In survival analysis, it is very common to assume that the lifetime variable and the censoring variable are independent. In this case, the product limit estimator is the standard non-parametric estimator for the distribution function of the lifetime variable. When the assumption of independence is not satisfied, Zheng and Klein (1995) proposed a copula-graphic estimator where the dependence between lifetime and censoring variable is described by a known copula. Rivest and Wells (2001) derived an explicit form for this estimator if the copula is Archimedean.In this paper, we extend the estimator of Rivest and Wells (2001) to the fixed design regression case. For our copula-graphic estimator, we find an asymptotic representation and prove weak convergence to a Gaussian limit. We perform a sensitivity analysis to assess the influence of a misspecified copula function on the estimator. Furthermore we illustrate the estimation method with a dataset on survival of Atlantic halibut.
Summary
For the analysis of clustered survival data, two different types of model that take the association into account are commonly used: frailty models and copula models. Frailty models assume that, conditionally on a frailty term for each cluster, the hazard functions of individuals within that cluster are independent. These unknown frailty terms with their imposed distribution are used to express the association between the different individuals in a cluster. Copula models in contrast assume that the joint survival function of the individuals within a cluster is given by a copula function, evaluated in the marginal survival function of each individual. It is the copula function which describes the association between the lifetimes within a cluster. A major disadvantage of the present copula models over the frailty models is that the size of the different clusters must be small and equal to set up manageable estimation procedures for the different model parameters. We describe a copula model for clustered survival data where the clusters are allowed to be moderate to large and varying in size by considering the class of Archimedean copulas with completely monotone generator. We develop both one‐ and two‐stage estimators for the copula parameters. Furthermore we show the consistency and asymptotic normality of these estimators. Finally, we perform a simulation study to investigate the finite sample properties of the estimators. We illustrate the method on a data set containing the time to first insemination in cows, with cows clustered in herds.
Both exhaled breath condensate interleukin-5 level and asthma control score were significant predictors of asthma exacerbations. These findings open up the possibility of assessing the potential of such parameters to titrate asthma treatment in future studies.
In this paper we extend the conditional Koziol-Green model of Veraverbeke and Cadarso Suárez (2000) to also accommodate for 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 on a real data set about the survival of Atlantic halibut.
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