Dynamic models for competing risks and relative survival

The thesis concerns regression models related to the competing risks setting in survival analysis and deals with both the case of known specific causes and the case of unknown (even if present) specific causes of the event of interest. In the first part, dealing with events whose specific cause is known, competing risks modelling has been applied to a breast cancer study and some of the dynamic aspects such as time-dependent variables are tackled within the context of the application. The aim of the application was to detect an optimal chemotherapy dosage for different typologies of patients with advanced breast cancer in order to control the risk of cardiotoxicity. The attention was concentrated on the cumulative incidence probability of getting cardiotoxicity in a well-defined time period, conditional on risk factors. This probability was estimated as a function of the time-dependent covariate dosage. Within the context of the application, some problems of goodness-of-fit related to time-dependent covariates are discussed. The previous application gave rise to investigating the role of time-dependent covariates in competing risks regression models. There exist various types of time-dependent covariates, which differ in their random or deterministic development in time. For so-called internal covariates, predictions based on the model are not allowed, or they meet with difficulties. We describe a general overview of the state of the art, problems and future directions. Moreover, a possible extension of the competing risks model, that allows us to include a simple random binary time-dependent variable, in a multi-state framework, is presented. Inclusion of the sojourn time of an individual in a certain state as a time-dependent covariate into the model, is also studied. In the second part of the thesis, dealing with events whose specific cause is unavailable, regression models for relative survival are discussed. We study the nonparametric additive excess hazards models, where the excess hazard is on additive form. We show how recent developments can be used to make inferential statements about this models, and especially to test the hypothesis that an excess risk effect is time-varying in contrast to being constant over time. We also show how a semiparametric additive risk model can be considered in the excess risk setting. These two additive models are easy to fit with estimators on explicit form and inference including tests for time-constant effects can be carried out based on a resampling scheme. We analyze a real dataset using different approaches and show the need for more flexible models in relative survival. Finally, we describe a new suggestion for goodness-of-fit of the additive and proportional models for relative survival, which avoids some disadvantages of recent proposals in the literature. The method consists of statistical and graphical tests based on cumulative martingale residuals and it is illustrated for testing the proportional hazards assumption in the semiparametric proportional excess hazards model.