Copula models for complex data: applications to dependent censoring and longitudinal data

Abstract

Throughout the many domains in statistics, traditional models often assume independence of the data. This typically comes with great theoretical simplification, but numerous real-life situations also violate this assumption. In such contexts, copula modelling is a particularly useful tool to enhance model validity. Specifically, copula functions link marginal quantities and capture any interdependence present, while preserving model interpretability due to a conceptual separation of this dependence from the rest of the model. Copula models for two dependence-invoking contexts are considered in this dissertation. We first work in the domain of survival analysis, that is inherently plagued by the phenomenon of censoring: whereas primary interest is in the time T until a specific event occurs, observation of the latter is sometimes precluded by an earlier competing event. Nonetheless, the censoring time C until this alternative event contains relevant information on T. Under independence of T and C, it only contributes to the available information about T by ensuring that T > C for that data subject. Frequently, however, T and C are connected through positive or negative dependence, such that knowing C increases the likeliness of T being in the near or further future, respectively, as compared to the case in which C would not have been observed yet. Such dependent censoring is often handled using copula models. Yet, as T and C are never simultaneously observed for one subject, it is theoretically impossible to distinguish independent from dependent censoring without imposing formal restrictions; this is referred to as nonidentifiability. Literature has therefore, initially, mostly focused on flexible (i.e. nonparametric) marginal modelling, while making the overly strict assumption of a completely known dependence structure. Over the past years, research has shown the possibility of less restrictive copula assumptions – viz. parametric rather than fully known copula functions – at the cost of likewise parametric margins. The first part of this thesis is situated in this area and further explores the boundary between flexibility and nonidentifiability