Flexible estimation in cure survival models using Bayesian P-splines

A common hypothesis in the analysis of survival data is that any observed unit will experience the monitored event if it is observed for a sufficient long time. Alternatively, one can explicitly acknowledge that an unknown and unidentified proportion of the population under study is cured and will never experience the event of interest. The promotion time model, which is motivated using biological mechanisms in the development of cancer, is one of the survival models taking this feature into account. The promotion time model assumes that the failure time of each subject is generated by the minimum of N latent event times which are independent with a common distribution F(t) = 1 - S(t) independent of N. We propose an extension which allows the covariates to influence simultaneously the probability of being cured and the latent distribution F(t). We estimate the latent distribution F(t) using a flexible Cox proportional hazard model where the logarithm of the baseline hazard function is specified using Bayesian P-splines. Introducing covariates in the latent distribution implies that the population hazard function might not have a proportional hazard structure. However, the use of the P-splines provides a smooth estimation of the population hazard ratio over time. When working with cure survival models, it is usually stressed that the follow up should be sufficiently long. We propose a restricted use of the model when that assumption is not respected. A simulation study evaluating the accuracy of our methodology is presented. The proposed model is illustrated on data from the phase III Melanoma e1684 clinical trial