a Flexible Baysian modelling of covariate effects on survival

Abstract

Proportional hazards models are commonly used in survival analysis. Typically a baseline hazard function is combined with hazard multipliers which depend on covariate values through a logarithmic link function and a linear predictor. Models have been developed which allow exibility in the form of the baseline hazard. However, the form of dependence of the hazard multipliers on covariates is usually speci ed. The aim of this research is to introduce exibility into the form of the dependence of the hazard function on the covariates by removing the assumptions of parametric forms which are usually made. Given su cient data, this will allow the model to adapt to the true form of the relationship and possibly uncover unexpected features. The Bayesian approach to inference is used. The choice of a suitable prior distribution allows a compromise which relaxes the assumption of a parametric form of relationship while imposing enough structure to exploit the information in nite data sets by specifying correlations in the prior distribution between log-hazards for neigbouring covariate pro les. The choice of prior distribution can therefore be important for obtaining useful posterior inferences. A generalised piecewise constant hazard model is introduced, in which quantitative covariates, as well as time, are categorised. Thus, the time and covariate space is divided into cells, within each of which the hazard is a constant. Two forms of prior distribution are considered, one based on a parametric model and the other using a Gaussian Markov random eld. When the number of covariates is large, this approach leads to a very large number of cells, many of which might not represent any observed cases. Therefore, we consider an alternative approach in which a Gaussian process prior for the log-hazards over the covariate space is used. The posterior distribution is computed only at the observed covariate pro les. The methodology developed is applicable to a wide range of survival data and is illustrated by applications to two data sets referring to patients with non-Hodgkin's lymphoma and leukaemia respectively.