Flexible parametric modelling of cause-specific hazards to estimate cumulative incidence functions

BACKGROUND

Competing risks are a common occurrence in survival analysis. They arise when a patient is at risk of more than one mutually exclusive event, such as death from different causes, and the occurrence of one of these may prevent any other event from ever happening.

METHODS

There are two main approaches to modelling competing risks: the first is to model the cause-specific hazards and transform these to the cumulative incidence function; the second is to model directly on a transformation of the cumulative incidence function. We focus on the first approach in this paper. This paper advocates the use of the flexible parametric survival model in this competing risk framework.

RESULTS

An illustrative example on the survival of breast cancer patients has shown that the flexible parametric proportional hazards model has almost perfect agreement with the Cox proportional hazards model. However, the large epidemiological data set used here shows clear evidence of non-proportional hazards. The flexible parametric model is able to adequately account for these through the incorporation of time-dependent effects.

CONCLUSION

A key advantage of using this approach is that smooth estimates of both the cause-specific hazard rates and the cumulative incidence functions can be obtained. It is also relatively easy to incorporate time-dependent effects which are commonly seen in epidemiological studies.

Proportional hazards model for competing risks data with missing cause of failure

We consider the semiparametric proportional hazards model for the cause-specific hazard function in analysis of competing risks data with missing cause of failure. The inverse probability weighted equation and augmented inverse probability weighted equation are proposed for estimating the regression parameters in the model, and their theoretical properties are established for inference. Simulation studies demonstrate that the augmented inverse probability weighted estimator is doubly robust and the proposed method is appropriate for practical use. The simulations also compare the proposed estimators with the multiple imputation estimator of Lu and Tsiatis (2001). The application of the proposed method is illustrated using data from a bone marrow transplant study.

The use and interpretation of competing risks regression models

PURPOSE

Competing risks observations, in which patients are subject to a number of potential failure events, are a feature of most clinical cancer studies. With competing risks, several modeling approaches are available to evaluate the relationship of covariates to cause-specific failures. We discuss the use and interpretation of commonly used competing risks regression models.

EXPERIMENTAL DESIGN

For competing risks analysis, the influence of covariate can be evaluated in relation to cause-specific hazard or on the cumulative incidence of the failure types. We present simulation studies to illustrate how covariate effects differ between these approaches. We then show the implications of model choice in an example from a Radiation Therapy Oncology Group (RTOG) clinical trial for prostate cancer.

RESULTS

The simulation studies illustrate that, depending on the relationship of a covariate to both the failure type of principal interest and the competing failure type, different models can result in substantially different effects. For example, a covariate that has no direct influence on the hazard of a primary event can still be significantly associated with the cumulative probability of that event, if the covariate influences the hazard of a competing event. This is a logical consequence of a fundamental difference between the model formulations. The example from RTOG similarly shows differences in the influence of age and tumor grade depending on the endpoint and the model type used.

CONCLUSIONS

Competing risks regression modeling requires that one considers the specific question of interest and subsequent choice of the best model to address it.

Regression analysis for cumulative incidence probability under competing risks and left-truncated sampling

The cumulative incidence function provides intuitive summary information about competing risks data. Via a mixture decomposition of this function, Chang and Wang (Statist. Sinca 19:391-408, 2009) study how covariates affect the cumulative incidence probability of a particular failure type at a chosen time point. Without specifying the corresponding failure time distribution, they proposed two estimators and derived their large sample properties. The first estimator utilized the technique of weighting to adjust for the censoring bias, and can be considered as an extension of Fine's method (J R Stat Soc Ser B 61: 817-830, 1999). The second used imputation and extends the idea of Wang (J R Stat Soc Ser B 65: 921-935, 2003) from a nonparametric setting to the current regression framework. In this article, when covariates take only discrete values, we extend both approaches of Chang and Wang (Statist Sinca 19:391-408, 2009) by allowing left truncation. Large sample properties of the proposed estimators are derived, and their finite sample performance is investigated through a simulation study. We also apply our methods to heart transplant survival data.

Predicting the absolute risk of dying from colorectal cancer and from other causes using population-based cancer registry data

This paper describes how population cancer registry data from the Surveillance, Epidemiology, and End Results program of the National Cancer Institute can be used to develop a prognostic model to predict the absolute risk of mortality from cancer and from other causes for an individual with specific covariates. It incorporates previously developed methods for competing risk modeling along with an imputation method to address missing cause of death information. We illustrate these approaches with colorectal cancer and evaluate the model discriminatory and calibration accuracy by time-dependent area under the receiver operating characteristic curve and calibration plot.

Evaluating prognostic accuracy of biomarkers under competing risk

To develop more targeted intervention strategies, an important research goal is to identify markers predictive of clinical events. A crucial step toward this goal is to characterize the clinical performance of a marker for predicting different types of events. In this article, we present statistical methods for evaluating the performance of a prognostic marker in predicting multiple competing events. To capture the potential time-varying predictive performance of the marker and incorporate competing risks, we define time- and cause-specific accuracy summaries by stratifying cases based on causes of failure. Such definition would allow one to evaluate the predictive accuracy of a marker for each type of event and compare its predictiveness across event types. Extending the nonparametric crude cause-specific receiver operating characteristics curve estimators by Saha and Heagerty (2010), we develop inference procedures for a range of cause-specific accuracy summaries. To estimate the accuracy measures and assess how covariates may affect the accuracy of a marker under the competing risk setting, we consider two forms of semiparametric models through the cause-specific hazard framework. These approaches enable a flexible modeling of the relationships between the marker and failure times for each cause, while efficiently accommodating additional covariates. We investigate the asymptotic property of the proposed accuracy estimators and demonstrate the finite sample performance of these estimators through simulation studies. The proposed procedures are illustrated with data from a prostate cancer prognostic study.

A proportional hazards regression model for the subdistribution with right-censored and left-truncated competing risks data

With competing risks failure time data, one often needs to assess the covariate effects on the cumulative incidence probabilities. Fine and Gray proposed a proportional hazards regression model to directly model the subdistribution of a competing risk. They developed the estimating procedure for right-censored competing risks data, based on the inverse probability of censoring weighting. Right-censored and left-truncated competing risks data sometimes occur in biomedical researches. In this paper, we study the proportional hazards regression model for the subdistribution of a competing risk with right-censored and left-truncated data. We adopt a new weighting technique to estimate the parameters in this model. We have derived the large sample properties of the proposed estimators. To illustrate the application of the new method, we analyze the failure time data for children with acute leukemia. In this example, the failure times for children who had bone marrow transplants were left truncated.

Early human cytomegalovirus replication after transplantation is associated with a decreased relapse risk: evidence for a putative virus-versus-leukemia effect in acute myeloid leukemia patients

The impact of early human cytomegalovirus (HCMV) replication on leukemic recurrence was evaluated in 266 consecutive adult (median age, 47 years; range, 18-73 years) acute myeloid leukemia patients, who underwent allogeneic stem cell transplantation (alloSCT) from 10 of 10 high-resolution human leukocyte Ag-identical unrelated (n = 148) or sibling (n = 118) donors. A total of 63% of patients (n = 167) were at risk for HCMV reactivation by patient and donor pretransplantation HCMV serostatus. In 77 patients, first HCMV replication as detected by pp65-antigenemia assay developed at a median of 46 days (range, 25-108 days) after alloSCT. Taking all relevant competing risk factors into account, the cumulative incidence of hematologic relapse at 10 years after alloSCT was 42% (95% confidence interval [CI], 35%-51%) in patients without opposed to 9% (95% CI, 4%-19%) in patients with early pp65-antigenemia (P < .0001). A substantial and independent reduction of the relapse risk associated with early HCMV replication was confirmed by multivariate analysis using time-dependent covariate functions for grades II to IV acute and chronic graft-versus-host disease, and pp65-antigenemia (hazard ratio = 0.2; 95% CI, 0.1-0.4, P < .0001). This is the first report that demonstrates an independent and substantial reduction of the leukemic relapse risk after early replicative HCMV infection in a homogeneous population of adult acute myeloid leukemia patients.

SAS macros for estimation of direct adjusted cumulative incidence curves under proportional subdistribution hazards models

The cumulative incidence function is commonly reported in studies with competing risks. The aim of this paper is to compute the treatment-specific cumulative incidence functions, adjusting for potentially imbalanced prognostic factors among treatment groups. The underlying regression model considered in this study is the proportional hazards model for a subdistribution function [1]. We propose estimating the direct adjusted cumulative incidences for each treatment using the pooled samples as the reference population. We develop two SAS macros for estimating the direct adjusted cumulative incidence function for each treatment based on two regression models. One model assumes the constant subdistribution hazard ratios between the treatments and the alternative model allows each treatment to have its own baseline subdistribution hazard function. The macros compute the standard errors for the direct adjusted cumulative incidence estimates, as well as the standard errors for the differences of adjusted cumulative incidence functions between any two treatments. Based on the macros' output, one can assess treatment effects at predetermined time points. A real bone marrow transplant data example illustrates the practical utility of the SAS macros.

NONPARAMETRIC ESTIMATION OF CONDITIONAL CUMULATIVE HAZARDS FOR MISSING POPULATION MARKS

A new function for the competing risks model, the conditional cumulative hazard function, is introduced, from which the conditional distribution of failure times of individuals failing due to cause j can be studied. The standard Nelson-Aalen estimator is not appropriate in this setting, as population membership (mark) information may be missing for some individuals owing to random right-censoring. We propose the use of imputed population marks for the censored individuals through fractional risk sets. Some asymptotic properties, including uniform strong consistency, are established. We study the practical performance of this estimator through simulation studies and apply it to a real data set for illustration.

Estimating subject-specific dependent competing risk profile with censored event time observations

In a longitudinal study, suppose that the primary endpoint is the time to a specific event. This response variable, however, may be censored by an independent censoring variable or by the occurrence of one of several dependent competing events. For each study subject, a set of baseline covariates is collected. The question is how to construct a reliable prediction rule for the future subject's profile of all competing risks of interest at a specific time point for risk-benefit decision making. In this article, we propose a two-stage procedure to make inferences about such subject-specific profiles. For the first step, we use a parametric model to obtain a univariate risk index score system. We then estimate consistently the average competing risks for subjects who have the same parametric index score via a nonparametric function estimation procedure. We illustrate this new proposal with the data from a randomized clinical trial for evaluating the efficacy of a treatment for prostate cancer. The primary endpoint for this study was the time to prostate cancer death, but had two types of dependent competing events, one from cardiovascular death and the other from death of other causes.

Importance of age of onset in pancreatic cancer kindreds

BACKGROUND

Young-onset cancer is a hallmark of many familial cancer syndromes, yet the implications of young-onset disease in predicting risk of pancreatic cancer among familial pancreatic cancer (FPC) kindred members remain unclear.

METHODS

To understand the relationship between age at onset of pancreatic cancer and risk of pancreatic cancer in kindred members, we compared the observed incidence of pancreatic cancer in 9040 individuals from 1718 kindreds enrolled in the National Familial Pancreas Tumor Registry with that observed in the general US population (Surveillance, Epidemiology, and End Results). Standardized incidence ratios (SIRs) were calculated for data stratified by familial vs sporadic cancer kindred membership, number of affected relatives, youngest age of onset among relatives, and smoking status. Competing risk survival analyses were performed to examine the risk of pancreatic cancer and risk of death from other causes according to youngest age of onset of pancreatic cancer in the family and the number of affected relatives.

RESULTS

Risk of pancreatic cancer was elevated in both FPC kindred members (SIR = 6.79, 95% confidence interval [CI] = 4.54 to 9.75, P < .001) and sporadic pancreatic cancer (SPC) kindred members (SIR = 2.41, 95% CI = 1.04 to 4.74, P = .04) compared with the general population. The presence of a young-onset patient (<50 years) in the family did not alter the risk for SPC kindred members (SIR = 2.74, 95% CI = 0.05 to 15.30, P = .59) compared with those without a young-onset case in the kindred (SIR = 2.36, 95% CI = 0.95 to 4.88, P = .06). However, risk was higher among members of FPC kindreds with a young-onset case in the kindred (SIR = 9.31, 95% CI = 3.42 to 20.28, P < .001) than those without a young-onset case in the kindred (SIR = 6.34, 95% CI = 4.02 to 9.51, P < .001). Competing risk survival analyses indicated that the lifetime risk of pancreatic cancer in FPC kindreds increased with decreasing age of onset in the kindred (hazard ratio = 1.55, 95% CI = 1.19 to 2.03 per year). However, youngest age of onset for pancreatic cancer in the kindred did not affect the risk among SPC kindred members.

CONCLUSIONS

Individuals with a family history of pancreatic cancer are at a statistically significantly increased risk of developing pancreatic cancer. Having a member of the family with a young-onset pancreatic cancer confers an added risk in FPC kindreds.

Assessing cumulative incidence functions under the semiparametric additive risk model

In analyzing competing risks data, a quantity of considerable interest is the cumulative incidence function. Often, the effect of covariates on the cumulative incidence function is modeled via the proportional hazards model for the cause-specific hazard function. As the proportionality assumption may be too restrictive in practice, we consider an alternative more flexible semiparametric additive hazards model of (Biometrika 1994; 81:501-514) for the cause-specific hazard. This model specifies the effect of covariates on the cause-specific hazard to be additive as well as allows the effect of some covariates to be fixed and that of others to be time varying. We present an approach for constructing confidence intervals as well as confidence bands for the cause-specific cumulative incidence function of subjects with given values of the covariates. Furthermore, we also present an approach for constructing confidence intervals and confidence bands for comparing two cumulative incidence functions given values of the covariates. The finite sample property of the proposed estimators is investigated through simulations. We conclude our paper with an analysis of the well-known malignant melanoma data using our method.

Competing risk regression models for epidemiologic data

Competing events can preclude the event of interest from occurring in epidemiologic data and can be analyzed by using extensions of survival analysis methods. In this paper, the authors outline 3 regression approaches for estimating 2 key quantities in competing risks analysis: the cause-specific relative hazard ((cs)RH) and the subdistribution relative hazard ((sd)RH). They compare and contrast the structure of the risk sets and the interpretation of parameters obtained with these methods. They also demonstrate the use of these methods with data from the Women's Interagency HIV Study established in 1993, treating time to initiation of highly active antiretroviral therapy or to clinical disease progression as competing events. In our example, women with an injection drug use history were less likely than those without a history of injection drug use to initiate therapy prior to progression to acquired immunodeficiency syndrome or death by both measures of association ((cs)RH = 0.67, 95% confidence interval: 0.57, 0.80 and (sd)RH = 0.60, 95% confidence interval: 0.50, 0.71). Moreover, the relative hazards for disease progression prior to treatment were elevated ((cs)RH = 1.71, 95% confidence interval: 1.37, 2.13 and (sd)RH = 2.01, 95% confidence interval: 1.62, 2.51). Methods for competing risks should be used by epidemiologists, with the choice of method guided by the scientific question.

Flexible competing risks regression modeling and goodness-of-fit

In this paper we consider different approaches for estimation and assessment of covariate effects for the cumulative incidence curve in the competing risks model. The classic approach is to model all cause-specific hazards and then estimate the cumulative incidence curve based on these cause-specific hazards. Another recent approach is to directly model the cumulative incidence by a proportional model (Fine and Gray, J Am Stat Assoc 94:496-509, 1999), and then obtain direct estimates of how covariates influences the cumulative incidence curve. We consider a simple and flexible class of regression models that is easy to fit and contains the Fine-Gray model as a special case. One advantage of this approach is that our regression modeling allows for non-proportional hazards. This leads to a new simple goodness-of-fit procedure for the proportional subdistribution hazards assumption that is very easy to use. The test is constructive in the sense that it shows exactly where non-proportionality is present. We illustrate our methods to a bone marrow transplant data from the Center for International Blood and Marrow Transplant Research (CIBMTR). Through this data example we demonstrate the use of the flexible regression models to analyze competing risks data when non-proportionality is present in the data.

Inference for outcome probabilities in multi-state models

In bone marrow transplantation studies, patients are followed over time and a number of events may be observed. These include both ultimate events like death and relapse and transient events like graft versus host disease and graft recovery. Such studies, therefore, lend themselves for using an analytic approach based on multi-state models. We will give a review of such methods with emphasis on regression models for both transition intensities and transition- and state occupation probabilities. Both semi-parametric models, like the Cox regression model, and parametric models based on piecewise constant intensities will be discussed.

Prediagnostic body-mass index, plasma C-peptide concentration, and prostate cancer-specific mortality in men with prostate cancer: a long-term survival analysis

BACKGROUND

Excess body-mass index (BMI) has been associated with adverse outcomes in prostate cancer, and hyperinsulinaemia is a candidate mediator, but prospective data are sparse. We assessed the effect of prediagnostic BMI and plasma C-peptide concentration (reflecting insulin secretion) on prostate cancer-specific mortality after diagnosis.

METHODS

This study involved men diagnosed with prostate cancer during the 24 years of follow-up in the Physicians' Health Study. BMI measurements were available at baseline in 1982 and eight years later in 1990 for 2546 men who developed prostate cancer. Baseline C-peptide concentration was available in 827 men. We used Cox proportional hazards regression models controlling for age, smoking, time between BMI measurement and prostate cancer diagnosis, and competing causes of death to assess the risk of prostate cancer-specific mortality according to BMI and C-peptide concentration.

FINDINGS

Of the 2546 men diagnosed with prostate cancer during the follow-up period, 989 (38.8%) were overweight (BMI 25.0-29.9 kg/m(2)) and 87 (3.4%) were obese (BMI >/=30 kg/m(2)). 281 men (11%) died from prostate cancer during this follow-up period. Compared with men of a healthy weight (BMI <25 kg/m(2)) at baseline, overweight men and obese men had a significantly higher risk of prostate cancer mortality (proportional hazard ratio [HR] 1.47 [95% CI 1.16-1.88] for overweight men and 2.66 [1.62-4.39] for obese men; p(trend)<0.0001). The trend remained significant after controlling for clinical stage and Gleason grade and was stronger for prostate cancer diagnosed during the PSA screening era (1991-2007) compared with during the pre-PSA screening era (1982-1990) or when using BMI measurements obtained in 1990 compared with those obtained in 1982. Of the 827 men with data available for baseline C-peptide concentration, 117 (14%) died from prostate cancer. Men with C-peptide concentrations in the highest quartile (high) versus the lowest quartile (low) had a higher risk of prostate cancer mortality (HR 2.38 [95% CI 1.31-4.30]; p(trend)=0.008). Compared with men with a BMI less than 25 kg/m(2) and low C-peptide concentrations, those with a BMI of 25 kg/m(2) or more and high C-peptide concentrations had a four-times higher risk of mortality (4.12 [1.97-8.61]; p(interaction)=0.001) independent of clinical predictors.

INTERPRETATION

Excess bodyweight and a high plasma concentration of C-peptide both predispose men with a subsequent diagnosis of prostate cancer to an increased likelihood of dying of their disease. Patients with both factors have the worst outcome. Further studies are now needed to confirm these findings.

Fetal growth and timing of parturition in humans

Animal studies indicate that either the fetus or the intrauterine environment, both of which set the pattern for fetal growth, may affect the timing of parturition. The authors examined the association between fetal growth and timing of spontaneous onset of labor in humans among low-risk white US women with singleton pregnancies (1987-1991). They restricted the data to pregnancies which had a reliable date of the last menstrual period, normal fetal growth in the first half of pregnancy, and no history of or current pregnancy complications that might have impaired fetal growth (n = 3,360). Subjects received ultrasound examinations at 15-22 and 31-35 weeks' gestation. Fetal growth was adjusted for parity, fetal sex, and maternal prepregnancy weight and height. Results showed that slower or faster fetal growth in the second half of pregnancy resulted in substantially lower or higher birth weight, respectively. However, fetal growth in the second half of pregnancy, even at extremes (2 standard deviations below or above the mean), did not have a meaningful impact on the timing of parturition; neither did fetal growth acceleration or deceleration in late pregnancy. Thus, in low-risk pregnancies where fetal growth is normal in early gestation, fetal growth in the second half of pregnancy does not affect the timing of normal parturition.

Evaluating cardiovascular mortality in type 2 diabetes patients: an analysis based on competing risks Markov chains and additive regression models

RATIONALE, AIMS AND OBJECTIVES

Type 2 diabetes represents a condition significantly associated with increased cardiovascular mortality. The aims of the study are: (i) to estimate the cumulative incidence function for cause-specific mortality using Cox and Aalen model; (ii) to describe how the prediction of cardiovascular or other causes mortality changes for patients with different pattern of covariates; (iii) to show if different statistical methods may give different results.

METHODS

Cox and Aalen additive regression model through the Markov chain approach, are used to estimate the cause-specific hazard for cardiovascular or other causes mortality in a cohort of 2865 type 2 diabetic patients without insulin treatment. The models are compared in the estimation of the risk of death for patients of different severity.

RESULTS

For younger patients with a better covariates profile, the Cumulative Incidence Function estimated by Cox and Aalen model was almost the same; for patients with the worst covariates profile, models gave different results: at the end of follow-up cardiovascular mortality rate estimated by Cox and Aalen model was 0.26 [95% confidence interval (CI) = 0.21-0.31] and 0.14 (95% CI = 0.09-0.18).

CONCLUSIONS

Standard Cox and Aalen model capture the risk process for patients equally well with average profiles of co-morbidities. The Aalen model, in addition, is shown to be better at identifying cause-specific risk of death for patients with more severe clinical profiles. This result is relevant in the development of analytic tools for research and resource management within diabetes care.

Estimation of distribution function in bivariate competing risk models

We consider lifetime data involving pairs of study individuals with more than one possible cause of failure for each individual. Non-parametric estimation of cause-specific distribution functions is considered under independent censoring. Properties of the estimators are discussed and an illustration of their application is given.

Modelling competing risks in cancer studies

Competing risks arise commonly in the analysis of cancer studies. Most common are the competing risks of relapse and death in remission. These two risks are the primary reason that patients fail treatment. In most medical papers the effects of covariates on the three outcomes (relapse, death in remission and treatment failure) are model by distinct proportional hazards regression models. Since the hazards of relapse and death in remission must add to that of treatment failure, we argue that this model leads to internal inconsistencies. We argue that additive models for either the hazard rates or the cumulative incidence functions are more natural and that these models properly partition the effect of a covariate on treatment failure into its component parts. We illustrate the use and interpretation of additive models for the hazard rate or for the cumulative incidence function using data from a study of the efficacy of two preparative regimes for hematopoietic stem cell transplantation.

A Semiparametric Mixture Model for Analyzing Clustered Competing Risks Data

A very general class of multivariate life distributions is considered for analyzing failure time clustered data that are subject to censoring and multiple modes of failure. Conditional on cluster-specific quantities, the joint distribution of the failure time and event indicator can be expressed as a mixture of the distribution of time to failure due to a certain type (or specific cause), and the failure type distribution. We assume here the marginal probabilities of various failure types are logistic functions of some covariates. The cluster-specific quantities are subject to some unknown distribution that causes frailty. The unknown frailty distribution is modeled nonparametrically using a Dirichlet process. In such a semiparametric setup, a hybrid method of estimation is proposed based on the i.i.d. Weighted Chinese Restaurant algorithm that helps us generate observations from the predictive distribution of the frailty. The Monte Carlo ECM algorithm plays a vital role for obtaining the estimates of the parameters that assess the extent of the effects of the causal factors for failures of a certain type. A simulation study is conducted to study the consistency of our methodology. The proposed methodology is used to analyze a real data set on HIV infection of a cohort of female prostitutes in Senegal.

Semiparametric models for cumulative incidence functions

In analyses of time-to-failure data with competing risks, cumulative incidence functions may be used to estimate the time-dependent cumulative probability of failure due to specific causes. These functions are commonly estimated using nonparametric methods, but in cases where events due to the cause of primary interest are infrequent relative to other modes of failure, nonparametric methods may result in rather imprecise estimates for the corresponding subdistribution. In such cases, it may be possible to model the cause-specific hazard of primary interest parametrically, while accounting for the other modes of failure using nonparametric estimators. The cumulative incidence estimators so obtained are simple to compute and are considerably more efficient than the usual nonparametric estimator, particularly with regard to interpolation of cumulative incidence at early or intermediate time points within the range of data used to fit the function. More surprisingly, they are often nearly as efficient as fully parametric estimators. We illustrate the utility of this approach in the analysis of patients treated for early stage breast cancer.

Extensions and applications of the Cox-Aalen survival model

Cox's regression model is the standard regression tool for survival analysis in most applications. Often, however, the model only provides a rough summary of the effect of some covariates. Therefore, if the aim is to give a detailed description of covariate effects and to consequently calculate predicted probabilities, more flexible models are needed. In another article, Scheike and Zhang (2002, Scandinavian Journal of Statistics 29, 75-88), we suggested a flexible extension of Cox's regression model, which aimed at extending the Cox model only for those covariates where additional flexibility are needed. One important advantage of the suggested approach is that even though covariates are allowed a nonparametric effect, the hassle and difficulty of finding smoothing parameters are not needed. We show how the extended model also leads to simple formulae for predicted probabilities and their standard errors, for example, in the competing risk framework.

MELD score as a predictor of pretransplant and posttransplant survival in OPTN/UNOS status 1 patients

The Model for End-Stage Liver Disease (MELD) score is predictive of survival and is used to prioritize patients with chronic liver disease patients for orthotopic liver transplantation (OLT). The aims of this study are (1) to assess the ability of MELD score at listing to predict pretransplant and posttransplant survival for nonchronic liver disease patients listed with the Organ Procurement and Transplantation Network/ United Network for Organ Sharing (OPTN/UNOS) as Status 1; and (2) to compare survival associated with 4 diagnostic groups within the Status 1 designation. The study population consisted of adult patients listed for OLT at Status 1 in the UNOS national database between November 1, 1999 and March 14, 2002 (N = 720). Events within 30 days of listing were analyzed using Kaplan-Meier and Cox regression methodology. Patients meeting criteria for fulminant hepatic failure without acetaminophen toxicity (FHF-NA, n = 312) had the poorest survival probability while awaiting OLT; this was negatively correlated with MELD score (P =.0001). These patients experienced the greatest survival benefit associated with OLT, with an estimated improvement of survival from about 58% to 91% (P <.0001). Patients listed for primary nonfunction within 7 days of OLT (n = 268) did not show mortality to be related to MELD score (P =.41) and did not show a significant association between survival and OLT (P =.68). In conclusion, liver allocation within the Status 1 designation may need to be further stratified by diagnosis, and MELD score may be useful for prioritizing FHF-NA candidates.

Fitting competing risks with an assumed copula

We propose a fully parametric model for the analysis of competing risks data where the types of failure may not be independent. We show how the dependence between the cause-specific survival times can be modelled with a copula function. Features include: identifiability of the problem; accessible understanding of the dependence structures; and flexibility in choosing marginal survival functions. The model is constructed in such a way that it allows us to adjust for concomitant variables and for a dependence parameter to assess the effects of these on each marginal survival model and on the relationship between the causes of death. The methods are applied to a prostate cancer data set. We find that, with the copula model, more accurate inferences are obtained than with the use of a simpler model such as the independent competing risks approach.

An EM-based semi-parametric mixture model approach to the regression analysis of competing-risks data

We consider a mixture model approach to the regression analysis of competing-risks data. Attention is focused on inference concerning the effects of factors on both the probability of occurrence and the hazard rate conditional on each of the failure types. These two quantities are specified in the mixture model using the logistic model and the proportional hazards model, respectively. We propose a semi-parametric mixture method to estimate the logistic and regression coefficients jointly, whereby the component-baseline hazard functions are completely unspecified. Estimation is based on maximum likelihood on the basis of the full likelihood, implemented via an expectation-conditional maximization (ECM) algorithm. Simulation studies are performed to compare the performance of the proposed semi-parametric method with a fully parametric mixture approach. The results show that when the component-baseline hazard is monotonic increasing, the semi-parametric and fully parametric mixture approaches are comparable for mildly and moderately censored samples. When the component-baseline hazard is not monotonic increasing, the semi-parametric method consistently provides less biased estimates than a fully parametric approach and is comparable in efficiency in the estimation of the parameters for all levels of censoring. The methods are illustrated using a real data set of prostate cancer patients treated with different dosages of the drug diethylstilbestrol.

Competing risks as a multi-state model

This paper deals with the competing risks model as a special case of a multi-state model. The properties of the model are reviewed and contrasted to the so-called latent failure time approach. The relation between the competing risks model and right-censoring is discussed and regression analysis of the cumulative incidence function briefly reviewed. Two real data examples are presented and a guide to the practitioner is given.

Competing approaches to analysis of failure times with competing risks

For the analysis of time to event data in contraceptive studies when individuals are subject to competing causes for discontinuation, some authors have recently advocated the use of the cumulative incidence rate as a more appropriate measure to summarize data than the complement of the Kaplan-Meier estimate of discontinuation. The former method estimates the rate of discontinuation in the presence of competing causes, while the latter is a hypothetical rate that would be observed if discontinuations for the other reasons could not occur. The difference between the two methods of analysis is the continuous time equivalent of a debate that took place in the contraceptive literature in the 1960s, when several authors advocated the use of net (adjusted or single decrement life table rates) rates in preference to crude rates (multiple decrement life table rates). A small simulation study illustrates the interpretation of the two types of estimate - the complement of the Kaplan-Meier estimate corresponds to a hypothetical rate where discontinuations for other reasons did not occur, while the cumulative incidence gives systematically lower estimates. The Kaplan-Meier estimates are more appropriate when estimating the effectiveness of a contraceptive method, but the cumulative incidence estimates are more appropriate when making programmatic decisions regarding contraceptive methods. Other areas of application, such as cancer studies, may prefer to use the cumulative incidence estimates, but their use should be determined according to the application.

Analysis of failure time hierarchical data in the presence of competing risks with application to oral contraceptive pill use in Egypt

Problems of practical interest in the analysis of data on contraceptive use, from Demographic and Health Surveys (DHS), include the estimation of the cause-specific probability of discontinuation by time t (the cumulative incidence function), in the presence of other competing causes and the evaluation of the effect of covariates on the cause-specific hazards of discontinuation. Methods of analysis of failure time data with competing risks are by now fairly well developed in the case of a simple random sample. However, the data from the DHS are clustered by geographical areas and include multiple episodes per woman. For a marginal (population average) approach, we propose using methods developed for simple random samples with standard errors calculated using a double bootstrap to take account of the clustered hierarchical nature of the data. In the conditional approach, the cause-specific hazards are modelled as log-linear functions of the covariates conditional on random effects of clusters and women, using a three-level multinomial discrete-time logit model. The methods are applied to data from Egypt 1992 DHS on the oral contraceptive pill use.

Omnibus tests for comparison of competing risks with adjustment for covariate effects

This article develops omnibus tests for comparing cause-specific hazard rates and cumulative incidence functions at specified covariate levels. Confidence bands for the difference and the ratio of two conditional cumulative incidence functions are also constructed. The omnibus test is formulated in terms of a test process given by a weighted difference of estimates of cumulative cause-specific hazard rates under Cox proportional hazards models. A simulation procedure is devised for sampling from the null distribution of the test process, leading to graphical and numerical technques for detecting significant differences in the risks. The approach is applied to a cohort study of type-specific HIV infection rates.

Nonparametric analysis of recurrent events and death

This article is concerned with the analysis of recurrent events in the presence of a terminal event such as death. We consider the mean frequency function, defined as the marginal mean of the cumulative number of recurrent events over time. A simple nonparametric estimator for this quantity is presented. It is shown that the estimator, properly normalized, converges weakly to a zero-mean Gaussian process with an easily estimable covariance function. Nonparametric statistics for comparing two mean frequency functions and for combining data on recurrent events and death are also developed. The asymptotic null distributions of these statistics, together with consistent variance estimators, are derived. The small-sample properties of the proposed estimators and test statistics are examined through simulation studies. An application to a cancer clinical trial is provided.

Confidence bands for cumulative incidence curves under the additive risk model

In the context of competing risks, the cumulative incidence function is often used to summarize the cause-specific failure-time data. As an alternative to the proportional hazards model, the additive risk model is used to investigate covariate effects by specifying that the subject-specific hazard function is the sum of a baseline hazard function and a regression function of covariates. Based on such a formulation, we present an approach to constructing simultaneous confidence intervals for the cause-specific cumulative incidence function of patients with given risk factors. A melanoma data set is used for the purpose of illustration.

Estimation of failure probabilities in the presence of competing risks: new representations of old estimators

A topic that has received attention in both the statistical and medical literature is the estimation of the probability of failure for endpoints that are subject to competing risks. Despite this, it is not uncommon to see the complement of the Kaplan-Meier estimate used in this setting and interpreted as the probability of failure. If one desires an estimate that can be interpreted in this way, however, the cumulative incidence estimate is the appropriate tool to use in such situations. We believe the more commonly seen representations of the Kaplan-Meier estimate and the cumulative incidence estimate do not lend themselves to easy explanation and understanding of this interpretation. We present, therefore, a representation of each estimate in a manner not ordinarily seen, each representation utilizing the concept of censored observations being 'redistributed to the right.' We feel these allow a more intuitive understanding of each estimate and therefore an appreciation of why the Kaplan-Meier method is inappropriate for estimation purposes in the presence of competing risks, while the cumulative incidence estimate is appropriate.