Copula-graphic Estimator Research Articles

A semiparametric model for the cause-specific hazard under risk proportionality

Semiparametric Cox proportional hazards models enjoy great popularity in empirical survival analysis. A semiparametric model for cause-specific hazards under a proportionality restriction across risks is considered, which has desired practical properties such as estimation by partial likelihood and an analytical solution for the copula-graphic estimator. The cause-specific and marginal hazards are shown to share functional form restrictions in this case. The model for the cause-specific hazard can be used for inference about parametric restrictions on the marginal hazard without the risk of misspecifying the latter and without knowing the risk dependence. After the class of parametric marginal hazards has been determined, it can be estimated in conjunction with the degree of risk dependence. Finite sample properties are investigated with simulations. An application to employment duration demonstrates the practicality of the approach.

Sensitivity Analysis for Survival Prognostic Prediction with Gene Selection: A Copula Method for Dependent Censoring.

Prognostic analysis for patient survival often employs gene expressions obtained from high-throughput screening for tumor tissues from patients. When dealing with survival data, a dependent censoring phenomenon arises, and thus the traditional Cox model may not correctly identify the effect of each gene. A copula-based gene selection model can effectively adjust for dependent censoring, yielding a multi-gene predictor for survival prognosis. However, methods to assess the impact of various types of dependent censoring on the multi-gene predictor have not been developed. In this article, we propose a sensitivity analysis method using the copula-graphic estimator under dependent censoring, and implement relevant methods in the R package "compound.Cox". The purpose of the proposed method is to investigate the sensitivity of the multi-gene predictor to a variety of dependent censoring mechanisms. In order to make the proposed sensitivity analysis practical, we develop a web application. We apply the proposed method and the web application to a lung cancer dataset. We provide a template file so that developers can modify the template to establish their own web applications.

Extension of Relative-Risk Power Estimator under Dependent Random Censored Data

In this paper, the considered problem consists in estimation of conditional survival function by right random censoring model in the presence of a covariate. We propose a new estimator of conditional survival function which is extension of relative-risk power estimator of independent censoring and study its large sample properties. We present result of asymptotic normality with the same limiting Gaussian process as for copula-graphic estimator.

Estimation of the Mann–Whitney effect in the two-sample problem under dependent censoring

The Mann–Whitney effect is a nonparametric measure for comparing the distribution between two groups, which can be estimated by right-censored data. However, the traditional estimator of the Mann–Whitney effect based on the Kaplan–Meier estimators can be inconsistent when the independent censoring assumption fails to hold. Investigation is made on the asymptotic bias of the traditional estimator of the Mann–Whitney effect when the independent censoring assumption is violated due to dependence between survival time and censoring time. A new estimator of the Mann–Whitney effect is proposed by applying the copula-graphic estimator to adjust for the effect of dependent censoring. The proposed estimator and test are consistent when the assumed copulas for the two groups are correct. Some consistency properties under misspecified copulas are also given. Simulations are conducted to verify the proposed method under possible misspecification on copulas. The method is illustrated by a real data set. We provide an R function “MW.test” to implement the proposed estimator and test.

Estimation of Conditional Survival Function under Dependent Right Random Censored Data

In this paper considered problem consist in estimation of conditional survival function by right random censoring model in the presence of covariate. We propose a new estimator of conditional survival function and study its large sample properties. We present result of asymptotic normality with the same limiting Gaussian process as for copula-graphic estimator.

Estimation of the survival function with redistribution algorithm under semi-competing risks data

This paper focuses on the estimation of the survival function of the non-terminal event time for semi-competing risks data. Without extra assumptions, we cannot make inference on the non-terminal event time because the non-terminal event time is dependently censored by the terminal event time. Thus, we utilize the Archimedean copula model to specify the dependency between the non-terminal event time and the terminal event time. Under the Archimedean copula assumption, we apply the redistribution method to estimate the survival function of the non-terminal event time and compare it with the copula-graphic estimator introduced by Lakhal et al. (2008). We also apply our suggested approach to analyze the Bone Marrow Transplant data.

Survival forests for data with dependent censoring.

Tree-based methods are very powerful and popular tools for analysing survival data with right-censoring. The existing methods assume that the true time-to-event and the censoring times are independent given the covariates. We propose different ways to build survival forests when dependent censoring is suspected, by using an appropriate estimator of the survival function when aggregating the individual trees and/or by modifying the splitting rule. The appropriate estimator used in this paper is the copula-graphic estimator. We also propose a new method for building survival forests, called p-forest, that may be used not only when dependent censoring is suspected, but also as a new survival forest method in general. The results from a simulation study indicate that these modifications improve greatly the estimation of the survival function in situations of dependent censoring. A real data example illustrates how the proposed methods can be used to perform a sensitivity analysis.

Copula-graphic estimation with left-truncated and right-censored data

ABSTRACTIn Survival Analysis and related fields of research right-censored and left-truncated data often appear. Usually, it is assumed that the right-censoring variable is independent of the lifetime of ultimate interest. However, in particular applications dependent censoring may be present; this is the case, for example, when there exist several competing risks acting on the same individual. In this paper we propose a copula-graphic estimator for such a situation. The estimator is based on a known Archimedean copula function which properly represents the dependence structure between the lifetime and the censoring time. Therefore, the current work extends the copula-graphic estimator in de Uña-Álvarez and Veraverbeke [Generalized copula-graphic estimator. Test. 2013;22:343–360] in the presence of left-truncation. An asymptotic representation of the estimator is derived. The performance of the estimator is investigated in an intensive Monte Carlo simulation study. An application to unemployment duration is included for illustration purposes.

Competing Risks Copula Models for Unemployment Duration: An Application to a German Hartz Reform

AbstractThe copula graphic estimator (CGE) for competing risks models has received little attention in empirical research, despite having been developed into a comprehensive research method. In this paper, we bridge the gap between theoretical developments and applied research by considering a general class of competing risks copula models, which nests popular models such as the Cox proportional hazards model, the semiparametric multivariate mixed proportional hazards model (MMPHM), and the CGE as special cases. Analyzing the effects of a German Hartz reform on unemployment duration, we illustrate that the CGE imposes fewer restrictions on partial covariate effects than standard methods do. Differences are less evident when a more flexible difference-in-differences estimator is applied. It is also found that the MMPHM estimates react more strongly to the choice of the copula than the CGE in terms of the shape of the treatment effect function over time. Thus, the MMPHM produces less robust results in our application.

Estimating Relapse Free Survival as a Net Probability: Regression Models and Graphical Representation. An Application of a Large Breast Cancer Case Series

In most clinical studies, the evaluation of the effect of a therapy and the impact of prognostic factors is based on relapse-free survival. Relapse free is a net survival, since it is interpreted as the relapse-free probability that would be observed if all patients experienced relapse sooner or later. Death without evidence of relapse prevents the subsequent observation of relapse, acting in a semi-competing risks framework. Relapse free survival is often estimated by standard regression models after censoring times to death. The association between relapse and death is thus accounted for. However, to better estimate relapse free survival, a bivariate distribution of times to events needs to be considered, for example by means of copula models. We concentrate here on the copula graphic estimator, for which a pertinent regression model has been developed. No direct parametric estimation of the regression coefficient for the covariates is available and the evaluation of the impact of covariates on relapse free survival is based on graphical representation for each covariate singularly. The advantage of this approach is based on the relationship between net survival, and crude cumulative incidences. Regression models can be fitted for the latter quantities and the estimates can be used to compute net survival through a copula structure. Our proposal is based on flexible regression transformation model on crude cumulative incidences based on pseudo-values. An overall view of the joint association among covariates and relapse free survival is obtained through Multiple Correspondence Analysis. Moreover cluster analysis on MCA coordinates was used to synthesize covariate patterns and to estimates the corresponding relapse free survival curve. This approach has been applied to a large “historical” case series of patients with breast cancer.

Properties of the marginal survival functions for dependent censored data under an assumed Archimedean copula

Abstract Given a random sample from a dependent censored variable ( X , δ ) = ( min ( T , C ) , 1 ( T C ) ) , general formulas are given for possible marginal survival functions of the failure time T and the censoring time C under the assumption that their underlying copula is Archimedean. These formulas are used to establish the relationship between these possible survival functions, along with useful identifiability results. A new estimator of the marginal survival functions is also proposed, under the assumption that the underlying Archimedean copula is known. Bias formulas for this estimator and other existing estimators are derived. Simulation studies show that the new estimator is comparable with the copula-graphic estimator proposed by Zheng and Klein (1995) and Rivest and Wells (2001), as well as with the estimator of Zheng and Klein (1994) under the Archimedean copula assumption.

A Regression Model for the Copula-Graphic Estimator

Abstract We suggest a pragmatic extension of the non-parametric copula-graphic estimator to a depending competing risks model with covariates. Our model is an attractive empirical approach for practitioners in many disciplines as it does not require knowledge of the marginal distributions. Although non-observable and only set-identifiable in most applications, classical duration models typically impose ad-hoc assumptions on their functional forms. Instead of directly estimating these distributions, we suggest a plug-in regression framework which utilises an estimator for the observable cumulative incidence curves which specification can be visually inspected. We perform simulations and estimate an unemployment duration model to demonstrate the advantages of our model compared to classical duration models such as the Cox proportional hazard model.

Generalized copula-graphic estimator

In this paper, a copula-graphic estimator is proposed for censored survival data. It is assumed that there is some dependent censoring acting on the variable of interest that may come from an existing competing risk. Furthermore, the full process is independently censored by some administrative censoring time. The dependent censoring is modeled through an Archimedean copula function, which is supposed to be known. An asymptotic representation of the estimator as a sum of independent and identically distributed random variables is obtained, and, consequently, a central limit theorem is established. We investigate the finite sample performance of the estimator through simulations. A real data illustration is included.

A Regression Model for the Copula Graphic Estimator

We consider a dependent competing risks model with many risks and many covariates. We show identifiability of the marginal distributions of latent variables for a given dependence structure. Instead of directly estimating these distributions, we suggest a plug-in regression framework for the Copula-Graphic estimator which utilizes a consistent estimator for the cumulative incidence curves. Our model is an attractive empirical approach as it does not require knowledge of the marginal distributions which are typically unknown in applications. We illustrate the applicability of our approach with the help of a parametric unemployment duration model with an unknown dependence structure. We construct identification bounds for the marginal distributions and partial effects in response to covariate changes. The bounds for the partial effects are surprisingly tight and often reveal the direction of the covariate effect.

Weak convergence for the conditional distribution function in a Koziol–Green model under dependent censoring

In this paper we consider the conditional Koziol–Green model of Braekers and Veraverbeke [2008. A conditional Koziol–Green model under dependent censoring. Statist. Probab. Lett., accepted for publication] in which they generalized the Koziol–Green model of Veraverbeke and Cadarso Suárez [2000. Estimation of the conditional distribution in a conditional Koziol–Green model. Test 9, 97–122] by assuming that the association between a censoring time and a time until an event is described by a known Archimedean copula function. They got in this way, an informative censoring model with two different types of informative censoring. Braekers and Veraverbeke [2008. A conditional Koziol–Green model under dependent censoring. Statist. Probab. Lett., accepted for publication] derived in this model a non-parametric Koziol–Green estimator for the conditional distribution function of the time until an event, for which they showed the uniform consistency and the asymptotic normality. In this paper we extend their results and prove the weak convergence of the process associated to this estimator. Furthermore we show that the conditional Koziol–Green estimator is asymptotically more efficient in this model than the general copula-graphic estimator of Braekers and Veraverbeke [2005. A copula-graphic estimator for the conditional survival function under dependent censoring. Canad. J. Statist. 33, 429–447]. As last result, we construct an asymptotic confidence band for the conditional Koziol–Green estimator. Through a simulation study, we investigate the small sample properties of this asymptotic confidence band. Afterwards we apply this estimator and its confidence band on a practical data set.

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 conditional Koziol–Green model under dependent censoring

In this paper we extend the conditional Koziol–Green model of Veraverbeke, N. and Cadarso Suárez, C. [2000. Estimation of the conditional distribution in a conditional Koziol–Green model. Test 9, 97–122] to also accommodate dependent censoring and in this way introduce a model with two different types of informative censoring. We derive in this model a copula-graphic estimator for the conditional distribution of the lifetime and establish an exponential bound and an almost sure asymptotic representation which serve as starting points for an almost sure consistency and an asymptotic normality result. Afterwards we apply this estimator to a real data set about the survival of Atlantic halibut.

Regression quantiles under dependent censoring

In a recent paper, Braekers and Veraverbeke [Braekers, R. and Veraverbeke, N., 2005, A copula-graphic estimator for the conditional survival function under dependent censoring. Canadian Journal of Statistics, 33, 429–447.] obtained asymptotic results for an estimator of the conditional distribution function of the lifetime, in a fixed design regression model with dependence between lifetime and censoring time. The author complements these findings by corresponding results for the conditional quantile estimator. The main results are uniform strong consistency and a Bahadur-type representation. The latter is an invaluable tool for deriving further asymptotic results, such as the weak convergence of the quantile process.

A Martingale Approach to the Copula-Graphic Estimator for the Survival Function under Dependent Censoring

The product limit estimator is arguably the most popular method of estimating survival probabilities in homogeneous samples. When the survival time and the censoring time are dependent, the product-limit estimator is an inconsistent estimator of the marginal survival function. Recently M. Zheng and J. P. Klein (1995, Biometrika82, 127–138) proposed a copula-graphic estimator that models the dependency between censoring and survival using a copula function. This work investigates their proposal. First it derives a closed form expression for the copula-graphic estimator when the joint survival function is modeled with an Archimedean copula. The copula-graphic estimator is then shown to be uniformly consistent and asymptotically normal. It is also equivalent to the usual product-limit estimator when the survival and censoring times are assumed to be independent. A sensitivity analysis of the specification of the copula model for the dependency is also presented.