Esimating relapse free survival as a net probability: regression models and graphical representation

In several experimental or observational clinical studies, the evaluation of the effect of a therapy and the impact of prognostic factors is based on relapse-free survival and the suited regression models. Relapse free survival is a net survival and needs to be interpreted as the survival 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 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 ignored. 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, with ad hoc estimating procedures. We concentrate here on the copula graphic estimator, for which a pertinent regression model has been developed (Lo and Wilke). The advantage of this approach is based on the relationship between net survival, overall survival and cause specific hazard. Regression models can be fitted for the latter quantity by standard statistical methods and the estimates can be used to compute net survival through a copula structure. Parametric models are preferred. To avoid the constraint of parametric distribution, we propose piecewise regression models. A consistent estimate of the association parameter for the copula model can be obtained by considering the semi-competing risks framework, because death can be observed after relapse. The drawback of the copula graphic regression model is that no direct parametric estimation of the regression coefficient for the covariates is available. To obtain an overall view of the association between covariate levels and net relapse free survival we propose a multivariate visualisation approach through Multiple Correspondence Analysis. This approach has been applied to two case series of patients with breast cancer and extremity soft tissue sarcoma respectively, in order to compare the results obtained by piecewise exponential model on cause specific hazard and net relapse free survival computed through copula graphic estimator.