Estimation of the incidence for non-terminal events in presence of a terminal event and evaluation of covariate effects: Sub-distribution and marginal distributions based on copulas. An application to disease progression on a breast cancer trial dataset

In clinical studies, during follow-up several kinds of events related to disease progression may be observed. In the semi-competing risks setting, some events, such as death, may prevent the observation of disease progression, thus acting as competitor for the event of interest. Methods of analysis specific for semi-competing risks data referring to marginal distribution of the non-competing events constitute a recent area of methodological research which has received a great impulse in latest years. However in clinical applications the analysis is traditionally based on crude cumulative incidences, and inference on marginal distributions is seldom considered, even when the principal aim concerns the probability of observing disease progression and death occurred without progression is a “nuisance”. Aim of this work is making a comparative review of semi-parametric marginal and sub-distribution methods of analysis, with particular reference to marginal regression models based on copulas. More specifically, two structures were considered for marginal models: in the first one all parameters are time-dependent, while in the second one parameters vary with covariates but does not depend on time. Applications to breast cancer clinical trial data and to a simulated dataset are reported, to show the differences and the similarities among marginal and sub-distribution approaches. Results highlight that, when the competing event acts during the whole follow-up, the marginal approach became essential for the correct estimation of marginal incidences and covariate effects. Regression methods based on copulas are promising, however there is a need of refinements concerning model building strategies, and, of standardised software routine for the practical application of these methods.