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This article proposes an approach to estimate and make inference on the parameters of copula link-based survival models. The methodology allows for the margins to be specified using flexible parametric formulations for time-to-event data, the baseline survival functions to be modeled using monotonic splines, and each parameter of the assumed joint survival distribution to depend on an additive predictor incorporating several types of covariate effects. All the model's coefficients as well as the smoothing parameters associated with the relevant components in the additive predictors are estimated using a carefully structured efficient and stable penalized likelihood algorithm. Some theoretical properties are also discussed. The proposed modeling framework is evaluated in a simulation study and illustrated using a real dataset. The relevant numerical computations can be easily carried out using the freely available GJRM R package. Supplementary materials for this article are available online.

Bivariate survival data arise frequently in familial association studies of chronic disease onset, as well as in clinical trials and observational studies with multiple time to event endpoints. The association between two event times is often scientifically important. In this article, we examine the association via a novel quantile association measure, which describes the dynamic association as a function of the quantile levels. The quantile association measure is free of marginal distributions, allowing direct evaluation of the underlying association pattern at different locations of the event times. We propose a nonparametric estimator for quantile association, as well as a semiparametric estimator that is superior in smoothness and efficiency. The proposed methods possess desirable asymptotic properties including uniform consistency and root-n convergence. They demonstrate satisfactory numerical performances under a range of dependence structures. An application of our methods suggests interesting association patterns between time to myocardial infarction and time to stroke in an atherosclerosis study.

Bivariate or multivariate survival data arise when a sample consists of clusters of two or more subjects which are correlated. This paper considers clustered bivariate survival data which is possibly censored. Two approaches are commonly used in modelling such type of correlated data: random effect models and marginal models. A random effect model includes a frailty model and assumes that subjects are independent within a cluster conditionally on a common non-negative random variable, the so-called frailty. In contrast, the marginal approach models the marginal distribution directly and then imposes a dependency structure through copula functions. In this manuscript, Bernstein copulas are used to account for the correlation in modelling bivariate survival data. A two-stage parametric estimation method is developed to estimate in the first stage the parameters in the marginal models and in the second stage the coefficients of the Bernstein polynomials in the association. Hereby we use a penalty parameter to make the fit desirably smooth. In this aspect linear constraints are introduced to ensure uniform univariate margins and we use quadratic programming to fit the model. We perform a Simulation study and illustrate the method on a real data set.

In many biomedicai applications, interest focuses on the occurrence of two or more consecutive failure events and the relationship between event times, such as age of disease onset and residual lifetime. Bivariate survival data with interval sampling arise frequently when disease registries or surveillance systems collect data based on disease incidence occurring within a specific calendar time interval. The initial event is then retrospectively confirmed and the subsequent failure event may be observed during follow-up. In life history studies, the initial and two consecutive failure events could correspond to birth, disease onset and death. The statistical features and bias of observed data in relation to interval sampling were discussed by Zhu & Wang (2012). Here we propose nonparametric estimation of the association between bivariate failure times based on Kendall's tau for data collected with interval sampling. A nonparametric estimator is given, where the contribution of each comparable and orderable pair is weighted by the inverse of the associated selection probability. Analysis methods for bivariate survival data with interval sampling rely on the assumption of quasi-independence, i.e., that bivariate failure times and the time of the initial event are independent in the observable region. This paper develops a nonparametric test of quasiindependence based on a bivariate conditional Kendall's tau for such data. Simulation studies demonstrate that the association estimator and testing procedure perform well with moderate sample sizes. Illustrations with two real datasets are provided.

To study disease association with risk factors in epidemiologie studies, cross-sectional sampling is often more focused and less costly for recruiting study subjects who have already experienced initiating events. For time-to-event outcome, however, such a sampling strategy may be length biased. Coupled with censoring, analysis of length-biased data can be quite challenging, due to induced informative censoring in which the survival time and censoring time are correlated through a common backward recurrence time. We propose to use the proportional mean residual life model of Oakes & Dasu (Biometrika 77, 409-10, 1990) for analysis of censored length-biased survival data. Several nonstandard data structures, including censoring of onset time and cross-sectional data without follow-up, can also be handled by the proposed methodology.

In disease registries, bivariate survival data are typically collected under interval sampling. It refers to a situation when entry into a registry is at the time of the first failure event (i.e., HIV infection) within a calendar time window. For all the cases in the registry, time of the initiating event (i.e., birth) is retrospectively identified, and subsequently the second failure event (i.e., death) is observed during follow-up. In this paper we discuss how interval sampling introduces bias into the data. Given the sampling design that the first event occurs within a specific time interval, the first failure time is doubly truncated, and the second failure time is possibly informatively right censored. Consider semi-stationary condition that the disease progression is independent of when the initiating event occurs. Under this condition, this paper adopts copula models to assess association between the bivariate survival times with interval sampling. We first obtain bias-corrected estimators of marginal survival functions, and estimate association parameter of copula model by a two-stage procedure. In the second part of the work, covariates are incorporated into the survival distributions via the proportional hazards models. Inference of the association measure in copula model is established, where the association is allowed to depend on covariates. Asymptotic properties of proposed estimators are established, and finite sample performance is evaluated by simulation studies. The method is applied to a community-based AIDS study in Rakai to investigate dependence between age at infection and residual lifetime without and with adjustment for HIV subtype.

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, Biometrics54, 1014-1029). The proposed methodology is illustrated with application to simulated data, as well as to data from a leukemia study.

In this paper, we propose two tests for parametric models belonging to the Archimedean copula family, one for uncensored bivariate data and the other one for right-censored bivariate data. Our test procedures are based on the Fisher transform of the correlation coefficient of a bivariate (U,V), which is a one-to-one transform of the original random pair (T1,T2) that can be modeled by an Archimedean copula model. A multiple imputation technique is applied to establish our test for censored data and its p value is computed by combining test statistics obtained from multiply imputed data sets. Simulation studies suggest that both procedures perform well when the sample size is large. The test for censored data is carried out for a medical data example.

This paper considers nonparametric inference for duration times of two successive events. Since the second duration process becomes observable only if the first event has occurred, the length of the first duration affects the probability of the second duration being censored. Dependent censoring arises if the two duration times are correlated, which is often the case. Standard approaches to this problem fail because of dependent censoring mechanism. A new product-limit estimator for the second duration variable and a pathdependent joint survival function estimator are proposed, both modified for the dependent censoring. Properties of the estimators are discussed. An example from Lawless (1982) is studied for illustrative purposes as well as a simulation study.

Survival data with censored initiating and terminating times have surfaced in some recent epidemiologic studies. Unlike standard survival analysis with known initiating times, analysis of data with both censored initiating and terminating times requires maximization of a complicated bivariate likelihood, which is often difficult to carry out. This article considers a missing-data formulation of the problem and focuses on the use of EM-type algorithms to simplify the computation of maximum likelihood estimates. This approach provides a feasible way of performing regression analysis with such bivariate survival data. Several illustrative examples are provided, including a real-data analysis application involving a cohort of HIV-infected hemophiliac patients.