Dependent censoring often arises in biomedical studies when time to tumour progression (e.g., relapse of cancer) is censored by an informative terminal event (e.g., death). For meta-analysis combining existing studies, a joint survival model between tumour progression and death has been considered under semicompeting risks, which induces dependence through the study-specific frailty. Our paper here utilizes copulas to generalize the joint frailty model by introducing additional source of dependence arising from intra-subject association between tumour progression and death. The practical value of the new model is particularly evident for meta-analyses in which only a few covariates are consistently measured across studies and hence there exist residual dependence. The covariate effects are formulated through the Cox proportional hazards model, and the baseline hazards are nonparametrically modeled on a basis of splines. The estimator is then obtained by maximizing a penalized log-likelihood function. We also show that the present methodologies are easily modified for the competing risks or recurrent event data, and are generalized to accommodate left-truncation. Simulations are performed to examine the performance of the proposed estimator. The method is applied to a meta-analysis for assessing a recently suggested biomarker CXCL12 for survival in ovarian cancer patients. We implement our proposed methods in R joint.Cox package.

During its 330 years of history, parametric distributions have been useful for survival and reliability analyses. In this paper, we comprehensively review the historical backgrounds and statistical properties of a number of parametric distributions used in survival and reliability analyses. We provide encyclopedic coverage of the important parametric distributions, which is more extensive than the existing textbooks on survival and reliability analyses. We also explain how these distributions have been adopted in survival and reliability analyses with original and state-of-the-art references. We cover the exponential, Weibull, Rayleigh, lognormal, log-logistic, gamma, generalized gamma, Pareto (types I, II, and IV), Hjorth, Burr (types III and XII), Dagum, exponential power, Gompertz, Birnbaum-Saunders, exponential-logarithmic, piecewise exponential, generalized exponential, exponentiated Weibull, generalized modified Weibull, and spline distributions. We analyze a real dataset for illustration.

For the analysis of competing risks data, three different types of hazard functions have been considered in the literature, namely the cause-specific hazard, the sub-distribution hazard, and the marginal hazard function. Accordingly, medical researchers can fit three different types of the Cox model to estimate the effect of covariates on each of the hazard function. While the relationship between the cause-specific hazard and the sub-distribution hazard has been extensively studied, the relationship to the marginal hazard function has not yet been analyzed due to the difficulties related to non-identifiability. In this paper, we adopt an assumed copula model to deal with the model identifiability issue, making it possible to establish a relationship between the sub-distribution hazard and the marginal hazard function. We then compare the two methods of fitting the Cox model to competing risks data. We also extend our comparative analysis to clustered competing risks data that are frequently used in medical studies. To facilitate the numerical comparison, we implement the computing algorithm for marginal Cox regression with clustered competing risks data in the R joint.Cox package and check its performance via simulations. For illustration, we analyze two survival datasets from lung cancer and bladder cancer patients.

Correlations among survival endpoints are important for exploring surrogate endpoints of the true endpoint. With a valid surrogate endpoint tightly correlated with the true endpoint, the efficacy of a new drug/treatment can be measurable on it. However, the existing methods for measuring correlation between two endpoints impose an invalid assumption: correlation structure is constant across different treatment arms. In this article, we reconsider the definition of Kendall's concordance measure (tau) in the context of individual patient data meta-analyses of randomized controlled trials. According to our new definition of Kendall's tau, its value depends on the treatment arms. We then suggest extending the existing copula (and frailty) models so that their Kendall's tau can vary across treatment arms. Our newly proposed model, a joint frailty-conditional copula model, is the implementation of the new definition of Kendall's tau in meta-analyses. In order to facilitate our approach, we develop an original R function condCox.reg(.) and make it available in the R package joint.Cox (https://CRAN.R-project.org/package=joint.Cox). We apply the proposed method to a gastric cancer dataset (3288 patients in 14 randomized trials from the GASTRIC group). This data analysis concludes that Kendall's tau has different values between the surgical treatment arm and the adjuvant chemotherapy arm (p-value<0.001), whereas disease-free survival remains a valid surrogate at individual level for overall survival in these trials.