0.10) and 0.58 (0.14) for females in a parametric analysis. A semi-parametric analysis shows very similar results. These analyses may indicate the existence of a strong genetic influence on individual frailty associated with mortality caused by heart diseases and CHD in both, males and females. The nature of genetic influences on frailty with respect to heart diseases and CHD is probably additive. No evidence for dominance and shared environment was found.     

A Note on Variance Estimation of the Aalen–Johansen Estimator of the Cumulative Incidence Function in Competing Risks,with a View towards Left‐Truncated Data

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.     

Regression with frailty in survival analysis

In studies of survival, the hazard function for each individual may depend on observed risk variables but usually not all such variables are known or measurable. This unknown factor of the hazard function is usually termed the individual heterogeneity or frailty. When survival is time to the occurrence of a particular type of event and more than one such time may be obtained for each individual, frailty is a common factor among such recurrence times. A model including frailty is fitted to such repeated measures of recurrence times.     

Semiparametric Analysis for Recurrent Event Data with Time-Dependent Covariates and Informative Censoring

Summary .  Recurrent event data analyses are usually conducted under the assumption that the censoring time is independent of the recurrent event process. In many applications the censoring time can be informative about the underlying recurrent event process, especially in situations where a correlated failure event could potentially terminate the observation of recurrent events. In this article, we consider a semiparametric model of recurrent event data that allows correlations between censoring times and recurrent event process via frailty. This flexible framework incorporates both time-dependent and time-independent covariates in the formulation, while leaving the distributions of frailty and censoring times unspecified. We propose a novel semiparametric inference procedure that depends on neither the frailty nor the censoring time distribution. Large sample properties of the regression parameter estimates and the estimated baseline cumulative intensity functions are studied. Numerical studies demonstrate that the proposed methodology performs well for realistic sample sizes. An analysis of hospitalization data for patients in an AIDS cohort study is presented to illustrate the proposed method.     

Concordance indices with left-truncated and right-censored data

In the context of time-to-event analysis, a primary objective is to model the risk of experiencing a particular event in relation to a set of observed predictors. The Concordance Index (C-Index) is a statistic frequently used in practice to assess how well such models discriminate between various risk levels in a population. However, the properties of conventional C-Index estimators when applied to left-truncated time-to-event data have not been well studied, despite the fact that left-truncation is commonly encountered in observational studies. We show that the limiting values of the conventional C-Index estimators depend on the underlying distribution of truncation times, which is similar to the situation with right-censoring as discussed in Uno et al. (2011) [On the C-statistics for evaluating overall adequacy of risk prediction procedures with censored survival data. Statistics in Medicine 30(10), 1105–1117]. We develop a new C-Index estimator based on inverse probability weighting (IPW) that corrects for this limitation, and we generalize this estimator to settings with left-truncated and right-censored data. The proposed IPW estimators are highly robust to the underlying truncation distribution and often outperform the conventional methods in terms of bias, mean squared error, and coverage probability. We apply these estimators to evaluate a predictive survival model for mortality among patients with end-stage renal disease.