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Flexible survival models are in need when modelling data from long term follow-up studies. In many cases, the assumption of proportionality imposed by a Cox model will not be valid. Instead, a model that can identify time varying effects of fixed covariates can be used. Although there are several approaches that deal with this problem, it is not always straightforward how to choose which covariates should be modelled having time varying effects and which not. At the same time, it is up to the researcher to define appropriate time functions that describe the dynamic pattern of the effects. In this work, we suggest a model that can deal with both fixed and time varying effects and uses simple hypotheses tests to distinguish which covariates do have dynamic effects. The model is an extension of the parsimonious reduced rank model of rank 1. As such, the number of parameters is kept low, and thus, a flexible set of time functions, such as b-splines, can be used. The basic theory is illustrated along with an efficient fitting algorithm. The proposed method is applied to a dataset of breast cancer patients and compared with a multivariate fractional polynomials approach for modelling time-varying effects. Copyright © 2016 John WileySons, Ltd.

The Cox proportional hazards model is the most common method to analyse survival data. However, the proportional hazards assumption might not hold. The natural extension of the Cox model is to introduce time-varying effects of the covariates. For some covariates such as (surgical)treatment non-proportionality could be expected beforehand. For some other covariates the non-proportionality only becomes apparent if the follow-up is long enough. It is often observed that all covariates show similar decaying effects over time. Such behaviour could be explained by the popular (gamma-) frailty model. However, the (marginal) effects of covariates in frailty models are not easy to interpret. In this paper we propose the reduced-rank model for time-varying effects of covariates. Starting point is a Cox model with p covariates and time-varying effects modelled by q time functions (constant included), leading to a pxq structure matrix that contains the regression coefficients for all covariate by time function interactions. By reducing the rank of this structure matrix a whole range of models is introduced, from the very flexible full-rank model (identical to a Cox model with time-varying effects) to the very rigid rank one model that mimics the structure of a gamma-frailty model, but is easier to interpret. We illustrate these models with an application to ovarian cancer patients.

Analysis of long-term follow-up survival studies require more sophisticated approaches than the proportional hazards model. To account for the dynamic behaviour of fixed covariates, penalized Cox models can be employed in models with interactions of the covariates and known time functions. In this work, I discuss some of the suggested methods and emphasize on the use of a ridge penalty in survival models. I review different strategies for choosing an optimal penalty weight and argue for the use of the computationally efficient restricted maximum likelihood (REML)-type method. A ridge penalty term can be subtracted from the likelihood when modelling time-varying effects in order to control the behaviour of the time functions. I suggest using flexible time functions such as B-splines and constrain the behaviour of these by adding proper penalties. I present the basic methods and illustrate different penalty weights in two different datasets.

In this work I consider models for survival data when the assumption of proportionality does not hold. The thesis consists of an Introduction, five papers, a Discussion and an Appendix. The Introduction presents technical information about the Cox model and introduces the ideas behind the extensions of the model proposed later on. In Chapter 2, reduced-rank methods for modelling non-proportional hazards are presented while Chapter 3 presents an algorithm for estimating Cox models with time varying effects of the covariates. The next Chapter deals with the gamma frailty (Burr) model and discusses alternative models with time dependent frailties. In Chapter 5 models with time varying effects of the covariates, frailty models and cure rate models are considered. The usefulness of each one of these models is discussed and their results are compared. The sixth Chapter of the thesis discusses ways of dealing with overdispersion when using generalized linear models. The Discussion is about future directions of the research presented in this thesis. Finally, there is an Appendix about the use of coxvc, a package written in R for fitting Cox models with time varying effects of the covariates.

Frailty models are used in univariate data to account for individual heterogeneity. In the popular gamma frailty model the marginal hazard has the form of a Burr model. Although the Burr model is very useful and can offer insight on the data, it is far from perfect. The estimation of the covariate effects is linked to the baseline hazard and this makes the model coefficients hard to interpret. At the same time, the frailties are assumed constant over time, while biological reasoning in some cases may indicate that frailties may be time dependent. In this paper we present a relaxation of the Burr model which is based on loosening the link between the estimation of the covariate effects and the baseline hazard. This can be achieved by replacing the cumulative baseline hazard in the Burr model by a set of time functions, and the frailty variance by a vector of coefficients directly estimated from the data using a partial likelihood. We illustrate the similarities of the model with the Burr model and a further extension of the latter, a model with an autoregressive stochastic process for the frailty. We compare the models on simulated data sets with constant and time-dependent frailties and show how the relaxed Burr models performs on two different real data sets. We show that the relaxed Burr model serves as a good approximation to the Burr model when the frailty is constant, and furthermore it gives better results when the frailty is time dependent.