The observation of time to tumour progression (TTP) or progression-free survival (PFS) may be terminated by a terminal event. In this context, deaths may be due to tumour progression, and the time to the major failure event (death) may be correlated with the TTP. The usual assumption of independence between the TTP process and death, required by many commonly used statistical methods, can be violated. Furthermore, although the relationship between TTP and time to death is most relevant to the anti-cancer drug development or to evaluation of TTP as a surrogate endpoint, statistical models that try to describe the dependence structure between these two characteristics are not frequently used. We propose a joint frailty model for the analysis of two survival endpoints, TTP and time to death, or PFS and time to death, in the context of data clustering (e.g. at the centre or trial level). This approach allows us to simultaneously evaluate the prognostic effects of covariates on the two survival endpoints, while accounting both for the relationship between the outcomes and for data clustering. We show how a maximum penalized likelihood estimation can be applied to a nonparametric estimation of the continuous hazard functions in a general joint frailty model with right censoring and delayed entry. The model was motivated by a large meta-analysis of randomized trials for head and neck cancers (Meta-Analysis of Chemotherapy in Head and Neck Cancers), in which the efficacy of chemotherapy on TTP or PFS and overall survival was investigated, as adjunct to surgery or radiotherapy or both.
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