The versatility of multi-state models for the analysis of longitudinal data with unobservable features

Multi-state models provide a convenient statistical framework for a wide variety of medical applications characterized by multiple events and longitudinal data. We illustrate this through four examples. The potential value of the incorporation of unobserved or partially observed states is highlighted. In addition, joint modelling of multiple processes is illustrated with application to potentially informative loss to follow-up, mis-measured or missclassified data and causal inference.

Bayesian gamma frailty models for survival data with semi-competing risks and treatment switching

Motivated from a colorectal cancer study, we propose a class of frailty semi-competing risks survival models to account for the dependence between disease progression time, survival time, and treatment switching. Properties of the proposed models are examined and an efficient Gibbs sampling algorithm using the collapsed Gibbs technique is developed. A Bayesian procedure for assessing the treatment effect is also proposed. The deviance information criterion (DIC) with an appropriate deviance function and Logarithm of the pseudomarginal likelihood (LPML) are constructed for model comparison. A simulation study is conducted to examine the empirical performance of DIC and LPML and as well as the posterior estimates. The proposed method is further applied to analyze data from a colorectal cancer study.

Gender differences in the effect of occupational endotoxin exposure on impaired lung function and death: the Shanghai Textile Worker Study

OBJECTIVE

Airborne endotoxin exposure has adverse and protective health effects. Studies show men have augmented acute inflammatory responses to endotoxin. In this longitudinal cohort study we investigated the effect of long-term exposure to endotoxin in cotton dust on health, and determined whether these effects differ by gender.

METHODS

In the Shanghai Textile Worker Study, 447 cotton and 472 control silk textile workers were followed from 1981 to 2011 with repeated measures of occupational endotoxin exposure, spirometry and health questionnaires. Impaired lung function was defined as a decline in forced expiratory volume in one second to less than the 5th centile of population predicted. Death was ascertained by death registries. We used Cox proportional hazards models to assess the effect of endotoxin exposure on the time to development of impaired lung function and death.

RESULTS

128 deaths and 164 diagnoses of impaired lung function were ascertained between 1981 and 2011. HRs for the composite end point of impaired lung function or death was 1.47 (95% CI 1.09 to 1.97) for cotton vs silk workers and 1.04 (95% CI 1.01 to 1.07) per 10 000 endotoxin units (EU)/m(3)-years increase in exposure. HRs for all-cause mortality was 1.36 (95% CI 0.93 to 1.99) for cotton vs silk workers and 1.04 (95% CI 0.99 to 1.08) per 10 000 EU/m(3)-years. The risk associated with occupational endotoxin exposure was elevated only in men.

CONCLUSIONS

Occupational endotoxin exposure is associated with an increase in the risk of impaired lung function and all-cause mortality in men.

Estimating parsimonious models of longitudinal causal effects using regressions on propensity scores

Parsimony is important for the interpretation of causal effect estimates of longitudinal treatments on subsequent outcomes. One method for parsimonious estimates fits marginal structural models by using inverse propensity scores as weights. This method leads to generally large variability that is uncommon in more likelihood-based approaches. A more recent method fits these models by using simulations from a fitted g-computation, but requires the modeling of high-dimensional longitudinal relations that are highly susceptible to misspecification. We propose a new method that, first, uses longitudinal propensity scores as regressors to reduce the dimension of the problem and then uses the approximate likelihood for the first estimates to fit parsimonious models. We demonstrate the methods by estimating the effect of anticoagulant therapy on survival for cancer and non-cancer patients who have inferior vena cava filters.

Landmark risk prediction of residual life for breast cancer survival

The importance of developing personalized risk prediction estimates has become increasingly evident in recent years. In general, patient populations may be heterogenous and represent a mixture of different unknown subtypes of disease. When the source of this heterogeneity and resulting subtypes of disease are unknown, accurate prediction of survival may be difficult. However, in certain disease settings, the onset time of an observable short-term event may be highly associated with these unknown subtypes of disease and thus may be useful in predicting long-term survival. One approach to incorporate short-term event information along with baseline markers for the prediction of long-term survival is through a landmark Cox model, which assumes a proportional hazards model for the residual life at a given landmark point. In this paper, we use this modeling framework to develop procedures to assess how a patient's long-term survival trajectory may change over time given good short-term outcome indications along with prognosis on the basis of baseline markers. We first propose time-varying accuracy measures to quantify the predictive performance of landmark prediction rules for residual life and provide resampling-based procedures to make inference about such accuracy measures. Simulation studies show that the proposed procedures perform well in finite samples. Throughout, we illustrate our proposed procedures by using a breast cancer dataset with information on time to metastasis and time to death. In addition to baseline clinical markers available for each patient, a chromosome instability genetic score, denoted by CIN25, is also available for each patient and has been shown to be predictive of survival for various types of cancer. We provide procedures to evaluate the incremental value of CIN25 for the prediction of residual life and examine how the residual life profile changes over time. This allows us to identify an informative landmark point, t(0) , such that accurate risk predictions of the residual life could be made for patients who survive past t(0) without metastasis.

Landmark Prediction of Long Term Survival Incorporating Short Term Event Time Information

In recent years, a wide range of markers have become available as potential tools to predict risk or progression of disease. In addition to such biological and genetic markers, short term outcome information may be useful in predicting long term disease outcomes. When such information is available, it would be desirable to combine this along with predictive markers to improve the prediction of long term survival. Most existing methods for incorporating censored short term event information in predicting long term survival focus on modeling the disease process and are derived under restrictive parametric models in a multi-state survival setting. When such model assumptions fail to hold, the resulting prediction of long term outcomes may be invalid or inaccurate. When there is only a single discrete baseline covariate, a fully non-parametric estimation procedure to incorporate short term event time information has been previously proposed. However, such an approach is not feasible for settings with one or more continuous covariates due to the curse of dimensionality. In this paper, we propose to incorporate short term event time information along with multiple covariates collected up to a landmark point via a flexible varying-coefficient model. To evaluate and compare the prediction performance of the resulting landmark prediction rule, we use robust non-parametric procedures which do not require the correct specification of the proposed varying coefficient model. Simulation studies suggest that the proposed procedures perform well in finite samples. We illustrate them here using a dataset of post-dialysis patients with end-stage renal disease.

A causal framework for surrogate endpoints with semi-competing risks data

In this note, we address the problem of surrogacy using a causal modelling framework that differs substantially from the potential outcomes model that pervades the biostatistical literature. The framework comes from econometrics and conceptualizes direct effects of the surrogate endpoint on the true endpoint. While this framework can incorporate the so-called semi-competing risks data structure, we also derive a fundamental non-identifiability result. Relationships to existing causal modelling frameworks are also discussed.

Analysis of multi-stage treatments for recurrent diseases

Patients with a non-curable disease such as many types of cancer usually go through the process of initial treatment, a various number of disease recurrences and salvage treatments, and eventually death. The analysis of the effects of initial and salvage treatments on overall survival is not trivial. One may try to use disease recurrences and salvage treatments as time-dependent covariates in a Cox proportional hazards model. However, because disease recurrence is an intermediate outcome between initial treatment and final survival, the interpretation of such an estimation result is awkward. It does not estimate the causal effects of treatments on overall survival. Nevertheless, such causal effect estimates are critical for treatment decision making. Our approach to address this issue is that, at any treatment stage, for each patient, we compute a potential survival time if he or she would receive the optimal subsequent treatments, and use this potential survival time to do comparison between current-stage treatment groups. This potential survival time is assumed to follow an accelerated failure time model at each treatment stage and calculated by backward induction, starting from the last stage of treatment. By doing that, the effects on survival of different treatments at each stage can be consistently estimated and fairly compared. Under suitable conditions, these estimated effects have a causal interpretation. We evaluated the proposed model and estimation method by simulation studies and illustrated using the motivating, real data set that describes initial and salvage treatments for patients with soft tissue sarcoma.

Joint modeling of progression-free survival and overall survival by a Bayesian normal induced copula estimation model

In cancer clinical trials, in addition to time to death (i.e., overall survival), progression-related measurements such as progression-free survival and time to progression are also commonly used to evaluate treatment efficacy. It is of scientific interest and importance to understand the correlations among these measurements. In this paper, we propose a Bayesian semi-competing risks approach to jointly model progression-related measurements and overall survival. This new model is referred to as the NICE model, which stands for the normal induced copula estimation model. Correlation among these variables can be directly derived from the joint model. In addition, when correlation exists, simulation shows that the joint model is able to borrow strength from correlated measurements, and as a consequence the NICE model improves inference on both variables. The proposed model is in a Bayesian framework that enables us to use it in various Bayesian contexts, such as Bayesian adaptive design and using posterior predictive samples to simulate future trials. We conducted simulation studies to demonstrate properties of the NICE model and applied this method to a data set from chemotherapy-naive patients with extensive-stage small-cell lung cancer.

Regression analysis based on conditional likelihood approach under semi-competing risks data

Medical studies often involve semi-competing risks data, which consist of two types of events, namely terminal event and non-terminal event. Because the non-terminal event may be dependently censored by the terminal event, it is not possible to make inference on the non-terminal event without extra assumptions. Therefore, this study assumes that the dependence structure on the non-terminal event and the terminal event follows a copula model, and lets the marginal regression models of the non-terminal event and the terminal event both follow time-varying effect models. This study uses a conditional likelihood approach to estimate the time-varying coefficient of the non-terminal event, and proves the large sample properties of the proposed estimator. Simulation studies show that the proposed estimator performs well. This study also uses the proposed method to analyze AIDS Clinical Trial Group (ACTG 320).

Investigating hospital heterogeneity with a multi-state frailty model: application to nosocomial pneumonia disease in intensive care units

Background

Multistate models have become increasingly useful to study the evolution of a patient’s state over time in intensive care units ICU (e.g. admission, infections, alive discharge or death in ICU). In addition, in critically-ill patients, data come from different ICUs, and because observations are clustered into groups (or units), the observed outcomes cannot be considered as independent. Thus a flexible multi-state model with random effects is needed to obtain valid outcome estimates.

Methods

We show how a simple multi-state frailty model can be used to study semi-competing risks while fully taking into account the clustering (in ICU) of the data and the longitudinal aspects of the data, including left truncation and right censoring. We suggest the use of independent frailty models or joint frailty models for the analysis of transition intensities. Two distinct models which differ in the definition of time t in the transition functions have been studied: semi-Markov models where the transitions depend on the waiting times and nonhomogenous Markov models where the transitions depend on the time since inclusion in the study. The parameters in the proposed multi-state model may conveniently be computed using a semi-parametric or parametric approach with an existing R package FrailtyPack for frailty models. The likelihood cross-validation criterion is proposed to guide the choice of a better fitting model.

Results

We illustrate the use of our approach though the analysis of nosocomial infections (ventilator-associated pneumonia infections: VAP) in ICU, with “alive discharge” and “death” in ICU as other endpoints. We show that the analysis of dependent survival data using a multi-state model without frailty terms may underestimate the variance of regression coefficients specific to each group, leading to incorrect inferences. Some factors are wrongly significantly associated based on the model without frailty terms. This result is confirmed by a short simulation study. We also present individual predictions of VAP underlining the usefulness of dynamic prognostic tools that can take into account the clustering of observations.

Conclusions

The use of multistate frailty models allows the analysis of very complex data. Such models could help improve the estimation of the impact of proposed prognostic features on each transition in a multi-centre study. We suggest a method and software that give accurate estimates and enables inference for any parameter or predictive quantity of interest.

Nonparametric multistate representations of survival and longitudinal data with measurement error

This paper proposes a nonparametric procedure to describe the progression of longitudinal cohorts over time from a population averaged perspective, leading to multistate probability curves with the states defined jointly by survival and longitudinal outcomes measured with error. To account for the challenges of informative dropout and nonlinear shapes of the longitudinal trajectories, we apply a bias corrected penalized spline regression to estimate the unobserved longitudinal trajectory for each subject. We then estimate the multistate probability curves on the basis of the survival data and the estimated longitudinal trajectories. We further use simulation-extrapolation method to reduce the estimation bias caused by the randomness of the estimated trajectories. We develop a bootstrap test to compare multistate probability curves between groups. We present theoretical justification of the estimation procedure along with a simulation study to demonstrate finite sample performance. We illustrate the procedure by data from the African American Study of Kidney Disease and Hypertension, and it can be widely applied in longitudinal studies.

Meta-analysis for surrogacy: accelerated failure time models and semicompeting risks modeling

There has been great recent interest in the medical and statistical literature in the assessment and validation of surrogate endpoints as proxies for clinical endpoints in medical studies. More recently, authors have focused on using metaanalytical methods for quantification of surrogacy. In this article, we extend existing procedures for analysis based on the accelerated failure time model to this setting. An advantage of this approach relative to proportional hazards model is that it allows for analysis in the semicompeting risks setting, where we model the region where the surrogate endpoint occurs before the true endpoint. Several estimation methods and attendant inferential procedures are presented. In addition, between- and within-trial methods for evaluating surrogacy are developed; a novel principal components procedure is developed for quantifying trial-level surrogacy. The methods are illustrated by application to data from several studies in colorectal cancer.

Cumulative Incidence Association Models for Bivariate Competing Risks Data

Association models, like frailty and copula models, are frequently used to analyze clustered survival data and evaluate within-cluster associations. The assumption of noninformative censoring is commonly applied to these models, though it may not be true in many situations. In this paper, we consider bivariate competing risk data and focus on association models specified for the bivariate cumulative incidence function (CIF), a nonparametrically identifiable quantity. Copula models are proposed which relate the bivariate CIF to its corresponding univariate CIFs, similarly to independently right censored data, and accommodate frailty models for the bivariate CIF. Two estimating equations are developed to estimate the association parameter, permitting the univariate CIFs to be estimated either parametrically or nonparametrically. Goodness-of-fit tests are presented for formally evaluating the parametric models. Both estimators perform well with moderate sample sizes in simulation studies. The practical use of the methodology is illustrated in an analysis of dementia associations.

Maximum likelihood analysis of semicompeting risks data with semiparametric regression models

The "semicompeting risks" include a terminal event and a non-terminal event. The terminal event may censor the non-terminal event but not vice versa. Because times to the two events are usually correlated, the non-terminal event is subject to dependent/informative censoring by the terminal event. We seek to conduct marginal regressions and joint association analyses for the two event times under semicompeting risks. The proposed method is based on the modeling setup where the semiparametric transformation models are assumed for marginal regressions, and a copula model is assumed for the joint distribution. We propose a nonparametric maximum likelihood approach for inferences, which provides a martingale representation for the score function and an analytical expression for the information matrix. Direct theoretical developments and computational implementation are allowed for the proposed approach. Simulations and a real data application demonstrate the utility of the proposed methodology.

Estimating progression-free survival in paediatric brain tumour patients when some progression statuses are unknown

In oncology, progression-free survival time, which is defined as the minimum of the times to disease progression or death, often is used to characterize treatment and covariate effects. We are motivated by the desire to estimate the progression time distribution on the basis of data from 780 paediatric patients with choroid plexus tumours, which are a rare brain cancer where disease progression always precedes death. In retrospective data on 674 patients, the times to death or censoring were recorded but progression times were missing. In a prospective study of 106 patients, both times were recorded but there were only 20 non-censored progression times and 10 non-censored survival times. Consequently, estimating the progression time distribution is complicated by the problems that, for most of the patients, either the survival time is known but the progression time is not known, or the survival time is right censored and it is not known whether the patient's disease progressed before censoring. For data with these missingness structures, we formulate a family of Bayesian parametric likelihoods and present methods for estimating the progression time distribution. The underlying idea is that estimating the association between the time to progression and subsequent survival time from patients having complete data provides a basis for utilizing covariates and partial event time data of other patients to infer their missing progression times. We illustrate the methodology by analysing the brain tumour data, and we also present a simulation study.

Interpretability and importance of functionals in competing risks and multistate models

The basic parameters in both survival analysis and more general multistate models, including the competing risks model and the illness-death model, are the transition hazards. It is often necessary to supplement the analysis of such models with other model parameters, which are all functionals of the transition hazards. Unfortunately, not all such functionals are equally meaningful in practical contexts, even though they may be mathematically well defined. We have found it useful to check whether the functionals satisfy three simple principles, which may be used as criteria for practical interpretability.

Frailty-based competing risks model for multivariate survival data

In this work, we provide a new class of frailty-based competing risks models for clustered failure times data. This class is based on expanding the competing risks model of Prentice et al. (1978, Biometrics 34, 541-554) to incorporate frailty variates, with the use of cause-specific proportional hazards frailty models for all the causes. Parametric and nonparametric maximum likelihood estimators are proposed. The main advantages of the proposed class of models, in contrast to the existing models, are: (1) the inclusion of covariates; (2) the flexible structure of the dependency among the various types of failure times within a cluster; and (3) the unspecified within-subject dependency structure. The proposed estimation procedures produce the most efficient parametric and semiparametric estimators and are easy to implement. Simulation studies show that the proposed methods perform very well in practical situations.

Another view on the analysis of cardiovascular morbidity/mortality trials

In many morbidity/mortality studies, composite endpoints are considered. Although the primary interest is to demonstrate that an invention delays death, the expected death rate is often that low that studies focussing on survival exclusively are not feasible. Components of the composite endpoint are chosen such that their occurrence is predictive for time to death. Therefore, if the time to non-fatal events is censored by death, censoring is no longer independent. As a consequence, the analysis of the components of a composite endpoint cannot be reasonably performed using classical methods for the analysis of survival times like Kaplan-Meier estimates or log-rank tests. In this paper we visualize the impact of disregarding dependent censoring during the analysis and discuss practicable alternatives for the analysis of morbidity/mortality studies. In the context of simulations we provide evidence that copula-based methods have the potential to deliver practically unbiased estimates of hazards of components of a composite endpoint. At the same time, they require minimal assumptions, which is important since not all assumptions are generally verifiable because of censoring. Therefore, there are alternative ways to analyze morbidity/mortality studies more appropriately by accounting for the dependencies among the components of composite endpoints. Despite the limitations mentioned, these alternatives can at minimum serve as sensitivity analyses to check the robustness of the currently used methods.

Joint modeling of progression-free survival and death in advanced cancer clinical trials

Progression-related endpoints (such as time to progression or progression-free survival) and time to death are common endpoints in cancer clinical trials. It is of interest to study the link between progression-related endpoints and time to death (e.g. to evaluate the degree of surrogacy). However, current methods ignore some aspects of the definitions of progression-related endpoints. We review those definitions and investigate their impact on modeling the joint distribution. Further, we propose a multi-state model in which the association between the endpoints is modeled through a frailty term. We also argue that interval-censoring needs to be taken into account to more closely match the latent disease evolution. The joint distribution and an expression for Kendall's tau are derived. The model is applied to data from a clinical trial in advanced metastatic ovarian cancer.

Incorporating short-term outcome information to predict long-term survival with discrete markers

In disease screening and prognosis studies, an important task is to determine useful markers for identifying high-risk subgroups. Once such markers are established, they can be incorporated into public health practice to provide appropriate strategies for treatment or disease monitoring based on each individual's predicted risk. In the recent years, genetic and biological markers have been examined extensively for their potential to signal progression or risk of disease. In addition to these markers, it has often been argued that short-term outcomes may be helpful in making a better prediction of disease outcomes in clinical practice. In this paper we propose model-free non-parametric procedures to incorporate short-term event information to improve the prediction of a long-term terminal event. We include the optional availability of a single discrete marker measurement and assess the additional information gained by including the short-term outcome. We focus on the semi-competing risk setting where the short-term event is an intermediate event that may be censored by the terminal event while the terminal event is only subject to administrative censoring. Simulation studies suggest that the proposed procedures perform well in finite samples. Our procedures are illustrated using a data set of post-dialysis patients with end-stage renal disease.

A semi-competing risks model for data with interval-censoring and informative observation: an application to the MRC cognitive function and ageing study

Semi-competing risks data occur frequently in medical research when interest is in simultaneous modelling of two or more processes, one of which may censor the others. We consider the analysis of semi-competing risks data in the presence of interval-censoring and informative loss-to-followup. The work is motivated by a data set from the MRC UK Cognitive Function and Ageing Study, which we use to model two processes, cognitive impairment and death. Analysis is carried out using a multi-state model, which is an extension of that used by Siannis et al. (Statist. Med. 2007; 26:426–442) to model semi-competing risks data with exact transition times, to data which is interval-censored. Model parameters are estimated using maximum likelihood. The role of a sensitivity parameter k, which influences the nature of informative censoring, is explored.

Quantile regression for left-truncated semicompeting risks data

Semicompeting risks is often encountered in biomedical studies where a terminating event censors a nonterminating event but not vice versa. In practice, left truncation on the terminating event may arise and can greatly complicate the regression analysis on the nonterminating event. In this work, we propose a quantile regression method for left-truncated semicompeting risks data, which provides meaningful interpretations as well as the flexibility to accommodate varying covariate effects. We develop estimation and inference procedures that can be easily implemented by existing statistical software. Asymptotic properties of the resulting estimators are established including uniform consistency and weak convergence. The finite-sample performance of the proposed method is evaluated via simulation studies. An application to a registry dataset provides an illustration of our proposals.

Assessing agreement in the timing of treatment initiation determined by repeated measurements of novel versus gold standard technologies with application to the monitoring of CD4 counts in HIV-infected patients

Repeated biomarker measurements are often taken over time to help assess risk of disease progression and guide clinical decision-making, such as whether to start treatment. Unfortunately, gold standard methodologies for measuring biomarkers are often prohibitively expensive or unavailable in resource-limited settings. For example, the costs of monitoring HIV-infected subjects to decide when to start or change treatments are a significant burden for many countries, often exceeding the costs of treatments. A major issue concerns how to evaluate changes in timing of key clinical decisions if a new, simpler or less expensive technology were used instead of the gold standard. We develop a framework for addressing this problem and apply it to the case of monitoring CD4 counts in HIV-infected patients. We focus on the practically important situation in which longitudinal natural history data are available for the gold standard (flow cytometry for CD4 counts), but where the first data expected for a new technology will come from a cross-sectional method comparison study, allowing for estimation of variability and systematic differences (bias) between the two technologies. In a case study, we illustrate how a combination of statistical modeling and simulation study might be used to evaluate the potential impact of using a new technology on treatment starting times in a population of HIV-infected subjects. This gives developers of new CD4 measurement technologies insight into what might constitute acceptable increases in variability and/or bias for novel methods. We finish with a discussion of our findings and some statistical problems that need further work.

On assessing surrogacy in a single trial setting using a semicompeting risks paradigm

SUMMARY

There has been a recent emphasis on the identification of biomarkers and other biologic measures that may be potentially used as surrogate endpoints in clinical trials. We focus on the setting of data from a single clinical trial. In this article, we consider a framework in which the surrogate must occur before the true endpoint. This suggests viewing the surrogate and true endpoints as semicompeting risks data; this approach is new to the literature on surrogate endpoints and leads to an asymmetrical treatment of the surrogate and true endpoints. However, such a data structure also conceptually complicates many of the previously considered measures of surrogacy in the literature. We propose novel estimation and inferential procedures for the relative effect and adjusted association quantities proposed by Buyse and Molenberghs (1998, Biometrics 54, 1014-1029). The proposed methodology is illustrated with application to simulated data, as well as to data from a leukemia study.

Multiple diseases in carrier probability estimation: accounting for surviving all cancers other than breast and ovary in BRCAPRO

Mendelian models can predict who carries an inherited deleterious mutation of known disease genes based on family history. For example, the BRCAPRO model is commonly used to identify families who carry mutations of BRCA1 and BRCA2, based on familial breast and ovarian cancers. These models incorporate the age of diagnosis of diseases in relatives and current age or age of death. We develop a rigorous foundation for handling multiple diseases with censoring. We prove that any disease unrelated to mutations can be excluded from the model, unless it is sufficiently common and dependent on a mutation-related disease time. Furthermore, if a family member has a disease with higher probability density among mutation carriers, but the model does not account for it, then the carrier probability is deflated. However, even if a family only has diseases the model accounts for, if the model excludes a mutation-related disease, then the carrier probability will be inflated. In light of these results, we extend BRCAPRO to account for surviving all non-breast/ovary cancers as a single outcome. The extension also enables BRCAPRO to extract more useful information from male relatives. Using 1500 families from the Cancer Genetics Network, accounting for surviving other cancers improves BRCAPRO's concordance index from 0.758 to 0.762 (p=0.046), improves its positive predictive value from 35 to 39 per cent (p<10(-6)) without impacting its negative predictive value, and improves its overall calibration, although calibration slightly worsens for those with carrier probability<10 per cent.

A joint frailty model for survival and gap times between recurrent events

Therapy for patients with a recurrent disease focuses on delaying disease recurrence and prolonging survival. A common analysis approach for such data is to estimate the distribution of disease-free survival, that is, the time to the first disease recurrence or death, whichever happens first. However, treating death similarly as disease recurrence may give misleading results. Also considering only the first recurrence and ignoring subsequent ones can result in loss of statistical power. We use a joint frailty model to simultaneously analyze disease recurrences and survival. Separate parameters for disease recurrence and survival are used in the joint model to distinguish treatment effects on these two types of events. The correlation between disease recurrences and survival is taken into account by a shared frailty. The effect of disease recurrence on survival can also be estimated by this model. The EM algorithm is used to fit the model, with Markov chain Monte Carlo simulations in the E-steps. The method is evaluated by simulation studies and illustrated through a study of patients with heart failure. Sensitivity analysis for the parametric assumption of the frailty distribution is assessed by simulations.

Semiparametric Inference for Surrogate Endpoints with Bivariate Censored Data

Considerable attention has been recently paid to the use of surrogate endpoints in clinical research. We deal with the situation where the two endpoints are both right censored. While proportional hazards analyses are typically used for this setting, their use leads to several complications. In this article, we propose the use of the accelerated failure time model for analysis of surrogate endpoints. Based on the model, we then describe estimation and inference procedures for several measures of surrogacy. A complication is that potentially both the independent and dependent variable are subject to censoring. We adapt the Theil-Sen estimator to this problem, develop the associated asymptotic results, and propose a novel resampling-based technique for calculating the variances of the proposed estimators. The finite-sample properties of the estimation methodology are assessed using simulation studies, and the proposed procedures are applied to data from an acute myelogenous leukemia clinical trial.

Estimating survival and association in a semicompeting risks model

In many follow-up studies, patients are subject to concurrent events. In this article, we consider semicompeting risks data as defined by Fine, Jiang, and Chappell (2001, Biometrika 88, 907-919) where one event is censored by the other but not vice versa. The proposed model involves marginal survival functions for the two events and a parametric family of copulas for their dependency. This article suggests a general method for estimating the dependence parameter when the dependency is modeled with an Archimedean copula. It uses the copula-graphic estimator of Zheng and Klein (1995, Biometrika 82, 127-138) for estimating the survival function of the nonterminal event, subject to dependent censoring. Asymptotic properties of these estimators are derived. Simulations show that the new methods work well with finite samples. The copula-graphic estimator is shown to be more accurate than the estimator proposed by Fine et al. (2001); its performances are similar to those of the self-consistent estimator of Jiang, Fine, Kosorok, and Chappell (2005, Scandinavian Journal of Statistics 33, 1-20). The analysis of a data set, emphasizing the estimation of characteristics of the observable region, is presented as an illustration.

A Random Time Interval Approach for Analysing the Impact of a Possible Intermediate Event on a Terminal Event

We consider the impact of a possible intermediate event on a terminal event in an illness-death model with states 'initial', 'intermediate' and 'terminal'. One aim is to unambiguously describe the occurrence of the intermediate event in terms of the observable data, the problem being that the intermediate event may not occur. We propose to consider a random time interval, whose length is the time spent in the intermediate state. We derive an estimator of the joint distribution of the left and right limit of the random time interval from the Aalen-Johansen estimator of the matrix of transition probabilities and study its asymptotic properties. We apply our approach to hospital infection data. Estimating the distribution of the random time interval will usually be only a first step of an analysis. We illustrate this by analysing change in length of hospital stay following an infection and derive the large sample properties of the respective estimator.

A missing data approach to semi-competing risks problems

For event time data involving multiple mutually exclusive competing causes of failure, classic competing risks results show that marginal survival distributions are not identifiable. In a related instance, one or more failure modes may be observed provided that the failure events occur in a specific order. In such situations, sometimes referred to as semi-competing risks problems, the observations may under realistic assumptions lend information about parameters of interest that would be nonidentifiable in the strict competing risks case. Here, we present an approach that makes use of partially observable multiple modes of failures to obtain an estimate of the marginal distribution of one event type that may occur prior to the occurrence of another event type or be precluded by it. We apply the proposed method to the problem of estimating the distribution of time to tumour recurrence at specific sites among breast cancer patients participating in randomized clinical trials.

A multi-state model for joint modelling of terminal and non-terminal events with application to Whitehall II

Serious coronary heart disease (CHD) is a primary outcome in the Whitehall II study, a large epidemiological study of British civil servants. Both fatal (F) and non-fatal (NF) CHD events are of interest and while essentially complete information is available on F events, the observation of NF events is subject to potentially informative censoring. A multi-state model with an unobserved state is introduced for the joint modelling of F and NF events. Two model-based assumptions ensure identifiability of the model and a parameter is introduced to allow sensitivity analyses concerning the assumption linked to informative censoring. Weibull transition rates, which include dependence on explanatory variables, are used in the analysis of Whitehall II data with a particular focus on the relationship between civil service grade and CHD events.

A computationally simple bivariate survival estimator for efficacy and safety

Both treatment efficacy and safety are typically the primary endpoints in Phase II, and even in some Phase III, clinical trials. Efficacy is frequently measured by time to response, death, or some other milestone event and thus is a continuous, possibly censored, outcome. Safety, however, is frequently measured on a discrete scale; in Eastern Cooperative Oncology Group clinical trial E2290, it was measured as the number of weekly rounds of chemotherapy that were tolerable to colorectal cancer patients. For the joint analysis of efficacy and safety, we propose a non-parametric, computationally simple estimator for the bivariate survival function when one time-to-event is continuous, one is discrete, and both are subject to right-censoring. The bivariate censoring times may depend on each other, but they are assumed to be independent of both event times. We derive a closed-form covariance estimator for the survivor function which allows for inference to be based on any of several possible statistics of interest. In addition, we derive its covariance with respect to calendar time of analysis, allowing for its use in sequential studies.

Regression Modeling of Semicompeting Risks Data

Semicompeting risks data are often encountered in clinical trials with intermediate endpoints subject to dependent censoring from informative dropout. Unlike with competing risks data, dropout may not be dependently censored by the intermediate event. There has recently been increased attention to these data, in particular inferences about the marginal distribution of the intermediate event without covariates. In this article, we incorporate covariates and formulate their effects on the survival function of the intermediate event via a functional regression model. To accommodate informative censoring, a time-dependent copula model is proposed in the observable region of the data which is more flexible than standard parametric copula models for the dependence between the events. The model permits estimation of the marginal distribution under weaker assumptions than in previous work on competing risks data. New nonparametric estimators for the marginal and dependence models are derived from nonlinear estimating equations and are shown to be uniformly consistent and to converge weakly to Gaussian processes. Graphical model checking techniques are presented for the assumed models. Nonparametric tests are developed accordingly, as are inferences for parametric submodels for the time-varying covariate effects and copula parameters. A novel time-varying sensitivity analysis is developed using the estimation procedures. Simulations and an AIDS data analysis demonstrate the practical utility of the methodology.

Semiparametric inferences for association with semi-competing risks data

In many biomedical studies, it is of interest to assess dependence between bivariate failure time data. We focus here on a special type of such data, referred to as semi-competing risks data. In this article, we develop methods for making inferences regarding dependence of semi-competing risks data across strata of a discrete covariate Z. A class of rank statistics for testing constancy of association across strata are proposed; its asymptotic properties are also derived. We develop a novel re-sampling-based technique for calculating the variances of the proposed test statistics. In addition, we develop methods for combining test statistics for assessing marginal effects of Z on the dependent censoring variable as well as its effects on association. The finite-sample properties of the proposed methodology are assessed using simulation studies, and they are applied to data from a leukaemia transplantation study.